GXWeb Polygon Playground 1

 

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Games 1 and 2: Dealing with Diagonals

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Challenge 1

Two teams (A and B) of $n$ players each are seated around a $2\cdot n$-gon polygonal table. Team A are allocated the odd-numbered seats, and team B, the even numbers. How do we set up the draw so that each member of team A competes against each member of team B exactly once? How many times must they meet?

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Challenge 2 Consider a clock face with $2\cdot n$ "hours" marked around the dial. Let $a(n) =$ number of ways to match the even hours to the odd hours, modulo rotations and reflections. OR... Given a $2\cdot n$-gon... Draw $n$ diagonals so that they don't share end points, and there are an odd number of polygon sides between the end-points.

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How many such variants do you think can be found for a hexagon ($n=3$)? For an octagon ($n=4$), decagon ($n=5$) - or the standard clock dodecagon ($n=6$)? And why stop there - try $n=7, 8, 9, 10$.

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Below you will find $n$ draggable segments. Use these to find as many solutions (variants) for these challenges as you can. (Vary the number of sides using the $n$ controls and browse some sample solutions using $Variants$)

(NOTE: To model Challenge 2, use negative values of $v$).

To begin, you might try to:

1. drag the segments to build three different variants for the hexagon.

   Press Check to check your models.

2. drag the segments to build a few different variants for the octagon.

   Press Check to check your models.

3. drag the segments to build some different variants for the decagon.

   Press Check to check your models.

4. Try a few more, browse the samples, and then try the Quiz!

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( - ) (n) ( + )
\[n\]							▶
		0		6	10
\[Variants\]						▶
			-80			0	10
( - ) \(Variants\) ( + )
r				▶
		0		10	10

Plot my variant ⇑Check ⇑Test yourself

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App generated by Geometry Expressions

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How many ways did you say?

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(Sneak Peek at the Decagon?)

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Show Challenge 1

Challenge 1

How do we set up the draw so that each member of team A (odd numbers) competes against each member of team B (even numbers) exactly once?

For n = 1, there is only one way. For n = 2, there are 2 ways. For n = 3, there are 3 ways. For n = 4, there are 4 ways. For n = 5, there are 5 ways. For n = 6, there are 6 ways. For n = 7, there are 7 ways. For n = 8, there are 8 ways. For n = 9, there are 9 ways. For n = 10, there are 10 ways.

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Show Challenge 2

Challenge 2

List the number of ways to match the even hours to the odd hours on a \(2\cdot n\)-gon "clock face", modulo rotations and reflections.

For n = 1, there is only one way. For n = 2, there is 1 way. For n = 3, there are 3 ways. For n = 4, there are 5 ways. For n = 5, there are 17 ways. For n = 6, there are 53 ways. For n = 7, there are 260 ways. For n = 8, there are 1466 ways. For n = 9, there are 10915 ways. For n = 10, there appear to be 93196 ways.

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Game 3: GXWeb Polygon Angles Theorem Generator

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Of course, polygons by their nature have TWO features which make them interesting: sides and angles. If we are going to play with polygons, it would be a shame to only play with their sides and neglect the many questions posed by their angle relationships!

For any given polygon, how many different angle theorems can be found?

Thanks to Phil Todd, creator of Geometry Expressions and GXWeb, this amazing GXWeb model automatically generates angle theorems for hexagons, octagons and more!

Explore... Try the challenges of the full GXWeb Polygon Theorem Generator!

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Game 4: GXWeb Cyclic Polygon Constructions

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Finally, our Polygonal Playground would hardly be complete without some thought given to actually constructing those polygons!

Two methods are shown here, both made relatively simple using GXWeb's constraint-based geometry system and computer algebra foundation. Study and play with both - then try them yourself using GXWeb

Method 1: Quick Cyclic Polygon Construction

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Method 2: General Cyclic Polygon Construction

Begin with a square ABCD, and mark the center point (4). Note that the side length (AB) of the square will also be the side length of each subsequent polygon. Draw an arc of radius AB from a base vertex of the square, and mark the point (6) such that the segment (46) is perpendicular to the base. Bisect the segment 46 (point 5). Use this distance 45 to continue placing the center points of subsequent polygons above.

All the ingredients are now in place: construct a circle at each of the center points, and then step around each circle in steps of length AB!

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Construct your own Model with GXWeb

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Share Your Results

 

Enter your name:	? Abel, Nils Agnesi, Maria Bernoulli, Daniel Cantor, Georg Descartes, Rene Euler, Leonhard Fermat, Pierre Galois, Evariste Germain, Sophie Hypatia Johnson, Katherine Liebniz, Gottfried Lovelace, Ada Monge, Gaspard Newton, Isaac Noether, Emmy Pythagoras Ramanujan, Srinivasa Riemann, Bernhard Sylvester, James Tao, Terence Weierstrass, Karl' Zeno
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Share with QR Code

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QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available. The default link here is the GXWeb Jigsaws and Quizzes, but you can use it as an alternative to sending your assessment data via email, web or share with others in your class. You can even use it to send your own messages!

GXWeb Jigsaws and Quizzes Send Your Own QR Code Reset QR Code

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Behind the Scenes

 

View JavaScript GXWeb Code (gxpoly.js)

 

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Learn How: Create your own GXWeb Assessment Task

Learn How 2: Take GXWeb to the Next Level

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©2023 Compass Learning Technologies ← Saltire Software → GXWeb Polygon Playground 1

 

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