https://mathimages.swarthmore.edu/api.php?action=feedcontributions&user=Rscott3&feedformat=atomMath Images - User contributions [en]2022-12-01T21:22:45ZUser contributionsMediaWiki 1.31.1https://mathimages.swarthmore.edu/index.php?title=Pages_Ready_for_Final_Review&diff=27068Pages Ready for Final Review2011-07-22T17:59:38Z<p>Rscott3: /* Swarthmore Pages */</p>
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'''<font color=darkred> AN IMPORTANT NOTE ABOUT PAGES PUT UP FOR FINAL REVIEW BETWEEN JULY 18TH AND 22ND'''<br />
<br />
'''Anna is going to a training which will keep her extremely busy during that week. Priority for editing will be given to Swarthmore students and any others for whom it is their last week of work on the project (please note if you are NOT at Swarthmore and it is your last week. ''' I will make a big effort to get through everything posted by this weekend, so there isn't a backlog. I will also be completely unavailable for editing August 6th-15th, and will not be reachable during that time. </font> <br />
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'''<font color=olive><font color=green>CHRIS T.</font>, <font color=blue>STEVE C.</font>, and <font color=slateblue>REBECCA</font> will be helping to review pages while Anna is busy.</font>'''<br />
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===Swarthmore Pages===<br />
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*[[Logistic Bifurcation]] by [[User:Dpatton1|Diana]] 17:00 7/7/11<br />
:<font color=green> Chris 7/16 I've put up comments from a "layman's" perspective. </font><br />
<br />
*[[Snell's Law]]<br />
addressed Anna and steve's comments. new draft. [[User:Ljeanlo1|Ljeanlo1]] 16:18, 22 July 2011 (UTC)<br />
<br />
*[[Witch of Agnesi]] <br />
addressed Chris' comment. new draft. [[User:Ljeanlo1|Ljeanlo1]] 16:18, 22 July 2011 (UTC)<br />
<br />
*[[Dimensions]] by [[User:Htasoff|Htasoff]]<br />
:<font color=green> SteveC 7/20 -- a couple of comments.</font><br />
::<font color=Mediumblue>Confused, Please explain.</font><br />
::<font color=green>I've tried to clarify the point.</font><br />
:::<font color=Mediumblue>Addressed the comment! 7/22</font><br />
<br />
===Sweet Briar Pages=== <br />
<br />
*[[Dandelin Spheres Theory]] by [[User:Flora1103|Flora Li]] 15:31, 17 July 2011 (UTC)<br />
:<font color=green> Chris 7.20.11 I've responded to some of your changes and comments. </font><br />
::<font color=plum>Flora 07.20 12:08 I responsed to your suggetions and made a little more changes.</font><br />
:::<font color=green> Chris 7.21.11 I've completed the rest of the page and made comments. </font><br />
::::<font color=plum>Thanks for your suggestion. I have made some change based on your comments. Flora 07.21 12:29</font><br />
<br />
*[[Steiner's Chain]] by [[User:Donko14|Anna]]<br />
:<font color=darkred> comments are up [[User:AnnaP|AnnaP]] 13:35, 21 July 2011 (UTC) </font><br />
::<font color=green> Thanks for your quick response with comments! I have made the necessary adjustments!!</font><br />
<font color=slateblue> <br />
::[[User:Rebecca|Rebecca]] 13:09, 22 July 2011 (UTC) This page is great! I left a few small comments, but overall very good! </font color><br />
:::<font color=green> Thanks Rebecca and Anna for your comments! I have made ALL the corrections this time and I left comments ( in green ) to your comments on the discussion page!</font> <br />
<br />
*[[Parabolic Integration]] by [[User:Donko14|Anna]]<br />
:<font color=darkred> I've suggested a pretty major revision of one section, and I actually think that it might be best to move that section to another page. See my notes [[User:AnnaP|AnnaP]] 05:07, 22 July 2011 (UTC) </font><br />
::<font color=green>Hi Anna, should I make the corrections to this section and wait for you to approve the page before I move it to the [[Parabola]] page?</font><br />
<br />
===Rensselaer Polytechnic Institute===<br />
[[Polar Equations]] by [[User:Chanj|Chanj]] 21:35, 7 July 2011 (UTC)<br />
:<font color=blue> I have edited my page by putting up new images, more detailed explanations and comments in the discussion reflecting those changes. --[[User:Chanj|Chanj]] 20:48, 15 July 2011 (UTC)</font><br /><br /><br />
<br />
[[Transformations and Matrices]] created by [[Users:Nordhr|Nordhr]], content written by Steve Cunningham. 11:43 22 July 2011<br /><br />
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<font color=green>A couple of minor comments added [[User:SteveC|SteveC]]</font><br />
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[[DU11]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Dimensions&diff=27067Dimensions2011-07-22T17:57:42Z<p>Rscott3: /* 4D */</p>
<hr />
<div>{{HelperPage|1=Cross-cap}}<br />
<br />
==Basic Description==<br />
<br />
A dimension can be thought of as a direction of motion. The first dimension has one direction, say "forward and backward". The second dimension has two: "forward and backward", "up and down". The third dimension has three directions: "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Tug-of-war is a one dimensional game; you can only go forward and backward. A game like Mario is two dimensional, which is quite apparent when you can't avoid obstacles by stepping out of the screen. Real life is 3D, you can go "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Dimensionality is a property of both objects and spaces, however the dimension of an object can be different than that of the space it is in.<br />
<br />
<br />
Additionally, there is the zeroth dimension and fractal dimensions. The zeroth dimension has no directions of motion. The zeroth dimension is a point without a "forward and backward", "up and down", or "right and left". Fractal dimensions are discussed in depth [[Fractal Dimension| here]].<br />
<br />
===Dimensions of Objects===<br />
<br />
<div id="Image1">[[Image:Line circle disk.png|Image 1.|thumb|300px|right]]</div><br />
<br />
<br />
Every object has an <balloon title="Belonging to a thing by its very nature.">intrinsic</balloon> dimensionality. Take the line; it has only one direction along it, meaning it is 1 dimensional. The circle is merely a line segment connected front to back. If you are on the circle, you can only go in one direction: around it. So the circle is 1 dimensional as well. On the disk (best thought of as a filled in circle), there are two directions akin to "up and down", "right and left", making the disk two dimensional. The image to the right displays these shapes.<br />
<br />
Whether the line is drawn on a 2 dimensional sheet of paper, or made in 3 dimensions as, say, a string, the line is always 1 dimensional. Similarly, the disk, drawn on paper, or made in 3 dimensions, like a flattened quarter, is always 2 dimensional. Althought a string and a flattened quarter are 3 dimensional because they do still have length, width, and height, they can be used as analogies for an ideal 1 dimensional line and 2 dimensional disk within 3 dimensional space.<br />
<br />
<br style="clear: both" /> <br />
===Dimensions of Spaces===<br />
<br />
Just as objects have dimensions, so do the spaces that objects are in. The dimensions of a space can, as discussed above, be thought of as a direction. We live in 3 dimensional space, meaning we have three directions in which we can move. Mario lives in 2 dimensional space; he can only "move up and down", "left and right". Humans are 3 dimensional objects; It takes three basic directions to describe our bodies:"upward and downward", "leftward and rightward", "forward and backward". Mario is 2 dimensional, and can be described with "upward and downward", "leftward and rightward". Thus, we have the same dimensions as the space where we live, as does Mario. Nevertheless, this is not always the case for objects. Ideally, a line drawn on a piece of paper is a 1 dimensional object that is embedded in the 2 dimensional space of the page. Thus, an object can exist in a space with a higher number of dimensions than that of the itself.<br />
<br />
<br style="clear: both" /> <br />
==Four Dimensions==<br />
===1D===<br />
Each dimension is represented mathematically by a coordinate variable. Any point specified by one variable, such as <math>x=8</math>, is located in 1 dimension, and can be plotted on the 1 dimensional number-line, also known as the x-axis, at the point (8), as shown below. <br />
<br />
::<div id="Image2">[[Image:8 nuberline 2.png|Image 2.|650px]]</div><br />
<br />
===2D===<br />
<br />
If the point is defined by two coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<br style="clear: both" /> <br />
then it can no longer be placed as a point solely on the number-line. It now has two coordinates, One stating that it is located at 8 in the x direction, and the other placing it at 7 in the y direction. Thus, we plot the point as (8,7) on the 2 dimensional xy-plane, which may be familiar from math class. We merely added another direction to the 1 dimensional number-line to arrive at the 2 dimensional plane.<br />
<br />
<br />
<div id="Image3">[[Image:8 7 xy axis 3.png|Image 3.|400px]]</div><br />
<br />
<br style="clear: both" /> <br />
===3D===<br />
<br />
If a point has three coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<math>z=6</math><br />
<br />
Then to capture all of this information on our graph, a third dimension, the z direction, must be added, say out of the paper the graph is drawn on. Our new point, (8,7,6), is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, as illustrated in The graph below.<br />
<br />
<div id="Image4">[[Image:Richard_did_this_for_harrison.jpg|Image 4.|300px]]</div><br />
<br />
===4D===<br />
<br />
Before directly discussing four dimensions, let us introduce an analogue in 2 and 3 dimensions. Notice that, in [[#Image5|Image 5]], (8,7,6) and (8,7,0) have the same x and y coordinates. If we look at them from directly along the z-axis, the two points overlap, appearing as though they were a single point at (8,7) on a 2 dimensional xy-plane ([[#Image6|Image 6]]). As such, they are at the same position in a two dimensions, but have different positions in a third. It is this third coordinate, their z coordinate, which distinguishes them as different points. <br />
<br />
{{{!}} align="center" style="text-align:center;" cellpadding=40 cellspacing=10<br />
{{!}}<div id="Image5"></div>[[Image:Richard_did_this_for_harrison_a_second_time.jpg|Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.|thumb|270px|right]]<br />
{{!}}<div id="Image6"></div>[[Image:And_richard_did_it_again.jpg|Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).|thumb|270px|right]]<br />
{{!}}}<br />
<br />
Just as in the previous examples, a 4<sup>th</sup> dimension can be thought of as merely adding anther coordinate variable describing the location of the point. A point (8,7,6,5) is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, and 5 in a fourth direction, which we will call the w direction. The two points (8,7,6,5) and (8,7,6,20) would appear in the same place as each other on a 3D graph in much the way as the points (8,7,6) and (8,7,0) appeared to be in the same place as each other on the 2 dimensional xy-plane. For (8,7,6,5) and (8,7,6,20), we must simply remember that they have an other direction in which they are not at the same place, even though in the three directions we know of, they are.<br />
<br />
One way to do this would be to give the points different coordinates in time. For instance, an object could be at the point (8,7,6) at 5 seconds after you start measuring, thus (8,7,6,5), and then another one is there at 20 seconds, thus (8,7,6,20).<br />
<br />
===''n'' Dimensions===<br />
<br />
Infinitely many dimensions exist, and when talking about and arbitrary dimension, the term "''n''-dimensional" is used. The use of multiple dimensions has many concrete applications as well.<br />
<br />
<br />
In Physics, multiple dimensions can describe the motion of ''n'' particles in 3 dimensions. Rather than considering ''n'' particles each with 3 dimensions, one can consider 1 system with 3''n'' dimensions, comprised of all the particles.<br />
<br />
<br />
*Particle 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub>)<br />
<br />
*Particle 2 is at (x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub>)<br />
<br />
*Particle 3 is at (x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
Or:<br />
<br />
*System 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub> , x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub> , x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
<br />
Similarly, in Economics, an economy is described interns of the number of products it produces. Each product has a quantity, so the economy as a whole has one value for each product it produces, leading to a multidimensional description of the state of the economy. The following, hypothetical economy has five attributes, or dimensions.<br />
<br />
<br />
*There are ''v'' units of product 1<br />
<br />
*There are ''w'' units of product 2<br />
<br />
*There are ''x'' units of product 3<br />
<br />
*There are ''y'' units of product 4<br />
<br />
*There are ''z'' units of product 5<br />
<br />
Thus:<br />
<br />
*The economy is in the state (v, w, x, y, z)<br />
<br />
<br />
==References==<br />
Image: http://wikieducator.org/MathGloss/C/Cartesian_plane</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=File:And_richard_did_it_again.jpg&diff=27066File:And richard did it again.jpg2011-07-22T17:57:33Z<p>Rscott3: </p>
<hr />
<div></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Dimensions&diff=27064Dimensions2011-07-22T17:55:43Z<p>Rscott3: /* 4D */</p>
<hr />
<div>{{HelperPage|1=Cross-cap}}<br />
<br />
==Basic Description==<br />
<br />
A dimension can be thought of as a direction of motion. The first dimension has one direction, say "forward and backward". The second dimension has two: "forward and backward", "up and down". The third dimension has three directions: "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Tug-of-war is a one dimensional game; you can only go forward and backward. A game like Mario is two dimensional, which is quite apparent when you can't avoid obstacles by stepping out of the screen. Real life is 3D, you can go "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Dimensionality is a property of both objects and spaces, however the dimension of an object can be different than that of the space it is in.<br />
<br />
<br />
Additionally, there is the zeroth dimension and fractal dimensions. The zeroth dimension has no directions of motion. The zeroth dimension is a point without a "forward and backward", "up and down", or "right and left". Fractal dimensions are discussed in depth [[Fractal Dimension| here]].<br />
<br />
===Dimensions of Objects===<br />
<br />
<div id="Image1">[[Image:Line circle disk.png|Image 1.|thumb|300px|right]]</div><br />
<br />
<br />
Every object has an <balloon title="Belonging to a thing by its very nature.">intrinsic</balloon> dimensionality. Take the line; it has only one direction along it, meaning it is 1 dimensional. The circle is merely a line segment connected front to back. If you are on the circle, you can only go in one direction: around it. So the circle is 1 dimensional as well. On the disk (best thought of as a filled in circle), there are two directions akin to "up and down", "right and left", making the disk two dimensional. The image to the right displays these shapes.<br />
<br />
Whether the line is drawn on a 2 dimensional sheet of paper, or made in 3 dimensions as, say, a string, the line is always 1 dimensional. Similarly, the disk, drawn on paper, or made in 3 dimensions, like a flattened quarter, is always 2 dimensional. Althought a string and a flattened quarter are 3 dimensional because they do still have length, width, and height, they can be used as analogies for an ideal 1 dimensional line and 2 dimensional disk within 3 dimensional space.<br />
<br />
<br style="clear: both" /> <br />
===Dimensions of Spaces===<br />
<br />
Just as objects have dimensions, so do the spaces that objects are in. The dimensions of a space can, as discussed above, be thought of as a direction. We live in 3 dimensional space, meaning we have three directions in which we can move. Mario lives in 2 dimensional space; he can only "move up and down", "left and right". Humans are 3 dimensional objects; It takes three basic directions to describe our bodies:"upward and downward", "leftward and rightward", "forward and backward". Mario is 2 dimensional, and can be described with "upward and downward", "leftward and rightward". Thus, we have the same dimensions as the space where we live, as does Mario. Nevertheless, this is not always the case for objects. Ideally, a line drawn on a piece of paper is a 1 dimensional object that is embedded in the 2 dimensional space of the page. Thus, an object can exist in a space with a higher number of dimensions than that of the itself.<br />
<br />
<br style="clear: both" /> <br />
==Four Dimensions==<br />
===1D===<br />
Each dimension is represented mathematically by a coordinate variable. Any point specified by one variable, such as <math>x=8</math>, is located in 1 dimension, and can be plotted on the 1 dimensional number-line, also known as the x-axis, at the point (8), as shown below. <br />
<br />
::<div id="Image2">[[Image:8 nuberline 2.png|Image 2.|650px]]</div><br />
<br />
===2D===<br />
<br />
If the point is defined by two coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<br style="clear: both" /> <br />
then it can no longer be placed as a point solely on the number-line. It now has two coordinates, One stating that it is located at 8 in the x direction, and the other placing it at 7 in the y direction. Thus, we plot the point as (8,7) on the 2 dimensional xy-plane, which may be familiar from math class. We merely added another direction to the 1 dimensional number-line to arrive at the 2 dimensional plane.<br />
<br />
<br />
<div id="Image3">[[Image:8 7 xy axis 3.png|Image 3.|400px]]</div><br />
<br />
<br style="clear: both" /> <br />
===3D===<br />
<br />
If a point has three coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<math>z=6</math><br />
<br />
Then to capture all of this information on our graph, a third dimension, the z direction, must be added, say out of the paper the graph is drawn on. Our new point, (8,7,6), is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, as illustrated in The graph below.<br />
<br />
<div id="Image4">[[Image:Richard_did_this_for_harrison.jpg|Image 4.|300px]]</div><br />
<br />
===4D===<br />
<br />
Before directly discussing four dimensions, let us introduce an analogue in 2 and 3 dimensions. Notice that, in [[#Image5|Image 5]], (8,7,6) and (8,7,0) have the same x and y coordinates. If we look at them from directly along the z-axis, the two points overlap, appearing as though they were a single point at (8,7) on a 2 dimensional xy-plane ([[#Image6|Image 6]]). As such, they are at the same position in a two dimensions, but have different positions in a third. It is this third coordinate, their z coordinate, which distinguishes them as different points. <br />
<br />
{{{!}} align="center" style="text-align:center;" cellpadding=40 cellspacing=10<br />
{{!}}<div id="Image5"></div>[[Image:Richard_did_this_for_harrison_a_second_time.jpg|Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.|thumb|270px|right]]<br />
{{!}}<div id="Image6"></div>[[Image:And_richard_id_it_again.jpg|Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).|thumb|270px|right]]<br />
{{!}}}<br />
<br />
Just as in the previous examples, a 4<sup>th</sup> dimension can be thought of as merely adding anther coordinate variable describing the location of the point. A point (8,7,6,5) is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, and 5 in a fourth direction, which we will call the w direction. The two points (8,7,6,5) and (8,7,6,20) would appear in the same place as each other on a 3D graph in much the way as the points (8,7,6) and (8,7,0) appeared to be in the same place as each other on the 2 dimensional xy-plane. For (8,7,6,5) and (8,7,6,20), we must simply remember that they have an other direction in which they are not at the same place, even though in the three directions we know of, they are.<br />
<br />
One way to do this would be to give the points different coordinates in time. For instance, an object could be at the point (8,7,6) at 5 seconds after you start measuring, thus (8,7,6,5), and then another one is there at 20 seconds, thus (8,7,6,20).<br />
<br />
===''n'' Dimensions===<br />
<br />
Infinitely many dimensions exist, and when talking about and arbitrary dimension, the term "''n''-dimensional" is used. The use of multiple dimensions has many concrete applications as well.<br />
<br />
<br />
In Physics, multiple dimensions can describe the motion of ''n'' particles in 3 dimensions. Rather than considering ''n'' particles each with 3 dimensions, one can consider 1 system with 3''n'' dimensions, comprised of all the particles.<br />
<br />
<br />
*Particle 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub>)<br />
<br />
*Particle 2 is at (x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub>)<br />
<br />
*Particle 3 is at (x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
Or:<br />
<br />
*System 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub> , x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub> , x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
<br />
Similarly, in Economics, an economy is described interns of the number of products it produces. Each product has a quantity, so the economy as a whole has one value for each product it produces, leading to a multidimensional description of the state of the economy. The following, hypothetical economy has five attributes, or dimensions.<br />
<br />
<br />
*There are ''v'' units of product 1<br />
<br />
*There are ''w'' units of product 2<br />
<br />
*There are ''x'' units of product 3<br />
<br />
*There are ''y'' units of product 4<br />
<br />
*There are ''z'' units of product 5<br />
<br />
Thus:<br />
<br />
*The economy is in the state (v, w, x, y, z)<br />
<br />
<br />
==References==<br />
Image: http://wikieducator.org/MathGloss/C/Cartesian_plane</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=File:And_richard_id_it_again.jpg&diff=27062File:And richard id it again.jpg2011-07-22T17:55:26Z<p>Rscott3: </p>
<hr />
<div></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Dimensions&diff=27055Dimensions2011-07-22T17:52:42Z<p>Rscott3: /* 4D */</p>
<hr />
<div>{{HelperPage|1=Cross-cap}}<br />
<br />
==Basic Description==<br />
<br />
A dimension can be thought of as a direction of motion. The first dimension has one direction, say "forward and backward". The second dimension has two: "forward and backward", "up and down". The third dimension has three directions: "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Tug-of-war is a one dimensional game; you can only go forward and backward. A game like Mario is two dimensional, which is quite apparent when you can't avoid obstacles by stepping out of the screen. Real life is 3D, you can go "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Dimensionality is a property of both objects and spaces, however the dimension of an object can be different than that of the space it is in.<br />
<br />
<br />
Additionally, there is the zeroth dimension and fractal dimensions. The zeroth dimension has no directions of motion. The zeroth dimension is a point without a "forward and backward", "up and down", or "right and left". Fractal dimensions are discussed in depth [[Fractal Dimension| here]].<br />
<br />
===Dimensions of Objects===<br />
<br />
<div id="Image1">[[Image:Line circle disk.png|Image 1.|thumb|300px|right]]</div><br />
<br />
<br />
Every object has an <balloon title="Belonging to a thing by its very nature.">intrinsic</balloon> dimensionality. Take the line; it has only one direction along it, meaning it is 1 dimensional. The circle is merely a line segment connected front to back. If you are on the circle, you can only go in one direction: around it. So the circle is 1 dimensional as well. On the disk (best thought of as a filled in circle), there are two directions akin to "up and down", "right and left", making the disk two dimensional. The image to the right displays these shapes.<br />
<br />
Whether the line is drawn on a 2 dimensional sheet of paper, or made in 3 dimensions as, say, a string, the line is always 1 dimensional. Similarly, the disk, drawn on paper, or made in 3 dimensions, like a flattened quarter, is always 2 dimensional. Althought a string and a flattened quarter are 3 dimensional because they do still have length, width, and height, they can be used as analogies for an ideal 1 dimensional line and 2 dimensional disk within 3 dimensional space.<br />
<br />
<br style="clear: both" /> <br />
===Dimensions of Spaces===<br />
<br />
Just as objects have dimensions, so do the spaces that objects are in. The dimensions of a space can, as discussed above, be thought of as a direction. We live in 3 dimensional space, meaning we have three directions in which we can move. Mario lives in 2 dimensional space; he can only "move up and down", "left and right". Humans are 3 dimensional objects; It takes three basic directions to describe our bodies:"upward and downward", "leftward and rightward", "forward and backward". Mario is 2 dimensional, and can be described with "upward and downward", "leftward and rightward". Thus, we have the same dimensions as the space where we live, as does Mario. Nevertheless, this is not always the case for objects. Ideally, a line drawn on a piece of paper is a 1 dimensional object that is embedded in the 2 dimensional space of the page. Thus, an object can exist in a space with a higher number of dimensions than that of the itself.<br />
<br />
<br style="clear: both" /> <br />
==Four Dimensions==<br />
===1D===<br />
Each dimension is represented mathematically by a coordinate variable. Any point specified by one variable, such as <math>x=8</math>, is located in 1 dimension, and can be plotted on the 1 dimensional number-line, also known as the x-axis, at the point (8), as shown below. <br />
<br />
::<div id="Image2">[[Image:8 nuberline 2.png|Image 2.|650px]]</div><br />
<br />
===2D===<br />
<br />
If the point is defined by two coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<br style="clear: both" /> <br />
then it can no longer be placed as a point solely on the number-line. It now has two coordinates, One stating that it is located at 8 in the x direction, and the other placing it at 7 in the y direction. Thus, we plot the point as (8,7) on the 2 dimensional xy-plane, which may be familiar from math class. We merely added another direction to the 1 dimensional number-line to arrive at the 2 dimensional plane.<br />
<br />
<br />
<div id="Image3">[[Image:8 7 xy axis 3.png|Image 3.|400px]]</div><br />
<br />
<br style="clear: both" /> <br />
===3D===<br />
<br />
If a point has three coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<math>z=6</math><br />
<br />
Then to capture all of this information on our graph, a third dimension, the z direction, must be added, say out of the paper the graph is drawn on. Our new point, (8,7,6), is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, as illustrated in The graph below.<br />
<br />
<div id="Image4">[[Image:Richard_did_this_for_harrison.jpg|Image 4.|300px]]</div><br />
<br />
===4D===<br />
<br />
Before directly discussing four dimensions, let us introduce an analogue in 2 and 3 dimensions. Notice that, in [[#Image5|Image 5]], (8,7,6) and (8,7,0) have the same x and y coordinates. If we look at them from directly along the z-axis, the two points overlap, appearing as though they were a single point at (8,7) on a 2 dimensional xy-plane ([[#Image6|Image 6]]). As such, they are at the same position in a two dimensions, but have different positions in a third. It is this third coordinate, their z coordinate, which distinguishes them as different points. <br />
<br />
{{{!}} align="center" style="text-align:center;" cellpadding=40 cellspacing=10<br />
{{!}}<div id="Image5"></div>[[Image:Richard_did_this_for_harrison_a_second_time.jpg|Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.|thumb|270px|right]]<br />
{{!}}<div id="Image6"></div>[[Image:876 870 2D2.png|Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).|thumb|270px|right]]<br />
{{!}}}<br />
<br />
Just as in the previous examples, a 4<sup>th</sup> dimension can be thought of as merely adding anther coordinate variable describing the location of the point. A point (8,7,6,5) is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, and 5 in a fourth direction, which we will call the w direction. The two points (8,7,6,5) and (8,7,6,20) would appear in the same place as each other on a 3D graph in much the way as the points (8,7,6) and (8,7,0) appeared to be in the same place as each other on the 2 dimensional xy-plane. For (8,7,6,5) and (8,7,6,20), we must simply remember that they have an other direction in which they are not at the same place, even though in the three directions we know of, they are.<br />
<br />
One way to do this would be to give the points different coordinates in time. For instance, an object could be at the point (8,7,6) at 5 seconds after you start measuring, thus (8,7,6,5), and then another one is there at 20 seconds, thus (8,7,6,20).<br />
<br />
===''n'' Dimensions===<br />
<br />
Infinitely many dimensions exist, and when talking about and arbitrary dimension, the term "''n''-dimensional" is used. The use of multiple dimensions has many concrete applications as well.<br />
<br />
<br />
In Physics, multiple dimensions can describe the motion of ''n'' particles in 3 dimensions. Rather than considering ''n'' particles each with 3 dimensions, one can consider 1 system with 3''n'' dimensions, comprised of all the particles.<br />
<br />
<br />
*Particle 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub>)<br />
<br />
*Particle 2 is at (x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub>)<br />
<br />
*Particle 3 is at (x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
Or:<br />
<br />
*System 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub> , x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub> , x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
<br />
Similarly, in Economics, an economy is described interns of the number of products it produces. Each product has a quantity, so the economy as a whole has one value for each product it produces, leading to a multidimensional description of the state of the economy. The following, hypothetical economy has five attributes, or dimensions.<br />
<br />
<br />
*There are ''v'' units of product 1<br />
<br />
*There are ''w'' units of product 2<br />
<br />
*There are ''x'' units of product 3<br />
<br />
*There are ''y'' units of product 4<br />
<br />
*There are ''z'' units of product 5<br />
<br />
Thus:<br />
<br />
*The economy is in the state (v, w, x, y, z)<br />
<br />
<br />
==References==<br />
Image: http://wikieducator.org/MathGloss/C/Cartesian_plane</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=File:Richard_did_this_for_harrison_a_second_time.jpg&diff=27052File:Richard did this for harrison a second time.jpg2011-07-22T17:49:49Z<p>Rscott3: </p>
<hr />
<div></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Dimensions&diff=27048Dimensions2011-07-22T17:43:59Z<p>Rscott3: /* 4D */</p>
<hr />
<div>{{HelperPage|1=Cross-cap}}<br />
<br />
==Basic Description==<br />
<br />
A dimension can be thought of as a direction of motion. The first dimension has one direction, say "forward and backward". The second dimension has two: "forward and backward", "up and down". The third dimension has three directions: "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Tug-of-war is a one dimensional game; you can only go forward and backward. A game like Mario is two dimensional, which is quite apparent when you can't avoid obstacles by stepping out of the screen. Real life is 3D, you can go "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Dimensionality is a property of both objects and spaces, however the dimension of an object can be different than that of the space it is in.<br />
<br />
<br />
Additionally, there is the zeroth dimension and fractal dimensions. The zeroth dimension has no directions of motion. The zeroth dimension is a point without a "forward and backward", "up and down", or "right and left". Fractal dimensions are discussed in depth [[Fractal Dimension| here]].<br />
<br />
===Dimensions of Objects===<br />
<br />
<div id="Image1">[[Image:Line circle disk.png|Image 1.|thumb|300px|right]]</div><br />
<br />
<br />
Every object has an <balloon title="Belonging to a thing by its very nature.">intrinsic</balloon> dimensionality. Take the line; it has only one direction along it, meaning it is 1 dimensional. The circle is merely a line segment connected front to back. If you are on the circle, you can only go in one direction: around it. So the circle is 1 dimensional as well. On the disk (best thought of as a filled in circle), there are two directions akin to "up and down", "right and left", making the disk two dimensional. The image to the right displays these shapes.<br />
<br />
Whether the line is drawn on a 2 dimensional sheet of paper, or made in 3 dimensions as, say, a string, the line is always 1 dimensional. Similarly, the disk, drawn on paper, or made in 3 dimensions, like a flattened quarter, is always 2 dimensional. Althought a string and a flattened quarter are 3 dimensional because they do still have length, width, and height, they can be used as analogies for an ideal 1 dimensional line and 2 dimensional disk within 3 dimensional space.<br />
<br />
<br style="clear: both" /> <br />
===Dimensions of Spaces===<br />
<br />
Just as objects have dimensions, so do the spaces that objects are in. The dimensions of a space can, as discussed above, be thought of as a direction. We live in 3 dimensional space, meaning we have three directions in which we can move. Mario lives in 2 dimensional space; he can only "move up and down", "left and right". Humans are 3 dimensional objects; It takes three basic directions to describe our bodies:"upward and downward", "leftward and rightward", "forward and backward". Mario is 2 dimensional, and can be described with "upward and downward", "leftward and rightward". Thus, we have the same dimensions as the space where we live, as does Mario. Nevertheless, this is not always the case for objects. Ideally, a line drawn on a piece of paper is a 1 dimensional object that is embedded in the 2 dimensional space of the page. Thus, an object can exist in a space with a higher number of dimensions than that of the itself.<br />
<br />
<br style="clear: both" /> <br />
==Four Dimensions==<br />
===1D===<br />
Each dimension is represented mathematically by a coordinate variable. Any point specified by one variable, such as <math>x=8</math>, is located in 1 dimension, and can be plotted on the 1 dimensional number-line, also known as the x-axis, at the point (8), as shown below. <br />
<br />
::<div id="Image2">[[Image:8 nuberline 2.png|Image 2.|650px]]</div><br />
<br />
===2D===<br />
<br />
If the point is defined by two coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<br style="clear: both" /> <br />
then it can no longer be placed as a point solely on the number-line. It now has two coordinates, One stating that it is located at 8 in the x direction, and the other placing it at 7 in the y direction. Thus, we plot the point as (8,7) on the 2 dimensional xy-plane, which may be familiar from math class. We merely added another direction to the 1 dimensional number-line to arrive at the 2 dimensional plane.<br />
<br />
<br />
<div id="Image3">[[Image:8 7 xy axis 3.png|Image 3.|400px]]</div><br />
<br />
<br style="clear: both" /> <br />
===3D===<br />
<br />
If a point has three coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<math>z=6</math><br />
<br />
Then to capture all of this information on our graph, a third dimension, the z direction, must be added, say out of the paper the graph is drawn on. Our new point, (8,7,6), is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, as illustrated in The graph below.<br />
<br />
<div id="Image4">[[Image:Richard_did_this_for_harrison.jpg|Image 4.|300px]]</div><br />
<br />
===4D===<br />
<br />
Before directly discussing four dimensions, let us introduce an analogue in 2 and 3 dimensions. Notice that, in [[#Image5|Image 5]], (8,7,6) and (8,7,0) have the same x and y coordinates. If we look at them from directly along the z-axis, the two points overlap, appearing as though they were a single point at (8,7) on a 2 dimensional xy-plane ([[#Image6|Image 6]]). As such, they are at the same position in a two dimensions, but have different positions in a third. It is this third coordinate, their z coordinate, which distinguishes them as different points. <br />
<br />
{{{!}} align="center" style="text-align:center;" cellpadding=40 cellspacing=10<br />
{{!}}<div id="Image5"></div>[[Image:876 870 3d graph2.png|Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.|thumb|270px|right]]<br />
{{!}}<div id="Image6"></div>[[Image:876 870 2D2.png|Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).|thumb|270px|right]]<br />
{{!}}}<br />
<br />
Just as in the previous examples, a 4<sup>th</sup> dimension can be thought of as merely adding anther coordinate variable describing the location of the point. A point (8,7,6,5) is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, and 5 in a fourth direction, which we will call the w direction. The two points (8,7,6,5) and (8,7,6,20) would appear in the same place as each other on a 3D graph in much the way as the points (8,7,6) and (8,7,0) appeared to be in the same place as each other on the 2 dimensional xy-plane. For (8,7,6,5) and (8,7,6,20), we must simply remember that they have an other direction in which they are not at the same place, even though in the three directions we know of, they are.<br />
<br />
One way to do this would be to give the points different coordinates in time. For instance, an object could be at the point (8,7,6) at 5 seconds after you start measuring, thus (8,7,6,5), and then another one is there at 20 seconds, thus (8,7,6,20).<br />
<br />
===''n'' Dimensions===<br />
<br />
Infinitely many dimensions exist, and when talking about and arbitrary dimension, the term "''n''-dimensional" is used. The use of multiple dimensions has many concrete applications as well.<br />
<br />
<br />
In Physics, multiple dimensions can describe the motion of ''n'' particles in 3 dimensions. Rather than considering ''n'' particles each with 3 dimensions, one can consider 1 system with 3''n'' dimensions, comprised of all the particles.<br />
<br />
<br />
*Particle 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub>)<br />
<br />
*Particle 2 is at (x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub>)<br />
<br />
*Particle 3 is at (x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
Or:<br />
<br />
*System 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub> , x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub> , x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
<br />
Similarly, in Economics, an economy is described interns of the number of products it produces. Each product has a quantity, so the economy as a whole has one value for each product it produces, leading to a multidimensional description of the state of the economy. The following, hypothetical economy has five attributes, or dimensions.<br />
<br />
<br />
*There are ''v'' units of product 1<br />
<br />
*There are ''w'' units of product 2<br />
<br />
*There are ''x'' units of product 3<br />
<br />
*There are ''y'' units of product 4<br />
<br />
*There are ''z'' units of product 5<br />
<br />
Thus:<br />
<br />
*The economy is in the state (v, w, x, y, z)<br />
<br />
<br />
==References==<br />
Image: http://wikieducator.org/MathGloss/C/Cartesian_plane</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Dimensions&diff=27045Dimensions2011-07-22T17:42:59Z<p>Rscott3: /* 3D */</p>
<hr />
<div>{{HelperPage|1=Cross-cap}}<br />
<br />
==Basic Description==<br />
<br />
A dimension can be thought of as a direction of motion. The first dimension has one direction, say "forward and backward". The second dimension has two: "forward and backward", "up and down". The third dimension has three directions: "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Tug-of-war is a one dimensional game; you can only go forward and backward. A game like Mario is two dimensional, which is quite apparent when you can't avoid obstacles by stepping out of the screen. Real life is 3D, you can go "forward and backward", "up and down", "right and left". <br />
<br />
<br />
Dimensionality is a property of both objects and spaces, however the dimension of an object can be different than that of the space it is in.<br />
<br />
<br />
Additionally, there is the zeroth dimension and fractal dimensions. The zeroth dimension has no directions of motion. The zeroth dimension is a point without a "forward and backward", "up and down", or "right and left". Fractal dimensions are discussed in depth [[Fractal Dimension| here]].<br />
<br />
===Dimensions of Objects===<br />
<br />
<div id="Image1">[[Image:Line circle disk.png|Image 1.|thumb|300px|right]]</div><br />
<br />
<br />
Every object has an <balloon title="Belonging to a thing by its very nature.">intrinsic</balloon> dimensionality. Take the line; it has only one direction along it, meaning it is 1 dimensional. The circle is merely a line segment connected front to back. If you are on the circle, you can only go in one direction: around it. So the circle is 1 dimensional as well. On the disk (best thought of as a filled in circle), there are two directions akin to "up and down", "right and left", making the disk two dimensional. The image to the right displays these shapes.<br />
<br />
Whether the line is drawn on a 2 dimensional sheet of paper, or made in 3 dimensions as, say, a string, the line is always 1 dimensional. Similarly, the disk, drawn on paper, or made in 3 dimensions, like a flattened quarter, is always 2 dimensional. Althought a string and a flattened quarter are 3 dimensional because they do still have length, width, and height, they can be used as analogies for an ideal 1 dimensional line and 2 dimensional disk within 3 dimensional space.<br />
<br />
<br style="clear: both" /> <br />
===Dimensions of Spaces===<br />
<br />
Just as objects have dimensions, so do the spaces that objects are in. The dimensions of a space can, as discussed above, be thought of as a direction. We live in 3 dimensional space, meaning we have three directions in which we can move. Mario lives in 2 dimensional space; he can only "move up and down", "left and right". Humans are 3 dimensional objects; It takes three basic directions to describe our bodies:"upward and downward", "leftward and rightward", "forward and backward". Mario is 2 dimensional, and can be described with "upward and downward", "leftward and rightward". Thus, we have the same dimensions as the space where we live, as does Mario. Nevertheless, this is not always the case for objects. Ideally, a line drawn on a piece of paper is a 1 dimensional object that is embedded in the 2 dimensional space of the page. Thus, an object can exist in a space with a higher number of dimensions than that of the itself.<br />
<br />
<br style="clear: both" /> <br />
==Four Dimensions==<br />
===1D===<br />
Each dimension is represented mathematically by a coordinate variable. Any point specified by one variable, such as <math>x=8</math>, is located in 1 dimension, and can be plotted on the 1 dimensional number-line, also known as the x-axis, at the point (8), as shown below. <br />
<br />
::<div id="Image2">[[Image:8 nuberline 2.png|Image 2.|650px]]</div><br />
<br />
===2D===<br />
<br />
If the point is defined by two coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<br style="clear: both" /> <br />
then it can no longer be placed as a point solely on the number-line. It now has two coordinates, One stating that it is located at 8 in the x direction, and the other placing it at 7 in the y direction. Thus, we plot the point as (8,7) on the 2 dimensional xy-plane, which may be familiar from math class. We merely added another direction to the 1 dimensional number-line to arrive at the 2 dimensional plane.<br />
<br />
<br />
<div id="Image3">[[Image:8 7 xy axis 3.png|Image 3.|400px]]</div><br />
<br />
<br style="clear: both" /> <br />
===3D===<br />
<br />
If a point has three coordinates:<br />
<br />
<math>x=8</math><br />
<br />
<math>y=7</math><br />
<br />
<math>z=6</math><br />
<br />
Then to capture all of this information on our graph, a third dimension, the z direction, must be added, say out of the paper the graph is drawn on. Our new point, (8,7,6), is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, as illustrated in The graph below.<br />
<br />
<div id="Image4">[[Image:Richard_did_this_for_harrison.jpg|Image 4.|300px]]</div><br />
<br />
===4D===<br />
<br />
Before directly discussing four dimensions, let us introduce an analogue in 2 and 3 dimensions. Notice that, in [[#Image5|Image 5]], (8,7,6) and (8,7,0) have the same x and y coordinates. If we look at them from directly along the z-axis, the two points overlap, appearing as though they were a single point at (8,7) on a 2 dimensional xy-plane ([[#Image6|Image 6]]). As such, they are at the same position in a two dimensions, but have different positions in a third. It is this third coordinate, their z coordinate, which distinguishes them as different points. <br />
<br />
{{{!}} align="center" style="text-align:center;" cellpadding=40 cellspacing=10<br />
{{!}}<div id="Image5"></div>[[Image:876 870 3d graph2.png|Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.|thumb|270px|right]]</div> <br />
{{!}}<div id="Image6"></div>[[Image:876 870 2D2.png|Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).|thumb|270px|right]]</div><br />
{{!}}}<br />
<br />
Just as in the previous examples, a 4<sup>th</sup> dimension can be thought of as merely adding anther coordinate variable describing the location of the point. A point (8,7,6,5) is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, and 5 in a fourth direction, which we will call the w direction. The two points (8,7,6,5) and (8,7,6,20) would appear in the same place as each other on a 3D graph in much the way as the points (8,7,6) and (8,7,0) appeared to be in the same place as each other on the 2 dimensional xy-plane. For (8,7,6,5) and (8,7,6,20), we must simply remember that they have an other direction in which they are not at the same place, even though in the three directions we know of, they are.<br />
<br />
One way to do this would be to give the points different coordinates in time. For instance, an object could be at the point (8,7,6) at 5 seconds after you start measuring, thus (8,7,6,5), and then another one is there at 20 seconds, thus (8,7,6,20).<br />
<br />
===''n'' Dimensions===<br />
<br />
Infinitely many dimensions exist, and when talking about and arbitrary dimension, the term "''n''-dimensional" is used. The use of multiple dimensions has many concrete applications as well.<br />
<br />
<br />
In Physics, multiple dimensions can describe the motion of ''n'' particles in 3 dimensions. Rather than considering ''n'' particles each with 3 dimensions, one can consider 1 system with 3''n'' dimensions, comprised of all the particles.<br />
<br />
<br />
*Particle 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub>)<br />
<br />
*Particle 2 is at (x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub>)<br />
<br />
*Particle 3 is at (x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
Or:<br />
<br />
*System 1 is at (x<sub>1</sub> , y<sub>1</sub> , z<sub>1</sub> , x<sub>2</sub> , y<sub>2</sub> , z<sub>2</sub> , x<sub>3</sub> , y<sub>3</sub> , z<sub>3</sub>)<br />
<br />
<br />
Similarly, in Economics, an economy is described interns of the number of products it produces. Each product has a quantity, so the economy as a whole has one value for each product it produces, leading to a multidimensional description of the state of the economy. The following, hypothetical economy has five attributes, or dimensions.<br />
<br />
<br />
*There are ''v'' units of product 1<br />
<br />
*There are ''w'' units of product 2<br />
<br />
*There are ''x'' units of product 3<br />
<br />
*There are ''y'' units of product 4<br />
<br />
*There are ''z'' units of product 5<br />
<br />
Thus:<br />
<br />
*The economy is in the state (v, w, x, y, z)<br />
<br />
<br />
==References==<br />
Image: http://wikieducator.org/MathGloss/C/Cartesian_plane</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=File:Richard_did_this_for_harrison.jpg&diff=27043File:Richard did this for harrison.jpg2011-07-22T17:42:19Z<p>Rscott3: </p>
<hr />
<div></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=SB11&diff=26964SB112011-07-22T14:59:31Z<p>Rscott3: /* Anna Donko */</p>
<hr />
<div>=<center> Sweet Briar 2011 </center>=<br />
<br />
==Discussion==<br />
<font color=navy>Please feel free to discuss math pages or leave a message for the creators!</font><br />
<br />
Hey girls, the discussion bar on the very top of every page next to "page" is really helpful and necessary for peer review or faculty review. If you guys go through other people's page and want to comment on their work, please do. [[User:PhoebeJiang|Phoebe]] 15:09, 11 June 2011 (UTC)<br />
<br />
==Summer 2011 Projects==<br />
<br />
===Anna Donko===<br />
*<big>[[Steiner's Chain]]</big><br />
:{{hide|1=<br />
:<font color=peach> Need to work on explaining algebraic formulas.</font> <br />
PLEASE look at this!! I would really appreciate comments and suggestions!!<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:27, 28 June 2011 (UTC): Just left you a lot of comments on the discussion page.</font><br />
:<font color=peach> Working on reading and using all of the comments on the discussion page, thanks all who left me some feedback! </font><br />
:<font color=orangered>Interesting topic! Left some comments! [[User:Rscott3|Richard]] 7/18</font>}}<br />
Submitted for final review <br />
:<font color=darkred> I put comments up [[User:AnnaP|AnnaP]] 13:36, 21 July 2011 (UTC) </font> <br />
::<font color=green>Thanks! I made the adjustments!</font><br />
:::<font color=darkred> I have a few more comments for you [[User:AnnaP|AnnaP]] 05:21, 22 July 2011 (UTC) </font><br />
<br />
*<big>[[Boy's Surface]]</big><br />
:{{hide|1=<br />
:<font color=peach> Leah and I are trying to figure out how we want to lay out our page...Please hold back with the comments while we continue to reorganize as well as add to our page!</font><br />
:<font color=red> Need to work on explanation of parametrization...??</font><br />
:<font color=salmon> Left some comments~ [[User:PhoebeJiang|Phoebe]] 23:56, 13 July 2011 (UTC)</font>}}<br />
::<font color=orangered> Put up some comments. Cool topic. [[User:Rscott3|Richard]] 7/22</font><br />
<br />
*<big>[[Parabolic Integration]]</big><br />
:{{hide|1=<br />
:<font color=peach>Just started this new page!! I am going to ink this to the already existing page, [[Parabola]]</font><br />
::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:51, 12 July 2011 (UTC) Great image! I like both pictures! The title is clear but I think you can make it shorter.</font><br />
:<font color=peach>Going to change my equations so they work out neater for the reader</font><br />
:<font color=black> Planning to add much more, including explanations of why/how parabolas are used in the physical world, as well as differentiate between parabolas and catenaries, lastly I will find the area of an actual life sized piece of parabolic architecture </font>.}}<br />
Submitted for final review<br />
<font color=darkred> You've done great work on this. Before I approve it, there are some significant revisions to be done, though. I'm also thinking that one of your sections really belongs on the parabola page, as opposed to this page. [[User:AnnaP|AnnaP]] 05:09, 22 July 2011 (UTC) </font><br />
<br />
===Phoebe Jiang===<br />
<br />
*<big>[[Application of the Euclidean Algorithm]]</big> <br />
:{{hide|1=<font color=salmon><br />
: Finish up with computer science section and knot theory.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page also. PLEASE leave anything you think will make it better!!! Thank you guys!!</font><br />
:<font color=red>I could really need comments on this page. Thanks in advance!</font><br />
::<font color=salmon> Put up for final review. [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC)</font>}}<br />
:::<font color=darkred> I can see that you've put a lot of good work into this page. Unfortunately, some parts need some heavy editing. I don't have the time to do a very thorough edit through the Chinese remainder theorem and later sections, but even if I did, I'm not sure you'd have time to do all of the edits. This may be one of those pages that you either don't totally finish or have to finish on your own time later on. That happens every year with some pages and isn't a big deal. For now, I'm removing it from the final review list. Let me know if you'd like me to review it again when I have more time (Saturday or later) [[User:AnnaP|AnnaP]] 19:48, 21 July 2011 (UTC) </font><br />
::::<font color=salmon> You are right. I'm so not confident with this page. It has a large amount of heavy math and they are not easy to explain. I'll probably edit it during my own time after this two month summer period. I think it's best for you to leave it for a while and review other pages first. Anyway, do you have any suggestion about future editing for me? Like, do I need to put more explanation or change some sections? No need to give me feedbacks very soon. Thanks! Phoebe 07/21</font><br />
<br />
*<big>[[Pappus Chain]]</big><br />
:{{hide|1=<br />
:<font color=salmon>This is used to be a subsection of [[Arbelos]]. Since arbelos is too long, I made Pappus chain a single page. I don't have time to finish it, so I think I'll leave the rest of the page for other people in the future. </font>}}<br />
:<font color=salmon>Leave it up there as work in progress. Hopefully somebody else will pick it up next year.</font><br />
<br />
*<big>[[Ramsey Number]]</big><br />
:<font color=salmon> This is a helper page. [[User:PhoebeJiang|Phoebe]] 20:09, 20 July 2011 (UTC)</font><br />
<br />
*{{HideThis|1=Completed Pages|2=<br />
*<big>[[Arbelos]]</big><br />
:{{hide|1=<br />
:<font color=salmon>Need to work on the two footnotes. And do a last revision, revise Pappus Chain and Bankoff Chain. <br />
:PLEASE look at this page. Comments are always welcome! </font>[[User:PhoebeJiang|Phoebe]] 19:49, 28 June 2011 (UTC)<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 13:21, 30 June 2011 (UTC): I've gone back through and responded to your comments or crossed things off where we both agree.</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Thank you Kate!!! <3 </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments for you. See my thoughts about a spin-off page. [[User:AnnaP|AnnaP]] 7/12 </font> <br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC) Thank you Anna and I've responded to your comments.</font><br />
::::<font color=darkred> One more round of comments is up [[User:AnnaP|AnnaP]] 15:38, 15 July 2011 (UTC) </font> <br />
:::::<font color=salmon> Responded to Anna's second round of the comments. </font>}}<br />
::::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 13:32, 19 July 2011 (UTC) </font> <br />
:::::::<font color=salmon> Thank you!! [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC) </font><br />
<br />
*<big>[[Euclidean Algorithm]] </big><br />
:{{hide|1=<br />
<font color=salmon><br />
:Need to work on Gabriel Lame and worst case of EA.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page and leave some commenst!!! Thank you guys!! </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 17:39, 15 July 2011 (UTC) </font> <br />
:::<font color=salmon> Thank you! Responded to Anna's comments. </font><br />
::::<font color=darkred> I put up one more comment [[User:AnnaP|AnnaP]] 13:39, 19 July 2011 (UTC) </font><br />
::::<font color=salmon> I moved the proof to the end. Thanks. [[User:PhoebeJiang|Phoebe]] 13:46, 19 July 2011 (UTC) </font>}}<br />
:::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 00:33, 20 July 2011 (UTC) </font><br />
::::::<font color=salmon> Thank you!!! [[User:PhoebeJiang|Phoebe]] 00:55, 20 July 2011 (UTC) </font><br />
<br />
*<big>[[Pigeonhole Principle]]</big><br />
:{{hide|1=<font color=salmon><br />
:Just started it :P </font>[[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
