Pythagorean Theorem Spiral
Pythagorean Theorem Proof:
- Using the diagram to the left, the Pythagorean Theorem (a² + b² = c²) can be proved. It consists of 4 of the same right triangle. When put together, each "c" side becomes the side of a square. The square has a hole in it, and the value of a side of the square is a - b. The area of the small square is (a - b)², and the sum of the area of the four triangles is 4·ab/2, or 2ab. Using the area of the triangles and square, we get: c²= (a - b)² + 2ab = a² - 2ab + b² + 2ab = a² + b². Therefore, the Pythagorean theorem, a² + b² = c² is true.
Reflection:
- Creating the spiral did not take too much time, and finding the lengths of the hypotenuses was probably the most time consuming part of this project. While I used the pythagorean theorem and filled in my table, I began to notice a pattern. I knew leg "a" would always be 3cm long, and leg "b" would be equal to the hypotenuse of the previous triangle, but I didn't know that any specific pattern would appear. Legs "a" and "b" of triangle 1 were 3cm long, and the hypotenuse was 3√2. On triangle 2, leg "a" was 3cm, leg "b" was 3√2, and the hypotenuse was 3√3. Each time I calculated the hypotenuse, it continued in that pattern: 3√4 (or 6) , 3√5, 3√6, 3√7, etc. I enjoyed this project and I was surprised that such a simple set of triangles could create such a pretty spiral. The most frustrating part of this activity was that I made the mistake of decorating it with sharpie and made mistakes and had to start over a few times. The Habit of a Mathematician that was most reflected in this activity was looking for patterns. As I used the pythagorean theorem to calculated side lengths and hypotenuses, I had to notice patterns and look for ways that the numbers were connected. Overall, I really liked this activity and being able to learn that there are so many ways to prove the pythagorean theorem.
- Creating the spiral did not take too much time, and finding the lengths of the hypotenuses was probably the most time consuming part of this project. While I used the pythagorean theorem and filled in my table, I began to notice a pattern. I knew leg "a" would always be 3cm long, and leg "b" would be equal to the hypotenuse of the previous triangle, but I didn't know that any specific pattern would appear. Legs "a" and "b" of triangle 1 were 3cm long, and the hypotenuse was 3√2. On triangle 2, leg "a" was 3cm, leg "b" was 3√2, and the hypotenuse was 3√3. Each time I calculated the hypotenuse, it continued in that pattern: 3√4 (or 6) , 3√5, 3√6, 3√7, etc. I enjoyed this project and I was surprised that such a simple set of triangles could create such a pretty spiral. The most frustrating part of this activity was that I made the mistake of decorating it with sharpie and made mistakes and had to start over a few times. The Habit of a Mathematician that was most reflected in this activity was looking for patterns. As I used the pythagorean theorem to calculated side lengths and hypotenuses, I had to notice patterns and look for ways that the numbers were connected. Overall, I really liked this activity and being able to learn that there are so many ways to prove the pythagorean theorem.