"There is no one way"

Monday, October 20, 2014

Pythagorean Proofs

I just added a second dynamic geometry proof of the Pythagorean theorem on my Web site.

Both are proofs "without words", which in reality means that you should use them to generate discussion. Indeed, many words are needed for students to fully grasp what they see, but the words should not come exclusively from the teacher. You can show the animation, and then replay each step and have students tell their neighbors what they see. Follow that with a class discussion of each step, and to wrap it up, have students write a summary of the proof. Of course, this works even better if the students can manipulate the applets themselves.
The applets are reasonably convincing,  the animations are entertaining, and the use of rotations and translations provide a nice connection to transformational geometry. The question is: what does it take to write a rigorous proof based on these applets? Students will tend to believe what they see without need for any further argumentation, but they would be wrong: what we see can be deceiving. To make that point, you may use page 8 of Introduction to Proof and/or this video.

The first ingredient in understanding these proofs is to break down how the figure is made from the original right triangle. In both cases, this involves a 90° rotation of the original triangle, such that image and pre-image share a vertex. SAS or LL similarity can be invoked to establish the triangles are congruent. (Or just the fact that the image was obtained by a rotation, but a rigorous proof of that would require knowing where the center would have to be to yield the image.)

The heart of the proof in both cases lies in comparing certain areas before and after the final translations. In the first figure, we start with a^2 and b^2, and end with c^2. In the second figure we go in the opposite direction. A rigorous proof would require arguments based on congruent triangles, or clarity on what the translation vectors are.

The Pythagorean theorem, of course, is a big deal in secondary math education, as it should be. It is beautiful, has many applications, and is completely within the conceptual reach of our students. Because it is important, I strongly advise that you teach it in more than one way. For one thing, the above two proofs can be previewed with scissors and graph paper, which would make the applets that much easier to understand.

In addition, check out Labs 8.4-8.5, and chapter 9 in my Geometry Labs (free download) for a hands-on approach on the geoboard or dot paper. In my view (and in my experience) hands-on work should precede any attempts at formal proof. (For a philosophical defense of that point of view, read this debate, and this article.)


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