:<font color=salmon>There is a problem I don't understand.</font> [[User:PhoebeJiang|Phoebe]] 02:06, 8 July 2011 (UTC)<br />
::<font color=orangered>Left some comments on the discussion page. Send us the problem you don't understand and we'll try it on our end of things (rscott3@swarthmore.edu). Let me know if you want me to read through the page again or anything. [[User:Rscott3|Richard]] 7/8</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 18:30, 8 July 2011 (UTC)Thank you Richard~ And I figured out the problem just now. :) </font><br />
::::<font color=orangered> Went back and looked over some of your changes. Left some responses to your comments. I'd be happy to read it again at some point if you'd like. [[User:Rscott3|Richard]] 7/12</font><br />
:::::<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC)Great! I'll check your responses. Thank you. </font><br />
:<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:32, 12 July 2011 (UTC)Finish writing the page. Will work on Richard's new comments. </font><br />
:<font color=salmon> Responded to Richard's comments. Appreciate other suggestions.</font><br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:59, 16 July 2011 (UTC) I left a few comments. Overall, I think this page is great! </font color><br />
::<font color=salmon> Thank you Becky! I really appreciate it. I made some changes to this page. [[User:PhoebeJiang|Phoebe]] 16:16, 16 July 2011 (UTC)</font><br />
:<font color=plum>Flora 00:28, 17 July 2011 (UTC)I leave a lot of comments for you. I'll finish '''Why Interesting''' part tomorrow.</font><br />
::<font color=salmon> Thank you very much!!! I've made some changes to the page and responded to your comments!! Is it better now? [[User:PhoebeJiang|Phoebe]] 04:19, 17 July 2011 (UTC)</font><br />
::<font color=plum>I left some comments for your '''Why Interesting''' section, and I responsed some of your changes. The rest are perfect, and I don't think you need any more work.</font><br />
:::<font color=salmon> Thanks! </font><br />
:<font color=salmon> Put up for final review. [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC)</font><br />
:<font color=orangered>Comments up! My new comments are in <font color=blue>blue</font>. Congrats on a good page! [[User:Rscott3|Richard]] 7/20</font><br />
::<font color=salmon> Lol~ Thank you Richard! Appreciate it! <3<3</font>}}<br />
}}<br />
<br />
===Flora Li===<br />
Please leave some comments for my pages, Thanks.<br />
*<big>[[Waves]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:[[User:Flora1103|Flora]] 19:45, 10 June 2011 (UTC)<font color=plum> Need to work on '''Sine Wave Generration'''. Need to add more detail explaination to the graphs and the equetions.</font><br />
:<font color=plum>Flora 16:34, 28 June 2011 (UTC)I changed the title. And try to re-organize it to more "waves".</font><br />
::<font color=orangered>Left some feedback. [[User:Rscott3|Richard]] 6/29</font><br />
::Flora 20:14, 29 June 2011 (UTC)<font color=plum>Thanks very much~</font><br />
:Flora 20:35, 13 July 2011 (UTC)<font color=plum>Send to final review</font><br />
::<font color=darkred> Flora, can you go through the [[Checklist for writing pages]] on the discussion page before I go through the review process for this page? That makes my job much, much easier, and helps me give the most appropriate feedback. You can find examples by looking at the discussion pages of other pages on the final review page. Thanks [[User:AnnaP|AnnaP]] 7/14 </font><br />
::<font color=plum>Flora 21:25, 14 July 2011 (UTC)Have already went through the check list and leave them in discussion page. Sorry about that.</font><br />
:<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:58, 14 July 2011 (UTC) Comments are addressed. </font><br />
::<font color=plum>Flora 02:41, 15 July 2011 (UTC)Have responsed to your suggestions. Thanks very much.</font><br />
}}<br />
:::<font color=darkred> Take a look at my comments on the page [[User:AnnaP|AnnaP]] 16:08, 15 July 2011 (UTC) </font> <br />
:::<font color=plum>Flora 18:46, 15 July 2011 (UTC)Thanks very much for your comments. I responsed already, and I'm working on Fourier images now. I will post them later.</font><br />
:::<font color=plum>Done with all changes. Hope getting approved for final review.</font><br />
:::::<font color=darkred> Just one ''tiny'' comment left [[User:AnnaP|AnnaP]] 19:42, 20 July 2011 (UTC) </font><br />
:::::<font color=plum>Fixed them.</font><br />
::::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 02:45, 21 July 2011 (UTC) </font><br />
::::::<font color=plum>Thank you very much.</font><br />
*<big>[[Dandelin Spheres Theory]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:Flora 22:20, 28 June 2011 (UTC)Have done most of it, still need to work on the references. I'm looking for any comments, plz~<br />
:<font color=plum>[[User:Flora1103|Flora]] 00:00, 11 June 2011 (UTC)I just start this topic. Working ont eh proofs</font><br />
:<font color=plum>Flora 20:50, 16 June 2011 (UTC) I find some useful information on a online school's webpage, but I couldn't open the page today. I don't know why. The url is [http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html].</font><br />
:[[User:PhoebeJiang|Phoebe]] 00:43, 19 June 2011 (UTC) <font color=salmon>I have no problem viewing it. Try a different browser I think. </font><br />
:<font color=plum>Flora 15:39, 20 June 2011 (UTC) I fixed that problem, thx. I now editting "Prove of conic section curve", and I change one image that showed before, so I still need to edit "Explore hyperbola spheres" later.</font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 21:12, 29 June 2011 (UTC): I left you a bunch of comments on this one, but I didn't get all the way through. It was slow reading and I ran out of time.</font><br />
:<font color=plum>Thanks very much.~</font><br />
<br />
:Flora 02:20, 14 July 2011 (UTC)<font color=plum>Start to revise it</font><br />
<br />
:Flora 21:55, 14 July 2011 (UTC)<font color=plum>Re-arranged the outlet, hope this is better.</font><br />
<br />
:<font color=plum>Flora 23:27, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Flora 20:42, 17 July 2011 (UTC)Responsed to Becky's suggestions. Submitted to final review.</font><br />
<br />
<br />
<br />
*<big>[[Dihedral Groups]]</big><br />
:Welcome comments XD<br />
:{{HideThis|1=More|2=<br />
:<font color=plum>Flora 20:54, 27 June 2011 (UTC)I add the main image but not start this page yet. Will work on it from next week. XD</font><br />
:<font color=plum>Flora 16:25, 13 July 2011 (UTC) Almost done with the page, still need to work on complex plane explanation, and subgroup part.</font><br />
:<font color=plum>Flora 01:58, 15 July 2011 (UTC)Finish this page, need to revise.</font><br />
:<font color=plum>Flora 23:26, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Send to final review</font><br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 13:46, 20 July 2011 (UTC)<br />
:::And a couple more small ones! [[User:AnnaP|AnnaP]] 19:35, 20 July 2011 (UTC) </font><br />
::::<font color=plum>Fixed and responsed to you.</font><br />
<br />
==Contact Us==<br />
<br />
===Faculty===<br />
'''Dr.Cammie Barnes''' <br />
<br />
*Email: cbarnes@sbc.edu<br />
<br />
*Skype: Cammie.Barnes<br />
<br />
===Students===<br />
'''Anna Donko'''<br />
<br />
*Email: donko14@sbc.edu<br />
<br />
*Skype: anna.k.donko<br />
<br />
'''Phoebe Jiang'''<br />
<br />
*Email: jiang14@sbc.edu<br />
<br />
*Skype: blacki2014<br />
<br />
'''Flora Li'''<br />
<br />
*Email: li14@sbc.edu<br />
<br />
*Skype: Flora Li<br />
<br />
==Links==<br />
[[Wiki Tricks]]<br />
<br />
[[Pages Ready for Final Review]]<br />
<br />
[[Site programming questions]]<br />
<br />
[[Feedback Requests]]<br />
<br />
[[Math Tools Requests]] ''This page is a place where students whose primary focus is writing pages can post requests for applets, animations, and new images that they'd like to see the computer science students create.'' <br />
<br />
[[User:PhoebeJiang|Phoebe]] 00:45, 19 June 2011 (UTC)<font color=salmon>Thanks for the info!</font><br />
<br />
[http://en.wikipedia.org/wiki/Help:Displaying_a_formula#Forced_PNG_rendering Displaying a Formula]<br />
<br />
[[Page Building Help]]<br />
<br />
[[S11]]<br />
<br />
[[RPI11]]<br />
<br />
[[DU11]]<br />
<br />
[[PartnerHome]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=S11&diff=26963S112011-07-22T14:58:52Z<p>Rscott3: /* Leah's Projects */</p>
<hr />
<div>__TOC__<br />
<br />
== Announcements ==<br />
For public-type help questions, see [[Help:Contents|Help]]. For Swat-specific ones, see [[Swarthmore summer research orientation]].<br />
<br />
<b>Remember to keep your projects sections short and up to date; only the last week of status changes should be mentioned. [[User:Smaurer1|Smaurer1]]</b><br />
<br />
===Group Discussion Questions===<br />
<br />
* [[Topics for conversations through Skype with RPI, SB, and/or Drexel]] Started 6/29.<br />
* [[Who are we writing for?]] Started 6/29.<br />
* [[Possibly expanding student Math Image roles]], to be addressed 7/6.<br />
<br />
===Questions:===<br />
*Things that are listed as Helper Pages but use the Image Page template<br />
:We need to decide whether these pages should be on the Helper Page template, the Image Page template, or both: {{Hide|1=<br />
:*[[Change of Coordinate Systems]] - Image template only; image template live<br />
:*[[Conic Section]] - both templates; image template not live<br />
:*[[Differentiability]] - both templates; image template not live<br />
:*[[Dot Product]] - Image template only; image template not live<br />
:*[[Gradients and Directional Derivatives]] - Image template only; image template not live<br />
:*[[Hyperbolic Geometry]] - Image template only; image template not live<br />
:*[[Inversion]] - Image template only; image template not live<br />
:*[[Iterated Functions]] - Image template only; image template not live<br />
:*[[Parametric Equations]] - Image template only; image template live<br />
:*[[Taylor Series]] - Image template only; image template not live<br />
:*[[Volume of Revolution]] - Image template only; image template live<br />
<br />
:<font color=dodgerblue>''(List complied by [[User:Kderosier|Kate]], June 16)''</font><br />
}}<br />
<br />
<br />
Older questions: {{Hide|1=<br />
*Harrison's question about text being cut off on Cross-cap: {{Hide|1=<br />
*[[User:Htasoff|Htasoff]] 14:16, 8 June 2011 (UTC)<br />
**The text in MME on the [[Cross-cap]] page is getting truncated when viewed through edit with form, but still appears on the page.<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 00:49, 14 June 2011 (UTC): When I've encountered this problem previously, it's been because a set of double curly brackets wasn't closed.</font><br />
}}<br />
<br />
*Harrison's question about creating a list of not-yet-existent Helper Pages: {{Hide|1=<br />
Harrison, 5/26/11:<br />
:*<s>We need a list of empty pages: Penrose Tiles is only linked to one, now two, pages. Empty pages like this could well fade into obscurity.</s><br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 17:09, 7 June 2011 (UTC): Such a list has been created. See [[Existing_Pages_Needing_Work#Empty_.28but_linked_to.29_Pages|here]].</font><br />
}}<br />
<br />
*Spam conversation: {{Hide|1=<br />
<font color=dodgerblue><br />
*[[User:Kderosier|Kate]] 14:38, 27 May 2011 (UTC): '''We have some new users who are creating a bunch of pages with links to illegally download or watch movies.''' At first, we though it might be someone from Sweet Briar practicing wiki-syntax, but now it's just starting to look like spam:<br />
**[[Watch_Sniper:_Reloaded_film_in_hd|This is the type of page I'm talking about]]<br />
**[[Special:Contributions/Calrivenick|List of pages created by Calrivenick]]<br />
**[[Special:Contributions/Cadedesi|List of pages created by Cadedesi]]<br />
</font><br />
:Let's talk about what to do this afternoon. [[User:Gene|Gene]] 15:19, 27 May 2011 (UTC)<br />
:<font color=dodgerblue>This problem has continued over the weekend. Someone spammed the talk page for Fun Topology with comments about buying Cialis and stuff. They also posted a lot more movie download pages under the Cadedesi username. I'm going to go through and delete again.</font><br />
:<font color=dodgerblue>The spam comments on Fun Topology were coming from this computer's IP address. </font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 14:34, 6 June 2011 (UTC): Haven't seen any more spam activity for a few days. I assume some computer people have handled the issue? I'm going to hide this conversation so that it's not taking up space on S11.</font><br />
}}<br />
<br />
*Citations/footnotes conversation: {{Hide|1=<br />
:I spent a really long time wandering around MediaWiki and Wikipedia this morning trying to figure out how to do footnotes/citations the way I wanted to - now that I know how, should I add instructions to one of the many help mages on Math Images? If so, which page? (Kate, 5/17)<br />
<br />
:<font color=red> Answer: put it here at [[Help:Wiki_Tricks|Wiki Tricks]] (XD, 5/17) </font><br />
<br />
:[[User:Smaurer1|Smaurer1]] 19:43, 17 May 2011 (UTC) Well, it's not clear that webarticles should have footnotes, although Wikipedia does. In text references may be better. This is surely something we should discuss as a group, and find out what last year's group decided, if they did. If there are footnotes, there has got to be a way to get back seamlessly to where you were before you jumped to the footnote.<br />
<br />
:Also, as for citations, we should be uniform in their format.<br />
<br />
:Finally, you can use 4 tildes to put your username and time stamp on your comments, and 3 tildes for just your username. <br />
<br />
:<font color=dodgerblue>Well, I wasn't doing comment-y type footnotes, I just wanted specific sentences to link to items in my References section. I think that format is better than full intext-citations, because it brings you to the source if that's what you're interested in, but it takes up less space if you're not. The references template we have comes equipped with little links to jump you back up, too- if you look at the [[Quipu#References|Quipu]] page, you can see that it lists each of the sections that reference is linked from, and those links bring you to the reference in that section. All in all, I think it's a clear and intuitive way to do references for both the writer and the reader (although, like most things, it doesn't play well with our hidden sections), and I'm going to go ahead and put the instructions up in WikiTricks. -[[User:Kderosier|Kate]] </font><br />
<br />
{{Hide|1=<br />
*What the help pages say now:<br />
**[[Tour_the_Math_Images_Project#Anatomy_of_a_page|The Tour page's "Anatomy of a page" section]]<br />
**[[Checklist_for_writing_pages#References_and_footnotes|The "References and footnotes" section on the checklist for writing pages]]<br />
*The way to do Wikipedia-like references:<br />
**[[Wiki_Tricks#How_to_do_Citations|Wiki Tricks - Citations section]]<br />
}}<br />
<br />
}}<br />
<br />
*Invisible Comments conversation: {{Hide|1= One of you asked "How do you put invisible comments in the source code?"<br />
<br />
:Answer: Same way you do in html, like this <nowiki><br />
:<!-- hidden stuff --><br />
:</nowiki><br />
:However, if you want to make comments about an article for its author, the comments are more likely to be seen if you either<br />
:* put it in the discussion page, or<br />
:* if it is important to put it right by the material commented on, put it in the article in color with your username and time stamp included.<br />
<br />
:Hidden comments in the source code are likely to be overlooked, except perhaps if they are written by the author him/herself, as a note for further development.<br />
}}<br />
}}<br />
<br />
== Current Individual Projects ==<br />
<br />
===Steve M (aka Prof Maurer)'s Role===<br />
{{Hide|1=<br />
My role is 2-fold:<br />
<br />
<ol><br />
<li> Come see me to sound me out (if you wish) on the mathematical appropriateness of an idea for a page, or for possible references.<br />
<br />
<li> Once you have a reasonable amount written, and want feedback on the quality and correctness of the mathematical exposition, ask me to look it over and then we will have a conference. (Abram and the oldies are as good as I am at discussion general organization and clarity issues.) <br />
</ol><br />
<br />
We have agreed to put a record on this S11 page of what we are doing and what help we want from others, but in addition tell me in person or by email if you want to conference with me.<br />
}}<br />
<br />
===Harrison's Projects===<br />
<br />
[[Harrison's detritus]]<br />
<br />
*[[User:Htasoff|Htasoff]] 23:44, 13 July 2011 (UTC) Pages will be submitted for final review in 1 - 3 days. A few, final comments are welcome. Real Projective Plane is still in the works, though.<br />
<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:19, 14 July 2011 (UTC): Hey Harrison, do you know if that one picture in inverse trig that you got from somewhere on the internet is a picture we can use? Can you check please?</font><br />
<br />
[[Rope around the Earth]]<br />
:<font color=darkred> Approved </font><br />
<br />
<font color=darkred> I have one small suggested wording comment for you on [[Congruent triangles]], then it's good to go. [[User:AnnaP|AnnaP]] 13:48, 21 July 2011 (UTC)</font><br />
<br />
===Richard's Projects===<br />
<br />
<br />
<font color=orangered> <br />
<br />
:APPLET INFO:<br />
:{{Hide|1=<br />
<br />
[[User:Alimurreza|Alimurreza]] 02:53, 6 July 2011 (UTC)I am working on your applet. Check this out here @([http://mathforum.org/mathimages/index.php/DU11 Reza's work]).<br />
:APPLET UPDATE:[[User:Alimurreza|Alimurreza]] 02:41, 15 July 2011 (UTC)I am done with Ambiguous Case applet. Please, check the applet. Feel free to send me any feedback or change-request.<br />
What i want mine to look like, but http://www.mccsc.edu/~aterwill/ambiguouscaseapplet/Ambiguous_Case_applet.html doesn't show the completed triangles.<br />
<br />
I like how this one shows the completed triangles<br />
http://www.mnwest.edu/fileadmin/static/website/dmatthews/Geogebra/AmbiguousCase01.html<br />
<br />
}}<br />
<br />
'''Completed Pages'''<br />
*[[Ambiguous Case]] [[User:Rscott3|Richard]] 7/19<br />
<br />
*[[Law of cosines]]<font color=orangered> [[User:Rscott3|Richard]] 6/23</font><br />
<br />
*[[Law of Sines]] [[User:AnnaP|AnnaP]] 7/7<br />
<br />
*[[Solving Triangles]] 7/12<br />
<br />
:Other ideas: {{Hide|1=<br />
*inscribed angles?<br />
<br />
*Normal Distribution?<br />
<br />
*Birthday Paradox?<br />
}}<br />
</font><br />
<br />
===Dayo's Projects===<br />
Current projects<br />
*[[Inscribed figures]] : there's a [[Compass & Straightedge Construction and the Impossible Constructions]] page, but I think that inscribed figures deserves its' own page. What do others think? <br />
::<font color=slateblue> [[User:Rebecca|Rebecca]] 01:07, 9 July 2011 (UTC) I left comments on the discussion page. </font color><br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 15:23, 18 July 2011 (UTC): I left you some comments!</font><br />
::<font color=slateblue> [[User:Rebecca|Rebecca]] 02:06, 22 July 2011 (UTC) Dayo has suggestions from me from Wednesday night </font color><br />
<br />
Future Projects: {{Hide|1=<br />
*[[Mathematics in architecture]]:make changes akin to [[Math for Computer Graphics and Computer Vision]], including:<br />
<br />
::*[[Cross sections]]: calculus application page, including examples of Tokyo international Forum, Suransuns Bridge, and other structures which could be thought of and put together easily in terms of their cross sections <br />
::*[[Torus]] edits, additions concerning the Torus in construction and architecture<br />
::*[[Domes]]:conic sections, arches, parabolas<br />
::*[[The Henderson Waves Bridge]]: sinusoids in architecture, parametric design<br />
::*[[Catenary]]: More real world examples: namely bridges<br />
::*[[Bridge of Peace]]: The equation(s) used to generate the surface, possibly words from the architect, very new, may be hard to get concrete technical information <br />
::*[[Kurilpa Bridge]]: Everyone have a look at the image and tell me what kind of actual subjects you could find in it, namely in the cables and tubes.<br />
::*Teaching Materials(6/30): ''growing up with science: projects'' could be the sort of activities we're looking for. I used these with a class, and think people should look at them.<br />
<br />
<br />
<br />
<br />
}}<br />
<br />
on hold: <br />
*[[Parametric Equations]]: integrating Xingda's page from [[S10]] into page.<br />
<br />
===Diana's Projects===<br />
<br />
====Current====<br />
*[[String Art Calculus]]<br />
<font color=green> Chris 7.21.11 I've put up a couple comments. </font> <br />
<br />
*[[Logistic Bifurcation]]<br />
:<font color=darkred> I've put up comments. There are a few places that could use some work. [[User:AnnaP|AnnaP]] 7/10 </font> <br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:26, 16 July 2011 (UTC) This is a very impressive page. I put up a few small suggestions on the discussion page. </font color><br />
<br />
*[[Markus-Lyapunov Fractals]]<br />
:<font color=darkred> Approved [[User:AnnaP|AnnaP]] 7/14 </font><br />
<br />
====Ideas for later projects====<br />
{{Hide|1=<br />
*Chirikov-Taylor Maps<br />
**This seems like a natural extension of the Markus-Lyapunov Fractals page, but maybe the math involved in the two is too similar?<br />
*This aspect of pendular motion:<br />
**[[http://www.youtube.com/watch?v=yVkdfJ9PkRQ&feature=player_embedded|Varied-Length Pendulums]]<br />
**I'm not sure how or whether to use this -- does a ''moving'' image count as a "math image"? -- But it's incredible, and I'd love to explore it.<br />
*Kuen Surface<br />
**It's just really cool.<br />
}}<br />
<br />
===Kate's Projects===<br />
*[[Anne Burns' Mathscapes]] (Scrapped out of [[Mountains In Spring|three]] [[Mathscape|other]] [[Fractal Scene I|pages]]):<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:29, 18 July 2011 (UTC): As soon as I get confirmation that Anne Burns doesn't mind us using her images, I'll submit this for final review.</font><br />
<br />
*Finished pages: {{Hide|1=<br />
*[[Perko pair knots]]:<br />
:<font color=darkred> Approved </font><br />
<br />
:*[[Critical Points]]:<br />
::<font color=darkred> Approved, but I did have one note on whether or not you intended to add something. It's fine as is, but I wanted to put up a suggestion. [[User:AnnaP|AnnaP]] 16:24, 15 July 2011 (UTC) </font><br />
:::<font color=dodgerblue>I responded to that note.</font><br />
<br />
:*[[Summation Notation]]:<br />
::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 16:24, 15 July 2011 (UTC) </font><br />
<br />
:*[[Bases]]:<br />
:: <font color=dodgerblue>[[User:Kderosier|Kate]] 17:31, 11 July 2011 (UTC): Changed things in response to Chris' comments.</font><br />
<br />
:* [[Quipu]]:<br />
::<font color=darkred> Put up as ready for the public 6/30 [[User:AnnaP|AnnaP]] </font><br />
<br />
:* [[Basic Trigonometric Functions]]:<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 18:11, 30 June 2011 (UTC): Changed the the things that were bolded.</font><br />
<br />
<br />
}}<br />
<br />
===Leah's Projects===<br />
*[[Bedsheet Problem]]<br />
going to do last edits [[User:Ljeanlo1|Ljeanlo1]] 23:57, 20 July 2011 (UTC) <br />
<br />
*[[Boy's Surface]] <br />
: take a look on this tomorrow. [[User:Ljeanlo1|Ljeanlo1]] 23:57, 20 July 2011 (UTC) <br />
: <font color=slateblue> [[User:Rebecca|Rebecca]] 01:34, 22 July 2011 (UTC) I left comments on the discussion page </font color><br />
::<font color=orangered> Put up some comments. Cool topic. [[User:Rscott3|Richard]] 7/22</font><br />
<br />
*[[Snell's Law]] <br />
-feedback page<br />
up for final review [[User:Ljeanlo1|Ljeanlo1]] 20:04, 21 July 2011 (UTC)<br />
<br />
*[[Witch of Agnesi]] <br />
up for final review [[User:Ljeanlo1|Ljeanlo1]] 20:05, 20 July 2011 (UTC) <br />
<br />
*[[Dot Product]] <font color=darkred> Approved, 7/14 </font><br />
<br />
*[[Vector]] <font color=darkred> Approved, 7/10 </font><br />
<br />
==Requests to S10 Students==<br />
<br />
<br />
*<s>Can XD do a demo for MATLAB?</s> Done - [[Demo of MATLAB using the example of Bifurcation]]<br />
<br />
==Useful Links==<br />
[[S10]]<br />
<br />
[[SB11]]<br />
<br />
[[RPI11]]<br />
<br />
[[DU11]]<br />
<br />
[[Pages Ready for Final Review]]<br />
<br />
[[Feedback Requests]]<br />
<br />
[[Sample discussion page]]<br />
<br />
[[Math Tools Requests]] ''This page is a place where students whose primary focus is writing pages can post requests for applets, animations, and new images that they'd like to see the computer science students create.''<br />
<br />
[[Page Building Help]]<br />
<br />
[[Help:Wiki Tricks|Wiki Tricks]]<br />
<br />
[[From a Bunch of Old Timers]]<br />
<br />
[[List of summer 2010 pages]]<br />
<br />
http://en.wikipedia.org/wiki/Help:Displaying_a_formula<br />
<br />
[[PartnerHome]]<br />
<br />
[[Existing Pages Needing Work]]<br />
<br />
[[Site programming questions]]<br />
<br />
== Current Group Projects ==</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Boy%27s_Surface&diff=26962Talk:Boy's Surface2011-07-22T14:57:40Z<p>Rscott3: /* General Comments */</p>
<hr />
<div>==General Comments==<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I ralise this page is very much still in the works. Nevertheless, considering the topic's relation to many of the pages I am working on, I wanted to check it just to make sure all our pages were being decently consistent. All my comments were written keeping in mind that the page is still in progress.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC)My main comment is that much of what is currently on the page is definitions and introductions to topological terms and concepts. I think that this would be better done on the Topology Glossary helper page that I just started based on Leah's suggestion. Many of the terms in topology have nuances in their definitions (ie. manifold, embedding, immersion) that can only be sufficiently explained in a section devoted to the term itself (I've found ''immersion'' to be a particularly tricky one, and am still constantly trying to straighten out what, exactly, characterizes it.)<br />
::*<font color=orangered> I second this. I don't think that this is the place for all of those definitions. I know that this page is still in the construction phase, but it seems like this page is more about topology than Boy's Surface specifically. Perhaps more images of the surface might help as well. [[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*I also tend to think that some of the definitions might be oversimplified. For example, I think there's a set of things that make something orientable or not (like continuity and differentiability) I think??? and parametrization is a bit simplified<br />
::In the same sense though, I think that there are one or two definitions that need a bit more simplification, like immersion.[[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*[[User:Rscott3|Richard]] 7/22 The tone of this page is very very conversational. Conversational is good, but there are some sentences and phrases here that just make the page longer and unnecessarily wordy. I think you can stay conversational with the way you guys explain things without some of these phrases. For example:<br />
::#"This probably sounds like a whole new language, but below Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!" in the BD<br />
::#"Before going into great detail with numerous definitions, we will layout Boy's Surface for you in bullet form:" in the BD<br />
::#"So what’s a manifold?" in the manifold section<br />
::#"Now we will tackle a few more vital terms associated with Boy's Surface." in the immersion section<br />
::#"Next, we come to embedding, it is important to understand this term because" and "Luckily, unlike some of the previous terms, the definition is straight forward." in the immersion section<br />
::#"First, I will explain what parametrization is." in the parametrization section.<br />
<br />
<br />
*You sometimes switch from "we" to "I". Personally, I avoid first person, but I think all this needs is some consistency. [[User:Rscott3|Richard]] 7/22<br />
</font><br />
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<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 23:18, 13 July 2011 (UTC) This is a cool image and fascinating page. You got a lot of interesting stuff in it. I'm really interested in this topic. I think there are several hard terms and conclusions needed more precise explanation (see my comments under each section). I understand they are really hard to explain and hard to understand. So try to add more contexts. :) Looking forward to the accomplished page!</font><br />
<br />
*<font color=salmon> I made several minor edits on your page. They are really minor... Plural form, an extra comma, something like that...</font><br />
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<font color=slateblue>[[User:Rebecca|Rebecca]] 01:10, 22 July 2011 (UTC) Nice collaboration! This page is a great addition to the site. </font color><br />
<br />
==Intro==<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 22:56, 13 July 2011 (UTC) Okay, before I went on looking your definitions of the terms, I'm overwhelmed by those fancy math terms in the intro and in the basic description. Maybe you can inform the readers that you are gonna explain every one of them shortly after. </font><br />
<br />
* [[User:Gene|Gene]] Good plan. It would be great if you could give some sort of an intuitive intro (this one is a bit too much like Wikipedia's, too). Wording: should be "<s>The</s>[this "The" ain't needed] Boy’s Surface is an immersion <s>on</s> of the projective plane in three-dimensional space."<br />
<font color=slateblue><br />
<br />
*[[User:Rebecca|Rebecca]] 01:01, 22 July 2011 (UTC) Why don't you link to the "real projective plane" page when you first mention it?<br />
* I think the first paragraph of this page is too complicated. I would hold off on mentioning the real projective plane until late in the page because it's a confusing topic. I think the image description should be cut down to "The object in this image is called Boy's Surface, which is a single sided surface with no edges." <br />
* "The model was constructed as well as donated by Mercedes-Benz." What about... "The model was constructed as well as donated by Mercedes-Benz, and it can be seen in the image below."<br />
</font color><br />
<br />
*<font color=orangered> I think you could even use the same stuff in the intro, just reword it to make it sound more exciting. Something like: "While trying to prove that immersion did not exist, Boy disproved his own theory in 1901 with his discovery of Boy's Surface." [[User:Rscott3|Richard]] 7/22</font><br />
<br />
==Basic Description==<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) "Topology focuses on objects that remain constant regardless how distorted the object is."<br />
** This is not correct; you may very well have meant "Topology is the study of properties that remain constant regardless how distorted the object is", which is correct.<br />
*<font color=salmon>It's kinda repetitive here. You mentioned Werner Boy and others things in the intro before. It's just a minor thing. </font><br />
<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:03, 22 July 2011 (UTC) I think you need to rearrange the basic description. My suggestions are below:<br />
:* I would start the section with ...<br />
::: Boy's Surface is:<br />
::::*A <b>non-orientable</b> surface. Nonorientable means (definition).<br />
::::* An <b> immersion </b> of [[the real projective plane]] in 3 dimensional space. This means..... (explanation).<br />
::::* One possible parametrization of the surface obtained by sewing a Mobius strip to the edge of a disk. <br />
* I think this a better way to do it than to explain things before you give the definitions- people wont be able to follow what you're saying without the definitions anyway.<br />
* Then I think you should move this to after the bullets. "Boy's surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. This probably sounds like a whole new language, but below the Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!<br />
::*<font color=orangered>I think you couldeven reword this to be:"Boy's surface is one possible parametrization of the real projective plane, the surface obtained by sewing a Möbius strip to the edge of a disk."[[User:Rscott3|Richard]] 7/22</font><br />
Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz."<br />
</font color><br />
<br />
<br />
<br />
===Manifolds/Surfaces===<br />
*<font color=orangered> Is "Topology" supposed to be capitalized? [[User:Rscott3|Richard]] 7/22<br />
<br />
*I'd avoid using "and other cool math properties". [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I laud your attempt to intuitively introduce the audience to the concept of a manifold, however the definition contains a quite a few nuances that make me think it would best be done on the Topology Glossary page, where it can be explained in better detail.<br />
<br />
*<font color=salmon> I like your analog of the square and tossed blanket, but I think you can explain what a manifold is better here. I still don't understand what exactly manifold is. Try to add a specific definition of manifold. Is it a certain shape? A general surface? Or a topological space?You can put pictures and tell the readers which one is a manifold and which one is not and why. Pictures and specific examples always help a lot. </font><br />
<br />
* [[User:Gene|Gene]] <s>The</s> Boy''''s''' surface is one of the shapes that is well known in Topology,<s>. Topology is</s> a branch of mathematics. You can think it as an abstract and more advanced version of geometry. Topology focuses on objects that remain constant regardless how distorted the object is. [How 'bout a simple example?]<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:04, 22 July 2011 (UTC) Manifold should be bolded.<br />
*The tossed blanket example is helpful and well explained. Nice work!<br />
</font color><br />
<br />
===Non Orientable===<br />
*<font color=orangered>Is "placed at every location" the right phrase to use? [[User:Rscott3|Richard]] 7/22<br />
<br />
*I think the Earth example is really good. What if you use it to describe manifold instead/as well?[[User:Rscott3|Richard]] 7/22</font><br />
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<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I don't understand the explanation of non-orientability. I like the earth analogy, but I can't follow the rest, especially how the image illustrates the concept.<br />
<br />
*<font color=salmon>I can understand this part well. Like the earth analogy and Mobius Strip. But I don't really understand how boy's surface is non-orientable. I guess it's because I can't see what's the back of the boy's surface from the main image. </font><br />
<br />
<font color=slateblue><br />
* If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right. </font color><br />
<br />
===Immersion, The Real Projective Plane, and Embedding===<br />
*<font color=orangered>I don't know what Euler Characteristics are. Maybe you could describe them? 7/22<br />
<br />
*You say that the symbol is the "Greek letter chi", but you don't explain what it represents. [[User:Rscott3|Richard]] 7/22<br />
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*I'm confused by the definition of immersion. [[User:Rscott3|Richard]] 7/22<br />
<br />
*You should define/make a mouseover for "injective" [[User:Rscott3|Richard]] 7/22</font><br />
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*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Though I fall victim to shortening "Real Projective Plane" to "Projective Plane" many-a-time, they are distinct things, and should not be introduced as synonymous.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Important: The Boy's surface is an immersion Not and embedding. The self intersections let you know that it is not.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Overall, I think this is a topic for the Topology Glossary helper page. Moving the discussions and definitions of these terms to the helper page will free up this page and let it focus more on Boy's surface. It will also allow a more in depth discussion of these terms, which can be extremely nuanced. On that note, I was very confused by the introductions to ''immersion'', ''embedding'', ect. that was provided.<br />
<br />
:<font color=salmon> Those definitions are necessary for this page. I think you can keep them on your page and then put them on the Topology Glossary as well.</font><br />
<br />
I noticed that you have this listed under geometry, but not topology (but in your page is says that this is a topology subject). I didn't want to add it to topology myself in case the form messed anything up (it's happened to me). <br /><br />
:: [[User:Nordhr|Nordhr] 18::44 27 Jun 2011 <br /><br /><br />
<br />
<br />
Hey Anna, what would you like to do with this: <br />
"The projective space is a modified Euclidean space where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. The projective space becomes the construction of the many circle with an additional circle at infinity. It is a fact that the real projective plane cannot be shown in three space without it passes through itself somewhere."<br />
<br />
Hey Leah, I think we are going to have to simplify that down a little and maybe provide a picture as an aid in understanding?<br />
[[User:Ljeanlo1|Ljeanlo1]] 17:32, 1 July 2011 (UTC)<br />
<br />
* <font color=salmon> About that algebraic equation, tell us what v and e represent. I know v stands for vertices and e for edges, but it's best if you make it clear.</font><br />
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* <font color=salmon> I got lost in understanding "immersion." I have no idea what that definition from WolframMathWorld means. Need more explanation. </font><br />
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<font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:08, 22 July 2011 (UTC) If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right.<br />
* "Straight forward" should be "straightforward"<br />
* You say that non-continuous means that every input has only one output. Your picture of non-continuous doesn't show this though. It shows two inputs mapping to one output, not one input mapping two outputs. I'm not sure which is actually true, but they should be consistent. <font color><br />
<br />
===Constructing Boy's Surface===<br />
<font color=orangered> <br />
*For the dscription, are you describing it like you're on the surface? [[User:Rscott3|Richard]] 7.22<br />
<br />
*Is "axis" plural? [[User:Rscott3|Richard]] 7/22</font><br />
<br />
==Constructing Boy's Surface==<br />
<br />
* <font color=salmon> For the last sentence in the second para: ''Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth, if you go far enough, most likely to infinity, you are going to return to your place of origin.'' <br />
<br />
::1. ''Like the graph.'' Which graph?<br />
::2. Found five commas in this sentence. You break this sentences into too many parts. <br />
::3. Despite those two points, you made it clear why you go back to the starting point.<br />
<br />
* Last sentence before the video: ''By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis. '' I don't get it. How they could be connected together? Need more illustrations here (pictures if possible). And does it have to be like these three combination pairs (aka. + x and - y, +y and -z and + z and -x )? <br />
<br />
* Need more explanation of the video. What I got from this video is that you can return to the origin. Then what happened after 0:50? Please forgive me if I'm getting picky....!!!! These are just my feelings and you don't have to agree to me. <br />
</font><br />
<br />
</font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:09, 22 July 2011 (UTC) I think that the video should definitely be moved up to the basic description. It was very helpful to be able to see Boy's Surface more clearly, and the video is much less complicated that the manifolds/surfaces section. We advise people to put the easier material up at the top of the page. <br />
* I agree with Harrison- The video could use a short explanation as well. This could be added to the basic description.<br />
</font color><br />
<br />
==Parametrization==<br />
<font color=salmon><br />
* Explain to us what complex number means via the mouse over in case other people are not familiar with it.<br />
<br />
* As for the parametrization equations, make sure you explain how you get <math>X = \frac{g_1}{g} </math>, <math>Y = \frac{g_2}{g}, </math> and <math>Z = \frac{g_3}{g} </math> and tell us what <math>Im, Re</math> mean. I just realize that you guys are not done with the page. Just make sure you make those equations clear.<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Boy%27s_Surface&diff=26960Talk:Boy's Surface2011-07-22T14:55:57Z<p>Rscott3: /* Basic Description */</p>
<hr />
<div>==General Comments==<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I ralise this page is very much still in the works. Nevertheless, considering the topic's relation to many of the pages I am working on, I wanted to check it just to make sure all our pages were being decently consistent. All my comments were written keeping in mind that the page is still in progress.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC)My main comment is that much of what is currently on the page is definitions and introductions to topological terms and concepts. I think that this would be better done on the Topology Glossary helper page that I just started based on Leah's suggestion. Many of the terms in topology have nuances in their definitions (ie. manifold, embedding, immersion) that can only be sufficiently explained in a section devoted to the term itself (I've found ''immersion'' to be a particularly tricky one, and am still constantly trying to straighten out what, exactly, characterizes it.)<br />
::*<font color=orangered> I second this. I don't think that this is the place for all of those definitions. I know that this page is still in the construction phase, but it seems like this page is more about topology than Boy's Surface specifically. [[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*I also tend to think that some of the definitions might be oversimplified. For example, I think there's a set of things that make something orientable or not (like continuity and differentiability) I think??? and parametrization is a bit simplified<br />
::In the same sense though, I think that there are one or two definitions that need a bit more simplification, like immersion.[[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*[[User:Rscott3|Richard]] 7/22 The tone of this page is very very conversational. Conversational is good, but there are some sentences and phrases here that just make the page longer and unnecessarily wordy. I think you can stay conversational with the way you guys explain things without some of these phrases. For example:<br />
::#"This probably sounds like a whole new language, but below Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!" in the BD<br />
::#"Before going into great detail with numerous definitions, we will layout Boy's Surface for you in bullet form:" in the BD<br />
::#"So what’s a manifold?" in the manifold section<br />
::#"Now we will tackle a few more vital terms associated with Boy's Surface." in the immersion section<br />
::#"Next, we come to embedding, it is important to understand this term because" and "Luckily, unlike some of the previous terms, the definition is straight forward." in the immersion section<br />
<br />
<br />
*You sometimes switch from "we" to "I". Personally, I avoid first person, but I think all this needs is some consistency. [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
<br />
<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 23:18, 13 July 2011 (UTC) This is a cool image and fascinating page. You got a lot of interesting stuff in it. I'm really interested in this topic. I think there are several hard terms and conclusions needed more precise explanation (see my comments under each section). I understand they are really hard to explain and hard to understand. So try to add more contexts. :) Looking forward to the accomplished page!</font><br />
<br />
*<font color=salmon> I made several minor edits on your page. They are really minor... Plural form, an extra comma, something like that...</font><br />
<br />
<font color=slateblue>[[User:Rebecca|Rebecca]] 01:10, 22 July 2011 (UTC) Nice collaboration! This page is a great addition to the site. </font color><br />
<br />
==Intro==<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 22:56, 13 July 2011 (UTC) Okay, before I went on looking your definitions of the terms, I'm overwhelmed by those fancy math terms in the intro and in the basic description. Maybe you can inform the readers that you are gonna explain every one of them shortly after. </font><br />
<br />
* [[User:Gene|Gene]] Good plan. It would be great if you could give some sort of an intuitive intro (this one is a bit too much like Wikipedia's, too). Wording: should be "<s>The</s>[this "The" ain't needed] Boy’s Surface is an immersion <s>on</s> of the projective plane in three-dimensional space."<br />
<font color=slateblue><br />
<br />
*[[User:Rebecca|Rebecca]] 01:01, 22 July 2011 (UTC) Why don't you link to the "real projective plane" page when you first mention it?<br />
* I think the first paragraph of this page is too complicated. I would hold off on mentioning the real projective plane until late in the page because it's a confusing topic. I think the image description should be cut down to "The object in this image is called Boy's Surface, which is a single sided surface with no edges." <br />
* "The model was constructed as well as donated by Mercedes-Benz." What about... "The model was constructed as well as donated by Mercedes-Benz, and it can be seen in the image below."<br />
</font color><br />
<br />
*<font color=orangered> I think you could even use the same stuff in the intro, just reword it to make it sound more exciting. Something like: "While trying to prove that immersion did not exist, Boy disproved his own theory in 1901 with his discovery of Boy's Surface." [[User:Rscott3|Richard]] 7/22</font><br />
<br />
==Basic Description==<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) "Topology focuses on objects that remain constant regardless how distorted the object is."<br />
** This is not correct; you may very well have meant "Topology is the study of properties that remain constant regardless how distorted the object is", which is correct.<br />
*<font color=salmon>It's kinda repetitive here. You mentioned Werner Boy and others things in the intro before. It's just a minor thing. </font><br />
<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:03, 22 July 2011 (UTC) I think you need to rearrange the basic description. My suggestions are below:<br />
:* I would start the section with ...<br />
::: Boy's Surface is:<br />
::::*A <b>non-orientable</b> surface. Nonorientable means (definition).<br />
::::* An <b> immersion </b> of [[the real projective plane]] in 3 dimensional space. This means..... (explanation).<br />
::::* One possible parametrization of the surface obtained by sewing a Mobius strip to the edge of a disk. <br />
* I think this a better way to do it than to explain things before you give the definitions- people wont be able to follow what you're saying without the definitions anyway.<br />
* Then I think you should move this to after the bullets. "Boy's surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. This probably sounds like a whole new language, but below the Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!<br />
::*<font color=orangered>I think you couldeven reword this to be:"Boy's surface is one possible parametrization of the real projective plane, the surface obtained by sewing a Möbius strip to the edge of a disk."[[User:Rscott3|Richard]] 7/22</font><br />
Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz."<br />
</font color><br />
<br />
<br />
<br />
===Manifolds/Surfaces===<br />
*<font color=orangered> Is "Topology" supposed to be capitalized? [[User:Rscott3|Richard]] 7/22<br />
<br />
*I'd avoid using "and other cool math properties". [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I laud your attempt to intuitively introduce the audience to the concept of a manifold, however the definition contains a quite a few nuances that make me think it would best be done on the Topology Glossary page, where it can be explained in better detail.<br />
<br />
*<font color=salmon> I like your analog of the square and tossed blanket, but I think you can explain what a manifold is better here. I still don't understand what exactly manifold is. Try to add a specific definition of manifold. Is it a certain shape? A general surface? Or a topological space?You can put pictures and tell the readers which one is a manifold and which one is not and why. Pictures and specific examples always help a lot. </font><br />
<br />
* [[User:Gene|Gene]] <s>The</s> Boy''''s''' surface is one of the shapes that is well known in Topology,<s>. Topology is</s> a branch of mathematics. You can think it as an abstract and more advanced version of geometry. Topology focuses on objects that remain constant regardless how distorted the object is. [How 'bout a simple example?]<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:04, 22 July 2011 (UTC) Manifold should be bolded.<br />
*The tossed blanket example is helpful and well explained. Nice work!<br />
</font color><br />
<br />
===Non Orientable===<br />
*<font color=orangered>Is "placed at every location" the right phrase to use? [[User:Rscott3|Richard]] 7/22<br />
<br />
*I think the Earth example is really good. What if you use it to describe manifold instead/as well?[[User:Rscott3|Richard]] 7/22</font><br />
<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I don't understand the explanation of non-orientability. I like the earth analogy, but I can't follow the rest, especially how the image illustrates the concept.<br />
<br />
*<font color=salmon>I can understand this part well. Like the earth analogy and Mobius Strip. But I don't really understand how boy's surface is non-orientable. I guess it's because I can't see what's the back of the boy's surface from the main image. </font><br />
<br />
<font color=slateblue><br />
* If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right. </font color><br />
<br />
===Immersion, The Real Projective Plane, and Embedding===<br />
*<font color=orangered>I don't know what Euler Characteristics are. Maybe you could describe them? 7/22<br />
<br />
*You say that the symbol is the "Greek letter chi", but you don't explain what it represents. [[User:Rscott3|Richard]] 7/22<br />
<br />
*I'm confused by the definition of immersion. [[User:Rscott3|Richard]] 7/22<br />
<br />
*You should define/make a mouseover for "injective" [[User:Rscott3|Richard]] 7/22</font><br />
<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Though I fall victim to shortening "Real Projective Plane" to "Projective Plane" many-a-time, they are distinct things, and should not be introduced as synonymous.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Important: The Boy's surface is an immersion Not and embedding. The self intersections let you know that it is not.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Overall, I think this is a topic for the Topology Glossary helper page. Moving the discussions and definitions of these terms to the helper page will free up this page and let it focus more on Boy's surface. It will also allow a more in depth discussion of these terms, which can be extremely nuanced. On that note, I was very confused by the introductions to ''immersion'', ''embedding'', ect. that was provided.<br />
<br />
:<font color=salmon> Those definitions are necessary for this page. I think you can keep them on your page and then put them on the Topology Glossary as well.</font><br />
<br />
I noticed that you have this listed under geometry, but not topology (but in your page is says that this is a topology subject). I didn't want to add it to topology myself in case the form messed anything up (it's happened to me). <br /><br />
:: [[User:Nordhr|Nordhr] 18::44 27 Jun 2011 <br /><br /><br />
<br />
<br />
Hey Anna, what would you like to do with this: <br />
"The projective space is a modified Euclidean space where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. The projective space becomes the construction of the many circle with an additional circle at infinity. It is a fact that the real projective plane cannot be shown in three space without it passes through itself somewhere."<br />
<br />
Hey Leah, I think we are going to have to simplify that down a little and maybe provide a picture as an aid in understanding?<br />
[[User:Ljeanlo1|Ljeanlo1]] 17:32, 1 July 2011 (UTC)<br />
<br />
* <font color=salmon> About that algebraic equation, tell us what v and e represent. I know v stands for vertices and e for edges, but it's best if you make it clear.</font><br />
<br />
* <font color=salmon> I got lost in understanding "immersion." I have no idea what that definition from WolframMathWorld means. Need more explanation. </font><br />
<br />
<font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:08, 22 July 2011 (UTC) If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right.<br />
* "Straight forward" should be "straightforward"<br />
* You say that non-continuous means that every input has only one output. Your picture of non-continuous doesn't show this though. It shows two inputs mapping to one output, not one input mapping two outputs. I'm not sure which is actually true, but they should be consistent. <font color><br />
<br />
===Constructing Boy's Surface===<br />
<font color=orangered> <br />
*For the dscription, are you describing it like you're on the surface? [[User:Rscott3|Richard]] 7.22<br />
<br />
*Is "axis" plural? [[User:Rscott3|Richard]] 7/22</font><br />
<br />
==Constructing Boy's Surface==<br />
<br />
* <font color=salmon> For the last sentence in the second para: ''Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth, if you go far enough, most likely to infinity, you are going to return to your place of origin.'' <br />
<br />
::1. ''Like the graph.'' Which graph?<br />
::2. Found five commas in this sentence. You break this sentences into too many parts. <br />
::3. Despite those two points, you made it clear why you go back to the starting point.<br />
<br />
* Last sentence before the video: ''By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis. '' I don't get it. How they could be connected together? Need more illustrations here (pictures if possible). And does it have to be like these three combination pairs (aka. + x and - y, +y and -z and + z and -x )? <br />
<br />
* Need more explanation of the video. What I got from this video is that you can return to the origin. Then what happened after 0:50? Please forgive me if I'm getting picky....!!!! These are just my feelings and you don't have to agree to me. <br />
</font><br />
<br />
</font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:09, 22 July 2011 (UTC) I think that the video should definitely be moved up to the basic description. It was very helpful to be able to see Boy's Surface more clearly, and the video is much less complicated that the manifolds/surfaces section. We advise people to put the easier material up at the top of the page. <br />
* I agree with Harrison- The video could use a short explanation as well. This could be added to the basic description.<br />
</font color><br />
<br />
==Parametrization==<br />
<font color=salmon><br />
* Explain to us what complex number means via the mouse over in case other people are not familiar with it.<br />
<br />
* As for the parametrization equations, make sure you explain how you get <math>X = \frac{g_1}{g} </math>, <math>Y = \frac{g_2}{g}, </math> and <math>Z = \frac{g_3}{g} </math> and tell us what <math>Im, Re</math> mean. I just realize that you guys are not done with the page. Just make sure you make those equations clear.<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Boy%27s_Surface&diff=26956Talk:Boy's Surface2011-07-22T14:45:56Z<p>Rscott3: /* Basic Description */</p>
<hr />
