Isosceles Right Triangle Spiral

Like the previous right triangle spiral (1.2), this spiral is based on a sequence of right triangles. This spiral, however, involves only isosceles right triangles. The numbers involved are very interesting, as students will discover when they do the worksheet that accompanies this activity. It is, again, a sequence of square roots that is involved.

This project can also be turned into a "reviewee" (small review sheet which folds up). The student should write definitions, postulates, and/or theorems in the triangles of the spiral. (Writing the theorems will provide excellent review in itself.) Then the student should score and fold on the line of each hypotenuse so that it folds up, somewhat like a fan, and will fit neatly in a shirt pocket. The student can review the information when waiting in the lunch line, passing the time while waiting for a ride home, or at other convenient moments.

Here are the steps in the construction:

Step 1: Construct an isosceles right triangle (ABC).

Step 2: Construct a second isosceles right triangle (ABD) on the hypotenuse of the original triangle, using the hypotenuse (AB) of the original triangle as the length of both legs of the new triangle.

Step 3: Constuct a new isosceles right triangle (BDE) on the hypotenuse of the original triangle, using the hypotenuse (BD) of the second triangle as the length of both legs of the new triangle.

Step 4: Constuct a new isosceles right triangle (BEF) on the hypotenuse of the original triangle, using the hypotenuse (BE) of the third triangle as the length of both legs of the new triangle.

Step 5: Constuct a new isosceles right triangle (BFG) on the hypotenuse of the original triangle, using the hypotenuse (BF) of the second triangle as the length of both legs of the new triangle.

Step 6 and 7: Continue this process to create your design:

Cut the figure out using scissors or an exacto knife, then score and fold in opposite directions on each consecutive hypotenuse to form a fold-up "toy".

You can write theorems on the triangles as shown above, and use your folded- up isosceles right triangle "toy" as a review sheet. And/or you could also color it any way you like. The example below shows it in solid colors, but students might like to try other methods of their own, or draw colored pictures within the triangles.

"The essence of mathematics is not to make simple things complicated, but to make complicated things simple." S. Gudder