# Pythagorean Theorem

### by: Heather Bridges

The following information is an outlined attempt as to how to present the Pythagorean Theorem to high school students. There are several proofs that have been used to explain why this theorem is true but, the best way to illustrate it to the students may be through their own hands on exploration.

Step 1: Create several different sizes of right triangles all of whose sides can be squared using one piece of construction paper. Then distribute these triangles to the students after they have been assigned to groups. The students have two assignments at this point:

1. The students need to create 3 squares using the construction paper provided in three different colors that have the side lengths of the 3 sides of their triangles. (Note: Have the students use the side of the triangle to measure for their squares. It is not necessary to measure exact length using a ruler. The students hopefully at this point are familiar with the areas of squares and so forth.)

The teacher should check to make sure each group has completed assignment #1 before proceeding on to the next assignment.

2. The students should then use their largest triangle as the game board and the two smaller triangles as their playing pieces in a matter of words. What the students need to do is see if they can cut and rearrange the smaller triangles on top of the larger to equal the same amount of area. (Note: remind the students to be very careful when cutting to not lose area.)

Hopefully, every group in the class will discover that the area of the big triangle is equal to the area of the sum of the two smaller triangles regardless of the size of the right triangle.

Alternative: It may even be helpful for a couple of the groups in the room to experiment with an acute or obtuse triangle to illustrate that this method works for only right triangles.

Step 2: Bring the students back together for a classroom discussion. See if anyone can interpret what the class together has just discovered given the following.

The students should see from the background dealing with the area of squares that the squares they created using the given lengths a, b, and c have the following areas.

And because of the class discoveries the students found that the area of the two smaller squares equals the area of the largest. Therefore they have single handedly discovered

The Pythagorean Theorem

Step 3: Reinforce the students understanding by having them to go to the computers and basically repeat what they just did using The Geometer's Sketchpad. Starting with constructing a right triangle properly. Then constructing the squares on each side of the right triangle. The students should then measure the areas of the squares to check to see if their previous assumptions still hold true with exact measurements. The students could then drag the vertices of their triangle to see if it works for any size right triangle.

Extension: Have the students to use the side lengths to construct other figures using sketches to see if the areas of the three figures still behave in the same manner. This exercise might provide some more room for creativity and student interest.