Pythagorean Theorem

Materials: graph paper (1/4" or larger grid); colored pencils; tape; scissors; chart paper; overhead projector; transparent grid; a second transparent grid (colored, if possible) cut into strips one unit wide and these lengths: 3, 3, 4, 5, 6, 6, 7, 8, 10, 12, 13; three squares cut from a transparent grid: 3 x 3, 4 x 4, and 5 x 5.

Preparation: Be sure there are enough materials for each pair of students. If you don't ask students to work in pairs, you'll also need scissors, colored paper, and tape for each student. Make a chart to hang at the front of the class. There should be at least 10 rows on the chart.

Prerequisite Skills and Concepts: Students must know how to square a number and what defines a right triangle, as well as the terms leg and hypotenuse.

• Say: Today, we are going to explore right triangles.

• Ask: Can you make a triangle with these sides: 6 units, 6 units, and 12 units?
Students may not know the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle cannot be less than or equal to the length of the third side. If this is the case, or if a significant number of students in the class are skeptical of it, use the overhead projector to demonstrate that the two shorter sides, in order to meet at all, must lie right on top of the third side. Try the same experiment with these lengths: 4, 5, and 12; 6, 7, and 12; etc. until you are satisfied that students believe that there are some combinations of side-lengths that will not form a triangle at all.

• Ask: Can you make a right triangle with sides of 3, 3, and 5 units?
Students will now know that you can make a triangle with sides of these lengths. If discussion doesn't bring out the fact that any group of 3 side-lengths will produce only one triangle, experiment on the overhead with some of your colored grid strips overlaid on the plain grid (to be sure you're creating right triangles). Continue until you are satisfied that students believe that only some special combinations of side-lengths will produce a right triangle. (With the grid strips specified above, you can make right triangles with these combinations: 3, 4, 5; 6, 8, 10; 5, 12, 13.)

• Say: Now, get together with your partner. I want you to cut out squares of grid paper with special requirements. You need to come up with groups of 3 squares that are related so that you can lay them out in a right triangle.
Demonstrate on the overhead with your 3 x 3, 4 x 4, and 5 x 5 squares.

Then, pass out materials and give pairs 15 minutes or so to model as many right triangles as possible.

• Say: Let's make a chart of the right triangles we have discovered. On this chart, I've used letters to refer to the lengths of the sides of our triangles. Why do you think I have some columns labeled a2, b2, and c2?
Relate the area of a square to the exponential notation. Fill in the chart you have prepared with all of the different triangles that students have found. Tape their triangles in the first column.

• Ask: What do you notice about the way the squares on the sides are related to the squares on the hypotenuse for each triangle?
This is a very important discussion. If students have not come up with enough examples to see the pattern, a2 + b2 = c2, then generate some more Pythagorean Triples and extend your chart with them.

• Say: This relationship, a2 + b2 = c2, holds for all right triangles. It's called the Pythagorean Theorem.