<div>==General Comments==<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I ralise this page is very much still in the works. Nevertheless, considering the topic's relation to many of the pages I am working on, I wanted to check it just to make sure all our pages were being decently consistent. All my comments were written keeping in mind that the page is still in progress.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC)My main comment is that much of what is currently on the page is definitions and introductions to topological terms and concepts. I think that this would be better done on the Topology Glossary helper page that I just started based on Leah's suggestion. Many of the terms in topology have nuances in their definitions (ie. manifold, embedding, immersion) that can only be sufficiently explained in a section devoted to the term itself (I've found ''immersion'' to be a particularly tricky one, and am still constantly trying to straighten out what, exactly, characterizes it.)<br />
::*<font color=orangered> I second this. I don't think that this is the place for all of those definitions. I know that this page is still in the construction phase, but it seems like this page is more about topology than Boy's Surface specifically. [[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*I also tend to think that some of the definitions might be oversimplified. For example, I think there's a set of things that make something orientable or not (like continuity and differentiability) I think??? and parametrization is a bit simplified<br />
::In the same sense though, I think that there are one or two definitions that need a bit more simplification, like immersion.[[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*[[User:Rscott3|Richard]] 7/22 The tone of this page is very very conversational. Conversational is good, but there are some sentences and phrases here that just make the page longer and unnecessarily wordy. I think you can stay conversational with the way you guys explain things without some of these phrases. For example:<br />
::#"This probably sounds like a whole new language, but below Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!" in the BD<br />
::#"Before going into great detail with numerous definitions, we will layout Boy's Surface for you in bullet form:" in the BD<br />
::#"So what’s a manifold?" in the manifold section<br />
::#"Now we will tackle a few more vital terms associated with Boy's Surface." in the immersion section<br />
::#"Next, we come to embedding, it is important to understand this term because" and "Luckily, unlike some of the previous terms, the definition is straight forward." in the immersion section<br />
<br />
<br />
*You sometimes switch from "we" to "I". Personally, I avoid first person, but I think all this needs is some consistency. [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
<br />
<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 23:18, 13 July 2011 (UTC) This is a cool image and fascinating page. You got a lot of interesting stuff in it. I'm really interested in this topic. I think there are several hard terms and conclusions needed more precise explanation (see my comments under each section). I understand they are really hard to explain and hard to understand. So try to add more contexts. :) Looking forward to the accomplished page!</font><br />
<br />
*<font color=salmon> I made several minor edits on your page. They are really minor... Plural form, an extra comma, something like that...</font><br />
<br />
<font color=slateblue>[[User:Rebecca|Rebecca]] 01:10, 22 July 2011 (UTC) Nice collaboration! This page is a great addition to the site. </font color><br />
<br />
==Intro==<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 22:56, 13 July 2011 (UTC) Okay, before I went on looking your definitions of the terms, I'm overwhelmed by those fancy math terms in the intro and in the basic description. Maybe you can inform the readers that you are gonna explain every one of them shortly after. </font><br />
<br />
* [[User:Gene|Gene]] Good plan. It would be great if you could give some sort of an intuitive intro (this one is a bit too much like Wikipedia's, too). Wording: should be "<s>The</s>[this "The" ain't needed] Boy’s Surface is an immersion <s>on</s> of the projective plane in three-dimensional space."<br />
<font color=slateblue><br />
<br />
*[[User:Rebecca|Rebecca]] 01:01, 22 July 2011 (UTC) Why don't you link to the "real projective plane" page when you first mention it?<br />
* I think the first paragraph of this page is too complicated. I would hold off on mentioning the real projective plane until late in the page because it's a confusing topic. I think the image description should be cut down to "The object in this image is called Boy's Surface, which is a single sided surface with no edges." <br />
* "The model was constructed as well as donated by Mercedes-Benz." What about... "The model was constructed as well as donated by Mercedes-Benz, and it can be seen in the image below."<br />
</font color><br />
<br />
*<font color=orangered> I think you could even use the same stuff in the intro, just reword it to make it sound more exciting. Something like: "While trying to prove that immersion did not exist, Boy disproved his own theory in 1901 with his discovery of Boy's Surface." [[User:Rscott3|Richard]] 7/22</font><br />
<br />
==Basic Description==<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) "Topology focuses on objects that remain constant regardless how distorted the object is."<br />
** This is not correct; you may very well have meant "Topology is the study of properties that remain constant regardless how distorted the object is", which is correct.<br />
*<font color=salmon>It's kinda repetitive here. You mentioned Werner Boy and others things in the intro before. It's just a minor thing. </font><br />
<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:03, 22 July 2011 (UTC) I think you need to rearrange the basic description. My suggestions are below:<br />
:* I would start the section with ...<br />
::: Boy's Surface is:<br />
::::*A <b>non-orientable</b> surface. Nonorientable means (definition).<br />
::::* An <b> immersion </b> of [[the real projective plane]] in 3 dimensional space. This means..... (explanation).<br />
::::* One possible parametrization of the surface obtained by sewing a Mobius strip to the edge of a disk. <br />
* I think this a better way to do it than to explain things before you give the definitions- people wont be able to follow what you're saying without the definitions anyway.<br />
* Then I think you should move this to after the bullets. "Boy's surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. This probably sounds like a whole new language, but below the Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!<br />
::*<font color=orangered>I think you couldeven reword this to be:"Boy's surface is one possible parametrization of the real projective plane, the surface obtained by sewing a Möbius strip to the edge of a disk."[[User:Rscott3|Richard]] 7/22</font><br />
Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz."<br />
</font color><br />
<br />
*<font color=orangered> Is "Topology" supposed to be capitalized? [[User:Rscott3|Richard]] 7/22</font><br />
<br />
===Manifolds/Surfaces===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I laud your attempt to intuitively introduce the audience to the concept of a manifold, however the definition contains a quite a few nuances that make me think it would best be done on the Topology Glossary page, where it can be explained in better detail.<br />
<br />
*<font color=salmon> I like your analog of the square and tossed blanket, but I think you can explain what a manifold is better here. I still don't understand what exactly manifold is. Try to add a specific definition of manifold. Is it a certain shape? A general surface? Or a topological space?You can put pictures and tell the readers which one is a manifold and which one is not and why. Pictures and specific examples always help a lot. </font><br />
<br />
* [[User:Gene|Gene]] <s>The</s> Boy''''s''' surface is one of the shapes that is well known in Topology,<s>. Topology is</s> a branch of mathematics. You can think it as an abstract and more advanced version of geometry. Topology focuses on objects that remain constant regardless how distorted the object is. [How 'bout a simple example?]<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:04, 22 July 2011 (UTC) Manifold should be bolded.<br />
*The tossed blanket example is helpful and well explained. Nice work!<br />
</font color><br />
<br />
===Non Orientable===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I don't understand the explanation of non-orientability. I like the earth analogy, but I can't follow the rest, especially how the image illustrates the concept.<br />
<br />
*<font color=salmon>I can understand this part well. Like the earth analogy and Mobius Strip. But I don't really understand how boy's surface is non-orientable. I guess it's because I can't see what's the back of the boy's surface from the main image. </font><br />
<br />
<font color=slateblue><br />
* If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right. </font color><br />
<br />
===Immersion, The Real Projective Plane, and Embedding===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Though I fall victim to shortening "Real Projective Plane" to "Projective Plane" many-a-time, they are distinct things, and should not be introduced as synonymous.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Important: The Boy's surface is an immersion Not and embedding. The self intersections let you know that it is not.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Overall, I think this is a topic for the Topology Glossary helper page. Moving the discussions and definitions of these terms to the helper page will free up this page and let it focus more on Boy's surface. It will also allow a more in depth discussion of these terms, which can be extremely nuanced. On that note, I was very confused by the introductions to ''immersion'', ''embedding'', ect. that was provided.<br />
<br />
:<font color=salmon> Those definitions are necessary for this page. I think you can keep them on your page and then put them on the Topology Glossary as well.</font><br />
<br />
I noticed that you have this listed under geometry, but not topology (but in your page is says that this is a topology subject). I didn't want to add it to topology myself in case the form messed anything up (it's happened to me). <br /><br />
:: [[User:Nordhr|Nordhr] 18::44 27 Jun 2011 <br /><br /><br />
<br />
<br />
Hey Anna, what would you like to do with this: <br />
"The projective space is a modified Euclidean space where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. The projective space becomes the construction of the many circle with an additional circle at infinity. It is a fact that the real projective plane cannot be shown in three space without it passes through itself somewhere."<br />
<br />
Hey Leah, I think we are going to have to simplify that down a little and maybe provide a picture as an aid in understanding?<br />
[[User:Ljeanlo1|Ljeanlo1]] 17:32, 1 July 2011 (UTC)<br />
<br />
* <font color=salmon> About that algebraic equation, tell us what v and e represent. I know v stands for vertices and e for edges, but it's best if you make it clear.</font><br />
<br />
* <font color=salmon> I got lost in understanding "immersion." I have no idea what that definition from WolframMathWorld means. Need more explanation. </font><br />
<br />
<font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:08, 22 July 2011 (UTC) If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right.<br />
* "Straight forward" should be "straightforward"<br />
* You say that non-continuous means that every input has only one output. Your picture of non-continuous doesn't show this though. It shows two inputs mapping to one output, not one input mapping two outputs. I'm not sure which is actually true, but they should be consistent. <font color><br />
<br />
==Constructing Boy's Surface==<br />
<br />
* <font color=salmon> For the last sentence in the second para: ''Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth, if you go far enough, most likely to infinity, you are going to return to your place of origin.'' <br />
<br />
::1. ''Like the graph.'' Which graph?<br />
::2. Found five commas in this sentence. You break this sentences into too many parts. <br />
::3. Despite those two points, you made it clear why you go back to the starting point.<br />
<br />
* Last sentence before the video: ''By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis. '' I don't get it. How they could be connected together? Need more illustrations here (pictures if possible). And does it have to be like these three combination pairs (aka. + x and - y, +y and -z and + z and -x )? <br />
<br />
* Need more explanation of the video. What I got from this video is that you can return to the origin. Then what happened after 0:50? Please forgive me if I'm getting picky....!!!! These are just my feelings and you don't have to agree to me. <br />
</font><br />
<br />
</font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:09, 22 July 2011 (UTC) I think that the video should definitely be moved up to the basic description. It was very helpful to be able to see Boy's Surface more clearly, and the video is much less complicated that the manifolds/surfaces section. We advise people to put the easier material up at the top of the page. <br />
* I agree with Harrison- The video could use a short explanation as well. This could be added to the basic description.<br />
</font color><br />
<br />
==Parametrization==<br />
<font color=salmon><br />
* Explain to us what complex number means via the mouse over in case other people are not familiar with it.<br />
<br />
* As for the parametrization equations, make sure you explain how you get <math>X = \frac{g_1}{g} </math>, <math>Y = \frac{g_2}{g}, </math> and <math>Z = \frac{g_3}{g} </math> and tell us what <math>Im, Re</math> mean. I just realize that you guys are not done with the page. Just make sure you make those equations clear.<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Boy%27s_Surface&diff=26952Talk:Boy's Surface2011-07-22T14:39:28Z<p>Rscott3: /* Intro */</p>
<hr />
<div>==General Comments==<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I ralise this page is very much still in the works. Nevertheless, considering the topic's relation to many of the pages I am working on, I wanted to check it just to make sure all our pages were being decently consistent. All my comments were written keeping in mind that the page is still in progress.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC)My main comment is that much of what is currently on the page is definitions and introductions to topological terms and concepts. I think that this would be better done on the Topology Glossary helper page that I just started based on Leah's suggestion. Many of the terms in topology have nuances in their definitions (ie. manifold, embedding, immersion) that can only be sufficiently explained in a section devoted to the term itself (I've found ''immersion'' to be a particularly tricky one, and am still constantly trying to straighten out what, exactly, characterizes it.)<br />
::*<font color=orangered> I second this. I don't think that this is the place for all of those definitions. I know that this page is still in the construction phase, but it seems like this page is more about topology than Boy's Surface specifically. [[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*I also tend to think that some of the definitions might be oversimplified. For example, I think there's a set of things that make something orientable or not (like continuity and differentiability) I think??? and parametrization is a bit simplified<br />
::In the same sense though, I think that there are one or two definitions that need a bit more simplification, like immersion.[[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*[[User:Rscott3|Richard]] 7/22 The tone of this page is very very conversational. Conversational is good, but there are some sentences and phrases here that just make the page longer and unnecessarily wordy. I think you can stay conversational with the way you guys explain things without some of these phrases. For example:<br />
::#"This probably sounds like a whole new language, but below Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!" in the BD<br />
::#"Before going into great detail with numerous definitions, we will layout Boy's Surface for you in bullet form:" in the BD<br />
::#"So what’s a manifold?" in the manifold section<br />
::#"Now we will tackle a few more vital terms associated with Boy's Surface." in the immersion section<br />
::#"Next, we come to embedding, it is important to understand this term because" and "Luckily, unlike some of the previous terms, the definition is straight forward." in the immersion section<br />
<br />
<br />
*You sometimes switch from "we" to "I". Personally, I avoid first person, but I think all this needs is some consistency. [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
<br />
<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 23:18, 13 July 2011 (UTC) This is a cool image and fascinating page. You got a lot of interesting stuff in it. I'm really interested in this topic. I think there are several hard terms and conclusions needed more precise explanation (see my comments under each section). I understand they are really hard to explain and hard to understand. So try to add more contexts. :) Looking forward to the accomplished page!</font><br />
<br />
*<font color=salmon> I made several minor edits on your page. They are really minor... Plural form, an extra comma, something like that...</font><br />
<br />
<font color=slateblue>[[User:Rebecca|Rebecca]] 01:10, 22 July 2011 (UTC) Nice collaboration! This page is a great addition to the site. </font color><br />
<br />
==Intro==<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 22:56, 13 July 2011 (UTC) Okay, before I went on looking your definitions of the terms, I'm overwhelmed by those fancy math terms in the intro and in the basic description. Maybe you can inform the readers that you are gonna explain every one of them shortly after. </font><br />
<br />
* [[User:Gene|Gene]] Good plan. It would be great if you could give some sort of an intuitive intro (this one is a bit too much like Wikipedia's, too). Wording: should be "<s>The</s>[this "The" ain't needed] Boy’s Surface is an immersion <s>on</s> of the projective plane in three-dimensional space."<br />
<font color=slateblue><br />
<br />
*[[User:Rebecca|Rebecca]] 01:01, 22 July 2011 (UTC) Why don't you link to the "real projective plane" page when you first mention it?<br />
* I think the first paragraph of this page is too complicated. I would hold off on mentioning the real projective plane until late in the page because it's a confusing topic. I think the image description should be cut down to "The object in this image is called Boy's Surface, which is a single sided surface with no edges." <br />
* "The model was constructed as well as donated by Mercedes-Benz." What about... "The model was constructed as well as donated by Mercedes-Benz, and it can be seen in the image below."<br />
</font color><br />
<br />
*<font color=orangered> I think you could even use the same stuff in the intro, just reword it to make it sound more exciting. Something like: "While trying to prove that immersion did not exist, Boy disproved his own theory in 1901 with his discovery of Boy's Surface." [[User:Rscott3|Richard]] 7/22</font><br />
<br />
==Basic Description==<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) "Topology focuses on objects that remain constant regardless how distorted the object is."<br />
** This is not correct; you may very well have meant "Topology is the study of properties that remain constant regardless how distorted the object is", which is correct.<br />
*<font color=salmon>It's kinda repetitive here. You mentioned Werner Boy and others things in the intro before. It's just a minor thing. </font><br />
<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:03, 22 July 2011 (UTC) I think you need to rearrange the basic description. My suggestions are below:<br />
:* I would start the section with ...<br />
::: Boy's Surface is:<br />
::::*A <b>non-orientable</b> surface. Nonorientable means (definition).<br />
::::* An <b> immersion </b> of [[the real projective plane]] in 3 dimensional space. This means..... (explanation).<br />
::::* One possible parametrization of the surface obtained by sewing a Mobius strip to the edge of a disk. <br />
* I think this a better way to do it than to explain things before you give the definitions- people wont be able to follow what you're saying without the definitions anyway.<br />
* Then I think you should move this to after the bullets. "Boy's surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. This probably sounds like a whole new language, but below the Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!<br />
Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz."<br />
</font color><br />
<br />
<br />
===Manifolds/Surfaces===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I laud your attempt to intuitively introduce the audience to the concept of a manifold, however the definition contains a quite a few nuances that make me think it would best be done on the Topology Glossary page, where it can be explained in better detail.<br />
<br />
*<font color=salmon> I like your analog of the square and tossed blanket, but I think you can explain what a manifold is better here. I still don't understand what exactly manifold is. Try to add a specific definition of manifold. Is it a certain shape? A general surface? Or a topological space?You can put pictures and tell the readers which one is a manifold and which one is not and why. Pictures and specific examples always help a lot. </font><br />
<br />
* [[User:Gene|Gene]] <s>The</s> Boy''''s''' surface is one of the shapes that is well known in Topology,<s>. Topology is</s> a branch of mathematics. You can think it as an abstract and more advanced version of geometry. Topology focuses on objects that remain constant regardless how distorted the object is. [How 'bout a simple example?]<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:04, 22 July 2011 (UTC) Manifold should be bolded.<br />
*The tossed blanket example is helpful and well explained. Nice work!<br />
</font color><br />
<br />
===Non Orientable===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I don't understand the explanation of non-orientability. I like the earth analogy, but I can't follow the rest, especially how the image illustrates the concept.<br />
<br />
*<font color=salmon>I can understand this part well. Like the earth analogy and Mobius Strip. But I don't really understand how boy's surface is non-orientable. I guess it's because I can't see what's the back of the boy's surface from the main image. </font><br />
<br />
<font color=slateblue><br />
* If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right. </font color><br />
<br />
===Immersion, The Real Projective Plane, and Embedding===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Though I fall victim to shortening "Real Projective Plane" to "Projective Plane" many-a-time, they are distinct things, and should not be introduced as synonymous.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Important: The Boy's surface is an immersion Not and embedding. The self intersections let you know that it is not.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Overall, I think this is a topic for the Topology Glossary helper page. Moving the discussions and definitions of these terms to the helper page will free up this page and let it focus more on Boy's surface. It will also allow a more in depth discussion of these terms, which can be extremely nuanced. On that note, I was very confused by the introductions to ''immersion'', ''embedding'', ect. that was provided.<br />
<br />
:<font color=salmon> Those definitions are necessary for this page. I think you can keep them on your page and then put them on the Topology Glossary as well.</font><br />
<br />
I noticed that you have this listed under geometry, but not topology (but in your page is says that this is a topology subject). I didn't want to add it to topology myself in case the form messed anything up (it's happened to me). <br /><br />
:: [[User:Nordhr|Nordhr] 18::44 27 Jun 2011 <br /><br /><br />
<br />
<br />
Hey Anna, what would you like to do with this: <br />
"The projective space is a modified Euclidean space where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. The projective space becomes the construction of the many circle with an additional circle at infinity. It is a fact that the real projective plane cannot be shown in three space without it passes through itself somewhere."<br />
<br />
Hey Leah, I think we are going to have to simplify that down a little and maybe provide a picture as an aid in understanding?<br />
[[User:Ljeanlo1|Ljeanlo1]] 17:32, 1 July 2011 (UTC)<br />
<br />
* <font color=salmon> About that algebraic equation, tell us what v and e represent. I know v stands for vertices and e for edges, but it's best if you make it clear.</font><br />
<br />
* <font color=salmon> I got lost in understanding "immersion." I have no idea what that definition from WolframMathWorld means. Need more explanation. </font><br />
<br />
<font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:08, 22 July 2011 (UTC) If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right.<br />
* "Straight forward" should be "straightforward"<br />
* You say that non-continuous means that every input has only one output. Your picture of non-continuous doesn't show this though. It shows two inputs mapping to one output, not one input mapping two outputs. I'm not sure which is actually true, but they should be consistent. <font color><br />
<br />
==Constructing Boy's Surface==<br />
<br />
* <font color=salmon> For the last sentence in the second para: ''Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth, if you go far enough, most likely to infinity, you are going to return to your place of origin.'' <br />
<br />
::1. ''Like the graph.'' Which graph?<br />
::2. Found five commas in this sentence. You break this sentences into too many parts. <br />
::3. Despite those two points, you made it clear why you go back to the starting point.<br />
<br />
* Last sentence before the video: ''By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis. '' I don't get it. How they could be connected together? Need more illustrations here (pictures if possible). And does it have to be like these three combination pairs (aka. + x and - y, +y and -z and + z and -x )? <br />
<br />
* Need more explanation of the video. What I got from this video is that you can return to the origin. Then what happened after 0:50? Please forgive me if I'm getting picky....!!!! These are just my feelings and you don't have to agree to me. <br />
</font><br />
<br />
</font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:09, 22 July 2011 (UTC) I think that the video should definitely be moved up to the basic description. It was very helpful to be able to see Boy's Surface more clearly, and the video is much less complicated that the manifolds/surfaces section. We advise people to put the easier material up at the top of the page. <br />
* I agree with Harrison- The video could use a short explanation as well. This could be added to the basic description.<br />
</font color><br />
<br />
==Parametrization==<br />
<font color=salmon><br />
* Explain to us what complex number means via the mouse over in case other people are not familiar with it.<br />
<br />
* As for the parametrization equations, make sure you explain how you get <math>X = \frac{g_1}{g} </math>, <math>Y = \frac{g_2}{g}, </math> and <math>Z = \frac{g_3}{g} </math> and tell us what <math>Im, Re</math> mean. I just realize that you guys are not done with the page. Just make sure you make those equations clear.<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Boy%27s_Surface&diff=26950Talk:Boy's Surface2011-07-22T14:39:12Z<p>Rscott3: /* Intro */</p>
<hr />
<div>==General Comments==<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I ralise this page is very much still in the works. Nevertheless, considering the topic's relation to many of the pages I am working on, I wanted to check it just to make sure all our pages were being decently consistent. All my comments were written keeping in mind that the page is still in progress.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC)My main comment is that much of what is currently on the page is definitions and introductions to topological terms and concepts. I think that this would be better done on the Topology Glossary helper page that I just started based on Leah's suggestion. Many of the terms in topology have nuances in their definitions (ie. manifold, embedding, immersion) that can only be sufficiently explained in a section devoted to the term itself (I've found ''immersion'' to be a particularly tricky one, and am still constantly trying to straighten out what, exactly, characterizes it.)<br />
::*<font color=orangered> I second this. I don't think that this is the place for all of those definitions. I know that this page is still in the construction phase, but it seems like this page is more about topology than Boy's Surface specifically. [[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*I also tend to think that some of the definitions might be oversimplified. For example, I think there's a set of things that make something orientable or not (like continuity and differentiability) I think??? and parametrization is a bit simplified<br />
::In the same sense though, I think that there are one or two definitions that need a bit more simplification, like immersion.[[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*[[User:Rscott3|Richard]] 7/22 The tone of this page is very very conversational. Conversational is good, but there are some sentences and phrases here that just make the page longer and unnecessarily wordy. I think you can stay conversational with the way you guys explain things without some of these phrases. For example:<br />
::#"This probably sounds like a whole new language, but below Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!" in the BD<br />
::#"Before going into great detail with numerous definitions, we will layout Boy's Surface for you in bullet form:" in the BD<br />
::#"So what’s a manifold?" in the manifold section<br />
::#"Now we will tackle a few more vital terms associated with Boy's Surface." in the immersion section<br />
::#"Next, we come to embedding, it is important to understand this term because" and "Luckily, unlike some of the previous terms, the definition is straight forward." in the immersion section<br />
<br />
<br />
*You sometimes switch from "we" to "I". Personally, I avoid first person, but I think all this needs is some consistency. [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
<br />
<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 23:18, 13 July 2011 (UTC) This is a cool image and fascinating page. You got a lot of interesting stuff in it. I'm really interested in this topic. I think there are several hard terms and conclusions needed more precise explanation (see my comments under each section). I understand they are really hard to explain and hard to understand. So try to add more contexts. :) Looking forward to the accomplished page!</font><br />
<br />
*<font color=salmon> I made several minor edits on your page. They are really minor... Plural form, an extra comma, something like that...</font><br />
<br />
<font color=slateblue>[[User:Rebecca|Rebecca]] 01:10, 22 July 2011 (UTC) Nice collaboration! This page is a great addition to the site. </font color><br />
<br />
==Intro==<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 22:56, 13 July 2011 (UTC) Okay, before I went on looking your definitions of the terms, I'm overwhelmed by those fancy math terms in the intro and in the basic description. Maybe you can inform the readers that you are gonna explain every one of them shortly after. </font><br />
<br />
* [[User:Gene|Gene]] Good plan. It would be great if you could give some sort of an intuitive intro (this one is a bit too much like Wikipedia's, too). Wording: should be "<s>The</s>[this "The" ain't needed] Boy’s Surface is an immersion <s>on</s> of the projective plane in three-dimensional space."<br />
<font color=slateblue><br />
<br />
*[[User:Rebecca|Rebecca]] 01:01, 22 July 2011 (UTC) Why don't you link to the "real projective plane" page when you first mention it?<br />
* I think the first paragraph of this page is too complicated. I would hold off on mentioning the real projective plane until late in the page because it's a confusing topic. I think the image description should be cut down to "The object in this image is called Boy's Surface, which is a single sided surface with no edges." <br />
* "The model was constructed as well as donated by Mercedes-Benz." What about... "The model was constructed as well as donated by Mercedes-Benz, and it can be seen in the image below."<br />
</font color><br />
<br />
*<font color=orangered> I think you could even use the same stuff in the intro, just reword it to make it sound more exciting. Something like: "While trying to prove that immersion did not exist, Boy disproved his own theory in 1901 with his discovery of Boy's Surface." [[User:Rscott3|Richard]] 7/22<br />
<br />
==Basic Description==<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) "Topology focuses on objects that remain constant regardless how distorted the object is."<br />
** This is not correct; you may very well have meant "Topology is the study of properties that remain constant regardless how distorted the object is", which is correct.<br />
*<font color=salmon>It's kinda repetitive here. You mentioned Werner Boy and others things in the intro before. It's just a minor thing. </font><br />
<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:03, 22 July 2011 (UTC) I think you need to rearrange the basic description. My suggestions are below:<br />
:* I would start the section with ...<br />
::: Boy's Surface is:<br />
::::*A <b>non-orientable</b> surface. Nonorientable means (definition).<br />
::::* An <b> immersion </b> of [[the real projective plane]] in 3 dimensional space. This means..... (explanation).<br />
::::* One possible parametrization of the surface obtained by sewing a Mobius strip to the edge of a disk. <br />
* I think this a better way to do it than to explain things before you give the definitions- people wont be able to follow what you're saying without the definitions anyway.<br />
* Then I think you should move this to after the bullets. "Boy's surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. This probably sounds like a whole new language, but below the Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!<br />
Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz."<br />
</font color><br />
<br />
<br />
===Manifolds/Surfaces===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I laud your attempt to intuitively introduce the audience to the concept of a manifold, however the definition contains a quite a few nuances that make me think it would best be done on the Topology Glossary page, where it can be explained in better detail.<br />
<br />
*<font color=salmon> I like your analog of the square and tossed blanket, but I think you can explain what a manifold is better here. I still don't understand what exactly manifold is. Try to add a specific definition of manifold. Is it a certain shape? A general surface? Or a topological space?You can put pictures and tell the readers which one is a manifold and which one is not and why. Pictures and specific examples always help a lot. </font><br />
<br />
* [[User:Gene|Gene]] <s>The</s> Boy''''s''' surface is one of the shapes that is well known in Topology,<s>. Topology is</s> a branch of mathematics. You can think it as an abstract and more advanced version of geometry. Topology focuses on objects that remain constant regardless how distorted the object is. [How 'bout a simple example?]<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:04, 22 July 2011 (UTC) Manifold should be bolded.<br />
*The tossed blanket example is helpful and well explained. Nice work!<br />
</font color><br />
<br />
===Non Orientable===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I don't understand the explanation of non-orientability. I like the earth analogy, but I can't follow the rest, especially how the image illustrates the concept.<br />
<br />
*<font color=salmon>I can understand this part well. Like the earth analogy and Mobius Strip. But I don't really understand how boy's surface is non-orientable. I guess it's because I can't see what's the back of the boy's surface from the main image. </font><br />
<br />
<font color=slateblue><br />
* If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right. </font color><br />
<br />
===Immersion, The Real Projective Plane, and Embedding===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Though I fall victim to shortening "Real Projective Plane" to "Projective Plane" many-a-time, they are distinct things, and should not be introduced as synonymous.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Important: The Boy's surface is an immersion Not and embedding. The self intersections let you know that it is not.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Overall, I think this is a topic for the Topology Glossary helper page. Moving the discussions and definitions of these terms to the helper page will free up this page and let it focus more on Boy's surface. It will also allow a more in depth discussion of these terms, which can be extremely nuanced. On that note, I was very confused by the introductions to ''immersion'', ''embedding'', ect. that was provided.<br />
<br />
:<font color=salmon> Those definitions are necessary for this page. I think you can keep them on your page and then put them on the Topology Glossary as well.</font><br />
<br />
I noticed that you have this listed under geometry, but not topology (but in your page is says that this is a topology subject). I didn't want to add it to topology myself in case the form messed anything up (it's happened to me). <br /><br />
:: [[User:Nordhr|Nordhr] 18::44 27 Jun 2011 <br /><br /><br />
<br />
<br />
Hey Anna, what would you like to do with this: <br />
"The projective space is a modified Euclidean space where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. The projective space becomes the construction of the many circle with an additional circle at infinity. It is a fact that the real projective plane cannot be shown in three space without it passes through itself somewhere."<br />
<br />
Hey Leah, I think we are going to have to simplify that down a little and maybe provide a picture as an aid in understanding?<br />
[[User:Ljeanlo1|Ljeanlo1]] 17:32, 1 July 2011 (UTC)<br />
<br />
* <font color=salmon> About that algebraic equation, tell us what v and e represent. I know v stands for vertices and e for edges, but it's best if you make it clear.</font><br />
<br />
* <font color=salmon> I got lost in understanding "immersion." I have no idea what that definition from WolframMathWorld means. Need more explanation. </font><br />
<br />
<font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:08, 22 July 2011 (UTC) If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right.<br />
* "Straight forward" should be "straightforward"<br />
* You say that non-continuous means that every input has only one output. Your picture of non-continuous doesn't show this though. It shows two inputs mapping to one output, not one input mapping two outputs. I'm not sure which is actually true, but they should be consistent. <font color><br />
<br />
==Constructing Boy's Surface==<br />
<br />
* <font color=salmon> For the last sentence in the second para: ''Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth, if you go far enough, most likely to infinity, you are going to return to your place of origin.'' <br />
<br />
::1. ''Like the graph.'' Which graph?<br />
::2. Found five commas in this sentence. You break this sentences into too many parts. <br />
::3. Despite those two points, you made it clear why you go back to the starting point.<br />
<br />
* Last sentence before the video: ''By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis. '' I don't get it. How they could be connected together? Need more illustrations here (pictures if possible). And does it have to be like these three combination pairs (aka. + x and - y, +y and -z and + z and -x )? <br />
<br />
* Need more explanation of the video. What I got from this video is that you can return to the origin. Then what happened after 0:50? Please forgive me if I'm getting picky....!!!! These are just my feelings and you don't have to agree to me. <br />
</font><br />
<br />
</font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:09, 22 July 2011 (UTC) I think that the video should definitely be moved up to the basic description. It was very helpful to be able to see Boy's Surface more clearly, and the video is much less complicated that the manifolds/surfaces section. We advise people to put the easier material up at the top of the page. <br />
* I agree with Harrison- The video could use a short explanation as well. This could be added to the basic description.<br />
</font color><br />
<br />
==Parametrization==<br />
<font color=salmon><br />
* Explain to us what complex number means via the mouse over in case other people are not familiar with it.<br />
<br />
* As for the parametrization equations, make sure you explain how you get <math>X = \frac{g_1}{g} </math>, <math>Y = \frac{g_2}{g}, </math> and <math>Z = \frac{g_3}{g} </math> and tell us what <math>Im, Re</math> mean. I just realize that you guys are not done with the page. Just make sure you make those equations clear.<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Boy%27s_Surface&diff=26949Talk:Boy's Surface2011-07-22T14:36:28Z<p>Rscott3: /* General Comments */</p>
<hr />
<div>==General Comments==<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I ralise this page is very much still in the works. Nevertheless, considering the topic's relation to many of the pages I am working on, I wanted to check it just to make sure all our pages were being decently consistent. All my comments were written keeping in mind that the page is still in progress.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC)My main comment is that much of what is currently on the page is definitions and introductions to topological terms and concepts. I think that this would be better done on the Topology Glossary helper page that I just started based on Leah's suggestion. Many of the terms in topology have nuances in their definitions (ie. manifold, embedding, immersion) that can only be sufficiently explained in a section devoted to the term itself (I've found ''immersion'' to be a particularly tricky one, and am still constantly trying to straighten out what, exactly, characterizes it.)<br />
::*<font color=orangered> I second this. I don't think that this is the place for all of those definitions. I know that this page is still in the construction phase, but it seems like this page is more about topology than Boy's Surface specifically. [[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*I also tend to think that some of the definitions might be oversimplified. For example, I think there's a set of things that make something orientable or not (like continuity and differentiability) I think??? and parametrization is a bit simplified<br />
::In the same sense though, I think that there are one or two definitions that need a bit more simplification, like immersion.[[User:Rscott3|Richard]] 7/22<br />
<br />
<br />
*[[User:Rscott3|Richard]] 7/22 The tone of this page is very very conversational. Conversational is good, but there are some sentences and phrases here that just make the page longer and unnecessarily wordy. I think you can stay conversational with the way you guys explain things without some of these phrases. For example:<br />
::#"This probably sounds like a whole new language, but below Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!" in the BD<br />
::#"Before going into great detail with numerous definitions, we will layout Boy's Surface for you in bullet form:" in the BD<br />
::#"So what’s a manifold?" in the manifold section<br />
::#"Now we will tackle a few more vital terms associated with Boy's Surface." in the immersion section<br />
::#"Next, we come to embedding, it is important to understand this term because" and "Luckily, unlike some of the previous terms, the definition is straight forward." in the immersion section<br />
<br />
<br />
*You sometimes switch from "we" to "I". Personally, I avoid first person, but I think all this needs is some consistency. [[User:Rscott3|Richard]] 7/22<br />
</font><br />
<br />
<br />
<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 23:18, 13 July 2011 (UTC) This is a cool image and fascinating page. You got a lot of interesting stuff in it. I'm really interested in this topic. I think there are several hard terms and conclusions needed more precise explanation (see my comments under each section). I understand they are really hard to explain and hard to understand. So try to add more contexts. :) Looking forward to the accomplished page!</font><br />
<br />
*<font color=salmon> I made several minor edits on your page. They are really minor... Plural form, an extra comma, something like that...</font><br />
<br />
<font color=slateblue>[[User:Rebecca|Rebecca]] 01:10, 22 July 2011 (UTC) Nice collaboration! This page is a great addition to the site. </font color><br />
<br />
==Intro==<br />
*<font color=salmon>[[User:PhoebeJiang|Phoebe]] 22:56, 13 July 2011 (UTC) Okay, before I went on looking your definitions of the terms, I'm overwhelmed by those fancy math terms in the intro and in the basic description. Maybe you can inform the readers that you are gonna explain every one of them shortly after. </font><br />
<br />
* [[User:Gene|Gene]] Good plan. It would be great if you could give some sort of an intuitive intro (this one is a bit too much like Wikipedia's, too). Wording: should be "<s>The</s>[this "The" ain't needed] Boy’s Surface is an immersion <s>on</s> of the projective plane in three-dimensional space."<br />
<font color=slateblue><br />
<br />
*[[User:Rebecca|Rebecca]] 01:01, 22 July 2011 (UTC) Why don't you link to the "real projective plane" page when you first mention it?<br />
* I think the first paragraph of this page is too complicated. I would hold off on mentioning the real projective plane until late in the page because it's a confusing topic. I think the image description should be cut down to "The object in this image is called Boy's Surface, which is a single sided surface with no edges." <br />
* "The model was constructed as well as donated by Mercedes-Benz." What about... "The model was constructed as well as donated by Mercedes-Benz, and it can be seen in the image below."<br />
</font color><br />
<br />
==Basic Description==<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) "Topology focuses on objects that remain constant regardless how distorted the object is."<br />
** This is not correct; you may very well have meant "Topology is the study of properties that remain constant regardless how distorted the object is", which is correct.<br />
*<font color=salmon>It's kinda repetitive here. You mentioned Werner Boy and others things in the intro before. It's just a minor thing. </font><br />
<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:03, 22 July 2011 (UTC) I think you need to rearrange the basic description. My suggestions are below:<br />
:* I would start the section with ...<br />
::: Boy's Surface is:<br />
::::*A <b>non-orientable</b> surface. Nonorientable means (definition).<br />
::::* An <b> immersion </b> of [[the real projective plane]] in 3 dimensional space. This means..... (explanation).<br />
::::* One possible parametrization of the surface obtained by sewing a Mobius strip to the edge of a disk. <br />
* I think this a better way to do it than to explain things before you give the definitions- people wont be able to follow what you're saying without the definitions anyway.<br />
* Then I think you should move this to after the bullets. "Boy's surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. This probably sounds like a whole new language, but below the Boy's surface will be described in much detail. We will provide definitions as well as examples of vital terms!<br />
Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz."<br />
</font color><br />
<br />
<br />
===Manifolds/Surfaces===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I laud your attempt to intuitively introduce the audience to the concept of a manifold, however the definition contains a quite a few nuances that make me think it would best be done on the Topology Glossary page, where it can be explained in better detail.<br />
<br />
*<font color=salmon> I like your analog of the square and tossed blanket, but I think you can explain what a manifold is better here. I still don't understand what exactly manifold is. Try to add a specific definition of manifold. Is it a certain shape? A general surface? Or a topological space?You can put pictures and tell the readers which one is a manifold and which one is not and why. Pictures and specific examples always help a lot. </font><br />
<br />
* [[User:Gene|Gene]] <s>The</s> Boy''''s''' surface is one of the shapes that is well known in Topology,<s>. Topology is</s> a branch of mathematics. You can think it as an abstract and more advanced version of geometry. Topology focuses on objects that remain constant regardless how distorted the object is. [How 'bout a simple example?]<br />
<br />
<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 01:04, 22 July 2011 (UTC) Manifold should be bolded.<br />
*The tossed blanket example is helpful and well explained. Nice work!<br />
</font color><br />
<br />
===Non Orientable===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) I don't understand the explanation of non-orientability. I like the earth analogy, but I can't follow the rest, especially how the image illustrates the concept.<br />
<br />
*<font color=salmon>I can understand this part well. Like the earth analogy and Mobius Strip. But I don't really understand how boy's surface is non-orientable. I guess it's because I can't see what's the back of the boy's surface from the main image. </font><br />
<br />
<font color=slateblue><br />
* If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right. </font color><br />
<br />
===Immersion, The Real Projective Plane, and Embedding===<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Though I fall victim to shortening "Real Projective Plane" to "Projective Plane" many-a-time, they are distinct things, and should not be introduced as synonymous.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Important: The Boy's surface is an immersion Not and embedding. The self intersections let you know that it is not.<br />
<br />
*[[User:Htasoff|Htasoff]] 20:27, 2 July 2011 (UTC) Overall, I think this is a topic for the Topology Glossary helper page. Moving the discussions and definitions of these terms to the helper page will free up this page and let it focus more on Boy's surface. It will also allow a more in depth discussion of these terms, which can be extremely nuanced. On that note, I was very confused by the introductions to ''immersion'', ''embedding'', ect. that was provided.<br />
<br />
:<font color=salmon> Those definitions are necessary for this page. I think you can keep them on your page and then put them on the Topology Glossary as well.</font><br />
<br />
I noticed that you have this listed under geometry, but not topology (but in your page is says that this is a topology subject). I didn't want to add it to topology myself in case the form messed anything up (it's happened to me). <br /><br />
:: [[User:Nordhr|Nordhr] 18::44 27 Jun 2011 <br /><br /><br />
<br />
<br />
Hey Anna, what would you like to do with this: <br />
"The projective space is a modified Euclidean space where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. The projective space becomes the construction of the many circle with an additional circle at infinity. It is a fact that the real projective plane cannot be shown in three space without it passes through itself somewhere."<br />
<br />
Hey Leah, I think we are going to have to simplify that down a little and maybe provide a picture as an aid in understanding?<br />
[[User:Ljeanlo1|Ljeanlo1]] 17:32, 1 July 2011 (UTC)<br />
<br />
* <font color=salmon> About that algebraic equation, tell us what v and e represent. I know v stands for vertices and e for edges, but it's best if you make it clear.</font><br />
<br />
* <font color=salmon> I got lost in understanding "immersion." I have no idea what that definition from WolframMathWorld means. Need more explanation. </font><br />
<br />
<font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:08, 22 July 2011 (UTC) If you could somehow make a sphere that looks 3D and show arrows going around it, that might be helpful.<br />
* I would suggest moving the mobius strip picture up next to the paragraph "The mobius strip is shown below...." * You could refer to it as the picture on the left or right.<br />
* "Straight forward" should be "straightforward"<br />
* You say that non-continuous means that every input has only one output. Your picture of non-continuous doesn't show this though. It shows two inputs mapping to one output, not one input mapping two outputs. I'm not sure which is actually true, but they should be consistent. <font color><br />
<br />
==Constructing Boy's Surface==<br />
<br />
* <font color=salmon> For the last sentence in the second para: ''Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth, if you go far enough, most likely to infinity, you are going to return to your place of origin.'' <br />
<br />
::1. ''Like the graph.'' Which graph?<br />
::2. Found five commas in this sentence. You break this sentences into too many parts. <br />
::3. Despite those two points, you made it clear why you go back to the starting point.<br />
<br />
* Last sentence before the video: ''By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis. '' I don't get it. How they could be connected together? Need more illustrations here (pictures if possible). And does it have to be like these three combination pairs (aka. + x and - y, +y and -z and + z and -x )? <br />
<br />
* Need more explanation of the video. What I got from this video is that you can return to the origin. Then what happened after 0:50? Please forgive me if I'm getting picky....!!!! These are just my feelings and you don't have to agree to me. <br />
</font><br />
<br />
</font color=slateblue><br />
*[[User:Rebecca|Rebecca]] 01:09, 22 July 2011 (UTC) I think that the video should definitely be moved up to the basic description. It was very helpful to be able to see Boy's Surface more clearly, and the video is much less complicated that the manifolds/surfaces section. We advise people to put the easier material up at the top of the page. <br />
* I agree with Harrison- The video could use a short explanation as well. This could be added to the basic description.<br />
</font color><br />
<br />
==Parametrization==<br />
<font color=salmon><br />
* Explain to us what complex number means via the mouse over in case other people are not familiar with it.<br />
<br />
* As for the parametrization equations, make sure you explain how you get <math>X = \frac{g_1}{g} </math>, <math>Y = \frac{g_2}{g}, </math> and <math>Z = \frac{g_3}{g} </math> and tell us what <math>Im, Re</math> mean. I just realize that you guys are not done with the page. Just make sure you make those equations clear.<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=S11&diff=26718S112011-07-21T14:02:42Z<p>Rscott3: /* Richard's Projects */</p>
<hr />
<div>__TOC__<br />
<br />
== Announcements ==<br />
For public-type help questions, see [[Help:Contents|Help]]. For Swat-specific ones, see [[Swarthmore summer research orientation]].<br />
<br />
<b>Remember to keep your projects sections short and up to date; only the last week of status changes should be mentioned. [[User:Smaurer1|Smaurer1]]</b><br />
<br />
===Group Discussion Questions===<br />
<br />
* [[Topics for conversations through Skype with RPI, SB, and/or Drexel]] Started 6/29.<br />
* [[Who are we writing for?]] Started 6/29.<br />
* [[Possibly expanding student Math Image roles]], to be addressed 7/6.<br />
<br />
===Questions:===<br />
*Things that are listed as Helper Pages but use the Image Page template<br />
