A neat geometry problem I saw from David Butler

I saw this problem today when it was re-tweeted by Matt Enlow:

It is a little advanced for my younger son, but I still thought it would be fun to turn into a mini project tonight with the boys.

We started by talking through the problem and taking a guess at what we thought the answer was -> Is there enough information to determine the side length of the square?

Although we didn’t really make any progress towards a solution in this initial discussion, I really like the ideas that we talked about. Specifically, I liked how much thought my older son put into how to label the diagram.

In this part of the project we began to discuss how to solve the problem. We found two equations, but had 3 variables. My older son began to think that we weren’t going to find a solution.

In trying to simplify one of our equations my younger son made a common algebra mistake. I spent most of the video slowly showing him how to tell that the algebra he thought was right was actually off.

At the end of the last problem we found an equation that seemed to be a step in the right direction of finding a solution to the problem. In this part of the project we explored that equation.

At the beginning my older son was really confused. I think he’s used to seeing problems where there is always a solution – the open endedness of this problem seemed to leave him puzzled.

We did get our sea legs back, though, exploring a few specific cases. The happy accident was that the two solutions we found to the problem gave us the same perimeter for the square – was a unique solution hiding here?

To wrap up the project we went up to the computer to look at our equation using Mathematica. We’d covered the important mathematical ideas already, but finding some of the exact solutions was going to be a chore and certainly finding the maximum perimeter wasn’t going to be in reach.

Nonetheless, there were a few fun surprises to be found 🙂

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