Talented math students will enjoy studying the theorem's various proofs. Euclid had one of the earliest proofs of the theorem, and the drawing he used has become known as the "Bride's Chair." A simple and elegant proof of this theorem was developed by an American president, James A. Garfield. Unlike most proofs, which are based on squares, this one was based on the trapezoid.
Students who understand similar triangles can use them to prove the theorem. Tell the students to draw a right triangle, ABC. Then tell them to draw the altitude, AD. This will produce three similar, right triangles. By using the fact that the ratios of corresponding sides of similar triangles are equal, they should, through algebraic manipulation, be able to show that if the sides have lengths a, b and c, then a^2 +b^2 = c^2.
If you have a classroom filled with boys who hate to stay in their seats and love to be class clowns, put them to work developing Pythagorean theorem skits. They can have Romeo use the Pythagorean theorem to determine how long his ladder has to be to reach Juliet's window. Ask them to develop games that require knowing how to calculate the lengths of legs and hypotenuses to win. Divide the student into groups and hold a contest to see who can come up with the best rap song explaining how to use this theorem. Have a treasure hunt with clues like, "What is the hypotenuse of a right triangle having legs 5 and 12? That is the horizontal distance you travel to reach the next spot in the hunt."
The Wheel of Theodorus will help your students learn about both the Pythagorean theorem and irrational numbers. Instruct your students to draw an isosceles right triangle with legs of 1. Ask them to find the hypotenuse, which will have lengths equaling √2, an irrational number. Now ask them to draw a right triangle adjacent to the first one but using √2 as the leg. Let them compute the length of this second hypotenuse, which will be 2. This project can continue until a spiral is formed. After completing the wheel, students can add extras to create pictures. In one classroom, students drew snails, turkeys, French horns and an elaborate hairstyle for women.
If all three sides of a right triangle have lengths that are integers, those integers are referred to as a Pythagorean triple. Common Pythagorean triples are 3, 4 and 5 or 5, 12 and 13. Ask the students to find as many Pythagorean triples as they can. After they have struggled, teach them the formula for finding them: a = n^2 - m^2, b = 2nm, c = n^2 + m^2, where m and n are integers and n is greater than m. If students want to check to see that this will always give correct values, they can prove it themselves by substituting these values into the Pythagorean theorem, a2 + b^2 = c^2:
Replace "a" with n^2 - m^2. Replace b with 2nm and c with n^2 + m^2:
(n^2 - m^2)^2 + (2mn)^2 = (n^2 + m^2)^2
Working with the left part of the equation gives n^4 - 2(mn)^2 + m^4 + 4(mn)^2. Combining like terms gives n^4 + 2(mn)^2 + n^4. Factoring completely gives (n^2 + n^2)^2. Notice that this is identical to the expression on the right side of the equal sign. Because the left side of the equation now equals the right side, the relationship has been proved.