In Geometry, we’ve been rocking out with various Pythagorean Theorem endeavors. Today, in addition to starting in on this Investigation, we made sure everyone was down with using the Pythagorean Theorem in three dimensions. One kid had worked on the sheet in question the previous class and was working on the additional, open-ended question at the bottom: “Find a box that has whole numbers for as many of its edges and diagonals as possible.”

The kid in question—who has a lot of prior mathematical experience, some sophistication, and is inclined toward quickness—interpreted “box” as being a cube, so that all edge lengths are the same. I realized this upon asking him, after he’d been giving me updates as I roamed the room checking in on kids. “It doesn’t work for anything less than 30.” “It doesn’t work for anything less than 80!”

The first thing I asked him was, “What do you know about the square root of two?” He said, “The square root of two is irrational.”

Then I asked him what his process was. He said that he was looking for a number that when you square it, cut it in half, and then take the square root, you get another whole number. And this is a strategy that makes sense if you know the Pythagorean Theorem and want to start with a whole-number face diagonal of a cube and end up with a whole-number edge.

He’d been checking every number on a calculator, one by one, to find one that worked.

So I asked him to use algebra to write out his calculation (pictured above). And then he saw that square root of two hanging out in the denominator.

I feel like the experience of doing those calculations and then running into this result might flesh out for him what is often the verbal veneer of “it’s irrational.” And I hope it was a lesson for him about the clarifying power of algebraic representation.

It certainly was for me.

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