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 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,

becauseit just leads me into, well, algebra. You’re trying to prove what’s fundamentally a nice geometric theorem aboutareas, and dragging in the question of “what we can calculate” seems like an unnatural complication. When you want toapplythe 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

numberswas 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

goodthing; 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.