Proof Behind the Pythagorean Theorem

Objectives and Goals:

Target Objective:

In a discussion group, the student will be able to justify the Pythagorean Theorem by explaining why the squares of the legs are equal to the square of the hypotenuse.

Activity Goals:

Given a blank right triangle, students will be able to classify each side by labeling "leg", "leg" and "hypotenuse" in the appropriate places.

Given a square, students will be able to prove a^2 +b^2 = c^2 by cutting, and rearranging the pieces of a square.

After watching a video, students will be able to imagine the wheel to explain why a^2 +b^2 = c^2.

NYS Standards:

MST 3.07.GE8.05: Students identify the right angle, hypotenuse, and legs of right triangle.

MST 3.07.GE8.06: Students explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem.

Activity Two: Part One: Learning to Label

Learn how to label a right triangle.

Things you need to know before starting: What is a right triangle?

Let's begin: Think about your body, have you ever noticed that your body is perpendicular(forms a 90 degree angle) to the ground? What part of your body keeps you standing up right (or up straight)? Hopefully, you agree that your legs hold up your body. Since your legs touch the ground let's say that your legs make a 90 degree angle with the ground. The same is true for a triangle. If you place the legs of a triangle so that they touch the ground, then the legs of a triangle support the body of the triangle. The third side of a triangle is called the hypotenuse of a triangle. The hypotenuse of a triangle is the longest side of the right triangle. It is easy to spot since, the hypotenuse can be found across from the right angle (or 90 degree angle).

Read more about the hypotenuse: http://www.answers.com/topic/hypotenuse#ixzz1eqWd8Krt

In your notes without looking, draw a picture of a right triangle. Label each of the three sides, using the appropriate name. Also label the 90 degree angle. After you are done, check back with the picture.

Take a look at a labeled triangle:

Activity Two: Part Two: Rearranging Triangles to Make Squares

Cut and Rearrange the Square. Before you start, walk through this activity page proving Pythagorean Theorem. We will be working on the proof.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=30

Let's begin. Taking your ruler, and construction page you need to cut a triangle that has legs with the measurements 3inches (base) and 4inches (height). To check that you are precise the hypotenuse should be 5inches. Then you will need to cut 3 squares. The squares should be 3x3, 4x4, and 5x5. Once all of your pieces are cut, watch the following video: http://www.youtube.com/watch?v=DQk_VrWdCr0&feature=player_embedded

Now, it is your turn to try to cut the 3x3 (smallest) square so that all of the pieces from the 3x3 will fit with the 4x4 square to completely cover the 5x5 (biggest) square.

Should there be any pieces left over? No, the areas are exactly the same!

Activity Two: Part Three: Visual Proof of Pythagorean Theorem

Watch the following video clip demonstrating a proof with water. Answer the questions: "Is there any extra water, or any missing water? Where does all of the water go?"

http://www.youtube.com/watch?v=CAkMUdeB06o

go back to Pythagorean Theorem

go back to Activity One Who is Pythagoras?

go on to Activity Three Finding the Unknown Side of a Right Triangle using Pythagorean Theorem