In this class project, students will need many colored, rather small squares of paper. The squares should be the same color on both sides. You may for example use a “note cube” which contains 550 3 ½ inch by 3 ½ inch squares in eleven bright colors.

A large sheet of packing paper taped on a wall may serve as a poster on which students tape their polygons.

THE PROJECT

Students cut their squares into four right triangles as shown in the figure above. If they use 3 ½ inch square paper, the legs will have lengths 1 ¾ and 3 ½ inches.

There are 3 tasks below.

Make as many different polygons as you can by joining two (task 1), three (task 2), or four (task 3) triangles (of the same color). Tape your figure together with scotch tape, and glue or tape it to the poster. Be sure to check that a polygon of the same shape and size is not already on the poster!

GROUND RULES

The triangles are joined by edges. The edges have to match exactly. One short leg of a triangle may be joined only with a short leg of another triangle. But you may turn triangles on the other side.

Here are two polygons, each made from two triangles:

Two polygons, a triangle and a parallelogram, each made from two triangles

REMARKS

How much geometric terminology is introduced is not very important, but the concepts of polygon and congruence are important.

1. Polygon
   Do not give a definition, but say that:
   • all sides of a polygon are straight segments.
   • some of the corners may be turned inwards (technically this means that a polygon does not have to be convex).
   • to determine the number of sides, it is better to count corners. Notice that the first polygon shown above is a triangle, even if one of its sides was made from two legs of two smaller triangles


2. Congruence
   We say two geometric figures are congruent when they have the same shape and size. In geometry we often shorten this to just one word: same.

   In order to check to see whether two plane figures are congruent, try to put one on top of the other. If you can make them fit exactly, they are congruent. In doing so, you may turn a figure on the other side.

   When the number of polygons on the poster increases, checking whether a new one is not congruent to (the same as) one already on the poster becomes difficult. Special attention should be given to this part of the project. And it is better if checking is done not by individual children, but by groups.

   Whether the design of the polygons is done individually or in groups does not matter, as long as all children participate in all phases of the project.

   If some care is taken about spacing the polygons on the poster, and about the pattern of colors that emerges, the display can be impressive.

TASK 1

How many can you make from two triangles? The rule is that two triangles only fit together if one side of one exactly fits one side of the other.

We found these six. Are there any more?

TASK 2

How many can you make from three triangles?

We found these ten. Are there any more?

TASK 3

How many can you make with four triangles?