Discovery fiction for the Pythagorean theorem

1jbash

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I don't think all that algebra (or symbolic arithmetic, or whatever you want to call it) is as intuitive to a high school student as it is to you. Frankly I find the "behold" proof really uncompelling, *because* it just leads me into, well, algebra. You're trying to prove what's fundamentally a nice geometric theorem about *areas*, and dragging in the question of "what we can calculate" seems like an unnatural complication. When you want to *apply* the theorem to get distances in the cartesian plane, then you can start calculating.

I also very much doubt that that's how the theorem was historically discovered; we're mostly talking about people whose notation for writing *numbers* was often really cumbersome, who totally lacked any notation at all for doing algebra, and who didn't necessarily identify areas with numbers as readily as we do.

The rearrangement proof lets you engage with the figures pretty directly, whereas all the others require you use a lot of extra concepts. Not only can you get to the Pythagorean theorem without doing algebra, and without engaging with cartesian coordinates, but you can get there without engaging with the concept of similar figures. That is a *good* thing; at the high school level you can't expect people to be able to manipulate all of those concepts with facility at the same time.

I think you will definitely lose them if you bring in the idea of generalizing it to other shapes. At their level, the concept of "proof" is shaky at best, and the instinct for abstraction hasn't taken hold. The idea of generalized or specialized versions of a theorem is going to be hard to explain all by itself.

I've been thinking recently about how to teach the Pythagorean theorem to high school students. As part of that thinking, I looked around to see how the topic was being taught in various textbooks, online videos, blog posts, etc. Typically, the discussion goes something like this:

First, the statement of the theorem is presented: For a right triangle with legs of lengths a and b and a hypotenuse of length c, we have a2+b2=c2.

Next, a picture like the following one is presented as a visual:

The student is told that the two smaller squares add up in area to the largest square.

Finally, any one of the typical proofs is presented. This could be a rearrangement proof, the "Behold!" proof, or Euclid's proof of proposition I.47.

One can improve the final step by using what is sometimes called Einstein's proof. (See also this post by Terence Tao, this video by Numberphile, this article by Alexander Givental, and in particular this comment by Tim Gowers for discussion and presentation of this proof.) This proof is an improvement over the typical presentation for a few reasons: it makes the theorem feel more intuitive, and (especially with the discussion in Gowers's comment) it gives some indication of how one might discover the proof.

It might seem like the "exposition problem" for the Pythagorean theorem is solved: we started with a bunch of proofs that made the theorem feel unintuitive that we didn't know how to discover ourselves, and now we have a good proof along with a story for how to discover it.

I claim that there is still some work left! I think the Pythagorean theorem is a case where even the theorem statement itself seems bizarre (rather than just the proofs being bizarre). Given an arbitrary right triangle, how would one guess that a2+b2=c2? And why would one even think this is a problem worth solving in the first place? I think this second question is easy to answer by pointing to the numerous applications of the theorem, so I will focus on the first question.

So with that background discussion out of the way, here is my proposal for how to teach the Pythagorean theorem, which is a kind of discovery fiction:

specializing/simplifyingthe problem by taking the two legs to be the same length. In this case, it becomes really obvious (by drawing the picture) that the "bigger square" made using the diagonal of the original square is double the area of the original square. This opens up two interesting threads: (1) it shows that the length of the diagonal of the original square is √2, and this then could lead to a lesson on its own about irrational numbers; (2) it gives an idea of how to generalize this approach to the case where the legs don't have the same length.needa square; we just needed some area that we could calculate using the hypotenuse. But actually, something to notice is that we don't even necessarily need to calculate it. Once we know the theorem statement, any three similar figures would work. So we can (1) generalize the Pythagorean theorem to be about any shape, and (2) try to specialize to some shape that is especially easy to work with. This is of course just the same idea Gowers discusses from Polya's book.Here are some things I like about this proposal:

Here are some questions I am wondering about, and I am interested in hearing people's thoughts: