This morning I talked through a 5th lesson from this problem. That lesson comes courtesy of this amazing video from Numberphile and Harvard math professor Barry Mazur:

So, perhaps the greatest surprise of all from the problem we were talking through last week – Problem #7 from the 2008 AMC 10 A – is that it helps you prove the Pythagorean theorem in a clever way.

The first step was reviewing how area scales when you increase the size of a 2 dimensional object:

Next we moved on to checking if area seemed to scale the same way for triangles:

Finally the punch line – how the combination of the original problem and the idea of how area scales proves the Pythagorean theorem:

I’m really happy about these two projects, but the 2nd one makes me really happy for a couple of reasons. First, the ideas are easy enough that I could talk through them with both kids. Yesterday’s lessons, while important, require a bit more background in algebra and geometry than my younger son has right now.

The second reason this lesson makes me happy builds on the first – the combination of the relatively easy ideas and the amazing result – the Pythagorean Theorem! – is a great example of mathematical reasoning. “Hard” theorems do not necessarily require “hard” ideas!