I saw a really neat tweet from Lior Patcher last week:

I thought it would be fun to share this proof with my younger son since the geometric ideas in it are both surprising and super interesting. Unfortunately this one didn’t go nearly as well as I’d hoped. I missed a good idea that he had and got caught up in a few details that weren’t that important. Oh well, even after 10 years of doing these projects, I don’t have a good feel ahead of time for how they’ll go.

That said, here’s what we did. I started by having him walk through what is probably the most common proof that the square root of 2 is irrational:

Next we looked at Apostol’s proof and talked about some of the geometric ideas, and I just 100% missed that he was absolutely on the right track:

Now we took a look at an algebraic approach to the problem using the Pythagorean theorem. This part also didn’t go as well as I hoped and I might revisit it tomorrow just to make sure that these algebraic ideas made sense:

Finally, we came back to the geometric ideas since I realized that he was on the right track. Unfortunately I spent way too much time at the end of this part on a minor point. But hopefully the main geometric idea that we talk through in the first half of this video came through ok.

It is always disappointing when these projects don’t go quite as planned – I definitely want to push the “try again” button on this one.