:We need to decide whether these pages should be on the Helper Page template, the Image Page template, or both: {{Hide|1=<br />
:*[[Change of Coordinate Systems]] - Image template only; image template live<br />
:*[[Conic Section]] - both templates; image template not live<br />
:*[[Differentiability]] - both templates; image template not live<br />
:*[[Dot Product]] - Image template only; image template not live<br />
:*[[Gradients and Directional Derivatives]] - Image template only; image template not live<br />
:*[[Hyperbolic Geometry]] - Image template only; image template not live<br />
:*[[Inversion]] - Image template only; image template not live<br />
:*[[Iterated Functions]] - Image template only; image template not live<br />
:*[[Parametric Equations]] - Image template only; image template live<br />
:*[[Taylor Series]] - Image template only; image template not live<br />
:*[[Volume of Revolution]] - Image template only; image template live<br />
<br />
:<font color=dodgerblue>''(List complied by [[User:Kderosier|Kate]], June 16)''</font><br />
}}<br />
<br />
<br />
Older questions: {{Hide|1=<br />
*Harrison's question about text being cut off on Cross-cap: {{Hide|1=<br />
*[[User:Htasoff|Htasoff]] 14:16, 8 June 2011 (UTC)<br />
**The text in MME on the [[Cross-cap]] page is getting truncated when viewed through edit with form, but still appears on the page.<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 00:49, 14 June 2011 (UTC): When I've encountered this problem previously, it's been because a set of double curly brackets wasn't closed.</font><br />
}}<br />
<br />
*Harrison's question about creating a list of not-yet-existent Helper Pages: {{Hide|1=<br />
Harrison, 5/26/11:<br />
:*<s>We need a list of empty pages: Penrose Tiles is only linked to one, now two, pages. Empty pages like this could well fade into obscurity.</s><br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 17:09, 7 June 2011 (UTC): Such a list has been created. See [[Existing_Pages_Needing_Work#Empty_.28but_linked_to.29_Pages|here]].</font><br />
}}<br />
<br />
*Spam conversation: {{Hide|1=<br />
<font color=dodgerblue><br />
*[[User:Kderosier|Kate]] 14:38, 27 May 2011 (UTC): '''We have some new users who are creating a bunch of pages with links to illegally download or watch movies.''' At first, we though it might be someone from Sweet Briar practicing wiki-syntax, but now it's just starting to look like spam:<br />
**[[Watch_Sniper:_Reloaded_film_in_hd|This is the type of page I'm talking about]]<br />
**[[Special:Contributions/Calrivenick|List of pages created by Calrivenick]]<br />
**[[Special:Contributions/Cadedesi|List of pages created by Cadedesi]]<br />
</font><br />
:Let's talk about what to do this afternoon. [[User:Gene|Gene]] 15:19, 27 May 2011 (UTC)<br />
:<font color=dodgerblue>This problem has continued over the weekend. Someone spammed the talk page for Fun Topology with comments about buying Cialis and stuff. They also posted a lot more movie download pages under the Cadedesi username. I'm going to go through and delete again.</font><br />
:<font color=dodgerblue>The spam comments on Fun Topology were coming from this computer's IP address. </font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 14:34, 6 June 2011 (UTC): Haven't seen any more spam activity for a few days. I assume some computer people have handled the issue? I'm going to hide this conversation so that it's not taking up space on S11.</font><br />
}}<br />
<br />
*Citations/footnotes conversation: {{Hide|1=<br />
:I spent a really long time wandering around MediaWiki and Wikipedia this morning trying to figure out how to do footnotes/citations the way I wanted to - now that I know how, should I add instructions to one of the many help mages on Math Images? If so, which page? (Kate, 5/17)<br />
<br />
:<font color=red> Answer: put it here at [[Help:Wiki_Tricks|Wiki Tricks]] (XD, 5/17) </font><br />
<br />
:[[User:Smaurer1|Smaurer1]] 19:43, 17 May 2011 (UTC) Well, it's not clear that webarticles should have footnotes, although Wikipedia does. In text references may be better. This is surely something we should discuss as a group, and find out what last year's group decided, if they did. If there are footnotes, there has got to be a way to get back seamlessly to where you were before you jumped to the footnote.<br />
<br />
:Also, as for citations, we should be uniform in their format.<br />
<br />
:Finally, you can use 4 tildes to put your username and time stamp on your comments, and 3 tildes for just your username. <br />
<br />
:<font color=dodgerblue>Well, I wasn't doing comment-y type footnotes, I just wanted specific sentences to link to items in my References section. I think that format is better than full intext-citations, because it brings you to the source if that's what you're interested in, but it takes up less space if you're not. The references template we have comes equipped with little links to jump you back up, too- if you look at the [[Quipu#References|Quipu]] page, you can see that it lists each of the sections that reference is linked from, and those links bring you to the reference in that section. All in all, I think it's a clear and intuitive way to do references for both the writer and the reader (although, like most things, it doesn't play well with our hidden sections), and I'm going to go ahead and put the instructions up in WikiTricks. -[[User:Kderosier|Kate]] </font><br />
<br />
{{Hide|1=<br />
*What the help pages say now:<br />
**[[Tour_the_Math_Images_Project#Anatomy_of_a_page|The Tour page's "Anatomy of a page" section]]<br />
**[[Checklist_for_writing_pages#References_and_footnotes|The "References and footnotes" section on the checklist for writing pages]]<br />
*The way to do Wikipedia-like references:<br />
**[[Wiki_Tricks#How_to_do_Citations|Wiki Tricks - Citations section]]<br />
}}<br />
<br />
}}<br />
<br />
*Invisible Comments conversation: {{Hide|1= One of you asked "How do you put invisible comments in the source code?"<br />
<br />
:Answer: Same way you do in html, like this <nowiki><br />
:<!-- hidden stuff --><br />
:</nowiki><br />
:However, if you want to make comments about an article for its author, the comments are more likely to be seen if you either<br />
:* put it in the discussion page, or<br />
:* if it is important to put it right by the material commented on, put it in the article in color with your username and time stamp included.<br />
<br />
:Hidden comments in the source code are likely to be overlooked, except perhaps if they are written by the author him/herself, as a note for further development.<br />
}}<br />
}}<br />
<br />
== Current Individual Projects ==<br />
<br />
===Steve M (aka Prof Maurer)'s Role===<br />
{{Hide|1=<br />
My role is 2-fold:<br />
<br />
<ol><br />
<li> Come see me to sound me out (if you wish) on the mathematical appropriateness of an idea for a page, or for possible references.<br />
<br />
<li> Once you have a reasonable amount written, and want feedback on the quality and correctness of the mathematical exposition, ask me to look it over and then we will have a conference. (Abram and the oldies are as good as I am at discussion general organization and clarity issues.) <br />
</ol><br />
<br />
We have agreed to put a record on this S11 page of what we are doing and what help we want from others, but in addition tell me in person or by email if you want to conference with me.<br />
}}<br />
<br />
===Harrison's Projects===<br />
<br />
[[Harrison's detritus]]<br />
<br />
*[[User:Htasoff|Htasoff]] 23:44, 13 July 2011 (UTC) Pages will be submitted for final review in 1 - 3 days. A few, final comments are welcome. Real Projective Plane is still in the works, though.<br />
<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:19, 14 July 2011 (UTC): Hey Harrison, do you know if that one picture in inverse trig that you got from somewhere on the internet is a picture we can use? Can you check please?</font><br />
<br />
[[Rope around the Earth]]<br />
:<font color=darkred> Approved </font><br />
<br />
<font color=darkred> I have one small suggested wording comment for you on [[Congruent triangles]], then it's good to go. [[User:AnnaP|AnnaP]] 13:48, 21 July 2011 (UTC)</font><br />
<br />
===Richard's Projects===<br />
<br />
<br />
<font color=orangered> <br />
<br />
:APPLET INFO:<br />
:{{Hide|1=<br />
<br />
[[User:Alimurreza|Alimurreza]] 02:53, 6 July 2011 (UTC)I am working on your applet. Check this out here @([http://mathforum.org/mathimages/index.php/DU11 Reza's work]).<br />
:APPLET UPDATE:[[User:Alimurreza|Alimurreza]] 02:41, 15 July 2011 (UTC)I am done with Ambiguous Case applet. Please, check the applet. Feel free to send me any feedback or change-request.<br />
What i want mine to look like, but http://www.mccsc.edu/~aterwill/ambiguouscaseapplet/Ambiguous_Case_applet.html doesn't show the completed triangles.<br />
<br />
I like how this one shows the completed triangles<br />
http://www.mnwest.edu/fileadmin/static/website/dmatthews/Geogebra/AmbiguousCase01.html<br />
<br />
}}<br />
<br />
'''Completed Pages'''<br />
*[[Ambiguous Case]] [[User:Rscott3|Richard]] 7/19<br />
<br />
*[[Law of cosines]]<font color=orangered> [[User:Rscott3|Richard]] 6/23</font><br />
<br />
*[[Law of Sines]] [[User:AnnaP|AnnaP]] 7/7<br />
<br />
*[[Solving Triangles]] 7/12<br />
<br />
:Other ideas: {{Hide|1=<br />
*inscribed angles?<br />
<br />
*Normal Distribution?<br />
<br />
*Birthday Paradox?<br />
}}<br />
</font><br />
<br />
===Dayo's Projects===<br />
Current projects<br />
*[[Inscribed figures]] : there's a [[Compass & Straightedge Construction and the Impossible Constructions]] page, but I think that inscribed figures deserves its' own page. What do others think? <br />
::<font color=slateblue> [[User:Rebecca|Rebecca]] 01:07, 9 July 2011 (UTC) I left comments on the discussion page. </font color><br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 15:23, 18 July 2011 (UTC): I left you some comments!</font><br />
<br />
Future Projects: {{Hide|1=<br />
*[[Mathematics in architecture]]:make changes akin to [[Math for Computer Graphics and Computer Vision]], including:<br />
<br />
::*[[Cross sections]]: calculus application page, including examples of Tokyo international Forum, Suransuns Bridge, and other structures which could be thought of and put together easily in terms of their cross sections <br />
::*[[Torus]] edits, additions concerning the Torus in construction and architecture<br />
::*[[Domes]]:conic sections, arches, parabolas<br />
::*[[The Henderson Waves Bridge]]: sinusoids in architecture, parametric design<br />
::*[[Catenary]]: More real world examples: namely bridges<br />
::*[[Bridge of Peace]]: The equation(s) used to generate the surface, possibly words from the architect, very new, may be hard to get concrete technical information <br />
::*[[Kurilpa Bridge]]: Everyone have a look at the image and tell me what kind of actual subjects you could find in it, namely in the cables and tubes.<br />
::*Teaching Materials(6/30): ''growing up with science: projects'' could be the sort of activities we're looking for. I used these with a class, and think people should look at them.<br />
<br />
<br />
<br />
<br />
}}<br />
<br />
on hold: <br />
*[[Parametric Equations]]: integrating Xingda's page from [[S10]] into page.<br />
<br />
===Diana's Projects===<br />
<br />
====Current====<br />
*[[String Art Calculus]]<br />
<br />
*[[Logistic Bifurcation]]<br />
:<font color=darkred> I've put up comments. There are a few places that could use some work. [[User:AnnaP|AnnaP]] 7/10 </font> <br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:26, 16 July 2011 (UTC) This is a very impressive page. I put up a few small suggestions on the discussion page. </font color><br />
<br />
*[[Markus-Lyapunov Fractals]]<br />
:<font color=darkred> Approved [[User:AnnaP|AnnaP]] 7/14 </font><br />
<br />
====Ideas for later projects====<br />
{{Hide|1=<br />
*Chirikov-Taylor Maps<br />
**This seems like a natural extension of the Markus-Lyapunov Fractals page, but maybe the math involved in the two is too similar?<br />
*This aspect of pendular motion:<br />
**[[http://www.youtube.com/watch?v=yVkdfJ9PkRQ&feature=player_embedded|Varied-Length Pendulums]]<br />
**I'm not sure how or whether to use this -- does a ''moving'' image count as a "math image"? -- But it's incredible, and I'd love to explore it.<br />
*Kuen Surface<br />
**It's just really cool.<br />
}}<br />
<br />
===Kate's Projects===<br />
*[[Anne Burns' Mathscapes]] (Scrapped out of [[Mountains In Spring|three]] [[Mathscape|other]] [[Fractal Scene I|pages]]):<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:29, 18 July 2011 (UTC): As soon as I get confirmation that Anne Burns doesn't mind us using her images, I'll submit this for final review.</font><br />
<br />
*Finished pages: {{Hide|1=<br />
*[[Perko pair knots]]:<br />
:<font color=darkred> Approved </font><br />
<br />
:*[[Critical Points]]:<br />
::<font color=darkred> Approved, but I did have one note on whether or not you intended to add something. It's fine as is, but I wanted to put up a suggestion. [[User:AnnaP|AnnaP]] 16:24, 15 July 2011 (UTC) </font><br />
:::<font color=dodgerblue>I responded to that note.</font><br />
<br />
:*[[Summation Notation]]:<br />
::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 16:24, 15 July 2011 (UTC) </font><br />
<br />
:*[[Bases]]:<br />
:: <font color=dodgerblue>[[User:Kderosier|Kate]] 17:31, 11 July 2011 (UTC): Changed things in response to Chris' comments.</font><br />
<br />
:* [[Quipu]]:<br />
::<font color=darkred> Put up as ready for the public 6/30 [[User:AnnaP|AnnaP]] </font><br />
<br />
:* [[Basic Trigonometric Functions]]:<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 18:11, 30 June 2011 (UTC): Changed the the things that were bolded.</font><br />
<br />
<br />
}}<br />
<br />
===Leah's Projects===<br />
*[[Bedsheet Problem]]<br />
going to do last edits [[User:Ljeanlo1|Ljeanlo1]] 23:57, 20 July 2011 (UTC) <br />
<br />
*[[Boy's Surface]] <br />
take a look on this tomorrow. [[User:Ljeanlo1|Ljeanlo1]] 23:57, 20 July 2011 (UTC) <br />
<br />
*[[Snell's Law]] <br />
-feedback page<br />
waiting on Anna's comments. [[User:Ljeanlo1|Ljeanlo1]] 20:38, 20 July 2011 (UTC)<br />
:<font color=darkred> I have one comment that you haven't addressed yet. I put it in bold [[User:AnnaP|AnnaP]] 13:53, 21 July 2011 (UTC) </font><br />
<br />
*[[Witch of Agnesi]] <br />
up for final review [[User:Ljeanlo1|Ljeanlo1]] 20:05, 20 July 2011 (UTC) <br />
<br />
*[[Dot Product]] <font color=darkred> Approved, 7/14 </font><br />
<br />
*[[Vector]] <font color=darkred> Approved, 7/10 </font><br />
<br />
==Requests to S10 Students==<br />
<br />
<br />
*<s>Can XD do a demo for MATLAB?</s> Done - [[Demo of MATLAB using the example of Bifurcation]]<br />
<br />
==Useful Links==<br />
[[S10]]<br />
<br />
[[SB11]]<br />
<br />
[[RPI11]]<br />
<br />
[[DU11]]<br />
<br />
[[Pages Ready for Final Review]]<br />
<br />
[[Feedback Requests]]<br />
<br />
[[Sample discussion page]]<br />
<br />
[[Math Tools Requests]] ''This page is a place where students whose primary focus is writing pages can post requests for applets, animations, and new images that they'd like to see the computer science students create.''<br />
<br />
[[Page Building Help]]<br />
<br />
[[Help:Wiki Tricks|Wiki Tricks]]<br />
<br />
[[From a Bunch of Old Timers]]<br />
<br />
[[List of summer 2010 pages]]<br />
<br />
http://en.wikipedia.org/wiki/Help:Displaying_a_formula<br />
<br />
[[PartnerHome]]<br />
<br />
[[Existing Pages Needing Work]]<br />
<br />
[[Site programming questions]]<br />
<br />
== Current Group Projects ==</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=File:Template.jpg&diff=26606File:Template.jpg2011-07-20T22:43:24Z<p>Rscott3: </p>
<hr />
<div></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=SB11&diff=26374SB112011-07-20T18:45:36Z<p>Rscott3: /* Phoebe Jiang */</p>
<hr />
<div>=<center> Sweet Briar 2011 </center>=<br />
<br />
==Discussion==<br />
<font color=navy>Please feel free to discuss math pages or leave a message for the creators!</font><br />
<br />
Hey girls, the discussion bar on the very top of every page next to "page" is really helpful and necessary for peer review or faculty review. If you guys go through other people's page and want to comment on their work, please do. [[User:PhoebeJiang|Phoebe]] 15:09, 11 June 2011 (UTC)<br />
<br />
==Summer 2011 Projects==<br />
<br />
===Anna Donko===<br />
*<big>[[Steiner's Chain]]</big><br />
:{{hide|1=<br />
:<font color=peach> Need to work on explaining algebraic formulas.</font> <br />
PLEASE look at this!! I would really appreciate comments and suggestions!!<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:27, 28 June 2011 (UTC): Just left you a lot of comments on the discussion page.</font><br />
:<font color=peach> Working on reading and using all of the comments on the discussion page, thanks all who left me some feedback! </font><br />
:<font color=orangered>Interesting topic! Left some comments! [[User:Rscott3|Richard]] 7/18</font>}}<br />
Submitted for final review <br />
<br />
*<big>[[Boy's Surface]]</big><br />
:{{hide|1=<br />
:<font color=peach> Leah and I are trying to figure out how we want to lay out our page...Please hold back with the comments while we continue to reorganize as well as add to our page!</font><br />
:<font color=red> Need to work on explanation of parametrization...??</font><br />
:<font color=salmon> Left some comments~ [[User:PhoebeJiang|Phoebe]] 23:56, 13 July 2011 (UTC)</font>}}<br />
<br />
*<big>[[Parabolic Integration]]</big><br />
:{{hide|1=<br />
:<font color=peach>Just started this new page!! I am going to ink this to the already existing page, [[Parabola]]</font><br />
::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:51, 12 July 2011 (UTC) Great image! I like both pictures! The title is clear but I think you can make it shorter.</font><br />
:<font color=peach>Going to change my equations so they work out neater for the reader</font><br />
:<font color=black> Planning to add much more, including explanations of why/how parabolas are used in the physical world, as well as differentiate between parabolas and catenaries, lastly I will find the area of an actual life sized piece of parabolic architecture </font>.}}<br />
<br />
===Phoebe Jiang===<br />
<br />
*<big>[[Arbelos]]</big><br />
:{{hide|1=<br />
:<font color=salmon>Need to work on the two footnotes. And do a last revision, revise Pappus Chain and Bankoff Chain. <br />
:PLEASE look at this page. Comments are always welcome! </font>[[User:PhoebeJiang|Phoebe]] 19:49, 28 June 2011 (UTC)<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 13:21, 30 June 2011 (UTC): I've gone back through and responded to your comments or crossed things off where we both agree.</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Thank you Kate!!! <3 </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments for you. See my thoughts about a spin-off page. [[User:AnnaP|AnnaP]] 7/12 </font> <br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC) Thank you Anna and I've responded to your comments.</font><br />
::::<font color=darkred> One more round of comments is up [[User:AnnaP|AnnaP]] 15:38, 15 July 2011 (UTC) </font> <br />
:::::<font color=salmon> Responded to Anna's second round of the comments. </font>}}<br />
::::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 13:32, 19 July 2011 (UTC) </font> <br />
:::::::<font color=salmon> Thank you!! [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC) </font><br />
<br />
*<big>[[Pappus Chain]]</big><br />
:{{hide|1=<br />
:<font color=salmon>This is used to be a subsection of [[Arbelos]]. Since arbelos is too long, I made Pappus chain a single page. I don't have time to finish it, so I think I'll leave the rest of the page for other people in the future. </font>}}<br />
<br />
*<big>[[Euclidean Algorithm]] </big><br />
:{{hide|1=<br />
<font color=salmon><br />
:Need to work on Gabriel Lame and worst case of EA.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page and leave some commenst!!! Thank you guys!! </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 17:39, 15 July 2011 (UTC) </font> <br />
:::<font color=salmon> Thank you! Responded to Anna's comments. </font><br />
::::<font color=darkred> I put up one more comment [[User:AnnaP|AnnaP]] 13:39, 19 July 2011 (UTC) </font><br />
::::<font color=salmon> I moved the proof to the end. Thanks. [[User:PhoebeJiang|Phoebe]] 13:46, 19 July 2011 (UTC) </font>}}<br />
:::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 00:33, 20 July 2011 (UTC) </font><br />
::::::<font color=salmon> Thank you!!! [[User:PhoebeJiang|Phoebe]] 00:55, 20 July 2011 (UTC) </font><br />
<br />
*<big>[[Application of the Euclidean Algorithm]]</big> <br />
:{{hide|1=<font color=salmon><br />
: Finish up with computer science section and knot theory.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page also. PLEASE leave anything you think will make it better!!! Thank you guys!!</font>}}<br />
:<font color=red>I could really need comments on this page. Thanks in advance!</font><br />
::<font color=salmon> Put up for final review. [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC)</font><br />
<br />
*<big>[[Pigeonhole Principle]]</big><br />
:{{hide|1=<font color=salmon><br />
:Just started it :P </font>[[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
:<font color=salmon>There is a problem I don't understand.</font> [[User:PhoebeJiang|Phoebe]] 02:06, 8 July 2011 (UTC)<br />
::<font color=orangered>Left some comments on the discussion page. Send us the problem you don't understand and we'll try it on our end of things (rscott3@swarthmore.edu). Let me know if you want me to read through the page again or anything. [[User:Rscott3|Richard]] 7/8</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 18:30, 8 July 2011 (UTC)Thank you Richard~ And I figured out the problem just now. :) </font><br />
::::<font color=orangered> Went back and looked over some of your changes. Left some responses to your comments. I'd be happy to read it again at some point if you'd like. [[User:Rscott3|Richard]] 7/12</font><br />
:::::<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC)Great! I'll check your responses. Thank you. </font><br />
:<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:32, 12 July 2011 (UTC)Finish writing the page. Will work on Richard's new comments. </font><br />
:<font color=salmon> Responded to Richard's comments. Appreciate other suggestions.</font><br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:59, 16 July 2011 (UTC) I left a few comments. Overall, I think this page is great! </font color><br />
::<font color=salmon> Thank you Becky! I really appreciate it. I made some changes to this page. [[User:PhoebeJiang|Phoebe]] 16:16, 16 July 2011 (UTC)</font><br />
:<font color=plum>Flora 00:28, 17 July 2011 (UTC)I leave a lot of comments for you. I'll finish '''Why Interesting''' part tomorrow.</font><br />
::<font color=salmon> Thank you very much!!! I've made some changes to the page and responded to your comments!! Is it better now? [[User:PhoebeJiang|Phoebe]] 04:19, 17 July 2011 (UTC)</font><br />
::<font color=plum>I left some comments for your '''Why Interesting''' section, and I responsed some of your changes. The rest are perfect, and I don't think you need any more work.</font><br />
:::<font color=salmon> Thanks! </font>}}<br />
:<font color=salmon> Put up for final review. [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC)</font><br />
:<font color=orangered>Comments up! My new comments are in <font color=blue>blue</font>. Congrats on a good page! [[User:Rscott3|Richard]] 7/20</font><br />
<br />
===Flora Li===<br />
Please leave some comments for my pages, Thanks.<br />
*<big>[[Waves]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:[[User:Flora1103|Flora]] 19:45, 10 June 2011 (UTC)<font color=plum> Need to work on '''Sine Wave Generration'''. Need to add more detail explaination to the graphs and the equetions.</font><br />
:<font color=plum>Flora 16:34, 28 June 2011 (UTC)I changed the title. And try to re-organize it to more "waves".</font><br />
::<font color=orangered>Left some feedback. [[User:Rscott3|Richard]] 6/29</font><br />
::Flora 20:14, 29 June 2011 (UTC)<font color=plum>Thanks very much~</font><br />
:Flora 20:35, 13 July 2011 (UTC)<font color=plum>Send to final review</font><br />
::<font color=darkred> Flora, can you go through the [[Checklist for writing pages]] on the discussion page before I go through the review process for this page? That makes my job much, much easier, and helps me give the most appropriate feedback. You can find examples by looking at the discussion pages of other pages on the final review page. Thanks [[User:AnnaP|AnnaP]] 7/14 </font><br />
::<font color=plum>Flora 21:25, 14 July 2011 (UTC)Have already went through the check list and leave them in discussion page. Sorry about that.</font><br />
:<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:58, 14 July 2011 (UTC) Comments are addressed. </font><br />
::<font color=plum>Flora 02:41, 15 July 2011 (UTC)Have responsed to your suggestions. Thanks very much.</font><br />
}}<br />
:::<font color=darkred> Take a look at my comments on the page [[User:AnnaP|AnnaP]] 16:08, 15 July 2011 (UTC) </font> <br />
:::<font color=plum>Flora 18:46, 15 July 2011 (UTC)Thanks very much for your comments. I responsed already, and I'm working on Fourier images now. I will post them later.</font><br />
:::<font color=plum>Done with all changes. Hope getting approved for final review.</font><br />
*<big>[[Dandelin Spheres Theory]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:Flora 22:20, 28 June 2011 (UTC)Have done most of it, still need to work on the references. I'm looking for any comments, plz~<br />
:<font color=plum>[[User:Flora1103|Flora]] 00:00, 11 June 2011 (UTC)I just start this topic. Working ont eh proofs</font><br />
:<font color=plum>Flora 20:50, 16 June 2011 (UTC) I find some useful information on a online school's webpage, but I couldn't open the page today. I don't know why. The url is [http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html].</font><br />
:[[User:PhoebeJiang|Phoebe]] 00:43, 19 June 2011 (UTC) <font color=salmon>I have no problem viewing it. Try a different browser I think. </font><br />
:<font color=plum>Flora 15:39, 20 June 2011 (UTC) I fixed that problem, thx. I now editting "Prove of conic section curve", and I change one image that showed before, so I still need to edit "Explore hyperbola spheres" later.</font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 21:12, 29 June 2011 (UTC): I left you a bunch of comments on this one, but I didn't get all the way through. It was slow reading and I ran out of time.</font><br />
:<font color=plum>Thanks very much.~</font><br />
<br />
:Flora 02:20, 14 July 2011 (UTC)<font color=plum>Start to revise it</font><br />
<br />
:Flora 21:55, 14 July 2011 (UTC)<font color=plum>Re-arranged the outlet, hope this is better.</font><br />
<br />
:<font color=plum>Flora 23:27, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Flora 20:42, 17 July 2011 (UTC)Responsed to Becky's suggestions. Submitted to final review.</font><br />
<br />
<br />
<br />
*<big>[[Dihedral Groups]]</big><br />
:Welcome comments XD<br />
:{{HideThis|1=More|2=<br />
:<font color=plum>Flora 20:54, 27 June 2011 (UTC)I add the main image but not start this page yet. Will work on it from next week. XD</font><br />
:<font color=plum>Flora 16:25, 13 July 2011 (UTC) Almost done with the page, still need to work on complex plane explanation, and subgroup part.</font><br />
:<font color=plum>Flora 01:58, 15 July 2011 (UTC)Finish this page, need to revise.</font><br />
:<font color=plum>Flora 23:26, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Send to final review</font><br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 13:46, 20 July 2011 (UTC) </font><br />
<br />
==Contact Us==<br />
<br />
===Faculty===<br />
'''Dr.Cammie Barnes''' <br />
<br />
*Email: cbarnes@sbc.edu<br />
<br />
*Skype: Cammie.Barnes<br />
<br />
===Students===<br />
'''Anna Donko'''<br />
<br />
*Email: donko14@sbc.edu<br />
<br />
*Skype: anna.k.donko<br />
<br />
'''Phoebe Jiang'''<br />
<br />
*Email: jiang14@sbc.edu<br />
<br />
*Skype: blacki2014<br />
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'''Flora Li'''<br />
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*Email: li14@sbc.edu<br />
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*Skype: Flora Li<br />
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[[User:PhoebeJiang|Phoebe]] 00:45, 19 June 2011 (UTC)<font color=salmon>Thanks for the info!</font><br />
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[[PartnerHome]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=SB11&diff=26372SB112011-07-20T18:44:45Z<p>Rscott3: /* Phoebe Jiang */</p>
<hr />
<div>=<center> Sweet Briar 2011 </center>=<br />
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==Discussion==<br />
<font color=navy>Please feel free to discuss math pages or leave a message for the creators!</font><br />
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Hey girls, the discussion bar on the very top of every page next to "page" is really helpful and necessary for peer review or faculty review. If you guys go through other people's page and want to comment on their work, please do. [[User:PhoebeJiang|Phoebe]] 15:09, 11 June 2011 (UTC)<br />
<br />
==Summer 2011 Projects==<br />
<br />
===Anna Donko===<br />
*<big>[[Steiner's Chain]]</big><br />
:{{hide|1=<br />
:<font color=peach> Need to work on explaining algebraic formulas.</font> <br />
PLEASE look at this!! I would really appreciate comments and suggestions!!<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:27, 28 June 2011 (UTC): Just left you a lot of comments on the discussion page.</font><br />
:<font color=peach> Working on reading and using all of the comments on the discussion page, thanks all who left me some feedback! </font><br />
:<font color=orangered>Interesting topic! Left some comments! [[User:Rscott3|Richard]] 7/18</font>}}<br />
Submitted for final review <br />
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*<big>[[Boy's Surface]]</big><br />
:{{hide|1=<br />
:<font color=peach> Leah and I are trying to figure out how we want to lay out our page...Please hold back with the comments while we continue to reorganize as well as add to our page!</font><br />
:<font color=red> Need to work on explanation of parametrization...??</font><br />
:<font color=salmon> Left some comments~ [[User:PhoebeJiang|Phoebe]] 23:56, 13 July 2011 (UTC)</font>}}<br />
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*<big>[[Parabolic Integration]]</big><br />
:{{hide|1=<br />
:<font color=peach>Just started this new page!! I am going to ink this to the already existing page, [[Parabola]]</font><br />
::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:51, 12 July 2011 (UTC) Great image! I like both pictures! The title is clear but I think you can make it shorter.</font><br />
:<font color=peach>Going to change my equations so they work out neater for the reader</font><br />
:<font color=black> Planning to add much more, including explanations of why/how parabolas are used in the physical world, as well as differentiate between parabolas and catenaries, lastly I will find the area of an actual life sized piece of parabolic architecture </font>.}}<br />
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===Phoebe Jiang===<br />
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*<big>[[Arbelos]]</big><br />
:{{hide|1=<br />
:<font color=salmon>Need to work on the two footnotes. And do a last revision, revise Pappus Chain and Bankoff Chain. <br />
:PLEASE look at this page. Comments are always welcome! </font>[[User:PhoebeJiang|Phoebe]] 19:49, 28 June 2011 (UTC)<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 13:21, 30 June 2011 (UTC): I've gone back through and responded to your comments or crossed things off where we both agree.</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Thank you Kate!!! <3 </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments for you. See my thoughts about a spin-off page. [[User:AnnaP|AnnaP]] 7/12 </font> <br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC) Thank you Anna and I've responded to your comments.</font><br />
::::<font color=darkred> One more round of comments is up [[User:AnnaP|AnnaP]] 15:38, 15 July 2011 (UTC) </font> <br />
:::::<font color=salmon> Responded to Anna's second round of the comments. </font>}}<br />
::::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 13:32, 19 July 2011 (UTC) </font> <br />
:::::::<font color=salmon> Thank you!! [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC) </font><br />
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*<big>[[Pappus Chain]]</big><br />
:{{hide|1=<br />
:<font color=salmon>This is used to be a subsection of [[Arbelos]]. Since arbelos is too long, I made Pappus chain a single page. I don't have time to finish it, so I think I'll leave the rest of the page for other people in the future. </font>}}<br />
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*<big>[[Euclidean Algorithm]] </big><br />
:{{hide|1=<br />
<font color=salmon><br />
:Need to work on Gabriel Lame and worst case of EA.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page and leave some commenst!!! Thank you guys!! </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 17:39, 15 July 2011 (UTC) </font> <br />
:::<font color=salmon> Thank you! Responded to Anna's comments. </font><br />
::::<font color=darkred> I put up one more comment [[User:AnnaP|AnnaP]] 13:39, 19 July 2011 (UTC) </font><br />
::::<font color=salmon> I moved the proof to the end. Thanks. [[User:PhoebeJiang|Phoebe]] 13:46, 19 July 2011 (UTC) </font>}}<br />
:::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 00:33, 20 July 2011 (UTC) </font><br />
::::::<font color=salmon> Thank you!!! [[User:PhoebeJiang|Phoebe]] 00:55, 20 July 2011 (UTC) </font><br />
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*<big>[[Application of the Euclidean Algorithm]]</big> <br />
:{{hide|1=<font color=salmon><br />
: Finish up with computer science section and knot theory.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page also. PLEASE leave anything you think will make it better!!! Thank you guys!!</font>}}<br />
:<font color=red>I could really need comments on this page. Thanks in advance!</font><br />
::<font color=salmon> Put up for final review. [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC)</font><br />
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*<big>[[Pigeonhole Principle]]</big><br />
:{{hide|1=<font color=salmon><br />
:Just started it :P </font>[[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
:<font color=salmon>There is a problem I don't understand.</font> [[User:PhoebeJiang|Phoebe]] 02:06, 8 July 2011 (UTC)<br />
::<font color=orangered>Left some comments on the discussion page. Send us the problem you don't understand and we'll try it on our end of things (rscott3@swarthmore.edu). Let me know if you want me to read through the page again or anything. [[User:Rscott3|Richard]] 7/8</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 18:30, 8 July 2011 (UTC)Thank you Richard~ And I figured out the problem just now. :) </font><br />
::::<font color=orangered> Went back and looked over some of your changes. Left some responses to your comments. I'd be happy to read it again at some point if you'd like. [[User:Rscott3|Richard]] 7/12</font><br />
:::::<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC)Great! I'll check your responses. Thank you. </font><br />
:<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:32, 12 July 2011 (UTC)Finish writing the page. Will work on Richard's new comments. </font><br />
:<font color=salmon> Responded to Richard's comments. Appreciate other suggestions.</font><br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:59, 16 July 2011 (UTC) I left a few comments. Overall, I think this page is great! </font color><br />
::<font color=salmon> Thank you Becky! I really appreciate it. I made some changes to this page. [[User:PhoebeJiang|Phoebe]] 16:16, 16 July 2011 (UTC)</font><br />
:<font color=plum>Flora 00:28, 17 July 2011 (UTC)I leave a lot of comments for you. I'll finish '''Why Interesting''' part tomorrow.</font><br />
::<font color=salmon> Thank you very much!!! I've made some changes to the page and responded to your comments!! Is it better now? [[User:PhoebeJiang|Phoebe]] 04:19, 17 July 2011 (UTC)</font><br />
::<font color=plum>I left some comments for your '''Why Interesting''' section, and I responsed some of your changes. The rest are perfect, and I don't think you need any more work.</font><br />
:::<font color=salmon> Thanks! </font>}}<br />
:<font color=salmon> Put up for final review. [[User:PhoebeJiang|Phoebe]] 21:46, 19 July 2011 (UTC)</font><br />
:<font color=orangered>Comments up! Congrats on a good page! [[User:Rscott3|Richard]] 7/20</font><br />
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===Flora Li===<br />
Please leave some comments for my pages, Thanks.<br />
*<big>[[Waves]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:[[User:Flora1103|Flora]] 19:45, 10 June 2011 (UTC)<font color=plum> Need to work on '''Sine Wave Generration'''. Need to add more detail explaination to the graphs and the equetions.</font><br />
:<font color=plum>Flora 16:34, 28 June 2011 (UTC)I changed the title. And try to re-organize it to more "waves".</font><br />
::<font color=orangered>Left some feedback. [[User:Rscott3|Richard]] 6/29</font><br />
::Flora 20:14, 29 June 2011 (UTC)<font color=plum>Thanks very much~</font><br />
:Flora 20:35, 13 July 2011 (UTC)<font color=plum>Send to final review</font><br />
::<font color=darkred> Flora, can you go through the [[Checklist for writing pages]] on the discussion page before I go through the review process for this page? That makes my job much, much easier, and helps me give the most appropriate feedback. You can find examples by looking at the discussion pages of other pages on the final review page. Thanks [[User:AnnaP|AnnaP]] 7/14 </font><br />
::<font color=plum>Flora 21:25, 14 July 2011 (UTC)Have already went through the check list and leave them in discussion page. Sorry about that.</font><br />
:<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:58, 14 July 2011 (UTC) Comments are addressed. </font><br />
::<font color=plum>Flora 02:41, 15 July 2011 (UTC)Have responsed to your suggestions. Thanks very much.</font><br />
}}<br />
:::<font color=darkred> Take a look at my comments on the page [[User:AnnaP|AnnaP]] 16:08, 15 July 2011 (UTC) </font> <br />
:::<font color=plum>Flora 18:46, 15 July 2011 (UTC)Thanks very much for your comments. I responsed already, and I'm working on Fourier images now. I will post them later.</font><br />
:::<font color=plum>Done with all changes. Hope getting approved for final review.</font><br />
*<big>[[Dandelin Spheres Theory]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:Flora 22:20, 28 June 2011 (UTC)Have done most of it, still need to work on the references. I'm looking for any comments, plz~<br />
:<font color=plum>[[User:Flora1103|Flora]] 00:00, 11 June 2011 (UTC)I just start this topic. Working ont eh proofs</font><br />
:<font color=plum>Flora 20:50, 16 June 2011 (UTC) I find some useful information on a online school's webpage, but I couldn't open the page today. I don't know why. The url is [http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html].</font><br />
:[[User:PhoebeJiang|Phoebe]] 00:43, 19 June 2011 (UTC) <font color=salmon>I have no problem viewing it. Try a different browser I think. </font><br />
:<font color=plum>Flora 15:39, 20 June 2011 (UTC) I fixed that problem, thx. I now editting "Prove of conic section curve", and I change one image that showed before, so I still need to edit "Explore hyperbola spheres" later.</font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 21:12, 29 June 2011 (UTC): I left you a bunch of comments on this one, but I didn't get all the way through. It was slow reading and I ran out of time.</font><br />
:<font color=plum>Thanks very much.~</font><br />
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:Flora 02:20, 14 July 2011 (UTC)<font color=plum>Start to revise it</font><br />
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:Flora 21:55, 14 July 2011 (UTC)<font color=plum>Re-arranged the outlet, hope this is better.</font><br />
<br />
:<font color=plum>Flora 23:27, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Flora 20:42, 17 July 2011 (UTC)Responsed to Becky's suggestions. Submitted to final review.</font><br />
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<br />
<br />
*<big>[[Dihedral Groups]]</big><br />
:Welcome comments XD<br />
:{{HideThis|1=More|2=<br />
:<font color=plum>Flora 20:54, 27 June 2011 (UTC)I add the main image but not start this page yet. Will work on it from next week. XD</font><br />
:<font color=plum>Flora 16:25, 13 July 2011 (UTC) Almost done with the page, still need to work on complex plane explanation, and subgroup part.</font><br />
:<font color=plum>Flora 01:58, 15 July 2011 (UTC)Finish this page, need to revise.</font><br />
:<font color=plum>Flora 23:26, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Send to final review</font><br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 13:46, 20 July 2011 (UTC) </font><br />
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'''Anna Donko'''<br />
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*Email: donko14@sbc.edu<br />
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*Skype: anna.k.donko<br />
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'''Phoebe Jiang'''<br />
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*Email: jiang14@sbc.edu<br />
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*Skype: blacki2014<br />
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'''Flora Li'''<br />
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*Email: li14@sbc.edu<br />
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*Skype: Flora Li<br />
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[[User:PhoebeJiang|Phoebe]] 00:45, 19 June 2011 (UTC)<font color=salmon>Thanks for the info!</font><br />
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[[PartnerHome]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Pigeonhole_Principle&diff=26364Talk:Pigeonhole Principle2011-07-20T18:41:38Z<p>Rscott3: </p>
<hr />
<div>=Response to Checklist=<br />
==Messages to the Future==<br />
*Made two suggestions for future editors.<br />
<br />
==References and footnotes==<br />
*Original sources of "borrowed" images are marked if you click them. <br />
*Direct quotes are cited. <br />
*References are listed with links at the bottom of the page.<br />
<br />
==Good writing==<br />
<br />
===Context===<br />
*This topic “Pigeonhole principle” is my personal favorite! The main image is appealing. "More Mathematical Explanation" is comprehensive. I gave a lot of examples for readers to know this principle better. I believe those examples make this page a lot more interesting than other pages.<br />
<br />
===Quality of prose and page structuring===<br />
*The beginning paragraph defines pigeonhole principle and provides another definition, name.<br />
*Each section is related to the main topic.<br />
*Real world applications and subsections in the mathematical explanations are listed from easy to hard, from fundamental to expanding. I hide some long proofs and just show the statements, in case people don't want to know the proof but just the examples themselves. The heaviest math is in “Why interesting.” Well, they are not heavy math but hard to understand.<br />
<br />
===Integration of images and text===<br />
*Every image is referred in the context. In every image, the denotations are noted and readers know what each symbol means. Some images are just for better structure; they are not really illustrative (i.e. birthday cake, socks, hair, cards).<br />
*Readers are clear about which picture they should look at while viewing this page. Sometimes they can get inspired by the picture and know how to prove the problem.<br />
*There are no large chunks of words. <br />
<br />
===Connections to other mathematical topics===<br />
*There is a link to another mathematical topic outside of Math Images. <br />
<br />
===Examples, Calculations, Applications, Proofs===<br />
*The equations, calculations, and examples are clear to readers.<br />
*Every statement or property has its proof. <br />
*I wrote a summary and some guiding text to help people know how to solve this sort of problems using pigeonhole principle. (especially the “how to construct pigeonholes” section)<br />
<br />
===Mathematical Accuracy and precision of language===<br />
*I try to make everything as clear as possible. Hopefully readers with any level of math will understand it.<br />
*I try to make everything error free. Corrections and suggestions are appreciated. <br />
*The definition of every mathematical term, theorem or rule readers might not know is either explained in the body text or via a mouse-over, or linked to another page.<br />
<br />
===Layout===<br />
*Texts are short, not very long, and broken up by images or broken in paragraphs.<br />
*Mathematical terms are boldfaced. <br />
*Hide and see is appropriately used.<br />
*No awkward white chunks. <br />
*No weird computer codes.<br />
<br />
===Thank You===<br />
<br />
=Comments from Chris 7.20.11=<br />
<br />
<font color=green> This is both a fun and interesting page. You've done an excellent job with this. My comments are mostly minor ones.<br />
<br />
<font color=salmon> Thank you very much Chris!! Very helpful comments! Phoebe 7/20</font><br />
<br />
Near the end of the Basic Description, you make an excellent statement that I think belongs instead in the intro: "The pigeonhole principle (PP) itself may seem simple but it is a powerful tool in mathematics." I'd make it the strong conclusion to your intro section.<br />
<br />
:<font color=salmon>I like your suggestion! You want me to move only this sentence or the whole paragraph this sentence is in? </font><br />
<br />
Basic Description<br />
*<s>Layout issue: It's clear on my browser that the last two paragraphs begin "The pigeonhole principle is also called the Dirichlet.." and "The PP itself may seem..." It's hard to tell. however, where the other paragraph(s) end(s) and begin. </s><br />
:<font color=salmon>I see. Fixed it. </font><br />
<br />
*Paragraph 1, Sentence 2 (P1S2) Is the principle really famous?<br />
:<font color=salmon>I think so... I've learnt it in middle school.</font><br />
<br />
*<s>P1S3 Add "that" between "principle is" and "for any."</s><br />
:<font color=salmon>Fixed it.</font><br />
<br />
*<s>Near the end of that long paragraph (if it is only one paragraph), there must be <i>at least</i> two pigeons in a pigeonhole. </s><br />
:<font color=salmon>Fixed it. </font><br />
<br />
Interesting Applications<br />
*<s>Birthdays: P1S1: Remove "the" from between "including" and "leap."</s><br />
:<font color=salmon>Fixed it. </font><br />
<br />
*<s>Hair: P1S2: Make "hair" into "hairs a human normally has." </s><br />
:<font color=salmon>Fixed it.</font><br />
<br />
*<s>Chairs: P2S1: Instead of "(see the two figures below)", add a new sentence that talks about how the first two examples show two ways of partitioning the chairs without two empty chairs next to each other.</s><br />
:<font color=salmon> Fixed it. Like your suggestion!</font><br />
<br />
*<s>Strangers or Friends: really well done!</s><br />
:<font color=salmon> Thanks! </font><br />
<br />
More Mathematical Examples<br />
*3. This example is very hard to follow. At the very least, give a concrete example to relate all the variables with subscripts to.<br />
:<font color=salmon> Good to know. Add a more concrete example. </font><br />
<br />
Summary<br />
<s>*P1, Subsection 3. You didn't let the limit be 50 or "want to have" the limit be 50; the limit ends up being 50 because only 50 distinct groups can be made. </s><br />
:<font color=salmon> Fixed it. </font><br />
<br />
How to Find the Bounds<br />
*<s>P1S1 Change the tense to past perfect; "we have only been told" and "whether we have reached" <br />
*P1S3 Either make the question are full sentence or say "This leads to the interesting question of how to find the bounds."</s><br />
<br />
:<font color=salmon> Fixed them. </font><br />
<br />
*Examples #1: Do you want a direct link to another website? (Ramsey Numbers) I don't think that's the convention for Math Images. I'll check tonight at Swarthmore; maybe you can also investigate this. <br />
:<font color=salmon> I'm sorry if I did anything inappropriate. I don't really want to link to another website either. It's just that I don't have time to make a page for Ramsey Number, but I put it as a future direction for other people. How about I create a helper page and link PP to [[Ramsey Number]]? I'll add more things to it if I have time and if I'm not able to do that, future editors could investigate it as well. What do you think?</font><br />
<br />
</font><br />
=Richard Comments 7/20=<br />
*<font color=blue>Phoebe, the page looks great! I know you've done a ton of editing and fixing already, so take these comments however you'd like. [[User:Rscott3|Richard]] 7/2<br />
:::After reading this a second or third time, I'm realizing more what I was having trouble saying last time. In the basic description, you outline these two great ways to think about the pigeonhole principle: (1)a very intuitive way of thinking in terms of pigeons fitting into pigeonholes (2)a more "mathy" way with average and maximum value. You do a great job of explaining why they are the same in the basic description, but I think you are selling yourself short on the math side of things especially with many of the examples that you use. I guess what I'm trying to say is that there's more math to this than your examples show. For example, you could use math to describe the suit of cards example as well as describing it in terms of pigeons and holes. you could say that the avg value=5/4 so the max must be at least that, and since we can't pick a fraction of a card, there must be at least two cards of the same suit. I think it could be helpful to describe each example using both ways (1) and (2) like you did in the basic description.<br />
:::::*This also provides the potential for a MME that is more than just example. I think you can more thoroughly describe avg value and max value pictorially and with words. By defining average, you can show mathematically that there must be a number in the non-empty finite set that is greater than the average.<br />
<br />
*For Interesting applications number 6 I'd reword the main statement to be something like "In a room with six people in it, there will always be either 3 people who know each other or 3 people who don't know each other.<br />
::*In the same example, I'd get rid of "This seems to be a chaotic problem but we need to find order in this mess." and "How should we prove that it is true all the time?" You can start the section off instead with something like "We can use a diagram to help us show that this statement is true. Note that these..."<br />
::*you sort of just say that by the pigeonhole principle, at least three lines must be the same color. I think it would be useful for you to show how you got that in terms of avg and max value here. <br />
<br />
*I'm a little confused still by MME example 3.<br />
:::And you used the wring less than symbol I think...try this one <math> \leq </math><br />
<br />
*For How to Construct example 2, I think you could maybe put this one in terms of pigeons and pigeonholes as well and put the factorial form of 15 choose 2 like you have for the example 11 choose 4.<br />
<br />
*In your summary section, you mention limit several times and I'm not exactly sure what you're talking about.<br />
<br />
*Same with bounds. Maybe define '''limits''' and '''bounds''' in terms of the principle.<br />
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*This is a really cool page, Phoebe. I really like its roots in application and that you use some pretty cool examples to teach the topic in the process.<br />
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[[User:Rscott3|Richard]] 7/20<br />
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</font><br />
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{{hide|1=<br />
=Comments from Flora 7.16=<br />
<font color=plum><br />
<br />
==General Comments==<br />
*This page is very good. All the examples are interesting and thoughtful.<br />
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*My suggestions sometimes are not all right. You can just ignore if they are wrong, or you can explain to me.<br />
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*The page is almost done, so all my suggestions are for tiny confusing points.<br />
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*The only one part I think you may need a big work is the 1st example in '''How to construct pigeonholes''', I could not get your answer from your example.<br />
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:<font color=salmon> Thank you Flora for looking at my page! You've made lots of important and helpful comments! </font><br />
<br />
==Intro==<br />
*The intro is interesting and attractive.But you may want to use fewer sentences to introduce the image.<br />
::<font color=salmon> I see your point, but I don't know which sentence should I leave out... All of them are very important...</font><br />
<br />
==Basic Description==<br />
*<s>The first sentence you say: "If more than n items are put into n pigeonholes, then at least one pigeonhole must contain more than one pigeon." First you say "items", then you change it to pigeon. If you want to introduce the pigeonhole theory in Basic Description, then you do not include the same thing in your Intro section.</s><br />
::<font color=salmon> I don't know what happened but I changed it before and somehow the old version showed up. Thanks for letting me know! </font><br />
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*In the green balloon, I saw the explaination of "non-empty" and "finite set". I'm wondering maybe you need to define "set" first. In "average value"'s balloon, I think your explaination is more confusing than "average value" itself.<br />
::<font color=salmon> Added the definition of "set". Yeah, when I was defining the average value, I found the definition more confusing than the original term. I put up a new definition. </font><br />
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*<s>I am not sure if "the average value of<math>\frac{pigeons}{pigeonhole}</math>" is right. I understand what you want to say, but <math>\frac{pigeons}{pigeonhole}</math> is already an average value. And in next sentence you say: "the maximum value of the number of pigeon per pigeonhole should be greater than one," same issue. Usually we say the maximum value of "something." The number of "Something" should be bigger than 1, than we can make the comparison.</s><br />
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::<font color=salmon>Good point. I'm gonna make it clear. </font><br />
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==Interesting Applications==<br />
*<s>This part is very clear and easy to understand.</s><br />
:<font color=salmon> <s>Thank you</s></font><br />
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*<s>The structure in section 2.Apair of socks is a little wierd, "a matching pair." is under the image, but not with the whole texts body. You may want to delete one blank line under the bold sentences to move "a matching pair." up one line.</s><br />
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:<font color=salmon> It is about the size of the user's browser. I move the text up, so this kind problem won't happen again. </font><br />
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*<s>In section 5. Chairs, it's a little confusing why you make the left figure in 2 rows. In your example, 9 people will seat in 1 row, where there are 12 chairs. But in the first figure, it seems that there are 2 rows. You could change your image to that 1 row with 12 triangles to show that there is 1 row with 12 empty chairs. And draw 9 circle beside to show that there are 9 persons who want to seat on those chair.</s><br />
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:<font color=salmon> Agree. I changed my image cause I found out that maybe more people would understand the updated image. </font><br />
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*<s>You can create more images like your right figure to show more possibles that at least consecutive set of 3 chairs are filled with people.</s><br />
