# Investigation: Pythagorean spiral

Lesson

## Objectives

• To practice finding the missing side length of the hypotenuse using the Pythagorean Theorem
• To understand some ways the Pythagorean Theorem is connected with nature

## Materials

• Protractor
• Metric Ruler
• Pencil
• 1 sheet of Computer Paper
• Crayons, Markers, or Colored Pencils

## Procedure

1. Using your protractor draw a right angle slightly down and left from the center of the paper. Make sure both sides of this angle are 1 inch long (track the side lengths on a separate piece of paper).
2. Connect the two legs to create a right triangle.
3. Use the Pythagorean Theorem to determine the length of the hypotenuse of the triangle you just created.
4. Use your protractor to create a 1 inch line perpendicular to the first triangle’s hypotenuse.

5. Connect the lines to create the hypotenuse of a second right triangle. Notice the hypotenuse of the old triangle now acts as one of the legs of the new triangle.
6. Calculate the hypotenuse of the new triangle using the Pythagorean Theorem.
7. Now use your protractor to create a 1 inch line perpendicular to the second triangle’s hypotenuse.
8. Draw in the triangle’s hypotenuse as you did for the second triangle.
9. Use the Pythagorean Theorem to determine the length of the new triangle’s hypotenuse.
10. Continue the pattern. Stop drawing triangles when you are about to overlap the start triangle (do not overlap triangles). In the end you should have 16 triangles. There will be a small gap between the first and last triangles. As you go, continue to find the length of each hypotenuse you create using the Pythagorean Theorem and record them.

11. Optional: Decorate the spiral using your choice of crayons, markers, or colored pencils.

## Discussion Questions

1. Which triangle had the longest hypotenuse? What was its length? Why did it have the longest hypotenuse?
2. Where in nature have you seen this pattern before? Think of some examples.