We returned from our vacation to Iceland yesterday. I didn’t have anything planned for a math project this morning, but fortunately ran across this fantastic puzzle from Catriona Shearer:
After seeing this puzzle my plan for the morning was to share it with the boys, see what they had to say, and then see what ideas they had for solving it.
Here’s their initial reaction. Their first idea was to try a few different configurations to test out the idea that the configurations didn’t change the area:
Now they solved for the area when the two squares had the same size:
The next idea they pursued fascinated me – they wanted to solve the puzzle using the assumption that the big square had twice the side length of the smaller square. Eventually this idea is going to lead to a big surprise!
It took a minute to get going with the algebra, but then they began to make progress. Seeing this progress happen live is why I love working through math problems with my kids
Sorry this video ends so abruptly – the memory card in the camera filled up.
Downloading the movies and clearing the memory card gave the boys a few minutes to think a bit more about the problem. When we restarted filming they had a plan. The algebra work was a little tricky for my younger son, I think, but we made it through and showed that the total area of the squares in this configuration was the same as the area in the last one.
They realized that their solution wasn’t a full solution to the problem, but I’m really happy with the work they did. After we finished with this last video I showed them the full solution off camera.
Mike's Math Page 
I like that the easiest case, where one of the squares is zero, involves having the other square poke outside of the circle. I expect that’s going to be a mental stumbling block for many.