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:<font color=salmon> Good point. </font><br />
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*Also in this example, it is not necessary to devide people into 4 groups. If there are only 2 groups, you can insert all 3 empty chairs together but still left at least 3 consecutive chairs filled with people. This example is interesting, and you can talk about more possibilities. For me this is an example that there are 12 holes, but you need to fill these hles with 3 empty chairs.<br />
:<font color=salmon> True, you can think of this question that way. I'd rather make it into four groups because it is easier to explain why one group definitely has 3 filled chairs. Also, I added a few lines to point out why there are four groups; because some groups may have zero filled chairs. What do you think about it now? </font> <br />
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*<s>6.Strangers or Friends. I fond this part is hard for me to understand. You say: "Draw lines joining every pair of two points," but I find that the highest point has no line with lower left point. And you say: "What the statement graphically means is that we can always find a pink triangle (3 mutual friends) or a green triangle (3 mutual strangers) in this hexagon." I find a triangle with all 3 sides in green. But it is hard to find, you may change your image a little bit to make it clear which triange is satisfied with the situation.</s><br />
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:<font color=salmon> Oops, my bad. Didn't realize that. Thanks! </font><br />
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==More Mathematical Examples==<br />
<s>*In the first example, it is hard to read for all notations. You could bring in a real example with numbers to say the example is true.<br />
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*In the second example, You could add one sentence to say that there are 50 pairs(subsets), to make sure the reader know what subset is.<br />
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:<font color=salmon> Fixed it. </font><br />
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</s><br />
<br />
==How to Construct Pigeonholes==<br />
===Examples===<br />
<s><br />
*I'm confused about the 1st example that you say "a square of side length 2," is it true? Or you mean the area is 2? If the side is 2, then the largest distance of random 2 points will be the side, which is also the hypotenuse of the right triangle, and it is 2, but not <math>\sqrt{2}</math>.<br />
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*The example you have in figure 5 is not very related to pigeonhole theory, because you simply put 4 points in the corner and 1 in center. And you calculate the largest distance with 2 points on the corner, but they are not in the same pigeonhole.<br />
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*Also if I put all 4 points on the corner, and 1 random. then the largest distance will be the diagonal of the biggest square, which is <math>2 \sqrt{2}</math>.<br />
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:<font color=salmon> I deleted the first example because it is not very good and I don't want to confuse the readers. Plus, three examples are enough for this section. </font><br />
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*I like the images you have in example 2. You could make a similar image for '''6.Strangers or Friends'''. But the hiden image is not easy to find. Could you move it down and say this is the solution?<br />
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:<font color=salmon> Sure. </font><br />
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*For your example 3. you may want to define the combination notation first or just avoid using them, since you did not use them again in the rest of you page.<br />
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:<font color=salmon> I may need the combination notations in order to get 105 and 6 but I'll definitely put up more explanation to it. </font><br />
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*Could you explain <math>C^4_2 = \frac{4\times3}{2} = 6</math> more clearly? It is confusing why choosing 2 over 4.<br />
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:<font color=salmon> Fixed it. </font></s><br />
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*Could you make "pair of courses" clearer? Is that you only make the same two courses into a pair or you make all 15 courses into pairs? Or you mean in 4 courses of each student can be make into pairs?<br />
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:<font color=salmon> I pair up all the 15 courses. And I pair up the 4 courses each student chooses. Good to know that I'm not being clear. Added more explanation. Is it better? </font><br />
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*Example 4 is really smart and interesting! But it may cause confusion why all the groups are disjoint. Could you explain that?<br />
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:<font color=salmon> Yes. They are disjoint because they don't have the same elements. They don't have the same elements because each group is expanded by adding multiples of a different odd number. </font><br />
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::That is more clear, you just need to add that sentence to your page.Flora 18:07, 17 July 2011 (UTC)<br />
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===Summary===<br />
*It is a little hard to follow your summary, since you talk all the four examples, but you don't have images there. Readers may be lost. For me I have to go back to each example to see what you talked about. Maybe you can move this section to the bottom of each example. You can also ignore this suggestion. It is not a big problem.<br />
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:<font color=salmon> I don't know. If the readers read those 3 examples carefully, they'll know what I'm talking about. And if I move them at the bottom of each example, it won't be a summary of "how to construct pigeonholes." I admit that this summary needs more work but this is the clearest summary I could do...== I appreciate it if other people could help me with this summary. </font><br />
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::It's fine if you just keep the oringin version.<br />
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==How to Find the Bounds==<br />
===Examples===<br />
*For the first example, it is confusing about the notation.<br />
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:<font color=salmon> I'm sorry but could you tell me which part is confusing? You mean you don't understand R (3, 3) ? </font><br />
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::Yes, that's the point. The explaination of this notation is not very clear.<br />
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*For the first example. The image only illustrate one possibility. But you do not provide a proof of your conclusion. If you trun any green line to pink, you can get a pink triangle. Turn other 2 pink to green you can get a green triangle also. I'm also confused if you can make all 4 lines from one point to a single color to make she or he know or does not know any other ones.<br />
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:<font color=salmon> ''Since we want to prove that a number is the best lower bound, the basic idea is showing that a smaller number leads to a contradiction or an error.'' As I stated earlier, I just need to give a counterexample to prove that 5 doesn't always work. No need to give a proof~ </font><br />
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::I understand your point. But as what I said, I think if you turn a green line to pink than you can get a pink triangle, so 5 works also.<br />
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==Why It's Interesting==<br />
*In the example of '''Taking Medicine''', you say this section may cause some confusion. I think your proof is very interesting. The confusing point may be that the first bold sentence is true, but it is not an argument. For normal people they will think any 14 days they take one pill each day will satisfy your condition. But I don't know how to help to improve this sentence, sorry.<br />
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*In the example of Domino, the figure is more under your statement but not left.<br />
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</font><br />
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=General Comments=<br />
<br />
<font color=orangered> Pictures will be so cool for this one! [[User:Rscott3|Richard]] 6/29</font><br />
:<font color=salmon> [[User:PhoebeJiang|Phoebe]] 18:35, 8 July 2011 (UTC)Yeah, I realized that too. I'm not finished with this page yet, so I'll work on it! Thanks!<br />
::Ps: We have pretty much the same font color~ lol </font><br />
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<font color=orangered><br />
[[User:Rscott3|Richard]] 7/8<br />
*This page is really really cool. You've got a ton of opportunities for applications and pictures.<br />
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*I'd make you Interesting Applications Section a "Why It's Interesting" section. I think this is more standard for math images. You can go to the edit with form tab to start that section.<br />
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*You really only use examples to go into the topic. Is there some graphical proof or derivation that you could include?</font><br />
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:[[User:PhoebeJiang|Phoebe]] 21:06, 8 July 2011 (UTC)<br />
<font color=purple><br />
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:*First of all, thank you for leaving comments!! I use a different color to distinguish us. <br />
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:*As you can see, I intend to use those 10 examples to lead in, giving readers a basic idea of what Pigeonhole principle is and how it works. The principle itself is pretty self-explanatary, so I didn't include the proof of this principle. Then, after you know how this simple principle helps in daily life, you will see the MME section with more examples and tricks about constructing pigeonholes. This is how I conceive this page, from easy to hard, from the description of pigeonhole principle, to applications (lead in), and eventually the hardest part, MME. So do you think I should put the applications under "Why interesting" after MME? If so, I'm afraid readers will feel overwhelmed by MME section. What do you think?<br />
::<font color=orangered> Okay, I see what you mean and I understand where you're coming from. To me, this page starts out with a very intuitive topic/description, and then gets more complicated in the current MME. What if there were some middle ground? I'm thinking like describing what average value is mathematically or something to put at the beginning of the MME to sort of segway into harder examples. There's a lot of examples on this page, and as the reader, I think it gets to be more difficult to pick out which examples are the most important ones. I don't know.... [[User:Rscott3|Richard]] 7/12</font><br />
:::<font color=purple> Richard! Check this out. I want some simple and interesting examples to lead in and I also want to put some hard applications under "why interesting" as you suggested. So, I'm thinking about splitting them into two sections, leaving the first 4-5 examples where they are now and put the rest of the examples after MME. Plus, those ten examples seem to be really long together. And people could apply those tricks mentioned in MME to see how the tricks work via the hard examples at the end of the page. What do you think about this idea? :) [[User:PhoebeJiang|Phoebe]] 20:38, 13 July 2011 (UTC) </font><br />
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:*Pigeonhole principle is basically about combinatorics, about numbers. This principle itself doesn't have much to do with pictures. I agree with you about the pictures. I personally like graphs than simply words and numbers, so I'm thinking about adding more illustrative pictures on it. <br />
</font><br />
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:*<font color=orangered> Phoebe, this page is looking great! I can read it through again if you want me to, just let me know when it's ready! [[User:Rscott3|Richard]] 7/12</font><br />
::*<font color=purple> [[User:PhoebeJiang|Phoebe]] 03:16, 14 July 2011 (UTC) Yeah actually could you read it through one more time, please? Tell me where I need to explain more precisely. Thanks a lot! </font><br />
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<br />
=Intro=<br />
<font color=orangered><br />
[[User:Rscott3|Richard]] 7/8<br />
*I'd reorganize this paragraph to something like this:<br />
::A pigeon is looking for a spot in the grid, but each box in the grid, or pigeonhole, is occupied. Where should the poor pigeon on the outside go? No matter which box he chooses, he must share with another pigeon. If all of the pigeons fit into the grid, here is definitely a pigeonhole contains more than one pigeon. This concept is known as the '''pigeonhole principle'''.<br />
*I think it's important to introduce the topic here in this section(I added a sentence at the end in my edit above).<br />
</font><br />
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::[[User:PhoebeJiang|Phoebe]] 21:05, 8 July 2011 (UTC)<br />
<font color=purple><br />
::*Agree. Fixed it! <br />
</font><br />
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=Basic Description=<br />
<font color=orangered><br />
[[User:Rscott3|Richard]] 7/8<br />
*In the first sentence, change "items" to "pigeons" or vice versa for consistency.<br />
<br />
*In the second paragraph, I think that this quote is more confusing than helpful. I feel like you generalizing the same thing in your own words would be waaaaaay more helpful.<br />
::*It might be helpful to define "average value" and "maximum value" (maybe with a green mouse-over)<br />
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*I'd delete "Although the two versions look very different, they are mathematically stating the same thing." and into the next paragraph, "To know why they are the same," and start with "Consider the main image instead". I think it's unnecessary to say.<br />
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*Maybe change "bigger" to "greater"???? That sounds more mathy to me.<br />
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*What's "its" referring to?<br />
::"Thus, '''its''' maximum value should be bigger than one as well, which means that there must be a pigeonhole contains more than one pigeons. Now we know that the two versions are actually talking about the same math principle."<br />
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*Be careful with words like "pretty obvious". You're assuming a lot of the reader.<br />
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*Move the sentence "Here are 10 exciting real life applications listed from easy to hard for you to have a better grasp on this famous principle." into the next section before the first example.<br />
</font><br />
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:[[User:PhoebeJiang|Phoebe]] 21:24, 8 July 2011 (UTC)<br />
:<font color=purple> <br />
:*You have a lot of good points. I totally agree with you. <br />
:*Fixed all of them. Are they better now?</font><br />
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<font color=slateblue> <br />
*[[User:Rebecca|Rebecca]] 02:35, 16 July 2011 (UTC) Phoebe, I'm Becky and I give feedback on many of the Math Images pages. I have a few suggestions for this page... <br />
* I would include mouseovers for "non empty" and "finite" because I think that if someone doesn't know what a maximum value or an average value is, they wont know either of those words either. <br />
* It's great that you introduce the general pigeonhole principle and then use explain the implications for the image. It's a logical order, and I think your approach is great! </font color><br />
::<font color=purple> Thank you Becky!!! Appreciate all of your comments! <3 Added the mouseover. </font><br />
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=Interesting Applications=<br />
<font color=orangered> [[User:Rscott3|Richard]] 7/8<br />
*This section is awesome awesome awesome.<br />
::<font color=purple>[[User:PhoebeJiang|Phoebe]] 23:04, 8 July 2011 (UTC)Thanks!</font><br />
*<font color=slateblue> [[User:Rebecca|Rebecca]] 02:58, 16 July 2011 (UTC) I agree- I like how you show people that these paradoxes must be true rather than formally proving them. It makes the page really interesting. </font color><br />
::<font color=purple> That's what I want!! Thanks!!! [[User:PhoebeJiang|Phoebe]] 16:13, 16 July 2011 (UTC)</font><br />
<br />
*Pictures would be really super cool here. The way its formatted right now makes it seem longer than it actually is, especially with the table of contents. One way that I'd think to format this is like the "More than just shadows" section of [[Solving Triangles]]. Assign a picture to each example, and make two columns where the picture and text alternate sides. The picture doesn't have to be super complicated. The birthday one could just be a cake or something. Make it look more visually appealing. <br />
::*This will also help the table of contents because you'd have to make each of the ten headings just bolded words, and not section titles.<br />
:::<font color=purple>[[User:PhoebeJiang|Phoebe]] 21:27, 8 July 2011 (UTC)Guess what! I'm thinking about the same thing!</font><br />
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*Some of the example descriptions get a bit wordy. For the purposes of these comments I'll just talk about the first one as an example and then say which ones were more confusing.<br />
::*For the birthday one, you could make it more clear by changing "Under the worst condition when the first 366 students have their birthdays from January 1st to December 31th, the 367th person has to be born on any day of the year. Thus, there are definitely two of the students who have their birthday falling on the same day." to "Under the worst condition when each of the first 366 students have their birthdays on different days from January 1st to December 31th, the birthday of the 367th person must be a repeat of one of those days. Thus, there are definitely two of the students who have their birthday falling on the same day."<br />
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::: <font color=purple>[[User:PhoebeJiang|Phoebe]] 22:34, 8 July 2011 (UTC)Yeah, yours is waaaay better. </font><br />
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*Numbers 5 and 9 could be a lot clearer with good pictures.<br />
::<font color=purple>[[User:PhoebeJiang|Phoebe]] 23:03, 8 July 2011 (UTC)Add some pictures. </font><br />
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*Number 7's main statement is confusing to me. <br />
::<font color=purple>[[User:PhoebeJiang|Phoebe]] 23:13, 8 July 2011 (UTC) Could you please be more specific about it? </font> <br />
:::<font color=orangered>I understand what you are trying to prove, but I think the statement is wordy. What if you say something like: " Some n number of people are at a party. There are always at least two people who shake hands with the same number of people."? But even with the way I worded it...It' awkward. [[User:Rscott3|Richard]] 7/12</font><br />
::*I really like how you relate this one back to the pigeon/pigeonhole metaphor.<br />
:::*<font color=purple> [[User:PhoebeJiang|Phoebe]] 23:03, 8 July 2011 (UTC) Thanks. I though it may be easier to understand if I do this. </font><br />
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*Number 8 is very confusing. That's the only one I don't understand.<br />
::<font color=purple> [[User:PhoebeJiang|Phoebe]] 23:55, 8 July 2011 (UTC) I made some changes. Try to make it as clear as possible. </font><br />
:::<font color=orangered>I'm still a bit confused by this. If he has to take at least one aspirin a day, does he take 14 consecutively in a period of 14 days? I think that's where I'm confused. What statement are you trying to make? Also, where did you get 59? [[User:Rscott3|Richard]] 7/12</font> <br />
::::<font color=purple> That is one possible answer. But he could also take 14 aspirin in less than 14 days, say he could take 14 aspirin in only 3 days. In other words, if he wants to try to avoid taking 14 aspirin during certain period, he can't do it. He may avoid taking 15 aspirin during a period of consecutive days. However, this statement is trying to say that you can always find a period of consecutive days where he takes exactly 14 aspirin. Am I being clear enough or not? Please ask me more questions if you are still confused. :P <br />
::::How did I get 59? Since you add 14 to every term in the first inequality, you'll get 45 + 14 = 59 for the very last term. <br />
::::Maybe I could generalize this statement instead of using numbers 45, 14, 30. (i.e. he has to take <math>n</math> aspirin over a <math>m</math> day period. Then there is definitely a period of consecutive days where he takes exactly <math>n - m -1 </math> aspirin.) Do it make more sense now? </font><br />
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*Number 10a is awesome! Great, clear description!<br />
::<font color=purple> [[User:PhoebeJiang|Phoebe]] 23:55, 8 July 2011 (UTC) Good to know. Thanks! </font><br />
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*For 10b, I didn't know exactly what you you were trying to show until you had already started to prove it. This one might be in need of some more clarity in the initial description of the proof and statement.<br />
::*Typo:"situation" should be "situations" in "Since there are over 32,000 possible colorings, we could not draw and check every one of the situation."<br />
</font><br />
:::<font color=purple> Fixed it. What do you think about this part now? </font><br />
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<font color=slateblue> <br />
* [[User:Rebecca|Rebecca]] 02:49, 16 July 2011 (UTC) "Here are 6 exciting real life applications for you to have a better grasp on this famous principle." I would say "real world applications to give you a better grasp of this famous principle."<br />
* "Under the worst condition when each of the first 366 students have their birthdays..." I would say "In the most extreme condition" <br />
* "This example may sound impossible to believe but mathematics, more precisely, the seemingly meaningless pigeonhole principle tells you it could happen." I would say " This example may sound impossible to believe, but the seemingly obvious pigeonhole principle tells us it must be true." </font color><br />
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::<font color=purple> Fixed all of them. Thank you. </font><br />
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=MME=<br />
==More Mathematical examples==<br />
<font color=orangered>[[User:Rscott3|Richard]] 7/8<br />
*Number 1 is confusing.<br />
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*For number 2, wouldn't it just be easier to choose all of the odd numbers, knowing that they are not consecutive and that there are only 50, then we know that the 51st must be even and next to an odd number.</font><br />
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::<font color=purple> [[User:PhoebeJiang|Phoebe]] 02:10, 9 July 2011 (UTC)Yes, you are right. </font><br />
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==How to Construct==<br />
<font color=orangered> [[User:Rscott3|Richard]] 7/8 <br />
*I was a bit confused by these examples.<br />
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*This might also be a good point to derive or prove the principle explicitly with the math for average and maximum values????</font><br />
::<font color=purple>[[User:PhoebeJiang|Phoebe]] 03:09, 9 July 2011 (UTC) I tried to fixed those confusing examples. Maybe some of them are still not very clear. I'll work on them later. Please feel free to tell me where should I improve. Thanks!</font><br />
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}}</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=SB11&diff=25996SB112011-07-19T14:07:38Z<p>Rscott3: /* Anna Donko */</p>
<hr />
<div>=<center> Sweet Briar 2011 </center>=<br />
<br />
==Discussion==<br />
<font color=navy>Please feel free to discuss math pages or leave a message for the creators!</font><br />
<br />
Hey girls, the discussion bar on the very top of every page next to "page" is really helpful and necessary for peer review or faculty review. If you guys go through other people's page and want to comment on their work, please do. [[User:PhoebeJiang|Phoebe]] 15:09, 11 June 2011 (UTC)<br />
<br />
==Summer 2011 Projects==<br />
<br />
===Anna Donko===<br />
<br />
*[[Steiner's Porism]]<br />
:<font color=peach> Need to work on explaining algebraic formulas.</font> <br />
PLEASE look at this!! I would really appreciate comments and suggestions!!<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:27, 28 June 2011 (UTC): Just left you a lot of comments on the discussion page.</font><br />
:<font color=peach> Working on reading and using all of the comments on the discussion page, thanks all who left me some feedback! </font><br />
:<font color=orangered>Interesting topic! Left some comments! [[User:Rscott3|Richard]] 7/18</font><br />
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*[[Boy's Surface]]<br />
:<font color=peach> Leah and I are trying to figure out how we want to lay out our page...Please hold back with the comments while we continue to reorganize as well as add to our page!</font><br />
:<font color=red> Need to work on explanation of parametrization...??</font><br />
:<font color=salmon> Left some comments~ [[User:PhoebeJiang|Phoebe]] 23:56, 13 July 2011 (UTC)</font><br />
<br />
<br />
*[[Using the Rectangular Method to Determine Area]]<br />
:<font color=peach>Just started this new page!! I am going to ink this to the already existing page, [[Parabola]]</font><br />
::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:51, 12 July 2011 (UTC) Great image! I like both pictures! The title is clear but I think you can make it shorter.</font><br />
:<font color=peach>Going to change my equations so they work out neater for the reader</font><br />
<br />
:<font color=black> Planning to add much more to Rectangular Method (Explanations of why/how parabolas are used in the physical world, as well as differentiate between parabolas and catenaries, lastly I will find the area of an actual life sized piece of parabolic architecture) Now, I am going to work on applying suggestions to my Steiner page so I can send it in for final review!<br />
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===Phoebe Jiang===<br />
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*<big>[[Arbelos]]</big><br />
:{{hide|1=<br />
:<font color=salmon>Need to work on the two footnotes. And do a last revision, revise Pappus Chain and Bankoff Chain. <br />
:PLEASE look at this page. Comments are always welcome! </font>[[User:PhoebeJiang|Phoebe]] 19:49, 28 June 2011 (UTC)<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 13:21, 30 June 2011 (UTC): I've gone back through and responded to your comments or crossed things off where we both agree.</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Thank you Kate!!! <3 </font><br />
:<font color=salmon> Submitted for final review. </font><br />
::<font color=darkred> I've put up comments for you. See my thoughts about a spin-off page. [[User:AnnaP|AnnaP]] 7/12 </font> <br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC) Thank you Anna and I've responded to your comments.</font>}}<br />
::::<font color=darkred> One more round of comments is up [[User:AnnaP|AnnaP]] 15:38, 15 July 2011 (UTC) </font> <br />
:::::<font color=salmon> Responded to Anna's second round of the comments. </font><br />
::::::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 13:32, 19 July 2011 (UTC) </font> <br />
<br />
*<big>[[Pappus Chain]]</big><br />
:{{hide|1=<br />
:<font color=salmon>This is used to be a subsection of [[Arbelos]]. Since arbelos is too long, I made Pappus chain a single page. I don't have time to finish it, so I think I'll leave the rest of the page for other people in the future. </font>}}<br />
<br />
*<big>[[Euclidean Algorithm]] </big><br />
:{{hide|1=<br />
<font color=salmon><br />
:Need to work on Gabriel Lame and worst case of EA.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page and leave some commenst!!! Thank you guys!! </font><br />
:<font color=salmon> Submitted for final review. </font>}}<br />
::<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 17:39, 15 July 2011 (UTC) </font> <br />
:::<font color=salmon> Thank you! Responded to Anna's comments. </font><br />
::::<font color=darkred> I put up one more comment [[User:AnnaP|AnnaP]] 13:39, 19 July 2011 (UTC) </font><br />
<br />
*<big>[[Application of the Euclidean Algorithm]]</big> <br />
:{{hide|1=<font color=salmon><br />
: Finish up with computer science section and knot theory.<br />
:Done. Need revision~ [[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
::[[User:PhoebeJiang|Phoebe]] 03:44, 8 July 2011 (UTC)Please look at this page also. PLEASE leave anything you think will make it better!!! Thank you guys!!</font>}}<br />
:<font color=red>I could really need comments on this page. Thanks in advance!</font><br />
<br />
*<big>[[Pigeonhole Principle]]</big><br />
:{{hide|1=<font color=salmon><br />
:Just started it :P </font>[[User:PhoebeJiang|Phoebe]] 19:48, 28 June 2011 (UTC)<br />
:<font color=salmon>There is a problem I don't understand.</font> [[User:PhoebeJiang|Phoebe]] 02:06, 8 July 2011 (UTC)<br />
::<font color=orangered>Left some comments on the discussion page. Send us the problem you don't understand and we'll try it on our end of things (rscott3@swarthmore.edu). Let me know if you want me to read through the page again or anything. [[User:Rscott3|Richard]] 7/8</font><br />
:::<font color=salmon>[[User:PhoebeJiang|Phoebe]] 18:30, 8 July 2011 (UTC)Thank you Richard~ And I figured out the problem just now. :) </font><br />
::::<font color=orangered> Went back and looked over some of your changes. Left some responses to your comments. I'd be happy to read it again at some point if you'd like. [[User:Rscott3|Richard]] 7/12</font><br />
:::::<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:35, 12 July 2011 (UTC)Great! I'll check your responses. Thank you. </font><br />
:<font color=salmon>[[User:PhoebeJiang|Phoebe]] 21:32, 12 July 2011 (UTC)Finish writing the page. Will work on Richard's new comments. </font><br />
:<font color=salmon> Responded to Richard's comments. Appreciate other suggestions.</font><br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:59, 16 July 2011 (UTC) I left a few comments. Overall, I think this page is great! </font color><br />
::<font color=salmon> Thank you Becky! I really appreciate it. I made some changes to this page. [[User:PhoebeJiang|Phoebe]] 16:16, 16 July 2011 (UTC)</font><br />
:<font color=plum>Flora 00:28, 17 July 2011 (UTC)I leave a lot of comments for you. I'll finish '''Why Interesting''' part tomorrow.</font><br />
::<font color=salmon> Thank you very much!!! I've made some changes to the page and responded to your comments!! Is it better now? [[User:PhoebeJiang|Phoebe]] 04:19, 17 July 2011 (UTC)</font><br />
::<font color=plum>I left some comments for your '''Why Interesting''' section, and I responsed some of your changes. The rest are perfect, and I don't think you need any more work.</font><br />
:::<font color=salmon> Thanks! </font>}}<br />
<br />
===Flora Li===<br />
Please leave some comments for my pages, Thanks.<br />
*<big>[[Waves]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:[[User:Flora1103|Flora]] 19:45, 10 June 2011 (UTC)<font color=plum> Need to work on '''Sine Wave Generration'''. Need to add more detail explaination to the graphs and the equetions.</font><br />
:<font color=plum>Flora 16:34, 28 June 2011 (UTC)I changed the title. And try to re-organize it to more "waves".</font><br />
::<font color=orangered>Left some feedback. [[User:Rscott3|Richard]] 6/29</font><br />
::Flora 20:14, 29 June 2011 (UTC)<font color=plum>Thanks very much~</font><br />
:Flora 20:35, 13 July 2011 (UTC)<font color=plum>Send to final review</font><br />
::<font color=darkred> Flora, can you go through the [[Checklist for writing pages]] on the discussion page before I go through the review process for this page? That makes my job much, much easier, and helps me give the most appropriate feedback. You can find examples by looking at the discussion pages of other pages on the final review page. Thanks [[User:AnnaP|AnnaP]] 7/14 </font><br />
::<font color=plum>Flora 21:25, 14 July 2011 (UTC)Have already went through the check list and leave them in discussion page. Sorry about that.</font><br />
:<font color=salmon> [[User:PhoebeJiang|Phoebe]] 21:58, 14 July 2011 (UTC) Comments are addressed. </font><br />
::<font color=plum>Flora 02:41, 15 July 2011 (UTC)Have responsed to your suggestions. Thanks very much.</font><br />
}}<br />
:::<font color=darkred> Take a look at my comments on the page [[User:AnnaP|AnnaP]] 16:08, 15 July 2011 (UTC) </font> <br />
:::<font color=plum>Flora 18:46, 15 July 2011 (UTC)Thanks very much for your comments. I responsed already, and I'm working on Fourier images now. I will post them later.</font><br />
:Flora 18:30, 18 July 2011 (UTC)<font color=plum>Cannot make some images, will re-edit Fourier Series section later today.</font><br />
*<big>[[Dandelin Spheres Theory]]</big><br />
:{{HideThis|1=More|2=<br />
:Flora 15:20, 29 June 2011 (UTC)<font color=plum>Have done with the references.~</font><br />
:Flora 22:20, 28 June 2011 (UTC)Have done most of it, still need to work on the references. I'm looking for any comments, plz~<br />
:<font color=plum>[[User:Flora1103|Flora]] 00:00, 11 June 2011 (UTC)I just start this topic. Working ont eh proofs</font><br />
:<font color=plum>Flora 20:50, 16 June 2011 (UTC) I find some useful information on a online school's webpage, but I couldn't open the page today. I don't know why. The url is [http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Imler/EMAT%206690%20Instruc%20Unit-%20Conics/L4D1ii.html].</font><br />
:[[User:PhoebeJiang|Phoebe]] 00:43, 19 June 2011 (UTC) <font color=salmon>I have no problem viewing it. Try a different browser I think. </font><br />
:<font color=plum>Flora 15:39, 20 June 2011 (UTC) I fixed that problem, thx. I now editting "Prove of conic section curve", and I change one image that showed before, so I still need to edit "Explore hyperbola spheres" later.</font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 21:12, 29 June 2011 (UTC): I left you a bunch of comments on this one, but I didn't get all the way through. It was slow reading and I ran out of time.</font><br />
:<font color=plum>Thanks very much.~</font><br />
<br />
:Flora 02:20, 14 July 2011 (UTC)<font color=plum>Start to revise it</font><br />
<br />
:Flora 21:55, 14 July 2011 (UTC)<font color=plum>Re-arranged the outlet, hope this is better.</font><br />
<br />
:<font color=plum>Flora 23:27, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
}}<br />
:<font color=plum>Flora 20:42, 17 July 2011 (UTC)Responsed to Becky's suggestions. Submitted to final review.</font><br />
<br />
<br />
<br />
*<big>[[Dihedral Groups]]</big><br />
:Welcome comments XD<br />
:{{HideThis|1=More|2=<br />
:<font color=plum>Flora 20:54, 27 June 2011 (UTC)I add the main image but not start this page yet. Will work on it from next week. XD</font><br />
<br />
:<font color=plum>Flora 16:25, 13 July 2011 (UTC) Almost done with the page, still need to work on complex plane explanation, and subgroup part.</font><br />
<br />
:<font color=plum>Flora 01:58, 15 July 2011 (UTC)Finish this page, need to revise.</font><br />
<br />
}}<br />
:<font color=plum>Flora 23:26, 15 July 2011 (UTC)Done with checklist. Will do one more revise before sending to final review.</font><br />
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[[PartnerHome]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25940Talk:Steiner's Chain2011-07-18T20:52:30Z<p>Rscott3: /* Annular Steiner Chains */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
<br />
*<font color=orangered>What's a cyclic quadrangle? <br />
*Try to add overlines to all of your formulas in this section? <br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
<font color=orangered>Do you have a figure 1 and 2? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
<font color=orangered>See my comment from the last time. And also where did you get these formulas? I'm not exactly sure how you got them.<br />
<br />
You've got an "is" "are" problem in the sentence starting with "The lengths of the radii of the tangent circles..."<br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
:::<font color=orangered>Agreed 100%. [[User:Rscott3|Richard]] 7/18</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font><br />
<br />
*<font color=orangered> This type reminds me of a gun cylinder/barrel thing!...Why It's Interesting maybe????? Steiner Chain in real life? http://www.pyramydair.com/blog/images/ar6-cylinder-web.jpg [[User:Rscott3|Richard]] 7/18<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25939Talk:Steiner's Chain2011-07-18T20:52:02Z<p>Rscott3: /* Types of Steiner Chains */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
<br />
*<font color=orangered>What's a cyclic quadrangle? <br />
*Try to add overlines to all of your formulas in this section? <br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
<font color=orangered>Do you have a figure 1 and 2? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
<font color=orangered>See my comment from the last time. And also where did you get these formulas? I'm not exactly sure how you got them.<br />
<br />
You've got an "is" "are" problem in the sentence starting with "The lengths of the radii of the tangent circles..."<br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
:::<font color=orangered>Agreed 100%. [[User:Rscott3|Richard]] 7/18</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font><br />
<br />
*<font color=orangered> This type reminds me of a gun cylinder/barrel thing!...Why It's Interesting maybe????? Steiner Chain in real life? http://www.pyramydair.com/blog/images/ar6-cylinder-web.jpg<br />
</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25937Talk:Steiner's Chain2011-07-18T20:48:30Z<p>Rscott3: /* Steiner Chain Construction via Inversion Formulas */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
<br />
*<font color=orangered>What's a cyclic quadrangle? <br />
*Try to add overlines to all of your formulas in this section? <br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
<font color=orangered>Do you have a figure 1 and 2? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
<font color=orangered>See my comment from the last time. And also where did you get these formulas? I'm not exactly sure how you got them.<br />
<br />
You've got an "is" "are" problem in the sentence starting with "The lengths of the radii of the tangent circles..."<br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25936Talk:Steiner's Chain2011-07-18T20:48:18Z<p>Rscott3: /* Steiner Chain Construction via Inversion Formulas */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
<br />
*<font color=orangered>What's a cyclic quadrangle? <br />
*Try to add overlines to all of your formulas in this section? <br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
<font color=orangered>Do you have a figure 1 and 2? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
<font color=orangered>See my comment from the last time. And also where did you get these formulas? I'm not exactly sure how you got them.<br />
<br />
You've got an "is" "are" problem in the sentence starting with "The lengths of the radii of the tangent circles..."<br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25931Talk:Steiner's Chain2011-07-18T20:44:52Z<p>Rscott3: /* Circle to Circle Inversion */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
<br />
*<font color=orangered>What's a cyclic quadrangle? <br />
*Try to add overlines to all of your formulas in this section? <br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25930Talk:Steiner's Chain2011-07-18T20:44:24Z<p>Rscott3: /* Circle to Circle Inversion */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
<br />
*<font color=orangered>What's a cyclic quadrangle? <br />
*<font color=orangered> Try to add overlines to all of your formulas in this section? <br />
[[User:Rscott3|Richard]] 7/18</font><br />
<br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25929Talk:Steiner's Chain2011-07-18T20:37:23Z<p>Rscott3: /* Formulas */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
:::<font color=orangered>I second Kate here. What's a Porism? It's sort of like this is just hanging out here. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25927Talk:Steiner's Chain2011-07-18T20:35:30Z<p>Rscott3: /* Basic Description */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
<font color=orangered>I'm not sure I understand your definition of inversion in this section. And you should say something about reflection fixing the points the same distance from the line as they start out. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
<br />
*<font color=orangered> By your definition, the main image is not concentric circles. You may want to make sure that this is just one particular way of making a Steiner chain. [[User:Rscott3|Richard]] 7/18</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
*<font color=orangered>I'm not sure that I understand all of the elements of this last picture. Nothing is tangent to the big circle. I assuming you can make a Steiner chain by inversion without being tangent? (I thought they were supposed to be) And why are there different sized circles within the smaller one? Is it a proportional thing? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Steiner%27s_Chain&diff=25926Talk:Steiner's Chain2011-07-18T20:22:52Z<p>Rscott3: /* Section-specific comments */</p>
<hr />
<div>=General Comments=<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:20, 28 June 2011 (UTC): You've got some great pictures, and it looks like you know what you're talking about, but you need to do a much better job of defining they key terms (Steiner's Porism, Steiner chains) and of explaining what you're doing and why you're doing it. I read the whole page carefully more than once, and I'm still confused about what's going on.</font><br />
<br />
<font color=orangered><br />
Hey Anna!<br />
<br />
Cool page! One comment that I got a lot from the people who worked on this last summer, was that when I have math writing, like you do with your proofs, to explain what happens from one line to the next.<br />
Like this:<br />
</font><br />
<br />
<math> \sin \frac{\pi}{n}= \frac{R-r}{R+r}</math><br />
<br />
Multiplying both sides by <math>R+r</math> gives us<br />
<br />
<math> \sin \frac{\pi}{n} (R+r)= R-r</math><br />
<br />
Distribution gives us<br />
<br />
<math> R \sin \frac{\pi}{n} + r\sin \frac{\pi}{n} = R - r</math><br />
<br />
After we subtract <math>R, r</math> from both sides, we have<br />
<br />
<font color=orangered><br />
...and so on and so forth.<br />
<br />
Becky left a ton of comments on the the discussion page for [[Law of Sines]] about this in hot pink if you need any reference. That page has a lot of math writing.<br />
<br />
Richard 6/13<br />
<br />
</font><br />
<br />
=Section-specific comments=<br />
==Intro==<br />
<font color=orangered>You mention circles, but your main picture has spheres. Maybe mention spheres in this part? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
==Basic Description==<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*You should link to [[Inversion]] in this section. It's not a perfect page, but what's there will definitely help people understand inversion.</font><br />
*<font color=dodgerblue>''A Steiner chain is a figure of tangent circles'' - sounds awkward. Might be better to say it's "made of" tangent circles.</font><br />
*<font color=dodgerblue>This section really doesn't help me understand what a Steiner chain or Steiner porisms are ''at all''. I see that you have a bit more of an explanation in the original caption, but you should really move that down here or at least say the same thing with different words here. Since the caption comes before the TOC, a lot of people tend to skip over it.</font><br />
*<font color=dodgerblue>Also, you need to offer some sort of explanation as to what a "porism" is - is "Steiner's porism" synonymous with "a Steiner chain"? Because the only term you've used so far is "Steiner chain", and I'm very confused as to why the page is called "Steiner's Porism" and not "Steiner chains".</font><br />
*<font color=dodgerblue>You end this bit by saying something about the "properties of a Steiner chain" - what are these properties? You should at least touch on them somewhere in the basic description. This will also help give context to the two construction sections that follow - give me something so that I can kind of see why we're doing what we're doing, and how we know when we've succeeded in creating a Steiner chain. To be honest, I read this whole section and still didn't really know what a Steiner chain was besides a picture involving circles. You need to give some sort of definition, and not just a description of how to make one example.</font><br />
<br />
===Creating a Steiner Chain===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*Unsure why you would say "regular triangle" - I think "equilateral triangle" is a much more familiar term for most people.</font><br />
*<font color=dodgerblue>''Using the points of <math>\triangle XYZ</math> as centers, construct tangent circles <math>X,Y,Z</math>…''<br />
::Upon re-reading, I'm pretty sure you mean that XYZ are tangent to each other, but at first I thought you meant that they were tangent to the original circle, which they're not. Consider rewording this sentence.</font><br />
*<font color=dodgerblue>Is the picture at the end of this section a Steiner chain? If so, please say so. If not, what is it? As it is, the section ends rather abruptly.</font><br />
<br />
===Creating a Steiner Chain using Inversion===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''Construct an inversion circle to reflect over the three tangent circles and the two concentric circles. ''<br />
::This sentence confused me. The way you've phrased it sounds as though the new circle you're creating is the one that's going to be reflected - "Construct an inversion circle that will then be reflected over the three tangent circles and the two concentric circles," whereas I'm pretty sure you mean to say "Construct an inversion circle to reflect the three tangent circles and the two concentric circles over." If you're uncomfortable with ending the sentence with a preposition, try "Construct an inversion circle over which the three tangent circles and the two concentric circles will be reflected." Also, there's no need to call that figure "the three tangent circles and the two concentric circles" - call it "the Steiner chain" or "our previous figure" or something.</font><br />
<br />
<br />
*<font color=dodgerblue>The last image in this section is kind of huge. Make sure it's on a different line than your text - it splits the paragraph up weird. And maybe make it a tiny bit smaller?</font><br />
<br />
*<font color=dodgerblue>This may just be part of my general confusion over what Steiner chains actually are, but why do we want to create a new Steiner chain by inversion? What is the purpose of doing this? Just for fun?</font><br />
<br />
==A More Mathematical Explanation==<br />
===Formulas===<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*I don't understand why you have this section heading. First of all, it's the wrong size - it's hidden inside the MME, but if you look at the TOC it's not listed as a sub-section of the MME - to fix that you need to add another equals sign on either side of the heading. But there's nothing in the MME that's not also inside Formulas, so it seems to me that all you've done is re-name your MME, which doesn't seem necessary - all of the subheadings say "Formula" in them…</font><br />
<br />
*<font color=dodgerblue>So Steiner's Porism is a ''theorem'' about Steiner chains? You NEED to explain this above, even if you don't want to state the full thing. You just can't have a page about something and not even show what that thing is except for inside a hidden section.</font><br />
<br />
*<font color=dodgerblue>Also, you need to actually ''explain'' the Porism - I'm not stupid, and I have almost no idea what it's actually saying. What is the "starting circle" of a Steiner chain? What does it mean for a Steiner chain to "close"? What are the "loops" of a Steiner chain? Try to rephrase this statement using just the terms you've introduced in the basic description (and then move that simpler statement ''to'' the basic description!). Also, why is it called a "Porism" and not a "theorem"?</font><br />
<br />
====Tangent Circles Formula====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*In your first image in this section, the ordered pairs for the points aren't close enough to the arrows showing which points they belong to.</font><br />
*<font color=dodgerblue>What is the point of this section? How are these equations about tangent circles relevant to the rest of the page? If you're just establishing them here so that you can use them later, you need to say that.</font><br />
====Concentric Circles Formula====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): Again, what on earth is the purpose of this section? Give me some context so I'm not so confused! :(</font><br />
====Circle to Circle Inversion====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):</font><br />
*<font color=dodgerblue>First of all, context! Context, context, context! I am so very very confused. Why are we doing any of this????</font><br />
*<font color=dodgerblue> ''Points (C, C'), (B, B') are inverses with respect to J \Rightarrow points C, C', B, B' are concyclic''<br />
::I've gotta question your notation here - I thought that C, C', B, and B' were all points in their own right. Why are you making ordered pairs out of points? Or are they not points after all? If they're not points, what are they?</font><br />
*<font color=dodgerblue>In this whole section, I think you need a little more explanation than just all these arrows - what you've got is technically correct as a proof, but it'd be a lot more readable if you sort of embedded it in explanatory sentences. (This is what Richard was talking about in his earlier comment.)</font><br />
*<font color=dodgerblue>Why did we stop where we did? What is the significance of that result? Also, fix that period that's just sorta chillin by itself at the end here.</font><br />
====Steiner Chain Construction via Inversion Formulas====<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC):<br />
*''An inversion can be done on a symmetrical arrangement of n circles (shown in Figure 3) in a region between two concentric circles, one with radiusR and the other with radius r. ''<br />
::I have no idea what this sentence means. Also, what is the purpose of this section? What are we trying to accomplish in it?</font><br />
*<font color=dodgerblue>''This arrangement is represented by: [math stuff]''<br />
::What arrangement? And where did that math come from? :( :( so confused.</font><br />
*<font color=dodgerblue>Again, you need to provide some explanation with your math. I have no idea what you're trying to do or how you're doing it, and also I don't like this format where you say "X=Y! Therefore, Z=W!" and ''then'' have an expandable section showing how you get from X=Y to Z=W - I think the explanation should come in the middle of the two statements.</font><br />
====Types of Steiner Chains====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): This section needs to be at the top of the page! These sections define important terms that I need to understand before I can understand the statement of Steiner's Porism. They should '''not''' be hidden down here! It should go at the top of the MME or even the bottom of the Basic Description - this is actually more important to understand what's going on with the page than the steps to constructing a Steiner Chain are, imo.</font><br />
=====Closed Steiner Chains=====<br />
=====Open Steiner Chains=====<br />
=====Multicyclic Steiner Chains=====<br />
=====Annular Steiner Chains=====<br />
*<font color=dodgerblue>[[User:Kderosier|Kate]] 18:17, 28 June 2011 (UTC): I'd like to see an example of a non-annular Steiner Chain in this section, just for contrast.</font></div>Rscott3https://mathimages.swarthmore.edu/index.php?title=S11&diff=25925S112011-07-18T20:20:09Z<p>Rscott3: /* Richard's Projects */</p>
<hr />
<div>__TOC__<br />
<br />
== Announcements ==<br />
For public-type help questions, see [[Help:Contents|Help]]. For Swat-specific ones, see [[Swarthmore summer research orientation]].<br />
<br />
<b>Remember to keep your projects sections short and up to date; only the last week of status changes should be mentioned. [[User:Smaurer1|Smaurer1]]</b><br />
<br />
===Group Discussion Questions===<br />
<br />
* [[Topics for conversations through Skype with RPI, SB, and/or Drexel]] Started 6/29.<br />
* [[Who are we writing for?]] Started 6/29.<br />
* [[Possibly expanding student Math Image roles]], to be addressed 7/6.<br />
<br />
===Questions:===<br />
*Things that are listed as Helper Pages but use the Image Page template<br />
:We need to decide whether these pages should be on the Helper Page template, the Image Page template, or both: {{Hide|1=<br />
:*[[Change of Coordinate Systems]] - Image template only; image template live<br />
:*[[Conic Section]] - both templates; image template not live<br />
:*[[Differentiability]] - both templates; image template not live<br />
:*[[Dot Product]] - Image template only; image template not live<br />
:*[[Gradients and Directional Derivatives]] - Image template only; image template not live<br />
:*[[Hyperbolic Geometry]] - Image template only; image template not live<br />
:*[[Inversion]] - Image template only; image template not live<br />
:*[[Iterated Functions]] - Image template only; image template not live<br />
:*[[Parametric Equations]] - Image template only; image template live<br />
:*[[Taylor Series]] - Image template only; image template not live<br />
:*[[Volume of Revolution]] - Image template only; image template live<br />
<br />
:<font color=dodgerblue>''(List complied by [[User:Kderosier|Kate]], June 16)''</font><br />
}}<br />
<br />
<br />
Older questions: {{Hide|1=<br />
*Harrison's question about text being cut off on Cross-cap: {{Hide|1=<br />
*[[User:Htasoff|Htasoff]] 14:16, 8 June 2011 (UTC)<br />
**The text in MME on the [[Cross-cap]] page is getting truncated when viewed through edit with form, but still appears on the page.<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 00:49, 14 June 2011 (UTC): When I've encountered this problem previously, it's been because a set of double curly brackets wasn't closed.</font><br />
}}<br />
<br />
*Harrison's question about creating a list of not-yet-existent Helper Pages: {{Hide|1=<br />
Harrison, 5/26/11:<br />
:*<s>We need a list of empty pages: Penrose Tiles is only linked to one, now two, pages. Empty pages like this could well fade into obscurity.</s><br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 17:09, 7 June 2011 (UTC): Such a list has been created. See [[Existing_Pages_Needing_Work#Empty_.28but_linked_to.29_Pages|here]].</font><br />
}}<br />
<br />
*Spam conversation: {{Hide|1=<br />
<font color=dodgerblue><br />
*[[User:Kderosier|Kate]] 14:38, 27 May 2011 (UTC): '''We have some new users who are creating a bunch of pages with links to illegally download or watch movies.''' At first, we though it might be someone from Sweet Briar practicing wiki-syntax, but now it's just starting to look like spam:<br />
**[[Watch_Sniper:_Reloaded_film_in_hd|This is the type of page I'm talking about]]<br />
**[[Special:Contributions/Calrivenick|List of pages created by Calrivenick]]<br />
**[[Special:Contributions/Cadedesi|List of pages created by Cadedesi]]<br />
</font><br />
:Let's talk about what to do this afternoon. [[User:Gene|Gene]] 15:19, 27 May 2011 (UTC)<br />
:<font color=dodgerblue>This problem has continued over the weekend. Someone spammed the talk page for Fun Topology with comments about buying Cialis and stuff. They also posted a lot more movie download pages under the Cadedesi username. I'm going to go through and delete again.</font><br />
