The other day, I came across the following problem: Given two squares, both with the same side length, where a vertex of one square is fixed at the center of the other, what is the maximum and minimum area of the overlapping region? I sat with it for a few minutes, thinking about the different ways I might draw the figure, like in the diagram here, and how I could represent the pieces that change, like the length of the part of a side of the red square that is inside the blue square, or some angle of rotation of a side of the red square. I anticipated using some trig function, which got me a little excited since I will probably be teaching trig next year. I soon realized the solution, which turned out to be something different from what I thought it would be, and recognized a connection I had not anticipated.
So I started thinking, would the same thing happen with other shapes? I tried equilateral triangles, and quickly realized there was a different thing happening here. Which has led to more questions about other polygons, polygons of different sizes, combinations of polygons … I am still exploring some of these. (I’ll post some of my thinking on my questions and solutions later.)
Some might think that after being a math teacher for a long time, the topics become a little old, a little stale. Not so! I am always finding problems that are interesting and often have surprising or elegant solutions. (That’s the best part. An unexpected solution or connection is like discovering a great restaurant or reading a good story.) Most often, solving these problems needs nothing beyond a little algebra or a little trig, which makes it even more fun, because I can share the problems and discoveries with my students. The follow up questions I think about are always an added bonus, and while I never seem to quite finish answering all the questions, it’s always nice to revisit them later.
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