:<font color=dodgerblue>The spam comments on Fun Topology were coming from this computer's IP address. </font><br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 14:34, 6 June 2011 (UTC): Haven't seen any more spam activity for a few days. I assume some computer people have handled the issue? I'm going to hide this conversation so that it's not taking up space on S11.</font><br />
}}<br />
<br />
*Citations/footnotes conversation: {{Hide|1=<br />
:I spent a really long time wandering around MediaWiki and Wikipedia this morning trying to figure out how to do footnotes/citations the way I wanted to - now that I know how, should I add instructions to one of the many help mages on Math Images? If so, which page? (Kate, 5/17)<br />
<br />
:<font color=red> Answer: put it here at [[Help:Wiki_Tricks|Wiki Tricks]] (XD, 5/17) </font><br />
<br />
:[[User:Smaurer1|Smaurer1]] 19:43, 17 May 2011 (UTC) Well, it's not clear that webarticles should have footnotes, although Wikipedia does. In text references may be better. This is surely something we should discuss as a group, and find out what last year's group decided, if they did. If there are footnotes, there has got to be a way to get back seamlessly to where you were before you jumped to the footnote.<br />
<br />
:Also, as for citations, we should be uniform in their format.<br />
<br />
:Finally, you can use 4 tildes to put your username and time stamp on your comments, and 3 tildes for just your username. <br />
<br />
:<font color=dodgerblue>Well, I wasn't doing comment-y type footnotes, I just wanted specific sentences to link to items in my References section. I think that format is better than full intext-citations, because it brings you to the source if that's what you're interested in, but it takes up less space if you're not. The references template we have comes equipped with little links to jump you back up, too- if you look at the [[Quipu#References|Quipu]] page, you can see that it lists each of the sections that reference is linked from, and those links bring you to the reference in that section. All in all, I think it's a clear and intuitive way to do references for both the writer and the reader (although, like most things, it doesn't play well with our hidden sections), and I'm going to go ahead and put the instructions up in WikiTricks. -[[User:Kderosier|Kate]] </font><br />
<br />
{{Hide|1=<br />
*What the help pages say now:<br />
**[[Tour_the_Math_Images_Project#Anatomy_of_a_page|The Tour page's "Anatomy of a page" section]]<br />
**[[Checklist_for_writing_pages#References_and_footnotes|The "References and footnotes" section on the checklist for writing pages]]<br />
*The way to do Wikipedia-like references:<br />
**[[Wiki_Tricks#How_to_do_Citations|Wiki Tricks - Citations section]]<br />
}}<br />
<br />
}}<br />
<br />
*Invisible Comments conversation: {{Hide|1= One of you asked "How do you put invisible comments in the source code?"<br />
<br />
:Answer: Same way you do in html, like this <nowiki><br />
:<!-- hidden stuff --><br />
:</nowiki><br />
:However, if you want to make comments about an article for its author, the comments are more likely to be seen if you either<br />
:* put it in the discussion page, or<br />
:* if it is important to put it right by the material commented on, put it in the article in color with your username and time stamp included.<br />
<br />
:Hidden comments in the source code are likely to be overlooked, except perhaps if they are written by the author him/herself, as a note for further development.<br />
}}<br />
}}<br />
<br />
== Current Individual Projects ==<br />
<br />
===Steve M (aka Prof Maurer)'s Role===<br />
{{Hide|1=<br />
My role is 2-fold:<br />
<br />
<ol><br />
<li> Come see me to sound me out (if you wish) on the mathematical appropriateness of an idea for a page, or for possible references.<br />
<br />
<li> Once you have a reasonable amount written, and want feedback on the quality and correctness of the mathematical exposition, ask me to look it over and then we will have a conference. (Abram and the oldies are as good as I am at discussion general organization and clarity issues.) <br />
</ol><br />
<br />
We have agreed to put a record on this S11 page of what we are doing and what help we want from others, but in addition tell me in person or by email if you want to conference with me.<br />
}}<br />
<br />
===Harrison's Projects===<br />
<br />
[[Harrison's detritus]]<br />
<br />
*[[User:Htasoff|Htasoff]] 23:44, 13 July 2011 (UTC) Pages will be submitted for final review in 1 - 3 days. A few, final comments are welcome. Real Projective Plane is still in the works, though.<br />
<br />
<font color=dodgerblue>[[User:Kderosier|Kate]] 18:19, 14 July 2011 (UTC): Hey Harrison, do you know if that one picture in inverse trig that you got from somewhere on the internet is a picture we can use? Can you check please?</font><br />
<br />
===Richard's Projects===<br />
<br />
<br />
<font color=orangered> <br />
<br />
This weekend, I'll be reading pages to give feedback: [[Steiner's Chain]] [[Snell's Law]] [[Pigeonhole Principle]] and an [[RPI11|RPI]] page<br />
<br />
*[[Ambiguous Case]]<br />
:*Put up for final review, but still working on lesson plan![[User:Rscott3|Richard]] 7/18 </font color><br />
<br />
:APPLET INFO:<br />
:{{Hide|1=<br />
<br />
[[User:Alimurreza|Alimurreza]] 02:53, 6 July 2011 (UTC)I am working on your applet. Check this out here @([http://mathforum.org/mathimages/index.php/DU11 Reza's work]).<br />
:APPLET UPDATE:[[User:Alimurreza|Alimurreza]] 02:41, 15 July 2011 (UTC)I am done with Ambiguous Case applet. Please, check the applet. Feel free to send me any feedback or change-request.<br />
What i want mine to look like, but http://www.mccsc.edu/~aterwill/ambiguouscaseapplet/Ambiguous_Case_applet.html doesn't show the completed triangles.<br />
<br />
I like how this one shows the completed triangles<br />
http://www.mnwest.edu/fileadmin/static/website/dmatthews/Geogebra/AmbiguousCase01.html<br />
<br />
}}<br />
<br />
'''Completed Pages'''<br />
<br />
*[[Law of cosines]]<font color=orangered> [[User:Rscott3|Richard]] 6/23</font><br />
<br />
*[[Law of Sines]] [[User:AnnaP|AnnaP]] 7/7<br />
<br />
*[[Solving Triangles]] 7/12<br />
<br />
:Other ideas: {{Hide|1=<br />
*inscribed angles?<br />
<br />
*Normal Distribution?<br />
<br />
*Birthday Paradox?<br />
}}<br />
<br />
===Dayo's Projects===<br />
Current projects<br />
*[[Inscribed figures]] : there's a [[Compass & Straightedge Construction and the Impossible Constructions]] page, but I think that inscribed figures deserves its' own page. What do others think? <br />
::<font color=slateblue> [[User:Rebecca|Rebecca]] 01:07, 9 July 2011 (UTC) I left comments on the discussion page. </font color><br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 15:23, 18 July 2011 (UTC): I left you some comments!</font><br />
<br />
Future Projects: {{Hide|1=<br />
*[[Mathematics in architecture]]:make changes akin to [[Math for Computer Graphics and Computer Vision]], including:<br />
<br />
::*[[Cross sections]]: calculus application page, including examples of Tokyo international Forum, Suransuns Bridge, and other structures which could be thought of and put together easily in terms of their cross sections <br />
::*[[Torus]] edits, additions concerning the Torus in construction and architecture<br />
::*[[Domes]]:conic sections, arches, parabolas<br />
::*[[The Henderson Waves Bridge]]: sinusoids in architecture, parametric design<br />
::*[[Catenary]]: More real world examples: namely bridges<br />
::*[[Bridge of Peace]]: The equation(s) used to generate the surface, possibly words from the architect, very new, may be hard to get concrete technical information <br />
::*[[Kurilpa Bridge]]: Everyone have a look at the image and tell me what kind of actual subjects you could find in it, namely in the cables and tubes.<br />
::*Teaching Materials(6/30): ''growing up with science: projects'' could be the sort of activities we're looking for. I used these with a class, and think people should look at them.<br />
<br />
<br />
<br />
<br />
}}<br />
<br />
on hold: <br />
*[[Parametric Equations]]: integrating Xingda's page from [[S10]] into page.<br />
<br />
===Diana's Projects===<br />
<br />
====Current====<br />
*[[Logistic Bifurcation]]<br />
<br />
:<font color=darkred> I've put up comments. There are a few places that could use some work. [[User:AnnaP|AnnaP]] 7/10 </font> <br />
:<font color=slateblue> [[User:Rebecca|Rebecca]] 02:26, 16 July 2011 (UTC) This is a very impressive page. I put up a few small suggestions on the discussion page. </font color><br />
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*[[Markus-Lyapunov Fractals]]<br />
:<font color=darkred> Approved [[User:AnnaP|AnnaP]] 7/14 </font><br />
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====Ideas for later projects====<br />
{{Hide|1=<br />
*Chirikov-Taylor Maps<br />
**This seems like a natural extension of the Markus-Lyapunov Fractals page, but maybe the math involved in the two is too similar?<br />
*This aspect of pendular motion:<br />
**[[http://www.youtube.com/watch?v=yVkdfJ9PkRQ&feature=player_embedded|Varied-Length Pendulums]]<br />
**I'm not sure how or whether to use this -- does a ''moving'' image count as a "math image"? -- But it's incredible, and I'd love to explore it.<br />
*Kuen Surface<br />
**It's just really cool.<br />
}}<br />
<br />
===Kate's Projects===<br />
*[[User:Kderosier#Applet_Testing|Java Applets]]<br />
<br />
*[[Anne Burns' Mathscapes]] (Scrapped out of [[Mountains In Spring|three]] [[Mathscape|other]] [[Fractal Scene I|pages]]):<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:29, 18 July 2011 (UTC): As soon as I get confirmation that Anne Burns doesn't mind us using her images, I'll submit this for final review.</font><br />
<br />
*[[Perko pair knots]]:<br />
:<font color=dodgerblue>[[User:Kderosier|Kate]] 18:29, 18 July 2011 (UTC): Made changes after talking to Abram. Would welcome feedback from others.</font><br />
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*Finished pages: {{Hide|1=<br />
:*[[Critical Points]]:<br />
::<font color=darkred> Approved, but I did have one note on whether or not you intended to add something. It's fine as is, but I wanted to put up a suggestion. [[User:AnnaP|AnnaP]] 16:24, 15 July 2011 (UTC) </font><br />
:::<font color=dodgerblue>I responded to that note.</font><br />
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:*[[Summation Notation]]:<br />
::<font color=darkred> Approved [[User:AnnaP|AnnaP]] 16:24, 15 July 2011 (UTC) </font><br />
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:*[[Bases]]:<br />
:: <font color=dodgerblue>[[User:Kderosier|Kate]] 17:31, 11 July 2011 (UTC): Changed things in response to Chris' comments.</font><br />
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:* [[Quipu]]:<br />
::<font color=darkred> Put up as ready for the public 6/30 [[User:AnnaP|AnnaP]] </font><br />
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:* [[Basic Trigonometric Functions]]:<br />
::<font color=dodgerblue>[[User:Kderosier|Kate]] 18:11, 30 June 2011 (UTC): Changed the the things that were bolded.</font><br />
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}}<br />
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===Leah's Projects===<br />
*[[Bedsheet Problem]]<br />
not sure what to do with this page. <br />
<br />
*[[Boy's Surface]] <br />
WIP<br />
<br />
*[[Snell's Law]] feedback page<br />
Waiting on the final applet by Reza. [[User:Ljeanlo1|Ljeanlo1]] 16:14, 18 July 2011 (UTC)<br />
<br />
*[[Witch of Agnesi]]<br />
addressing comments from chris. [[User:Ljeanlo1|Ljeanlo1]] 16:14, 18 July 2011 (UTC) <br />
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*[[Dot Product]] <font color=darkred> Approved, 7/14 </font><br />
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*[[Vector]] <font color=darkred> Approved, 7/10 </font><br />
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==Requests to S10 Students==<br />
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<br />
*<s>Can XD do a demo for MATLAB?</s> Done - [[Demo of MATLAB using the example of Bifurcation]]<br />
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== Current Group Projects ==</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Pages_Ready_for_Final_Review&diff=25924Pages Ready for Final Review2011-07-18T20:17:23Z<p>Rscott3: /* Swarthmore Pages */</p>
<hr />
<div>=Pages Ready for Final Review=<br />
Once your page has gone through several cycles of feedback and you (the author) feel that it is ready for the public:<br />
*Go through either the [[Checklist for writing pages]] or the [[Checklist for writing helper pages]] one last time<br />
*Take a few minutes and type up a response to the issues brought up by the checklist by writing 1-3 sentences in response to the bold headings on the checklist. (Check out the [[Sample Response to Checklist]])<br />
*Link the page here. <br />
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Your page will be reviewed by one of the designated final reviewers who will either give you further feedback or mark the page as "ready for the public."<br />
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Please put your name by your page so that we can contact you with questions.<br />
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'''<font color=darkred> AN IMPORTANT NOTE ABOUT PAGES PUT UP FOR FINAL REVIEW BETWEEN JULY 18TH AND 22ND'''<br />
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'''Anna is going to a training which will keep her extremely busy during that week. Priority for editing will be given to Swarthmore students and any others for whom it is their last week of work on the project (please note if you are NOT at Swarthmore and it is your last week. ''' I will make a big effort to get through everything posted by this weekend, so there isn't a backlog. I will also be completely unavailable for editing August 6th-15th, and will not be reachable during that time. </font> <br />
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:<font color=salmon> Next week is gonna be the last week for Sweet Briar students as well. Phoebe 07/15</font><br />
<br />
===Swarthmore Pages===<br />
<br />
*[[Logistic Bifurcation]] by [[User:Dpatton1|Diana]] 17:00 7/7/11<br />
:<font color=green> Chris 7/16 I've put up comments from a "layman's" perspective. </font><br />
:<font color=darkred> I've put up comments. There are a few places that could use some work. [[User:AnnaP|AnnaP]] 7/10 </font> <br />
<br />
*[[Rope around the Earth]]<br />
:[[User:Htasoff|Htasoff]] 23:20, 29 June 2011 (UTC) Comments implemented and/ or responded to.<br />
:: <font color=darkred>I've written again on some of your comments. I feel strongly about addressing those points.[[User:AnnaP|AnnaP]] 7/7</font><br />
:::[[User:Htasoff|Htasoff]] 20:52, 15 July 2011 (UTC) I've addressed them now.<br />
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*[[Radians]] by [[User:Htasoff|Htasoff]]<br />
:<font color=Mediumblue>The programmers at the Math Forum are still working on the Helper Page template. The template on this page is the most current version.</font><br />
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[[Snell's Law]] [[User:Ljeanlo1|Ljeanlo1]] 16:14, 18 July 2011 (UTC) <br />
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*[[Witch of Agnesi]] [[User:Ljeanlo1|Ljeanlo1]] [[User:Ljeanlo1|Ljeanlo1]] 16:14, 18 July 2011 (UTC)<br />
working on chris' comments. <br />
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*[[Congruent triangles]] by [[User:Htasoff|Htasoff]]<br />
:Is fully developed and has instructions for future editors based on Steve's comments.<br />
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*[[Dimensions]] by [[User:Htasoff|Htasoff]]<br />
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*[[Ambiguous Case]] by [[User:Rscott3|Richard]]<br />
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===Sweet Briar Pages===<br />
<br />
*[[Arbelos]] by [[User:PhoebeJiang|Phoebe]] 04:03, 11 July 2011 (UTC)<br />
:{{hide|1=<br />
:<font color=darkred> I've put up some comments. You've done a great job! [[User:AnnaP|AnnaP]] 7/12 </font><br />
::<font color=salmon>Responded to the comments. And here is the spin-off page [[Pappus Chain]]. </font>}}<br />
:::<font color=darkred> One last round of comments is up! [[User:AnnaP|AnnaP]] 15:37, 15 July 2011 (UTC)</font><br />
:::<font color=salmon> Thank you!!! I've responded to the second round of comments. Btw, do you think I should submit [[Pappus Chain]] for final review as well or just leave it there? </font><br />
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*[[Waves]] by [[User:Flora1103|Flora Li]] 20:25, 13 July 2011 (UTC)<br />
:<font color=darkred> Make sure you go through the checklist! 7/14 [[User:AnnaP|AnnaP]] </font><br />
::<font color=plum>Flora 21:25, 14 July 2011 (UTC)Have already went through the check list and leave them in discussion page. Sorry about that.</font><br />
:::<font color=darkred> It's all good :) . I've put up a bunch of comments for you to go through. [[User:AnnaP|AnnaP]] 16:06, 15 July 2011 (UTC)</font> <br />
:::<font color=plum>Flora 18:48, 15 July 2011 (UTC)Thanks very much for your comments. I responsed already, and I'm working on Fourier images now. I will post them later.</font><br />
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*[[Euclidean Algorithm]] by [[User:PhoebeJiang|Phoebe]] 02:05, 15 July 2011 (UTC)<br />
:<font color=darkred> I've put up comments [[User:AnnaP|AnnaP]] 17:38, 15 July 2011 (UTC) </font><br />
::<font color=salmon> Responded to the comments. Thanks! Phoebe</font><br />
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*[[Dandelin Spheres Theory]] by [[User:Flora1103|Flora Li]] 15:31, 17 July 2011 (UTC)<br />
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*[[Dihedral Groups]] by [[User:Flora1103|Flora Li]] 18:27, 18 July 2011 (UTC)<br />
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===Rensselaer Polytechnic Institute===<br />
[[Polar Equations]] by [[User:Chanj|Chanj]] 21:35, 7 July 2011 (UTC)<br />
: <font color=darkred> I've put comments up [[User:AnnaP|AnnaP]] 7/10 </font><br />
::<font color=blue> I have edited my page by putting up new images, more detailed explanations and comments in the discussion reflecting those changes. --[[User:Chanj|Chanj]] 20:48, 15 July 2011 (UTC)</font><br />
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[[DU11]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25919Talk:Ambiguous Case2011-07-18T19:57:45Z<p>Rscott3: /* Page Comments */</p>
<hr />
<div>=Response to Checklist=<br />
This page is a helper page. I therefore used the [[Checklist for writing helper pages]].<br />
<br />
NOTE: This response to the checklist just addresses the content part of the page up through Teaching Materials. This week I hope to be working closely with Chris, Ann, Diana, etc. on the actual Teaching Materials section. This is the site's first page to include such materials, so one goal is to make this a sort of an experiment. Anna/Chris/Steve, I ask that you only review the content part of this page for right now.<br />
<br />
==Messages for Future==<br />
I think that the real value of this page is the Teaching Materials section that is going to happen. Other than that, I think that the content of the page itself is a solid, thorough description of the ambiguous case and I don't see much that could be added content-wise.<br />
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==References==<br />
All images were made by me and the applet was made by Reza.<br />
<br />
==Quality of prose and page structuring==<br />
*This page is set up so that the top part and description outlines the rest of the page and introduces the topic thoroughly.<br />
*The main goal of this page is to set up a geometric perspective for a topic grounded in trigonometry.<br />
*The sections are strategically set up so that the most difficult scenarios are at the end. Each section individually is also easy to follow and as clear and spelled out as possible.<br />
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==Integration of Images and Text==<br />
The images, table, and applet are all necessary and useful parts of this page. They are clear and used in the description of the content.<br />
<br />
==Links to other Pages==<br />
All of the pages that link to it are listed.<br />
<br />
==Examples, Calculations, Applications, Proofs==<br />
Examples, visuals, and proofs are given throughout the page. This page is really at a basic level discussing a geometric proof for the ambiguous case--why there are different solutions for different side lengths.<br />
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==Mathematical Accuracy and precision of language==<br />
The math is clearly explained when necessary. It has been checked for errors time and time again.<br />
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==Layout==<br />
The paragraphs are short and the page effectively includes bold phrases and several balloons.<br />
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The page has been tested in different window sizes and has been tested to reduce white spaces and appropriately fit the titles and pictures.<br />
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I've gotten a lot of feedback on this page and I've really taken the time to sit down with this and make sure it is clear and concise. I'm really excited to get to work on this page as a teaching material. I think there's a lot of potential in this page for that reason.<br />
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=Teaching Material Comments=<br />
{{Hide|1=<br />
<br />
<font color=mediumseagreen><br />
Comments from Steve Weimar, the Math Forum 6/28<br />
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*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
</font><br />
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<br />
<font color=mediumseagreen><br />
*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
</font><br />
::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
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*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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<font color=orangered><br />
*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
</font><br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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<font color=orangered><br />
*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
</font><br />
}}<br />
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=Page Comments=<br />
{{Hide|1=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
<font color=mediumseagreen><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
</font><br />
::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
</font><br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
</font><br />
::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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<br />
<br />
*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
</font><br />
:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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<br />
*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
{{Hide|1=<br />
I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
<br />
SSA Postulate: what we can't know<br />
<br />
Why is that? Explanation/Informal proof<br />
<br />
What can be known from SSA?<br />
<br />
How can one figure out which case it is?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
::::<font color=orangered>I think these comments have more to do with basic trig functions page. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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::Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example).<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font> <br />
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::The table is a very good idea, though it seems large for the page.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::Remove "to compute height" in the very last sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::First Scenario: Two "the"s in sentence 3.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram.<br />
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::Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
::::<font color=orangered> Both solutions don't necessarily have to be oblique. Say angle C is 90 degrees... [[User:Rscott3|Richard]] 7/18</font><br />
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::Determining Both Solutions: Put a link to "Law of Sines" in third sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::There is no angle labeled B to correspond to your equation. <br />
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::Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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::::<font color=orangered>I tried to explain that within the paragraph that there is only one solution of two completed triangles, and not two solutions but only one that works. (I had trouble wording that...I think I make sense????? [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
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The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
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{{Hide|1=<br />
===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
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*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
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* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
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* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
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*The end of this section is very clear!<br />
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[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
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====First Scenario: No solution====<br />
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<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
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* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
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<font color=orangered>edited first sentence Richard 5/25</font><br />
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----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
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<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
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2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
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2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
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<font color=orangered> <br />
*I added a few words to show that the swinging side compared to the height is the determinant<br />
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*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
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Richard 5/25</font><br />
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==Extra Picture==<br />
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[[Image:Ambig_cas_act1.jpg]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25918Talk:Ambiguous Case2011-07-18T19:57:15Z<p>Rscott3: /* Teaching Material Comments */</p>
<hr />
<div>=Response to Checklist=<br />
This page is a helper page. I therefore used the [[Checklist for writing helper pages]].<br />
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NOTE: This response to the checklist just addresses the content part of the page up through Teaching Materials. This week I hope to be working closely with Chris, Ann, Diana, etc. on the actual Teaching Materials section. This is the site's first page to include such materials, so one goal is to make this a sort of an experiment. Anna/Chris/Steve, I ask that you only review the content part of this page for right now.<br />
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==Messages for Future==<br />
I think that the real value of this page is the Teaching Materials section that is going to happen. Other than that, I think that the content of the page itself is a solid, thorough description of the ambiguous case and I don't see much that could be added content-wise.<br />
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==References==<br />
All images were made by me and the applet was made by Reza.<br />
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==Quality of prose and page structuring==<br />
*This page is set up so that the top part and description outlines the rest of the page and introduces the topic thoroughly.<br />
*The main goal of this page is to set up a geometric perspective for a topic grounded in trigonometry.<br />
*The sections are strategically set up so that the most difficult scenarios are at the end. Each section individually is also easy to follow and as clear and spelled out as possible.<br />
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==Integration of Images and Text==<br />
The images, table, and applet are all necessary and useful parts of this page. They are clear and used in the description of the content.<br />
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==Links to other Pages==<br />
All of the pages that link to it are listed.<br />
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==Examples, Calculations, Applications, Proofs==<br />
Examples, visuals, and proofs are given throughout the page. This page is really at a basic level discussing a geometric proof for the ambiguous case--why there are different solutions for different side lengths.<br />
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==Mathematical Accuracy and precision of language==<br />
The math is clearly explained when necessary. It has been checked for errors time and time again.<br />
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==Layout==<br />
The paragraphs are short and the page effectively includes bold phrases and several balloons.<br />
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The page has been tested in different window sizes and has been tested to reduce white spaces and appropriately fit the titles and pictures.<br />
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I've gotten a lot of feedback on this page and I've really taken the time to sit down with this and make sure it is clear and concise. I'm really excited to get to work on this page as a teaching material. I think there's a lot of potential in this page for that reason.<br />
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=Teaching Material Comments=<br />
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Comments from Steve Weimar, the Math Forum 6/28<br />
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*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
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*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
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::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
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*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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<font color=orangered><br />
*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
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=Page Comments=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
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::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
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::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
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:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
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I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
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SSA Postulate: what we can't know<br />
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Why is that? Explanation/Informal proof<br />
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What can be known from SSA?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
::::<font color=orangered>I think these comments have more to do with basic trig functions page. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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::Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example).<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font> <br />
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::The table is a very good idea, though it seems large for the page.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::Remove "to compute height" in the very last sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::First Scenario: Two "the"s in sentence 3.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram.<br />
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::Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
::::<font color=orangered> Both solutions don't necessarily have to be oblique. Say angle C is 90 degrees... [[User:Rscott3|Richard]] 7/18</font><br />
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::Determining Both Solutions: Put a link to "Law of Sines" in third sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::There is no angle labeled B to correspond to your equation. <br />
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::Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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::::<font color=orangered>I tried to explain that within the paragraph that there is only one solution of two completed triangles, and not two solutions but only one that works. (I had trouble wording that...I think I make sense????? [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
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The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
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{{Hide|1=<br />
===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
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*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
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* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
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* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
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*The end of this section is very clear!<br />
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[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
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====First Scenario: No solution====<br />
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<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
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* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
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<font color=orangered>edited first sentence Richard 5/25</font><br />
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----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
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<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
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2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
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2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
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*I added a few words to show that the swinging side compared to the height is the determinant<br />
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*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
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Richard 5/25</font><br />
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==Extra Picture==<br />
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[[Image:Ambig_cas_act1.jpg]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25917Talk:Ambiguous Case2011-07-18T19:56:40Z<p>Rscott3: /* Response to Checklist */</p>
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<div>=Response to Checklist=<br />
This page is a helper page. I therefore used the [[Checklist for writing helper pages]].<br />
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NOTE: This response to the checklist just addresses the content part of the page up through Teaching Materials. This week I hope to be working closely with Chris, Ann, Diana, etc. on the actual Teaching Materials section. This is the site's first page to include such materials, so one goal is to make this a sort of an experiment. Anna/Chris/Steve, I ask that you only review the content part of this page for right now.<br />
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==Messages for Future==<br />
I think that the real value of this page is the Teaching Materials section that is going to happen. Other than that, I think that the content of the page itself is a solid, thorough description of the ambiguous case and I don't see much that could be added content-wise.<br />
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==References==<br />
All images were made by me and the applet was made by Reza.<br />
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==Quality of prose and page structuring==<br />
*This page is set up so that the top part and description outlines the rest of the page and introduces the topic thoroughly.<br />
*The main goal of this page is to set up a geometric perspective for a topic grounded in trigonometry.<br />
*The sections are strategically set up so that the most difficult scenarios are at the end. Each section individually is also easy to follow and as clear and spelled out as possible.<br />
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==Integration of Images and Text==<br />
The images, table, and applet are all necessary and useful parts of this page. They are clear and used in the description of the content.<br />
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==Links to other Pages==<br />
All of the pages that link to it are listed.<br />
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==Examples, Calculations, Applications, Proofs==<br />
Examples, visuals, and proofs are given throughout the page. This page is really at a basic level discussing a geometric proof for the ambiguous case--why there are different solutions for different side lengths.<br />
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==Mathematical Accuracy and precision of language==<br />
The math is clearly explained when necessary. It has been checked for errors time and time again.<br />
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==Layout==<br />
The paragraphs are short and the page effectively includes bold phrases and several balloons.<br />
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The page has been tested in different window sizes and has been tested to reduce white spaces and appropriately fit the titles and pictures.<br />
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I've gotten a lot of feedback on this page and I've really taken the time to sit down with this and make sure it is clear and concise. I'm really excited to get to work on this page as a teaching material. I think there's a lot of potential in this page for that reason.<br />
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=Teaching Material Comments=<br />
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Comments from Steve Weimar, the Math Forum 6/28<br />
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*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
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*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
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::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
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*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
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=Page Comments=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
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::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
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::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
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:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
{{Hide|1=<br />
I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
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SSA Postulate: what we can't know<br />
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Why is that? Explanation/Informal proof<br />
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What can be known from SSA?<br />
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How can one figure out which case it is?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
::::<font color=orangered>I think these comments have more to do with basic trig functions page. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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::Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example).<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font> <br />
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::The table is a very good idea, though it seems large for the page.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::Remove "to compute height" in the very last sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::First Scenario: Two "the"s in sentence 3.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram.<br />
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::Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
::::<font color=orangered> Both solutions don't necessarily have to be oblique. Say angle C is 90 degrees... [[User:Rscott3|Richard]] 7/18</font><br />
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::Determining Both Solutions: Put a link to "Law of Sines" in third sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::There is no angle labeled B to correspond to your equation. <br />
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::Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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::::<font color=orangered>I tried to explain that within the paragraph that there is only one solution of two completed triangles, and not two solutions but only one that works. (I had trouble wording that...I think I make sense????? [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
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The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
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===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
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*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
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* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
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* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
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*The end of this section is very clear!<br />
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[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
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====First Scenario: No solution====<br />
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<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
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* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
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<font color=orangered>edited first sentence Richard 5/25</font><br />
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----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
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<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
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2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
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2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
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*I added a few words to show that the swinging side compared to the height is the determinant<br />
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*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
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Richard 5/25</font><br />
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==Extra Picture==<br />
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[[Image:Ambig_cas_act1.jpg]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25888Talk:Ambiguous Case2011-07-18T18:27:13Z<p>Rscott3: </p>
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<div>=Response to Checklist=<br />
This page is a helper page. I therefore used the [[Checklist for writing helper pages]].<br />
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NOTE: This response to the checklist just addresses the content part of the page up through Teaching Materials. This week I hope to be working closely with Chris, Ann, Diana, etc. on the actual Teaching Materials section. This is the site's first page to include such materials, so one goal is to make this a sort of an experiment. Anna/Chris/Steve, I ask that you only review the content part of this page for right now.<br />
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==Messages for Future==<br />
I think that the real value of this page is the Teaching Materials section that is going to happen. Other than that, I think that the content of the page itself is a solid, thorough description of the ambiguous case and I don't see much that could be added content-wise.<br />
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==References==<br />
All images were made by me and the applet was made by Reza.<br />
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==Quality of prose and page structuring==<br />
*This page is set up so that the top part and description outlines the rest of the page and introduces the topic thoroughly.<br />
*The main goal of this page is to set up a geometric perspective for a topic grounded in trigonometry.<br />
*The sections are strategically <br />
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=Teaching Material Comments=<br />
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Comments from Steve Weimar, the Math Forum 6/28<br />
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*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
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*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
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::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
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*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
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=Page Comments=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
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::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
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::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
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:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
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I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
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SSA Postulate: what we can't know<br />
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Why is that? Explanation/Informal proof<br />
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What can be known from SSA?<br />
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How can one figure out which case it is?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
::::<font color=orangered>I think these comments have more to do with basic trig functions page. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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::Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example).<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font> <br />
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::The table is a very good idea, though it seems large for the page.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::Remove "to compute height" in the very last sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::First Scenario: Two "the"s in sentence 3.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram.<br />
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::Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
::::<font color=orangered> Both solutions don't necessarily have to be oblique. Say angle C is 90 degrees... [[User:Rscott3|Richard]] 7/18</font><br />
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::Determining Both Solutions: Put a link to "Law of Sines" in third sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::There is no angle labeled B to correspond to your equation. <br />
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::Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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::::<font color=orangered>I tried to explain that within the paragraph that there is only one solution of two completed triangles, and not two solutions but only one that works. (I had trouble wording that...I think I make sense????? [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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<br />
=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
<br />
The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
<br />
{{Hide|1=<br />
===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
<font color=hotpink> <br />
<br />
*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
<br />
* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
<br />
* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
<br />
*The end of this section is very clear!<br />
<br />
[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
<br />
</font color><br />
<br />
====First Scenario: No solution====<br />
<font color=hotpink><br />
<br />
<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
<br />
* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
</font color><br />
<br />
<font color=orangered>edited first sentence Richard 5/25</font><br />
<br />
----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
<br />
<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
<br />
2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
<br />
2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
<br />
<br />
<font color=orangered> <br />
*I added a few words to show that the swinging side compared to the height is the determinant<br />
<br />
*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
<br />
<br />
<br />
Richard 5/25</font><br />
}}<br />
<br />
==Extra Picture==<br />
<br />
[[Image:Ambig_cas_act1.jpg]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Ambiguous_Case&diff=25881Ambiguous Case2011-07-18T18:14:34Z<p>Rscott3: </p>
<hr />
<div>{{HelperPage|1=Law of Sines|2=Congruent triangles|3=Solving Triangles}}<br />
<br />
__TOC__<br />
<br />
Given two adjacent side lengths and an angle opposite one of them, there is no definite completion of a triangle. According to [[triangle congruence postulates]], two triangles cannot be proved congruent given these three elements. This configuration is commonly referred to as '''side-side-angle (SSA)'''. <br />
<br />
With the SSA configuration, there is a fixed angle connecting the base of the triangle and one of the adjacent sides. The length of the base is unknown, denoted below by a dashed line. The length of the third side length is also fixed, but neither angle adjacent to that side is known. This means that this third side can swing from the upper vertex in any way that connects this vertex and any point along the indefinitely sized base so that all three sides of the triangle are connected.<br />
<br />
[[Image:Triangle_template1.jpg|center]]<br />
<br />
Given any SSA configuration, there are several outcomes that can occur when solving a triangle: no solution, one solution, or two solutions. Which of the three scenarios occurs for any SSA configuration depends on the length of the swinging side compared to the height of the triangle.<br />
<br />
In any SSA configuration, we can draw the height of the triangle even though we don't know the length of the base. The height is the perpendicular distance from the upper vertex to the base. Because there is a right angle between the height and base, we can always use the fixed angle and the length of the fixed side to determine the height. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging side to this known height of the triangle.<br />
<br />
[[Image:Height2.jpg|center]]<br />
<br />
<br />
Let <math>a</math> be the length of the side opposite <math>A</math>, an acute angle. The table below describes the different solutions for different scenarios.<br />
<br />
<br />
{{{!}}border="1" cellspacing="3" cellpadding="8" align="center"<br />
{{!}}'''Scenario'''{{!}}{{!}}'''Number of Solutions'''{{!}}{{!}}'''Type of Triangle'''<br />
{{!}}-<br />
{{!}}<math>a < h</math>{{!}}{{!}}no solution{{!}}{{!}}none<br />
{{!}}-<br />
{{!}}<math>a = h</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}right triangle<br />
{{!}}-<br />
{{!}}<math>b > a > h</math><br />
{{!}}{{!}}two solutions<br />
{{!}}{{!}}oblique<br />
{{!}}-<br />
{{!}}<math>a \geq b</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}when <math>a = b</math>, equilateral/isosocles<br />
<br />
when <math>a > b</math>, obtuse<br />
{{!}}}<br />
<br />
<br />
<br />
Because the SSA configuration can prompt different numbers of solutions for for different scenarios, it is often referred to as the '''ambiguous case'''.<br />
<br />
<br />
We can use [[Basic Trigonometric Functions|trigonometry]] to determine the value of height <math>h</math>. Click below to see how.<br />
<br />
{{Hide|1=<br />
<br />
::<math>\sin A =\frac{opposite}{hypotenuse}</math><br />
<br />
<br />
Substituting the appropriate variables,<br />
<br />
::<math>\sin A = \frac{h}{b}</math><br />
<br />
<br />
Multiply both sides by <math>b</math> to get<br />
<br />
::<math>h = b \sin A</math><br />
<br />
}}<br />
<br />
===First Scenario: No Solution===<br />
<br />
In the first scenario, the length of the swinging side is shorter than the height, <math>h</math>. Because this side of the triangle is shorter than the height, there is no solution. The shortest distance between a point and a given line is the line segment that is perpendicular to the given line and goes through that point, which in this case is height <math> h</math>. Since the side length is shorter than the shortest possible distance between the base and the upper vertex of the triangle, the side opposite the fixed angle will never be able to reach the base of the triangle. Thus, there are no solutions when the swinging side length <math>a</math> is less than <math> h</math> or <math> b \sin A</math>.<br />
<br />
In the picture below with numbers we've chosen, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete.<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:ambig case no solution 1.jpg|center]]<br />
{{!}}}<br />
<br />
<br />
Since the other given side length is <math>4</math> and since <math>4<5</math>, there is no solution.<br />
<br />
In summary, when <math>a < b \sin A</math>, there is no solution for a SSA configuration.<br />
<br />
===Second Scenario: One Right Solution===<br />
<br />
In the second scenario, the length of the swinging side is equal to the height, <math>h</math>. Because this side of the triangle is equal to the height, there is only one solution. The height, as explained above, is the single shortest possible distance from the upper vertex to the base of the triangle. Since the swinging side of the triangle is the same length as the height, there is only one way to orient this leg to make the triangle complete: perpendicular to the base and through the upper vertex. Thus, there is only one solution when the length <math>a</math> is equal to <math> h</math> or <math> b \sin A</math>, and this automatically forms a right triangle.<br />
<br />
<br />
In the picture below, no matter how the green side swings, it will only touch the base of the triangle once. This triangle will only be complete when the triangle becomes a right triangle.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_one_solution.jpg|center]]<br />
<br />
{{!}}}<br />
Since the other given side length is <math>5</math> and since <math>5=5</math>, there is only one solution which is a right triangle.<br />
<br />
In summary, when <math>a = b \sin A</math>, there is just one solution for a SSA configuration.<br />
<br />
===Third Scenario: Two Solutions===<br />
<br />
In the third scenario, the length of the swinging side is greater than the height, <math>h</math>. Because this side of the triangle is greater than the height, there are two solutions to complete the triangle. The swinging side will complete the triangle at exactly two points: one in which the swinging side and the fixed side form an acute angle, and one where those same two sides form an obtuse angle. Thus, there are two solutions when the swinging side length <math>a</math> is greater than <math> h</math> or <math> b \sin A</math><br />
<br />
In the picture below, no matter how the blue side swings, it's endpoint will touch the base of the triangle only twice. This SSA configuration will complete two separate triangles.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_two_solutions.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
Since the other given side length is <math>6</math> and since <math>6>5</math>, there is are two unique triangular completions.<br />
<br />
In summary, when <math>a > b \sin A</math>, there are two solutions for a SSA configuration.<br />
<br />
<br />
====Determining Both Solutions====<br />
<br />
The ambiguous case often produce two possible completions of the triangle. In these two potential triangles, the corresponding angles between the swinging sides and the unknown sides are <balloon title="two angles whose measures sum to 180 degrees">supplementary</balloon>. To find both triangles, use the [[Law of Sines|law of sines]] to solve for the first triangle, then find the supplement of the measure of the angle between the swinging side and the base and solve using that angle.<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case1.jpg|center]]<br />
<br />
{{!}}{{!}}<br />
Because of the cyclic nature of [[sine]] as a periodic function, the sine of a given angle is the same as the sine of its supplement. There are two solutions by the law of sines since <br />
::<math> \frac{b}{\sin B} = \frac{b}{\sin (180^\circ -B)}</math>.<br />
{{!}}}<br />
<br />
===Fourth Scenario: One Oblique Solution===<br />
In this final scenario, the length of the swinging side is greater than the length of the fixed side. In this scenario, two possible triangles can be formed by swinging the side, but only one contains an angle of the given measure. To get one of the possible triangles, the lower endpoint of the swinging side swings past (to the left of) the vertex that connects the base and the fixed side so that the fixed angle is not included in the solution. This creates a triangle that includes the supplement of the fixed angle, but not the fixed angle. Because this fixed angle is missing from the completed triangle, this possibility is not a viable solution. Thus, there is only one solution when length <math>a</math> is greater than length <math>b</math>.<br />
<br />
If <math>a=b</math>, at least two of the sides of the triangle will be the same length and the solution will be either an <balloon title="a triangle with two sides of equal length">isosceles</balloon> or <balloon title="a regular triangle with three sides of equal length and three 60&deg; angles">equilateral</balloon> triangle.<br />
<br />
In the picture below, the swinging pink side forms two triangles. The one on the left, however, does not include the fixed angle with a measure of 30 &deg;, and is therefore not a solution for this SSA configuration.<br />
<br />
<br />
[[Image:One_solution_oblique1.jpg|center]]<br />
<br />
In summary, when <math> a \geq b</math>, there is one solutions for a SSA configuration.<br />
<br />
===Ambiguous Case Applet===<br />
{{{!}}<br />
{{!}}Grab the point E to adjust the length of the swinging side. Grab the point B to move it around and change the height. Blue and purple triangles are solutions given the fixed angle A, the length of side AB, and the length of the swinging side. Red triangles are not viable solutions, like in the fourth scenario. <br />
<br />
Pay attention to the numerical values on the side. You'll notice that the number of solutions depends on the length of BE compared to the height.<br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
{{!}}{{!}}<br />
{{#iframe:http://www.cs.drexel.edu/~mar343/MathImages/AmbiguousCase/|600|500}}<br />
{{!}}}<br />
<br />
=Teaching Materials=<br />
==Ambiguous Case Demonstration==<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case_acitvity.jpg|300px|left]]<br />
{{!}}{{!}}During lessons about the ambiguous case, it can often be tricky to visualize the different solutions that a particular SSA configuration produces. Sometimes, it can be easy to forget that the length of the base is not fixed, which makes it easy to forget that the two different solutions can have two different-sized bases.<br />
<br />
This activity will show the solutions for each of the scenarios when the swinging side length is changed. Students will be able to physically see the different possible solutions given an SSA configuration.<br />
<br />
===Materials Needed===<br />
:*Chalk/Chalkboard<br />
:*Yarn<br />
:*Tape<br />
:*Scissors<br />
{{!}}}<br />
<br />
<br style="clear: both" /><br />
<br />
<br />
===Instructions===<br />
{{{!}}<br />
{{!}}<br />
[[Image:ACTM_diagram.jpg|right|300px]]<br />
{{!}}{{!}}<br />
*Draw a base of a triangle on the chalkboard, parallel to the ground. Indicate that it is not of a given length, perhaps by making a dotted or colored line all the way across the board. <br />
*Tape one end of the yarn at the end of the base. Continue to tape up the yarn so that the yarn follows what would be first fixed side of a triangle. Make sure that there is a big enough angle between these first two sides so that you can mark it. Now we have drawn the base of unknown length, the fixed angle, and the first fixed side, adjacent to the angles--the A and first S of our ASS triangle.<br />
*Tape the yarn to the board at some point above the dotted line. This will be the vertex between the two sides of fixed length. Let the leftover yarn hang straight down from the taped vertex. Make sure that the hanging yarn can swing back and forth. This will be the swinging side of fixed length. Be sure to leave enough yarn so that you can show all four scenarios.<br />
{{!}}-<br />
{{!}}<br />
*Grab the end of the yarn and move it so that the swinging side rotates from the taped vertex. Wherever the end of the yarn touches the line, a triangle is completed. <br />
*First, show the scenario where the swinging side length is longer than the length of the other fixed side. Show how the fixed angle is not included in one of the completed triangles. <br />
*Now cut the yarn so that it is long enough to show the next case, when there are two unique solutions. <br />
*Cut the yarn again to show the scenario with one solution and a right triangle, and then again to show no solution.<br />
{{!}}{{!}}<br />
[[Image:ACTM_demonstration.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
<br />
<br style="clear: both" /><br />
<br />
===Sample Pictures===<br />
Click here to see sample pictures of the activity in action!<br />
<br />
{{Hide|1=<br />
[[Image:Nosolution.jpg|center|180px]]<br />
[[Image:No_solutions_demo.jpg|center|700px]]<br />
[[Image:One_solution.jpg|center|180px]]<br />
[[Image:One solution right demo.jpg|center|350px]]<br />
[[Image:One_solution_demo.jpg|center|700px]]<br />
[[Image:Two_solutions.jpg|center|180px]]<br />
[[Image:Two solutions demo.jpg|center|700px]]<br />
<br />
}}<br />
<br />
==References==<br />
All of the images on this page were made by the creator/author, [[User:Rscott3|Richard]].<br />
The applet was made by [[User:alimurreza|Reza]].</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Ambiguous_Case&diff=25847Ambiguous Case2011-07-18T15:34:56Z<p>Rscott3: </p>
<hr />
<div>{{HelperPage|1=Law of Sines|2=Congruent triangles|3=Solving Triangles}}<br />
<br />
__TOC__<br />
<br />
Given two adjacent side lengths and an angle opposite one of them, there is no definite completion of a triangle. According to [[triangle congruence postulates]], two triangles cannot be proved congruent given these three elements. This configuration is commonly referred to as '''side-side-angle (SSA)'''. <br />
<br />
With the SSA configuration, there is a fixed angle connecting the base of the triangle and one of the adjacent sides. The length of the base is unknown, denoted below by a dashed line. The length of the third side length is also fixed, but neither angle adjacent to that side is known. This means that this third side can swing from the upper vertex in any way that connects this vertex and any point along the indefinitely sized base so that all three sides of the triangle are connected.<br />
<br />
[[Image:Triangle_template1.jpg|center]]<br />
<br />
Given any SSA configuration, there are several outcomes that can occur when solving a triangle: no solution, one solution, or two solutions. Which of the three scenarios occurs for any SSA configuration depends on the length of the swinging side compared to the height of the triangle.<br />
<br />
In any SSA configuration, we can draw the height of the triangle even though we don't know the length of the base. The height is the perpendicular distance from the upper vertex to the base. Because there is a right angle between the height and base, we can always use the fixed angle and the length of the fixed side to determine the height. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging side to this known height of the triangle.<br />
<br />
[[Image:Height2.jpg|center]]<br />
<br />
<br />
Let <math>a</math> be the length of the side opposite <math>A</math>, an acute angle. The table below describes the different solutions for different scenarios.<br />
<br />
<br />
{{{!}}border="1" cellspacing="3" cellpadding="8" align="center"<br />
{{!}}'''Scenario'''{{!}}{{!}}'''Number of Solutions'''{{!}}{{!}}'''Type of Triangle'''<br />
{{!}}-<br />
{{!}}<math>a < h</math>{{!}}{{!}}no solution{{!}}{{!}}none<br />
{{!}}-<br />
{{!}}<math>a = h</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}right triangle<br />
{{!}}-<br />
{{!}}<math>b > a > h</math><br />
{{!}}{{!}}two solutions<br />
{{!}}{{!}}oblique<br />
{{!}}-<br />
{{!}}<math>a \geq b</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}when <math>a = b</math>, equilateral/isosocles<br />
<br />
when <math>a > b</math>, obtuse<br />
{{!}}}<br />
<br />
<br />
<br />
Because the SSA configuration can prompt different numbers of solutions for for different scenarios, it is often referred to as the '''ambiguous case'''.<br />
<br />
<br />
We can use [[Basic Trigonometric Functions|trigonometry]] to determine the value of height <math>h</math>. Click below to see how.<br />
<br />
{{Hide|1=<br />
<br />
::<math>\sin A =\frac{opposite}{hypotenuse}</math><br />
<br />
<br />
Substituting the appropriate variables,<br />
<br />
::<math>\sin A = \frac{h}{b}</math><br />
<br />
<br />
Multiply both sides by <math>b</math> to get<br />
<br />
::<math>h = b \sin A</math><br />
<br />
}}<br />
<br />
===First Scenario: No Solution===<br />
<br />
In the first scenario, the length of the swinging side is shorter than the height, <math>h</math>. Because this side of the triangle is shorter than the height, there is no solution. The shortest distance between a point and a given line is the line segment that is perpendicular to the given line and goes through that point, which in this case is height <math> h</math>. Since the side length is shorter than the shortest possible distance between the base and the upper vertex of the triangle, the side opposite the fixed angle will never be able to reach the base of the triangle. Thus, there are no solutions when the swinging side length <math>a</math> is less than <math> h</math> or <math> b \sin A</math>.<br />
<br />
In the picture below with numbers we've chosen, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete.<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:ambig case no solution 1.jpg|center]]<br />
{{!}}}<br />
<br />
<br />
Since the other given side length is <math>4</math> and since <math>4<5</math>, there is no solution.<br />
<br />
In summary, when <math>a < b \sin A</math>, there is no solution for a SSA configuration.<br />
<br />
===Second Scenario: One Right Solution===<br />
<br />
In the second scenario, the length of the swinging side is equal to the height, <math>h</math>. Because this side of the triangle is equal to the height, there is only one solution. The height, as explained above, is the single shortest possible distance from the upper vertex to the base of the triangle. Since the swinging side of the triangle is the same length as the height, there is only one way to orient this leg to make the triangle complete: perpendicular to the base and through the upper vertex. Thus, there is only one solution when the length <math>a</math> is equal to <math> h</math> or <math> b \sin A</math>, and this automatically forms a right triangle.<br />
<br />
<br />
In the picture below, no matter how the green side swings, it will only touch the base of the triangle once. This triangle will only be complete when the triangle becomes a right triangle.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_one_solution.jpg|center]]<br />
<br />
{{!}}}<br />
Since the other given side length is <math>5</math> and since <math>5=5</math>, there is only one solution which is a right triangle.<br />
<br />
In summary, when <math>a = b \sin A</math>, there is just one solution for a SSA configuration.<br />
<br />
===Third Scenario: Two Solutions===<br />
<br />
In the third scenario, the length of the swinging side is greater than the height, <math>h</math>. Because this side of the triangle is greater than the height, there are two solutions to complete the triangle. The swinging side will complete the triangle at exactly two points: one in which the swinging side and the fixed side form an acute angle, and one where those same two sides form an obtuse angle. Thus, there are two solutions when the swinging side length <math>a</math> is greater than <math> h</math> or <math> b \sin A</math><br />
<br />
In the picture below, no matter how the blue side swings, it's endpoint will touch the base of the triangle only twice. This SSA configuration will complete two separate triangles.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_two_solutions.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
Since the other given side length is <math>6</math> and since <math>6>5</math>, there is are two unique triangular completions.<br />
<br />
In summary, when <math>a > b \sin A</math>, there are two solutions for a SSA configuration.<br />
<br />
<br />
====Determining Both Solutions====<br />
<br />
The ambiguous case often produce two possible completions of the triangle. In these two potential triangles, the corresponding angles between the swinging sides and the unknown sides are <balloon title="two angles whose measures sum to 180 degrees">supplementary</balloon>. To find both triangles, use the [[Law of Sines|law of sines]] to solve for the first triangle, then find the supplement of the measure of the angle between the swinging side and the base and solve using that angle.<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case1.jpg|center]]<br />
<br />
{{!}}{{!}}<br />
Because of the cyclic nature of [[sine]] as a periodic function, the sine of a given angle is the same as the sine of its supplement. There are two solutions by the law of sines since <br />
::<math> \frac{b}{\sin B} = \frac{b}{\sin (180^\circ -B)}</math>.<br />
{{!}}}<br />
<br />
===Fourth Scenario: One Oblique Solution===<br />
In this final scenario, the length of the swinging side is greater than the length of the fixed side. In this scenario, two possible triangles can be formed by swinging the side, but only one contains an angle of the given measure. To get one of the possible triangles, the lower endpoint of the swinging side swings past (to the left of) the vertex that connects the base and the fixed side so that the fixed angle is not included in the solution. This creates a triangle that includes the supplement of the fixed angle, but not the fixed angle. Because this fixed angle is missing from the completed triangle, this possibility is not a viable solution. Thus, there is only one solution when length <math>a</math> is greater than length <math>b</math>.<br />
<br />
If <math>a=b</math>, at least two of the sides of the triangle will be the same length and the solution will be either an <balloon title="a triangle with two sides of equal length">isosceles</balloon> or <balloon title="a regular triangle with three sides of equal length and three 60&deg; angles">equilateral</balloon> triangle.<br />
<br />
In the picture below, the swinging pink side forms two triangles. The one on the left, however, does not include the fixed angle with a measure of 30 &deg;, and is therefore not a solution for this SSA configuration.<br />
<br />
<br />
[[Image:One_solution_oblique1.jpg|center]]<br />
<br />
In summary, when <math> a \geq b</math>, there is one solutions for a SSA configuration.<br />
<br />
===Ambiguous Case Applet===<br />
{{{!}}<br />
{{!}}Grab the point E to adjust the length of the swinging side. Grab the point B to move it around and change the height. Blue and purple triangles are solutions given the fixed angle A, the length of side AB, and the length of the swinging side. Red triangles are not viable solutions, like in the fourth scenario. <br />
<br />
Pay attention to the numerical values on the side. You'll notice that the number of solutions depends on the length of BE compared to the height.<br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
<br />
<br><br />
{{!}}{{!}}<br />
{{#iframe:http://www.cs.drexel.edu/~mar343/MathImages/AmbiguousCase/|600|500}}<br />
{{!}}}<br />
<br />
=Teaching Materials=<br />
==Ambiguous Case Demonstration==<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case_acitvity.jpg|300px|left]]<br />
{{!}}{{!}}During lessons about the ambiguous case, it can often be tricky to visualize the different solutions that a particular SSA configuration produces. Sometimes, it can be easy to forget that the length of the base is not fixed, which makes it easy to forget that the two different solutions can have two different-sized bases.<br />
<br />
This activity will show the solutions for each of the scenarios when the swinging side length is changed. Students will be able to physically see the different possible solutions given an SSA configuration.<br />
<br />
===Materials Needed===<br />
:*Chalk/Chalkboard<br />
:*Yarn<br />
:*Tape<br />
:*Scissors<br />
{{!}}}<br />
<br />
<br style="clear: both" /><br />
<br />
<br />
===Instructions===<br />
{{{!}}<br />
{{!}}<br />
[[Image:ACTM_diagram.jpg|right|300px]]<br />
{{!}}{{!}}<br />
*Draw a base of a triangle on the chalkboard, parallel to the ground. Indicate that it is not of a given length, perhaps by making a dotted or colored line all the way across the board. <br />
*Tape one end of the yarn at the end of the base. Continue to tape up the yarn so that the yarn follows what would be first fixed side of a triangle. Make sure that there is a big enough angle between these first two sides so that you can mark it. Now we have drawn the base of unknown length, the fixed angle, and the first fixed side, adjacent to the angles--the A and first S of our ASS triangle.<br />
*Tape the yarn to the board at some point above the dotted line. This will be the vertex between the two sides of fixed length. Let the leftover yarn hang straight down from the taped vertex. Make sure that the hanging yarn can swing back and forth. This will be the swinging side of fixed length. Be sure to leave enough yarn so that you can show all four scenarios.<br />
{{!}}-<br />
{{!}}<br />
*Grab the end of the yarn and move it so that the swinging side rotates from the taped vertex. Wherever the end of the yarn touches the line, a triangle is completed. <br />
*First, show the scenario where the swinging side length is longer than the length of the other fixed side. Show how the fixed angle is not included in one of the completed triangles. <br />
*Now cut the yarn so that it is long enough to show the next case, when there are two unique solutions. <br />
*Cut the yarn again to show the scenario with one solution and a right triangle, and then again to show no solution.<br />
{{!}}{{!}}<br />
[[Image:ACTM_demonstration.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
<br />
<br style="clear: both" /><br />
<br />
===Sample Pictures===<br />
Click here to see sample pictures of the activity in action!<br />
<br />
{{Hide|1=<br />
[[Image:Nosolution.jpg|center|180px]]<br />
[[Image:No_solutions_demo.jpg|center|700px]]<br />
[[Image:One_solution.jpg|center|180px]]<br />
[[Image:One solution right demo.jpg|center|350px]]<br />
[[Image:One_solution_demo.jpg|center|700px]]<br />
[[Image:Two_solutions.jpg|center|180px]]<br />
[[Image:Two solutions demo.jpg|center|700px]]<br />
<br />
}}</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Ambiguous_Case&diff=25845Ambiguous Case2011-07-18T15:31:22Z<p>Rscott3: /* Ambiguous Case Applet */</p>
<hr />
<div>{{HelperPage|1=Law of Sines|2=Congruent triangles|3=Solving Triangles}}<br />
<br />
__TOC__<br />
<br />
Given two adjacent side lengths and an angle opposite one of them, there is no definite completion of a triangle. According to [[triangle congruence postulates]], two triangles cannot be proved congruent given these three elements. This configuration is commonly referred to as '''side-side-angle (SSA)'''. <br />
<br />
With the SSA configuration, there is a fixed angle connecting the base of the triangle and one of the adjacent sides. The length of the base is unknown, denoted below by a dashed line. The length of the third side length is also fixed, but neither angle adjacent to that side is known. This means that this third side can swing from the upper vertex in any way that connects this vertex and any point along the indefinitely sized base so that all three sides of the triangle are connected.<br />
<br />
[[Image:Triangle_template1.jpg|center]]<br />
<br />
Given any SSA configuration, there are several outcomes that can occur when solving a triangle: no solution, one solution, or two solutions. Which of the three scenarios occurs for any SSA configuration depends on the length of the swinging side compared to the height of the triangle.<br />
<br />
In any SSA configuration, we can draw the height of the triangle even though we don't know the length of the base. The height is the perpendicular distance from the upper vertex to the base. Because there is a right angle between the height and base, we can always use the fixed angle and the length of the fixed side to determine the height. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging side to this known height of the triangle.<br />
<br />
[[Image:Height2.jpg|center]]<br />
<br />
<br />
Let <math>a</math> be the length of the side opposite <math>A</math>, an acute angle. The table below describes the different solutions for different scenarios.<br />
<br />
<br />
{{{!}}border="1" cellspacing="3" cellpadding="8" align="center"<br />
{{!}}'''Scenario'''{{!}}{{!}}'''Number of Solutions'''{{!}}{{!}}'''Type of Triangle'''<br />
{{!}}-<br />
{{!}}<math>a < h</math>{{!}}{{!}}no solution{{!}}{{!}}none<br />
{{!}}-<br />
{{!}}<math>a = h</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}right triangle<br />
{{!}}-<br />
{{!}}<math>b > a > h</math><br />
{{!}}{{!}}two solutions<br />
{{!}}{{!}}oblique<br />
{{!}}-<br />
{{!}}<math>a \geq b</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}when <math>a = b</math>, equilateral/isosocles<br />
<br />
when <math>a > b</math>, obtuse<br />
{{!}}}<br />
<br />
<br />
<br />
Because the SSA configuration can prompt different numbers of solutions for for different scenarios, it is often referred to as the '''ambiguous case'''.<br />
<br />
<br />
We can use [[Basic Trigonometric Functions|trigonometry]] to determine the value of height <math>h</math>. Click below to see how.<br />
<br />
{{Hide|1=<br />
<br />
::<math>\sin A =\frac{opposite}{hypotenuse}</math><br />
<br />
<br />
Substituting the appropriate variables,<br />
<br />
::<math>\sin A = \frac{h}{b}</math><br />
<br />
<br />
Multiply both sides by <math>b</math> to get<br />
<br />
::<math>h = b \sin A</math><br />
<br />
}}<br />
<br />
===First Scenario: No Solution===<br />
<br />
In the first scenario, the length of the swinging side is shorter than the height, <math>h</math>. Because this side of the triangle is shorter than the height, there is no solution. The shortest distance between a point and a given line is the line segment that is perpendicular to the given line and goes through that point, which in this case is height <math> h</math>. Since the side length is shorter than the shortest possible distance between the base and the upper vertex of the triangle, the side opposite the fixed angle will never be able to reach the base of the triangle. Thus, there are no solutions when the swinging side length <math>a</math> is less than <math> h</math> or <math> b \sin A</math>.<br />
<br />
In the picture below with numbers we've chosen, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete.<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:ambig case no solution 1.jpg|center]]<br />
{{!}}}<br />
<br />
<br />
Since the other given side length is <math>4</math> and since <math>4<5</math>, there is no solution.<br />
<br />
In summary, when <math>a < b \sin A</math>, there is no solution for a SSA configuration.<br />
<br />
===Second Scenario: One Right Solution===<br />
<br />
In the second scenario, the length of the swinging side is equal to the height, <math>h</math>. Because this side of the triangle is equal to the height, there is only one solution. The height, as explained above, is the single shortest possible distance from the upper vertex to the base of the triangle. Since the swinging side of the triangle is the same length as the height, there is only one way to orient this leg to make the triangle complete: perpendicular to the base and through the upper vertex. Thus, there is only one solution when the length <math>a</math> is equal to <math> h</math> or <math> b \sin A</math>, and this automatically forms a right triangle.<br />
<br />
<br />
In the picture below, no matter how the green side swings, it will only touch the base of the triangle once. This triangle will only be complete when the triangle becomes a right triangle.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_one_solution.jpg|center]]<br />
<br />
{{!}}}<br />
Since the other given side length is <math>5</math> and since <math>5=5</math>, there is only one solution which is a right triangle.<br />
<br />
In summary, when <math>a = b \sin A</math>, there is just one solution for a SSA configuration.<br />
<br />
===Third Scenario: Two Solutions===<br />
<br />
In the third scenario, the length of the swinging side is greater than the height, <math>h</math>. Because this side of the triangle is greater than the height, there are two solutions to complete the triangle. The swinging side will complete the triangle at exactly two points: one in which the swinging side and the fixed side form an acute angle, and one where those same two sides form an obtuse angle. Thus, there are two solutions when the swinging side length <math>a</math> is greater than <math> h</math> or <math> b \sin A</math><br />
<br />
In the picture below, no matter how the blue side swings, it's endpoint will touch the base of the triangle only twice. This SSA configuration will complete two separate triangles.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_two_solutions.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
Since the other given side length is <math>6</math> and since <math>6>5</math>, there is are two unique triangular completions.<br />
<br />
In summary, when <math>a > b \sin A</math>, there are two solutions for a SSA configuration.<br />
<br />
<br />
====Determining Both Solutions====<br />
<br />
The ambiguous case often produce two possible completions of the triangle. In these two potential triangles, the corresponding angles between the swinging sides and the unknown sides are <balloon title="two angles whose measures sum to 180 degrees">supplementary</balloon>. To find both triangles, use the [[Law of Sines|law of sines]] to solve for the first triangle, then find the supplement of the measure of the angle between the swinging side and the base and solve using that angle.<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case1.jpg|center]]<br />
<br />
{{!}}{{!}}<br />
Because of the cyclic nature of [[sine]] as a periodic function, the sine of a given angle is the same as the sine of its supplement. There are two solutions by the law of sines since <br />
::<math> \frac{b}{\sin B} = \frac{b}{\sin (180^\circ -B)}</math>.<br />
{{!}}}<br />
<br />
===Fourth Scenario: One Oblique Solution===<br />
In this final scenario, the length of the swinging side is greater than the length of the fixed side. In this scenario, two possible triangles can be formed by swinging the side, but only one contains an angle of the given measure. To get one of the possible triangles, the lower endpoint of the swinging side swings past (to the left of) the vertex that connects the base and the fixed side so that the fixed angle is not included in the solution. This creates a triangle that includes the supplement of the fixed angle, but not the fixed angle. Because this fixed angle is missing from the completed triangle, this possibility is not a viable solution. Thus, there is only one solution when length <math>a</math> is greater than length <math>b</math>.<br />
<br />
If <math>a=b</math>, at least two of the sides of the triangle will be the same length and the solution will be either an <balloon title="a triangle with two sides of equal length">isosceles</balloon> or <balloon title="a regular triangle with three sides of equal length and three 60&deg; angles">equilateral</balloon> triangle.<br />
<br />
In the picture below, the swinging pink side forms two triangles. The one on the left, however, does not include the fixed angle with a measure of 30 &deg;, and is therefore not a solution for this SSA configuration.<br />
<br />
<br />
[[Image:One_solution_oblique1.jpg|center]]<br />
<br />
In summary, when <math> a \geq b</math>, there is one solutions for a SSA configuration.<br />
<br />
===Ambiguous Case Applet===<br />
{{{!}}<br />
{{!}}Grab the point E to adjust the length of the swinging side. Blue and purple triangles are solutions given the fixed angle A, the length of side AB, and the length of the swinging side. Red triangles are not viable solutions, like in the fourth scenario. Grab the point B to move it around and change the height. <br />
<br />
Pay attention to the numerical values on the side. You'll notice that the number of solutions depends on the length of BE compared to the height.<br />
{{!}}{{!}}<br />
{{#iframe:http://www.cs.drexel.edu/~mar343/MathImages/AmbiguousCase/|600|500}}<br />
{{!}}}<br />
<br />
=Teaching Materials=<br />
==Ambiguous Case Demonstration==<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case_acitvity.jpg|300px|left]]<br />
{{!}}{{!}}During lessons about the ambiguous case, it can often be tricky to visualize the different solutions that a particular SSA configuration produces. Sometimes, it can be easy to forget that the length of the base is not fixed, which makes it easy to forget that the two different solutions can have two different-sized bases.<br />
<br />
This activity will show the solutions for each of the scenarios when the swinging side length is changed. Students will be able to physically see the different possible solutions given an SSA configuration.<br />
<br />
===Materials Needed===<br />
:*Chalk/Chalkboard<br />
:*Yarn<br />
:*Tape<br />
:*Scissors<br />
{{!}}}<br />
<br />
<br style="clear: both" /><br />
<br />
<br />
===Instructions===<br />
{{{!}}<br />
{{!}}<br />
[[Image:ACTM_diagram.jpg|right|300px]]<br />
{{!}}{{!}}<br />
*Draw a base of a triangle on the chalkboard, parallel to the ground. Indicate that it is not of a given length, perhaps by making a dotted or colored line all the way across the board. <br />
*Tape one end of the yarn at the end of the base. Continue to tape up the yarn so that the yarn follows what would be first fixed side of a triangle. Make sure that there is a big enough angle between these first two sides so that you can mark it. Now we have drawn the base of unknown length, the fixed angle, and the first fixed side, adjacent to the angles--the A and first S of our ASS triangle.<br />
*Tape the yarn to the board at some point above the dotted line. This will be the vertex between the two sides of fixed length. Let the leftover yarn hang straight down from the taped vertex. Make sure that the hanging yarn can swing back and forth. This will be the swinging side of fixed length. Be sure to leave enough yarn so that you can show all four scenarios.<br />
{{!}}-<br />
{{!}}<br />
*Grab the end of the yarn and move it so that the swinging side rotates from the taped vertex. Wherever the end of the yarn touches the line, a triangle is completed. <br />
*First, show the scenario where the swinging side length is longer than the length of the other fixed side. Show how the fixed angle is not included in one of the completed triangles. <br />
*Now cut the yarn so that it is long enough to show the next case, when there are two unique solutions. <br />
*Cut the yarn again to show the scenario with one solution and a right triangle, and then again to show no solution.<br />
{{!}}{{!}}<br />
[[Image:ACTM_demonstration.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
<br />
<br style="clear: both" /><br />
<br />
===Sample Pictures===<br />
Click here to see sample pictures of the activity in action!<br />
<br />
{{Hide|1=<br />
[[Image:Nosolution.jpg|center|180px]]<br />
[[Image:No_solutions_demo.jpg|center|700px]]<br />
[[Image:One_solution.jpg|center|180px]]<br />
[[Image:One solution right demo.jpg|center|350px]]<br />
[[Image:One_solution_demo.jpg|center|700px]]<br />
[[Image:Two_solutions.jpg|center|180px]]<br />
[[Image:Two solutions demo.jpg|center|700px]]<br />
<br />
}}</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Ambiguous_Case&diff=25839Ambiguous Case2011-07-18T15:21:15Z<p>Rscott3: </p>
<hr />
<div>{{HelperPage|1=Law of Sines|2=Congruent triangles|3=Solving Triangles}}<br />
<br />
__TOC__<br />
<br />
Given two adjacent side lengths and an angle opposite one of them, there is no definite completion of a triangle. According to [[triangle congruence postulates]], two triangles cannot be proved congruent given these three elements. This configuration is commonly referred to as '''side-side-angle (SSA)'''. <br />
<br />
With the SSA configuration, there is a fixed angle connecting the base of the triangle and one of the adjacent sides. The length of the base is unknown, denoted below by a dashed line. The length of the third side length is also fixed, but neither angle adjacent to that side is known. This means that this third side can swing from the upper vertex in any way that connects this vertex and any point along the indefinitely sized base so that all three sides of the triangle are connected.<br />
<br />
[[Image:Triangle_template1.jpg|center]]<br />
<br />
Given any SSA configuration, there are several outcomes that can occur when solving a triangle: no solution, one solution, or two solutions. Which of the three scenarios occurs for any SSA configuration depends on the length of the swinging side compared to the height of the triangle.<br />
<br />
In any SSA configuration, we can draw the height of the triangle even though we don't know the length of the base. The height is the perpendicular distance from the upper vertex to the base. Because there is a right angle between the height and base, we can always use the fixed angle and the length of the fixed side to determine the height. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging side to this known height of the triangle.<br />
<br />
[[Image:Height2.jpg|center]]<br />
<br />
<br />
Let <math>a</math> be the length of the side opposite <math>A</math>, an acute angle. The table below describes the different solutions for different scenarios.<br />
<br />
<br />
{{{!}}border="1" cellspacing="3" cellpadding="8" align="center"<br />
{{!}}'''Scenario'''{{!}}{{!}}'''Number of Solutions'''{{!}}{{!}}'''Type of Triangle'''<br />
{{!}}-<br />
{{!}}<math>a < h</math>{{!}}{{!}}no solution{{!}}{{!}}none<br />
{{!}}-<br />
{{!}}<math>a = h</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}right triangle<br />
{{!}}-<br />
{{!}}<math>b > a > h</math><br />
{{!}}{{!}}two solutions<br />
{{!}}{{!}}oblique<br />
{{!}}-<br />
{{!}}<math>a \geq b</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}when <math>a = b</math>, equilateral/isosocles<br />
<br />
when <math>a > b</math>, obtuse<br />
{{!}}}<br />
<br />
<br />
<br />
Because the SSA configuration can prompt different numbers of solutions for for different scenarios, it is often referred to as the '''ambiguous case'''.<br />
<br />
<br />
We can use [[Basic Trigonometric Functions|trigonometry]] to determine the value of height <math>h</math>. Click below to see how.<br />
<br />
{{Hide|1=<br />
<br />
::<math>\sin A =\frac{opposite}{hypotenuse}</math><br />
<br />
<br />
Substituting the appropriate variables,<br />
<br />
::<math>\sin A = \frac{h}{b}</math><br />
<br />
<br />
Multiply both sides by <math>b</math> to get<br />
<br />
::<math>h = b \sin A</math><br />
<br />
}}<br />
<br />
===First Scenario: No Solution===<br />
<br />
In the first scenario, the length of the swinging side is shorter than the height, <math>h</math>. Because this side of the triangle is shorter than the height, there is no solution. The shortest distance between a point and a given line is the line segment that is perpendicular to the given line and goes through that point, which in this case is height <math> h</math>. Since the side length is shorter than the shortest possible distance between the base and the upper vertex of the triangle, the side opposite the fixed angle will never be able to reach the base of the triangle. Thus, there are no solutions when the swinging side length <math>a</math> is less than <math> h</math> or <math> b \sin A</math>.<br />
<br />
In the picture below with numbers we've chosen, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete.<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:ambig case no solution 1.jpg|center]]<br />
{{!}}}<br />
<br />
<br />
Since the other given side length is <math>4</math> and since <math>4<5</math>, there is no solution.<br />
<br />
In summary, when <math>a < b \sin A</math>, there is no solution for a SSA configuration.<br />
<br />
===Second Scenario: One Right Solution===<br />
<br />
In the second scenario, the length of the swinging side is equal to the height, <math>h</math>. Because this side of the triangle is equal to the height, there is only one solution. The height, as explained above, is the single shortest possible distance from the upper vertex to the base of the triangle. Since the swinging side of the triangle is the same length as the height, there is only one way to orient this leg to make the triangle complete: perpendicular to the base and through the upper vertex. Thus, there is only one solution when the length <math>a</math> is equal to <math> h</math> or <math> b \sin A</math>, and this automatically forms a right triangle.<br />
<br />
<br />
In the picture below, no matter how the green side swings, it will only touch the base of the triangle once. This triangle will only be complete when the triangle becomes a right triangle.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_one_solution.jpg|center]]<br />
<br />
{{!}}}<br />
Since the other given side length is <math>5</math> and since <math>5=5</math>, there is only one solution which is a right triangle.<br />
<br />
In summary, when <math>a = b \sin A</math>, there is just one solution for a SSA configuration.<br />
<br />
===Third Scenario: Two Solutions===<br />
<br />
In the third scenario, the length of the swinging side is greater than the height, <math>h</math>. Because this side of the triangle is greater than the height, there are two solutions to complete the triangle. The swinging side will complete the triangle at exactly two points: one in which the swinging side and the fixed side form an acute angle, and one where those same two sides form an obtuse angle. Thus, there are two solutions when the swinging side length <math>a</math> is greater than <math> h</math> or <math> b \sin A</math><br />
<br />
In the picture below, no matter how the blue side swings, it's endpoint will touch the base of the triangle only twice. This SSA configuration will complete two separate triangles.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_two_solutions.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
Since the other given side length is <math>6</math> and since <math>6>5</math>, there is are two unique triangular completions.<br />
<br />
In summary, when <math>a > b \sin A</math>, there are two solutions for a SSA configuration.<br />
<br />
<br />
====Determining Both Solutions====<br />
<br />
The ambiguous case often produce two possible completions of the triangle. In these two potential triangles, the corresponding angles between the swinging sides and the unknown sides are <balloon title="two angles whose measures sum to 180 degrees">supplementary</balloon>. To find both triangles, use the [[Law of Sines|law of sines]] to solve for the first triangle, then find the supplement of the measure of the angle between the swinging side and the base and solve using that angle.<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case1.jpg|center]]<br />
<br />
{{!}}{{!}}<br />
Because of the cyclic nature of [[sine]] as a periodic function, the sine of a given angle is the same as the sine of its supplement. There are two solutions by the law of sines since <br />
::<math> \frac{b}{\sin B} = \frac{b}{\sin (180^\circ -B)}</math>.<br />
{{!}}}<br />
<br />
===Fourth Scenario: One Oblique Solution===<br />
In this final scenario, the length of the swinging side is greater than the length of the fixed side. In this scenario, two possible triangles can be formed by swinging the side, but only one contains an angle of the given measure. To get one of the possible triangles, the lower endpoint of the swinging side swings past (to the left of) the vertex that connects the base and the fixed side so that the fixed angle is not included in the solution. This creates a triangle that includes the supplement of the fixed angle, but not the fixed angle. Because this fixed angle is missing from the completed triangle, this possibility is not a viable solution. Thus, there is only one solution when length <math>a</math> is greater than length <math>b</math>.<br />
<br />
If <math>a=b</math>, at least two of the sides of the triangle will be the same length and the solution will be either an <balloon title="a triangle with two sides of equal length">isosceles</balloon> or <balloon title="a regular triangle with three sides of equal length and three 60&deg; angles">equilateral</balloon> triangle.<br />
<br />
In the picture below, the swinging pink side forms two triangles. The one on the left, however, does not include the fixed angle with a measure of 30 &deg;, and is therefore not a solution for this SSA configuration.<br />
<br />
<br />
[[Image:One_solution_oblique1.jpg|center]]<br />
<br />
In summary, when <math> a \geq b</math>, there is one solutions for a SSA configuration.<br />
<br />
===Ambiguous Case Applet===<br />
<br />
{{#iframe:http://www.cs.drexel.edu/~mar343/MathImages/AmbiguousCase/|700|500}}<br />
<br />
<br />
=Teaching Materials=<br />
==Ambiguous Case Demonstration==<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case_acitvity.jpg|300px|left]]<br />
{{!}}{{!}}During lessons about the ambiguous case, it can often be tricky to visualize the different solutions that a particular SSA configuration produces. Sometimes, it can be easy to forget that the length of the base is not fixed, which makes it easy to forget that the two different solutions can have two different-sized bases.<br />
<br />
This activity will show the solutions for each of the scenarios when the swinging side length is changed. Students will be able to physically see the different possible solutions given an SSA configuration.<br />
<br />
===Materials Needed===<br />
:*Chalk/Chalkboard<br />
:*Yarn<br />
:*Tape<br />
:*Scissors<br />
{{!}}}<br />
<br />
<br style="clear: both" /><br />
<br />
<br />
===Instructions===<br />
{{{!}}<br />
{{!}}<br />
[[Image:ACTM_diagram.jpg|right|300px]]<br />
{{!}}{{!}}<br />
*Draw a base of a triangle on the chalkboard, parallel to the ground. Indicate that it is not of a given length, perhaps by making a dotted or colored line all the way across the board. <br />
*Tape one end of the yarn at the end of the base. Continue to tape up the yarn so that the yarn follows what would be first fixed side of a triangle. Make sure that there is a big enough angle between these first two sides so that you can mark it. Now we have drawn the base of unknown length, the fixed angle, and the first fixed side, adjacent to the angles--the A and first S of our ASS triangle.<br />
*Tape the yarn to the board at some point above the dotted line. This will be the vertex between the two sides of fixed length. Let the leftover yarn hang straight down from the taped vertex. Make sure that the hanging yarn can swing back and forth. This will be the swinging side of fixed length. Be sure to leave enough yarn so that you can show all four scenarios.<br />
{{!}}-<br />
{{!}}<br />
*Grab the end of the yarn and move it so that the swinging side rotates from the taped vertex. Wherever the end of the yarn touches the line, a triangle is completed. <br />
*First, show the scenario where the swinging side length is longer than the length of the other fixed side. Show how the fixed angle is not included in one of the completed triangles. <br />
*Now cut the yarn so that it is long enough to show the next case, when there are two unique solutions. <br />
*Cut the yarn again to show the scenario with one solution and a right triangle, and then again to show no solution.<br />
{{!}}{{!}}<br />
[[Image:ACTM_demonstration.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
<br />
<br style="clear: both" /><br />
<br />
===Sample Pictures===<br />
Click here to see sample pictures of the activity in action!<br />
<br />
{{Hide|1=<br />
[[Image:Nosolution.jpg|center|180px]]<br />
[[Image:No_solutions_demo.jpg|center|700px]]<br />
[[Image:One_solution.jpg|center|180px]]<br />
[[Image:One solution right demo.jpg|center|350px]]<br />
[[Image:One_solution_demo.jpg|center|700px]]<br />
[[Image:Two_solutions.jpg|center|180px]]<br />
[[Image:Two solutions demo.jpg|center|700px]]<br />
<br />
}}</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=DU11&diff=25838DU112011-07-18T15:20:42Z<p>Rscott3: /* Md. Alimoor Reza (alimurreza) */</p>
<hr />
<div><font style="position:relative;top:-10px" size="2">[[PartnerHome| << Back to Partner Home]]</font><br><br />
<center><font size="5"><b>[http://www.drexel.edu/ Drexel University]</b></font><br><br />
Welcome to Drexel's Summer 2011 page.</center><br />
<br />
__TOC__<br />
<br />
==Latest News & Works in Progress ==<br />
<br />
<br />
<br><br />
<br />
<br />
==Calvin Morrison ([http://mathforum.org/mathimages/index.php/User:CalvinMorrison CalvinMorrison])==<br />
<br />
=== Future Plans ===<br />
This is what I am planning on working on over the summer<br />
<br />
==== Ideas ====<br />
Working on the organization bit of the project:<br />
<br />
*Left over from other summers, we have a lot of pages that need to be reviewed, fixed up, then added.<br />
*Updating the main website. If I am having trouble finding something - Imagine someone who doesn't even work for the project! Remember, the project isn't for us. it's for people to utilize. that can't do it if it's not effective at organizing itself.<br />
*From the Homepage, we should be able to get everywhere within around 3 clicks. It's taken me up to 7 clicks to get to a specific page, that's to many!<br />
<br />
==== Unfinished from 2009 ====<br />
<br />
This of course only includes ones that were clearly unfinished or marked as WIP, obviously these need to be reviewed using the checklist: [[Checklist for writing pages]]<br />
<br />
*[[Fourier Transform]]<br />
*[[Bump Mapping]]<br />
*[[Gaussian Pyramid]]<br />
*[[Metaballs]]<br />
*[[Change Of Coordinate Transformations]]<br />
*[[Cardioid]]<br />
*[[Bedsheet Problem]] - Definitely interested in this one, I think it needs to be reworked as the better known "can you fold a piece of paper 7 times, not a bed sheet, Also the mathematical explanation is a bit hard to follow as well as the original explanation.<br />
:*<font color=dodgerblue>[[User:Kderosier|Kate]] 13:30, 27 June 2011 (UTC): Hi, sorry to butt in, but I was looking at this page and thought I should let you know that this page is not from 2009, nor is it abandoned. [[lUser:Ljeanlo1|Leah]] created it at the beginning of this summer and is still working on it. See [[S11#Leah's Projects|S11]].</font><br />
<br />
==== Helpful Pages ====<br />
<br />
*[[Special:Allpages]]: so all the wiki pages are listed here (so if you lost something, it's bound to be here!)<br />
*http://mathforum.org/mathimages/index.php/Image:Inprogress.png: At the bottom it will tell you what pages are tagged with this image (therefore all the unfinished pages!<br />
<br />
==Md. Alimoor Reza ([http://mathforum.org/mathimages/index.php/User:alimurreza alimurreza])==<br />
<br /><br />
Current and completed tasks here.<br /><br />
*[[Epitrochoid]] Completed. Check out the applet([http://cs.drexel.edu/~mar343/MathImages/Epitrochoid Epitrochoid]).<br />
*[[Epicycloid]] Completed. Check out the applet([http://cs.drexel.edu/~mar343/MathImages/Epicycloid Epicycloid]).<br />
*[[3D Viewer]] Completed. Check out the applet([http://cs.drexel.edu/~mar343/MathImages/Crosscap Cross-cap]). This is an Interactive Applet that imports a crosscap in JavaView .jvx format and displays it. The model can be rotated/translated/scaled around any axis in 3D. Additionally, JavaView allows for the other information(vertex, vertex normal, etc.) of the model. It was requested by Harrison from Swarthmore College. Check out three applets here:<br/><br />
:*([http://www.cs.drexel.edu/~mar343/MathImages/Crosscap/cc_cut.html Cross-capped disk sliced open])<br/><br />
:*([http://www.cs.drexel.edu/~mar343/MathImages/Crosscap/cc_p1.html Cross-cap model-1])<br/><br />
:*([http://www.cs.drexel.edu/~mar343/MathImages/Crosscap/cc_p2.html Cross-cap model-2])<br/><br />
<br />
*[[Ambiguous Case]] Completed. Check out the new version.([http://www.cs.drexel.edu/~mar343/MathImages/AmbiguousCase/ AmbiguousCase]). Working on an Interactive Applet that shows the ambiguity. It was requested by Richard from Swarthmore College.<br />
::Status and interaction with Richard: In progress until 7/14.<br />
::<font color=orangered>Looks great! What still needs to be done? [[User:Rscott3|Richard]] 7/7 </font><br />
::<font color=orangered>Thanks. Need to add the arc denoting the angle etc. [[User:alimurreza|Reza]] 7/8 </font><br />
::::<font color=orangered>Reza, This is sooooooooooooo great. Thanks thanks thanks. Is there a way for me to embed it on the page? [[User:Rscott3|Richard]] 7/18<br />
:::::Figured it out!</font><br />
<br />
<br />
<br />
*[[Snell's Law]] In progress. Working on an Interactive Applet that exhibits Snell's Law. It was requested by Leah from Swarthmore College. Check out the current version.([http://www.cs.drexel.edu/~mar343/MathImages/SnellsLaw/ Snell's Law])<br />
<br />
Future Plans: <br /><br />
*[[Nephroid]]<br />
*[[Hypocycloid]]<br />
*[[Cycloid]]<br />
Cardioid, Hypotrochoid already implemented. So the proposed curves would complement these similar patterns.<br />
<br />
====Other Math Image Pages====<br />
:* Existing pages need work ([http://mathforum.org/mathimages/index.php/Existing_Pages_Needing_Work])<br /><br />
:* Pages with most categories ([http://mathforum.org/mathimages/index.php/Special:Mostcategories])<br /><br />
:* Special pages ([http://mathforum.org/mathimages/index.php/Special:Specialpages])<br /><br />
<br />
==Michelle Guo ([http://mathforum.org/mathimages/index.php/User:Rguo Rguo])==<br />
<br /><br />
<br />
==== Currently Working On ====<br />
Unfinished Pages from Previous Years<br />
:* [[Platonic Solid]]<br />
::<font color=darkgreen>Please comment on this page if you have the time! -- [[User:Rguo|Rguo]] 21:26, 15 July 2011 (UTC)</font><br />
:* Flash versions of the programs for [[Ambiguous Case]] and [[Snell's Law]] that Reza is working on<br />
:* [[Three Cottages Problem]] flash app<br />
:* A flash [[Towers of Hanoi]] app due to Calvin's comments on the refresh rate of the Java app on the site<br />
<br />
<br /><br />
<br />
==== Future Plans ==== <br />
<br />
<br />
<br /><br /><br />
<br />
==Who We Are==<br />
<br />
David Breen, [[User:David]] <br /><br />
Michelle Guo, [[User: Rguo]] (undergraduate student)<br /><br />
Calvin Morrison, [[User: CalvinMorrison]] (undergraduate student)<br /><br />
Md. Alimoor Reza, [[User: alimurreza]] (graduate student)<br />
<br />
[[PartnerHome/Drexel]]<br />
<br />
==Links==<br />
<br />
[[Pages Ready for Final Review]]<br />
<br />
[[Feedback Requests]]<br />
<br />
[[Site programming questions]]<br />
<br />
[[SB11]]<br />
<br />
[[S11]]<br />
<br />
[[RPI11]]<br />
<br />
[[Math Tools Requests]] ''This page is a place where students whose primary focus is writing pages can post requests for applets, animations, and new images that they'd like to see the computer science students create.''</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=DU11&diff=25828DU112011-07-18T15:08:30Z<p>Rscott3: /* Md. Alimoor Reza (alimurreza) */</p>
<hr />
<div><font style="position:relative;top:-10px" size="2">[[PartnerHome| << Back to Partner Home]]</font><br><br />
<center><font size="5"><b>[http://www.drexel.edu/ Drexel University]</b></font><br><br />
Welcome to Drexel's Summer 2011 page.</center><br />
<br />
__TOC__<br />
<br />
==Latest News & Works in Progress ==<br />
<br />
<br />
<br><br />
<br />
<br />
==Calvin Morrison ([http://mathforum.org/mathimages/index.php/User:CalvinMorrison CalvinMorrison])==<br />
<br />
=== Future Plans ===<br />
This is what I am planning on working on over the summer<br />
<br />
==== Ideas ====<br />
Working on the organization bit of the project:<br />
<br />
*Left over from other summers, we have a lot of pages that need to be reviewed, fixed up, then added.<br />
*Updating the main website. If I am having trouble finding something - Imagine someone who doesn't even work for the project! Remember, the project isn't for us. it's for people to utilize. that can't do it if it's not effective at organizing itself.<br />
*From the Homepage, we should be able to get everywhere within around 3 clicks. It's taken me up to 7 clicks to get to a specific page, that's to many!<br />
<br />
==== Unfinished from 2009 ====<br />
<br />
This of course only includes ones that were clearly unfinished or marked as WIP, obviously these need to be reviewed using the checklist: [[Checklist for writing pages]]<br />
<br />
*[[Fourier Transform]]<br />
*[[Bump Mapping]]<br />
*[[Gaussian Pyramid]]<br />
*[[Metaballs]]<br />
*[[Change Of Coordinate Transformations]]<br />
*[[Cardioid]]<br />
*[[Bedsheet Problem]] - Definitely interested in this one, I think it needs to be reworked as the better known "can you fold a piece of paper 7 times, not a bed sheet, Also the mathematical explanation is a bit hard to follow as well as the original explanation.<br />
:*<font color=dodgerblue>[[User:Kderosier|Kate]] 13:30, 27 June 2011 (UTC): Hi, sorry to butt in, but I was looking at this page and thought I should let you know that this page is not from 2009, nor is it abandoned. [[lUser:Ljeanlo1|Leah]] created it at the beginning of this summer and is still working on it. See [[S11#Leah's Projects|S11]].</font><br />
<br />
==== Helpful Pages ====<br />
<br />
*[[Special:Allpages]]: so all the wiki pages are listed here (so if you lost something, it's bound to be here!)<br />
*http://mathforum.org/mathimages/index.php/Image:Inprogress.png: At the bottom it will tell you what pages are tagged with this image (therefore all the unfinished pages!<br />
<br />
==Md. Alimoor Reza ([http://mathforum.org/mathimages/index.php/User:alimurreza alimurreza])==<br />
<br /><br />
Current and completed tasks here.<br /><br />
*[[Epitrochoid]] Completed. Check out the applet([http://cs.drexel.edu/~mar343/MathImages/Epitrochoid Epitrochoid]).<br />
*[[Epicycloid]] Completed. Check out the applet([http://cs.drexel.edu/~mar343/MathImages/Epicycloid Epicycloid]).<br />
*[[3D Viewer]] Completed. Check out the applet([http://cs.drexel.edu/~mar343/MathImages/Crosscap Cross-cap]). This is an Interactive Applet that imports a crosscap in JavaView .jvx format and displays it. The model can be rotated/translated/scaled around any axis in 3D. Additionally, JavaView allows for the other information(vertex, vertex normal, etc.) of the model. It was requested by Harrison from Swarthmore College. Check out three applets here:<br/><br />
:*([http://www.cs.drexel.edu/~mar343/MathImages/Crosscap/cc_cut.html Cross-capped disk sliced open])<br/><br />
:*([http://www.cs.drexel.edu/~mar343/MathImages/Crosscap/cc_p1.html Cross-cap model-1])<br/><br />
:*([http://www.cs.drexel.edu/~mar343/MathImages/Crosscap/cc_p2.html Cross-cap model-2])<br/><br />
<br />
*[[Ambiguous Case]] Completed. Check out the new version.([http://www.cs.drexel.edu/~mar343/MathImages/AmbiguousCase/ AmbiguousCase]). Working on an Interactive Applet that shows the ambiguity. It was requested by Richard from Swarthmore College.<br />
::Status and interaction with Richard: In progress until 7/14.<br />
::<font color=orangered>Looks great! What still needs to be done? [[User:Rscott3|Richard]] 7/7 </font><br />
::<font color=orangered>Thanks. Need to add the arc denoting the angle etc. [[User:alimurreza|Reza]] 7/8 </font><br />
::::<font color=orangered>Reza, This is sooooooooooooo great. Thanks thanks thanks. Is there a way for me to embed it on the page? [[User:Rscott3|Richard]] 7/18</font><br />
<br />
<br />
*[[Snell's Law]] In progress. Working on an Interactive Applet that exhibits Snell's Law. It was requested by Leah from Swarthmore College. Check out the current version.([http://www.cs.drexel.edu/~mar343/MathImages/SnellsLaw/ Snell's Law])<br />
<br />
Future Plans: <br /><br />
*[[Nephroid]]<br />
*[[Hypocycloid]]<br />
*[[Cycloid]]<br />
Cardioid, Hypotrochoid already implemented. So the proposed curves would complement these similar patterns.<br />
<br />
====Other Math Image Pages====<br />
:* Existing pages need work ([http://mathforum.org/mathimages/index.php/Existing_Pages_Needing_Work])<br /><br />
:* Pages with most categories ([http://mathforum.org/mathimages/index.php/Special:Mostcategories])<br /><br />
:* Special pages ([http://mathforum.org/mathimages/index.php/Special:Specialpages])<br /><br />
<br />
==Michelle Guo ([http://mathforum.org/mathimages/index.php/User:Rguo Rguo])==<br />
<br /><br />
<br />
==== Currently Working On ====<br />
Unfinished Pages from Previous Years<br />
:* [[Platonic Solid]]<br />
::<font color=darkgreen>Please comment on this page if you have the time! -- [[User:Rguo|Rguo]] 21:26, 15 July 2011 (UTC)</font><br />
:* Flash versions of the programs for [[Ambiguous Case]] and [[Snell's Law]] that Reza is working on<br />
:* [[Three Cottages Problem]] flash app<br />
:* A flash [[Towers of Hanoi]] app due to Calvin's comments on the refresh rate of the Java app on the site<br />
<br />
<br /><br />
<br />
==== Future Plans ==== <br />
<br />
<br />
<br /><br /><br />
<br />
==Who We Are==<br />
<br />
David Breen, [[User:David]] <br /><br />
Michelle Guo, [[User: Rguo]] (undergraduate student)<br /><br />
Calvin Morrison, [[User: CalvinMorrison]] (undergraduate student)<br /><br />
Md. Alimoor Reza, [[User: alimurreza]] (graduate student)<br />
<br />
[[PartnerHome/Drexel]]<br />
<br />
==Links==<br />
<br />
[[Pages Ready for Final Review]]<br />
<br />
[[Feedback Requests]]<br />
<br />
[[Site programming questions]]<br />
<br />
[[SB11]]<br />
<br />
[[S11]]<br />
<br />
[[RPI11]]<br />
<br />
[[Math Tools Requests]] ''This page is a place where students whose primary focus is writing pages can post requests for applets, animations, and new images that they'd like to see the computer science students create.''</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25812Talk:Ambiguous Case2011-07-18T14:49:05Z<p>Rscott3: /* Page Comments */</p>
<hr />
<div>=Teaching Material Comments=<br />
<font color=mediumseagreen><br />
Comments from Steve Weimar, the Math Forum 6/28<br />
<br />
*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
</font><br />
<br />
<br />
<font color=mediumseagreen><br />
*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
</font><br />
::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
<br />
<br />
*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
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=Page Comments=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
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::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
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::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
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:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
{{Hide|1=<br />
I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
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SSA Postulate: what we can't know<br />
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Why is that? Explanation/Informal proof<br />
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What can be known from SSA?<br />
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How can one figure out which case it is?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
::::<font color=orangered>I think these comments have more to do with basic trig functions page. [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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::Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example).<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font> <br />
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::The table is a very good idea, though it seems large for the page.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::Remove "to compute height" in the very last sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::First Scenario: Two "the"s in sentence 3.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram.<br />
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::Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
::::<font color=orangered> Both solutions don't necessarily have to be oblique. Say angle C is 90 degrees... [[User:Rscott3|Richard]] 7/18</font><br />
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::Determining Both Solutions: Put a link to "Law of Sines" in third sentence.<br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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::There is no angle labeled B to correspond to your equation. <br />
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::Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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::::<font color=orangered>I tried to explain that within the paragraph that there is only one solution of two completed triangles, and not two solutions but only one that works. (I had trouble wording that...I think I make sense????? [[User:Rscott3|Richard]] 7/18</font><br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
::::<font color=orangered>Comment Addressed. [[User:Rscott3|Richard]] 7/18</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
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The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
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===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
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*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
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* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
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* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
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*The end of this section is very clear!<br />
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[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
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====First Scenario: No solution====<br />
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<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
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* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
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<font color=orangered>edited first sentence Richard 5/25</font><br />
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----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
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<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
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2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
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2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
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*I added a few words to show that the swinging side compared to the height is the determinant<br />
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*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
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Richard 5/25</font><br />
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==Extra Picture==<br />
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[[Image:Ambig_cas_act1.jpg]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Ambiguous_Case&diff=25811Ambiguous Case2011-07-18T14:48:47Z<p>Rscott3: </p>
<hr />
<div>{{HelperPage|1=Law of Sines|2=Congruent triangles|3=Solving Triangles}}<br />
<br />
__TOC__<br />
<br />
Given two adjacent side lengths and an angle opposite one of them, there is no definite completion of a triangle. According to [[triangle congruence postulates]], two triangles cannot be proved congruent given these three elements. This configuration is commonly referred to as '''side-side-angle (SSA)'''. <br />
<br />
With the SSA configuration, there is a fixed angle connecting the base of the triangle and one of the adjacent sides. The length of the base is unknown, denoted below by a dashed line. The length of the third side length is also fixed, but neither angle adjacent to that side is known. This means that this third side can swing from the upper vertex in any way that connects this vertex and any point along the indefinitely sized base so that all three sides of the triangle are connected.<br />
<br />
[[Image:Triangle_template1.jpg|center]]<br />
<br />
Given any SSA configuration, there are several outcomes that can occur when solving a triangle: no solution, one solution, or two solutions. Which of the three scenarios occurs for any SSA configuration depends on the length of the swinging side compared to the height of the triangle.<br />
<br />
In any SSA configuration, we can draw the height of the triangle even though we don't know the length of the base. The height is the perpendicular distance from the upper vertex to the base. Because there is a right angle between the height and base, we can always use the fixed angle and the length of the fixed side to determine the height. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging side to this known height of the triangle.<br />
<br />
[[Image:Height2.jpg|center]]<br />
<br />
<br />
Let <math>a</math> be the length of the side opposite <math>A</math>, an acute angle. The table below describes the different solutions for different scenarios.<br />
<br />
<br />
{{{!}}border="1" cellspacing="3" cellpadding="8" align="center"<br />
{{!}}'''Scenario'''{{!}}{{!}}'''Number of Solutions'''{{!}}{{!}}'''Type of Triangle'''<br />
{{!}}-<br />
{{!}}<math>a < h</math>{{!}}{{!}}no solution{{!}}{{!}}none<br />
{{!}}-<br />
{{!}}<math>a = h</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}right triangle<br />
{{!}}-<br />
{{!}}<math>b > a > h</math><br />
{{!}}{{!}}two solutions<br />
{{!}}{{!}}oblique<br />
{{!}}-<br />
{{!}}<math>a \geq b</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}when <math>a = b</math>, equilateral/isosocles<br />
<br />
when <math>a > b</math>, obtuse<br />
{{!}}}<br />
<br />
<br />
<br />
Because the SSA configuration can prompt different numbers of solutions for for different scenarios, it is often referred to as the '''ambiguous case'''.<br />
<br />
<br />
We can use [[Basic Trigonometric Functions|trigonometry]] to determine the value of height <math>h</math>. Click below to see how.<br />
<br />
{{Hide|1=<br />
<br />
::<math>\sin A =\frac{opposite}{hypotenuse}</math><br />
<br />
<br />
Substituting the appropriate variables,<br />
<br />
::<math>\sin A = \frac{h}{b}</math><br />
<br />
<br />
Multiply both sides by <math>b</math> to get<br />
<br />
::<math>h = b \sin A</math><br />
<br />
}}<br />
<br />
===First Scenario: No Solution===<br />
<br />
In the first scenario, the length of the swinging side is shorter than the height, <math>h</math>. Because this side of the triangle is shorter than the height, there is no solution. The shortest distance between a point and a given line is the line segment that is perpendicular to the given line and goes through that point, which in this case is height <math> h</math>. Since the side length is shorter than the shortest possible distance between the base and the upper vertex of the triangle, the side opposite the fixed angle will never be able to reach the base of the triangle. Thus, there are no solutions when the swinging side length <math>a</math> is less than <math> h</math> or <math> b \sin A</math>.<br />
<br />
In the picture below with numbers we've chosen, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete.<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:ambig case no solution 1.jpg|center]]<br />
{{!}}}<br />
<br />
<br />
Since the other given side length is <math>4</math> and since <math>4<5</math>, there is no solution.<br />
<br />
In summary, when <math>a < b \sin A</math>, there is no solution for a SSA configuration.<br />
<br />
===Second Scenario: One Right Solution===<br />
<br />
In the second scenario, the length of the swinging side is equal to the height, <math>h</math>. Because this side of the triangle is equal to the height, there is only one solution. The height, as explained above, is the single shortest possible distance from the upper vertex to the base of the triangle. Since the swinging side of the triangle is the same length as the height, there is only one way to orient this leg to make the triangle complete: perpendicular to the base and through the upper vertex. Thus, there is only one solution when the length <math>a</math> is equal to <math> h</math> or <math> b \sin A</math>, and this automatically forms a right triangle.<br />
<br />
<br />
In the picture below, no matter how the green side swings, it will only touch the base of the triangle once. This triangle will only be complete when the triangle becomes a right triangle.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_one_solution.jpg|center]]<br />
<br />
{{!}}}<br />
Since the other given side length is <math>5</math> and since <math>5=5</math>, there is only one solution which is a right triangle.<br />
<br />
In summary, when <math>a = b \sin A</math>, there is just one solution for a SSA configuration.<br />
<br />
===Third Scenario: Two Solutions===<br />
<br />
In the third scenario, the length of the swinging side is greater than the height, <math>h</math>. Because this side of the triangle is greater than the height, there are two solutions to complete the triangle. The swinging side will complete the triangle at exactly two points: one in which the swinging side and the fixed side form an acute angle, and one where those same two sides form an obtuse angle. Thus, there are two solutions when the swinging side length <math>a</math> is greater than <math> h</math> or <math> b \sin A</math><br />
<br />
In the picture below, no matter how the blue side swings, it's endpoint will touch the base of the triangle only twice. This SSA configuration will complete two separate triangles.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_two_solutions.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
Since the other given side length is <math>6</math> and since <math>6>5</math>, there is are two unique triangular completions.<br />
<br />
In summary, when <math>a > b \sin A</math>, there are two solutions for a SSA configuration.<br />
<br />
<br />
====Determining Both Solutions====<br />
<br />
The ambiguous case often produce two possible completions of the triangle. In these two potential triangles, the corresponding angles between the swinging sides and the unknown sides are <balloon title="two angles whose measures sum to 180 degrees">supplementary</balloon>. To find both triangles, use the [[Law of Sines|law of sines]] to solve for the first triangle, then find the supplement of the measure of the angle between the swinging side and the base and solve using that angle.<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case1.jpg|center]]<br />
<br />
{{!}}{{!}}<br />
Because of the cyclic nature of [[sine]] as a periodic function, the sine of a given angle is the same as the sine of its supplement. There are two solutions by the law of sines since <br />
::<math> \frac{b}{\sin B} = \frac{b}{\sin (180^\circ -B)}</math>.<br />
{{!}}}<br />
<br />
===Fourth Scenario: One Oblique Solution===<br />
In this final scenario, the length of the swinging side is greater than the length of the fixed side. In this scenario, two possible triangles can be formed by swinging the side, but only one contains an angle of the given measure. To get one of the possible triangles, the lower endpoint of the swinging side swings past (to the left of) the vertex that connects the base and the fixed side so that the fixed angle is not included in the solution. This creates a triangle that includes the supplement of the fixed angle, but not the fixed angle. Because this fixed angle is missing from the completed triangle, this possibility is not a viable solution. Thus, there is only one solution when length <math>a</math> is greater than length <math>b</math>.<br />
<br />
If <math>a=b</math>, at least two of the sides of the triangle will be the same length and the solution will be either an <balloon title="a triangle with two sides of equal length">isosceles</balloon> or <balloon title="a regular triangle with three sides of equal length and three 60&deg; angles">equilateral</balloon> triangle.<br />
<br />
In the picture below, the swinging pink side forms two triangles. The one on the left, however, does not include the fixed angle with a measure of 30 &deg;, and is therefore not a solution for this SSA configuration.<br />
<br />
<br />
[[Image:One_solution_oblique1.jpg|center]]<br />
<br />
In summary, when <math> a \geq b</math>, there is one solutions for a SSA configuration.<br />
<br />
=Teaching Materials=<br />
==Ambiguous Case Demonstration==<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case_acitvity.jpg|300px|left]]<br />
{{!}}{{!}}During lessons about the ambiguous case, it can often be tricky to visualize the different solutions that a particular SSA configuration produces. Sometimes, it can be easy to forget that the length of the base is not fixed, which makes it easy to forget that the two different solutions can have two different-sized bases.<br />
<br />
This activity will show the solutions for each of the scenarios when the swinging side length is changed. Students will be able to physically see the different possible solutions given an SSA configuration.<br />
<br />
===Materials Needed===<br />
:*Chalk/Chalkboard<br />
:*Yarn<br />
:*Tape<br />
:*Scissors<br />
{{!}}}<br />
<br />
<br style="clear: both" /><br />
<br />
<br />
===Instructions===<br />
{{{!}}<br />
{{!}}<br />
[[Image:ACTM_diagram.jpg|right|300px]]<br />
{{!}}{{!}}<br />
*Draw a base of a triangle on the chalkboard, parallel to the ground. Indicate that it is not of a given length, perhaps by making a dotted or colored line all the way across the board. <br />
*Tape one end of the yarn at the end of the base. Continue to tape up the yarn so that the yarn follows what would be first fixed side of a triangle. Make sure that there is a big enough angle between these first two sides so that you can mark it. Now we have drawn the base of unknown length, the fixed angle, and the first fixed side, adjacent to the angles--the A and first S of our ASS triangle.<br />
*Tape the yarn to the board at some point above the dotted line. This will be the vertex between the two sides of fixed length. Let the leftover yarn hang straight down from the taped vertex. Make sure that the hanging yarn can swing back and forth. This will be the swinging side of fixed length. Be sure to leave enough yarn so that you can show all four scenarios.<br />
{{!}}-<br />
{{!}}<br />
*Grab the end of the yarn and move it so that the swinging side rotates from the taped vertex. Wherever the end of the yarn touches the line, a triangle is completed. <br />
*First, show the scenario where the swinging side length is longer than the length of the other fixed side. Show how the fixed angle is not included in one of the completed triangles. <br />
*Now cut the yarn so that it is long enough to show the next case, when there are two unique solutions. <br />
*Cut the yarn again to show the scenario with one solution and a right triangle, and then again to show no solution.<br />
{{!}}{{!}}<br />
[[Image:ACTM_demonstration.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
<br />
<br style="clear: both" /><br />
<br />
===Sample Pictures===<br />
Click here to see sample pictures of the activity in action!<br />
<br />
{{Hide|1=<br />
[[Image:Nosolution.jpg|center|180px]]<br />
[[Image:No_solutions_demo.jpg|center|700px]]<br />
[[Image:One_solution.jpg|center|180px]]<br />
[[Image:One solution right demo.jpg|center|350px]]<br />
[[Image:One_solution_demo.jpg|center|700px]]<br />
[[Image:Two_solutions.jpg|center|180px]]<br />
[[Image:Two solutions demo.jpg|center|700px]]<br />
<br />
}}</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Ambiguous_Case&diff=25795Ambiguous Case2011-07-18T14:36:17Z<p>Rscott3: </p>
<hr />
<div>{{HelperPage|1=Law of Sines|2=Congruent triangles|3=Solving Triangles}}<br />
<br />
__TOC__<br />
<br />
Given two adjacent side lengths and an angle opposite one of them, there is no definite completion of a triangle. According to [[triangle congruence postulates]], two triangles cannot be proved congruent given these three elements. This configuration is commonly referred to as '''side-side-angle (SSA)'''. <br />
<br />
With the SSA configuration, there is a fixed angle connecting the base of the triangle and one of the adjacent sides. The length of the base is unknown, denoted below by a dashed line. The length of the third side length is also fixed, but neither angle adjacent to that side is known. This means that this third side can swing from the upper vertex in any way that connects this <balloon title="the point at which two sides of a triangle meet">vertex</balloon> and any point along the indefinitely sized base so that all three sides of the triangle are connected.<br />
<br />
[[Image:Triangle_template1.jpg|center]]<br />
<br />
Given any SSA configuration, there are several outcomes that can occur when solving a triangle: no solution, one solution, or two solutions. Which of the three scenarios occurs for any SSA configuration depends on the length of the swinging side compared to the height of the triangle.<br />
<br />
In any SSA configuration, we can draw the height of the triangle even though we don't know the length of the base. The height is the perpendicular distance from the upper vertex to the base. Because there is a right angle between the height and base, we can always use the fixed angle and the length of the fixed side to determine the height. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging side to this known height of the triangle.<br />
<br />
[[Image:Height2.jpg|center]]<br />
<br />
<br />
Let <math>a</math> be the length of the side opposite <math>A</math>, an acute angle. The table below describes the different solutions for different scenarios.<br />
<br />
<br />
{{{!}}border="1" cellspacing="3" cellpadding="8" align="center"<br />
{{!}}'''Scenario'''{{!}}{{!}}'''Number of Solutions'''{{!}}{{!}}'''Type of Triangle'''<br />
{{!}}-<br />
{{!}}<math>a < h</math>{{!}}{{!}}no solution{{!}}{{!}}none<br />
{{!}}-<br />
{{!}}<math>a = h</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}right triangle<br />
{{!}}-<br />
{{!}}<math>b > a > h</math><br />
{{!}}{{!}}two solutions<br />
{{!}}{{!}}oblique<br />
{{!}}-<br />
{{!}}<math>a \geq b</math><br />
{{!}}{{!}}one solution<br />
{{!}}{{!}}when <math>a = b</math>, equilateral/isosocles<br />
<br />
when <math>a > b</math>, obtuse<br />
{{!}}}<br />
<br />
<br />
<br />
Because the SSA configuration can prompt different numbers of solutions for for different scenarios, it is often referred to as the '''ambiguous case'''.<br />
<br />
<br />
We can use [[Basic Trigonometric Functions|trigonometry]] to determine the value of height <math>h</math>. Click below to see how.<br />
<br />
{{Hide|1=<br />
<br />
::<math>\sin A =\frac{opposite}{hypotenuse}</math><br />
<br />
<br />
Substituting the appropriate variables,<br />
<br />
::<math>\sin A = \frac{h}{b}</math><br />
<br />
<br />
Multiply both sides by <math>b</math> to get<br />
<br />
::<math>h = b \sin A</math><br />
<br />
}}<br />
<br />
===First Scenario: No Solution===<br />
<br />
In the first scenario, the length of the swinging side is shorter than the height, <math>h</math>. Because this side of the triangle is shorter than the height, there is no solution. The shortest distance between a point and a given line is the line segment that is perpendicular to the given line and goes through that point, which in this case is height <math> h</math>. Since the side length is shorter than the shortest possible distance between the base and the upper vertex of the triangle, the side opposite the fixed angle will never be able to reach the base of the triangle. Thus, there are no solutions when the swinging side length <math>a</math> is less than <math> h</math> or <math> b \sin A</math>.<br />
<br />
In the picture below with numbers we've chosen, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete.<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:ambig case no solution 1.jpg|center]]<br />
{{!}}}<br />
<br />
<br />
Since the other given side length is <math>4</math> and since <math>4<5</math>, there is no solution.<br />
<br />
In summary, when <math>a < b \sin A</math>, there is no solution for a SSA configuration.<br />
<br />
===Second Scenario: One Right Solution===<br />
<br />
In the second scenario, the length of the swinging side is equal to the height, <math>h</math>. Because this side of the triangle is equal to the height, there is only one solution. The height, as explained above, is the single shortest possible distance from the upper vertex to the base of the triangle. Since the swinging side of the triangle is the same length as the height, there is only one way to orient this leg to make the triangle complete: perpendicular to the base and through the upper vertex. Thus, there is only one solution when the length <math>a</math> is equal to <math> h</math> or <math> b \sin A</math>, and this automatically forms a right triangle.<br />
<br />
<br />
In the picture below, no matter how the green side swings, it will only touch the base of the triangle once. This triangle will only be complete when the triangle becomes a right triangle.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_one_solution.jpg|center]]<br />
<br />
{{!}}}<br />
Since the other given side length is <math>5</math> and since <math>5=5</math>, there is only one solution which is a right triangle.<br />
<br />
In summary, when <math>a = b \sin A</math>, there is just one solution for a SSA configuration.<br />
<br />
===Third Scenario: Two Solutions===<br />
<br />
In the third scenario, the length of the swinging side is greater than the height, <math>h</math>. Because this side of the triangle is greater than the height, there are two solutions to complete the triangle. The swinging side will complete the triangle at exactly two points: one in which the swinging side and the fixed side form an acute angle, and one where those same two sides form an obtuse angle. Thus, there are two solutions when the swinging side length <math>a</math> is greater than <math> h</math> or <math> b \sin A</math><br />
<br />
In the picture below, no matter how the blue side swings, it's endpoint will touch the base of the triangle only twice. This SSA configuration will complete two separate triangles.<br />
<br />
<br />
{{{!}}<br />
{{!}}<br />
::<math>h = b \sin A</math><br />
<br />
<br />
Substituting in the appropriate measures, <br />
<br />
<br />
::<math>h = 10 \sin 30^\circ</math><br />
<br />
<br />
::<math>h = 10 (\frac{1}{2})</math><br />
<br />
::<math>h = 5</math><br />
{{!}}{{!}}<br />
[[Image:Ambig_case_two_solutions.jpg|center]]<br />
<br />
{{!}}}<br />
<br />
Since the other given side length is <math>6</math> and since <math>6>5</math>, there is are two unique triangular completions.<br />
<br />
In summary, when <math>a > b \sin A</math>, there are two solutions for a SSA configuration.<br />
<br />
<br />
====Determining Both Solutions====<br />
<br />
The ambiguous case often produce two possible completions of the triangle. In these two potential triangles, the corresponding angles between the swinging sides and the unknown sides are <balloon title="two angles whose measures sum to 180 degrees">supplementary</balloon>. To find both triangles, use the [[Law of Sines|law of sines]] to solve for the first triangle, then find the supplement of the measure of the angle between the swinging side and the base and solve using that angle.<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case1.jpg|center]]<br />
<br />
{{!}}{{!}}<br />
Because of the cyclic nature of [[sine]] as a periodic function, the sine of a given angle is the same as the sine of its supplement. There are two solutions by the law of sines since <br />
::<math> \frac{b}{\sin B} = \frac{b}{\sin (180^\circ -B)}</math>.<br />
{{!}}}<br />
<br />
===Fourth Scenario: One Oblique Solution===<br />
In this final scenario, the length of the swinging side is greater than the length of the fixed side. In this scenario, two possible triangles can be formed by swinging the side, but only one contains an angle of the given measure. To get one of the possible triangles, the lower endpoint of the swinging side swings past (to the left of) the vertex that connects the base and the fixed side so that the fixed angle is not included in the solution. This creates a triangle that includes the supplement of the fixed angle, but not the fixed angle. Because this fixed angle is missing from the completed triangle, this possibility is not a viable solution. Thus, there is only one solution when length <math>a</math> is greater than length <math>b</math>.<br />
<br />
If <math>a=b</math>, at least two of the sides of the triangle will be the same length and the solution will be either an <balloon title="a triangle with two sides of equal length">isosceles</balloon> or <balloon title="a regular triangle with three sides of equal length and three 60&deg; angles">equilateral</balloon> triangle.<br />
<br />
In the picture below, the swinging pink side forms two triangles. The one on the left, however, does not include the fixed angle with a measure of 30 &deg;, and is therefore not a solution for this SSA configuration.<br />
<br />
<br />
[[Image:One_solution_oblique1.jpg|center]]<br />
<br />
In summary, when <math> a \geq b</math>, there is one solutions for a SSA configuration.<br />
<br />
=Teaching Materials=<br />
==Ambiguous Case Demonstration==<br />
<br />
{{{!}}<br />
{{!}}<br />
[[Image:Ambiguous_case_acitvity.jpg|300px|left]]<br />
{{!}}{{!}}During lessons about the ambiguous case, it can often be tricky to visualize the different solutions that a particular SSA configuration produces. Sometimes, it can be easy to forget that the length of the base is not fixed, which makes it easy to forget that the two different solutions can have two different-sized bases.<br />
<br />
This activity will show the solutions for each of the scenarios when the swinging side length is changed. Students will be able to physically see the different possible solutions given an SSA configuration.<br />
<br />
===Materials Needed===<br />
:*Chalk/Chalkboard<br />
:*Yarn<br />
:*Tape<br />
:*Scissors<br />
{{!}}}<br />
<br />
<br style="clear: both" /><br />
<br />
<br />
===Instructions===<br />
{{{!}}<br />
{{!}}<br />
[[Image:ACTM_diagram.jpg|right|300px]]<br />
{{!}}{{!}}<br />
*Draw a base of a triangle on the chalkboard, parallel to the ground. Indicate that it is not of a given length, perhaps by making a dotted or colored line all the way across the board. <br />
*Tape one end of the yarn at the end of the base. Continue to tape up the yarn so that the yarn follows what would be first fixed side of a triangle. Make sure that there is a big enough angle between these first two sides so that you can mark it. Now we have drawn the base of unknown length, the fixed angle, and the first fixed side, adjacent to the angles--the A and first S of our ASS triangle.<br />
*Tape the yarn to the board at some point above the dotted line. This will be the vertex between the two sides of fixed length. Let the leftover yarn hang straight down from the taped vertex. Make sure that the hanging yarn can swing back and forth. This will be the swinging side of fixed length. Be sure to leave enough yarn so that you can show all four scenarios.<br />
{{!}}-<br />
{{!}}<br />
*Grab the end of the yarn and move it so that the swinging side rotates from the taped vertex. Wherever the end of the yarn touches the line, a triangle is completed. <br />
*First, show the scenario where the swinging side length is longer than the length of the other fixed side. Show how the fixed angle is not included in one of the completed triangles. <br />
*Now cut the yarn so that it is long enough to show the next case, when there are two unique solutions. <br />
*Cut the yarn again to show the scenario with one solution and a right triangle, and then again to show no solution.<br />
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===Sample Pictures===<br />
Click here to see sample pictures of the activity in action!<br />
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[[Image:Nosolution.jpg|center|180px]]<br />
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}}</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25788Talk:Ambiguous Case2011-07-18T14:22:08Z<p>Rscott3: /* Page Comments */</p>
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<div>=Teaching Material Comments=<br />
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Comments from Steve Weimar, the Math Forum 6/28<br />
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*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
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*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
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::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
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*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
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=Page Comments=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
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::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
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::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
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:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
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I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
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SSA Postulate: what we can't know<br />
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Why is that? Explanation/Informal proof<br />
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What can be known from SSA?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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::Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example). <br />
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::The table is a very good idea, though it seems large for the page.<br />
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::Remove "to compute height" in the very last sentence.<br />
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::First Scenario: Two "the"s in sentence 3. <br />
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::In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram. <br />
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::Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
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::Determining Both Solutions: Put a link to "Law of Sines" in third sentence. <br />
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::There is no angle labeled B to correspond to your equation. <br />
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::Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
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The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
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===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
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*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
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* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
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* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
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*The end of this section is very clear!<br />
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[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
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====First Scenario: No solution====<br />
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<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
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* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
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<font color=orangered>edited first sentence Richard 5/25</font><br />
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----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
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<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
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2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
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2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
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*I added a few words to show that the swinging side compared to the height is the determinant<br />
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*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
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==Extra Picture==<br />
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[[Image:Ambig_cas_act1.jpg]]</div>Rscott3https://mathimages.swarthmore.edu/index.php?title=Talk:Ambiguous_Case&diff=25787Talk:Ambiguous Case2011-07-18T14:20:33Z<p>Rscott3: /* Page Comments */</p>
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<div>=Teaching Material Comments=<br />
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Comments from Steve Weimar, the Math Forum 6/28<br />
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*It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?<br />
::<font color=orangered> I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? [[User:Rscott3|Richard]] 6/29 </font><br />
:::<font color=black>Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.</font><br />
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*tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.<br />
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::<font color=orangered>Ideally, this page will ultimately include an applet. maybe magnets would work too? [[User:Rscott3|Richard]] 6/29</font><br />
:::<font color=black>Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.</font><br />
::::<font color=darkgoldenrod>[[User:Gene|Gene]] 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.</font><br />
:::::<font color=orangered>Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. [[User:Rscott3|Richard]] 7/5</font><br />
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*[[User:AnnaP|AnnaP]] <font color=darkred> 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have ''students'' do this on poster board to be used later on in the class to remind kids about the discussion. </font><br />
::<font color=firebrick>Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to ''do'' with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.</font><br />
:::<font color=orangered> Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. [[User:Rscott3|Richard]] 6/29 </font><br />
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*Conversation with Ann 6/29:<br />
::*Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html<br />
::*Goal: Make this become a more interactive activity. Make students ask the questions.<br />
::*Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people<br />
::::Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.<br />
::*Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.<br />
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*<font color=red>xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great</font><br />
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*<font color=mediumseagreen>Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?</font><br />
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*<font color=firebrick>Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.</font><br />
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*Notes on Suzanne Alejandre Lessons 7/6<br />
:*Puts up lessons/demos/activities/all different types of resources<br />
::*A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.<br />
:*Her Lessons are really focused on problem solving, having the students do the problem solving<br />
:*Some ideas for ambiguous case<br />
::*an applet will provide the technological interactive alternative like <br />
::*The teaching materials section can really be about the hands on activity that promotes inquiry based learning<br />
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*Conversation with Ann 7/6: http://mathforum.org/mathtools/cell/g,ALL,ALL,ALL<br />
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*Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site<br />
:*Need to change the one solution oblique picture...<br />
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=Page Comments=<br />
*<font color=orangered>So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? [[User:Rscott3|Richard]] 6/30</font><br />
::Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.<br />
::: <font color=purple> ''Abram, 7/5/11'':It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute. </font><br />
::::<font color=orangered>Mentioned that A is an acute angle for all of the scenarios in the top general description. [[User:Rscott3|Richard]] 19:44, 12 July 2011 (UTC) 7/12</font><br />
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*This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?<br />
::<font color=orangered> This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? [[User:Rscott3|Richard]] 6/29 </font><br />
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*In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".<br />
::[[User:AnnaP|AnnaP]]<font color=darkred>6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.<br />
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::<font color=black>Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.</font><br />
:::<font color=orangered>This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.[[User:Rscott3|Richard]]6/29<br />
::::I (at least partially) explained this case in a fourth scenario section. [[User:Rscott3|Richard]] 7/5<br />
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*The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.<br />
::<font color=orangered>This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? [[User:Rscott3|Richard]]6/29<br />
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::<font color=purple> ''Abram, 7/5/11'': In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base". </font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."<br />
::*<font color=orangered>Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario. <br />
::*In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something [[User:Rscott3|Richard]]6/29<br />
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:::<font color=purple> ''Abram, 7/5/11'': It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal. </font><br />
::::<font color=orangered>Addressed this comment. Let me know if you think I should do more. [[User:Rscott3|Richard]] 7/13</font><br />
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*the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.<br />
::<font color=orangered> Asked for clarification 7/5</font><br />
:::<font color=mediumseagreen>E-mail response from Steve W 7/5<br />
{{Hide|1=<br />
I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:<br />
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SSA Postulate: what we can't know<br />
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Why is that? Explanation/Informal proof<br />
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What can be known from SSA?<br />
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How can one figure out which case it is?<br />
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I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.<br />
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The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?<br />
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:::<font color=orangered> Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. [[User:Rscott3|Richard]] 7/14</font><br />
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*in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.<br />
::::<font color=orangered>Changed to variables. [[User:Rscott3|Richard]] 7/13</font><br />
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*in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.<br />
::::<font color=orangered>Changed the labels. [[User:Rscott3|Richard]] 7/13</font><br />
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*"just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".<br />
::<font color=orangered>This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.[[User:Rscott3|Richard]]6/29<br />
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::::<font color=orangered>Changed the text. [[User:Rscott3|Richard]] 7/13</font><br />
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*Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity. <br />
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:Strong points of the page:<br />
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::1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.<br />
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::2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases. <br />
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::3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible. <br />
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*One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11<br />
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::<font color=orangered> I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the [[Law of Sines]] page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. [[User:Rscott3|Richard]] 6/29<br />
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:::<font color=black>Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.</font><br />
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*Dayo made an edit to the page to fix a grammatical error on 6/29<br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.</s> </font><br />
::<font color=orangered> I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.</font><br />
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*<font color=dodgerblue>'''[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.'''</font><br />
::<font color=purple> ''Abram, 7/5/11'': One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?</font><br />
:::<font color=orangered>Kate showed me how to do that today. Fixed. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''This means that this third side can be positioned in whatever way connects the upper point of the other side''<br />
::I think I'd say "in any way that connects" just to be clear that it's not just one way.</s></font><br />
:::<font color=orangered> Coolio! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=dodgerblue>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): ''# If a > h, there are two solutions. ''<br />
::You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.</font><br />
::<font color=orangered> I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. [[User:Rscott3|Richard]] 7/5</font><br />
:::<font color=dodgerblue>I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually ''talk'' about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. ([[User:Kderosier|Kate]] 17:48, 6 July 2011 (UTC))</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*<font color=dodgerblue><s>[[User:Kderosier|Kate]] 15:30, 1 July 2011 (UTC): [[User:Kderosier|Kate]] 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- ''To find both triangle, just solve for the'' (Should be both triange'''s''')</s></font><br />
::<font color=orangered> Thanks!!! Comment Addressed [[User:Rscott3|Richard]] 7/5</font><br />
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*<font color=purple> ''Abram, 7/5/11'': You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).<br />
::::<font color=orangered>I keep ending up with these kinds of comments...oh boy. [[User:Rscott3|Richard]] 7/6</font><br />
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*Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text. <br />
::<font color=orangered>I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. [[User:Rscott3|Richard]] 7/6</font><br />
::::<font color=orangered>Addressed this comment. [[User:Rscott3|Richard]] 7/13</font><br />
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*My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".<br />
::<font color=orangered>My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) [[User:Rscott3|Richard]] 7/6</font> <br />
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*Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.<br />
::<font color=orangered>The page is set up so that the picture will always be below. [[User:Rscott3|Richard]] 7/6</font><br />
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*<font color=hotpink>[[User:Rebecca]] 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page. </font><br />
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*<font color=mediumspringgreen>Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11<br />
::1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =.... <br />
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::2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.</font><br />
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*<font color=black>Chris 7.16.11 The page is very strong. Here are some final edits:<br />
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Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example). <br />
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The table is a very good idea, though it seems large for the page.<br />
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Remove "to compute height" in the very last sentence.<br />
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First Scenario: Two "the"s in sentence 3. <br />
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In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram. <br />
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Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?<br />
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Determining Both Solutions: Put a link to "Law of Sines" in third sentence. <br />
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There is no angle labeled B to correspond to your equation. <br />
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Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.<br />
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*<font color=darkgoldenrod>[[User:Gene]] 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.</font><br />
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=Applet Comments=<br />
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* <font color="darkgreen">For the applet you suggested for this page, [[User:Alimurreza|Reza]] said he would begin working on a Java applet as per the specifications listed on the S11 page - [[User:Rguo|Rguo]] - 6/30 </font><br />
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*<font color=cadetblue>[[User:Alimurreza|Alimurreza]] from Drexel is working on an applet for the page!!!!!! 7/5 ([http://mathforum.org/mathimages/index.php/DU11 Reza's work])</font><br />
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=Older Comments=<br />
Originally, the page was a subsection of the [[Law of Sines]] page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the [[Congruent triangles]] page that the ambiguous case could be a helper page for the two pages. <br />
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The following comments were (and still are) on the [[Talk:Law of Sines|Law of Sines Discussion Page]]:<br />
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===The ambiguous case===<br />
<font color=orangered> I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24</font><br />
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*I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear. <br />
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* You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case. <br />
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* I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."<br />
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*The end of this section is very clear!<br />
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[[User:Rebecca|Rebecca]] 01:43, 25 May 2011 (UTC) <br />
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====First Scenario: No solution====<br />
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<s>* First sentence is very confusing. You have too many fragmented thoughts. <br />
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* Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!<br />
[[User:Rebecca|Rebecca]] 01:44, 25 May 2011 (UTC)</s><br />
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<font color=orangered>edited first sentence Richard 5/25</font><br />
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----xd 02:02, 25 May 2011 (UTC)<br />
1. You need better transition between the previous section to this section. This should not be an independent section by itself.<br />
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<font color=orangered>I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25</font><br />
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2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t<br />
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2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.<br />
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*I added a few words to show that the swinging side compared to the height is the determinant<br />
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*but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?<br />
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Richard 5/25</font><br />
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==Extra Picture==<br />
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[[Image:Ambig_cas_act1.jpg]]</div>Rscott3