Today was day 2 for school, but day 1 for my Honors Precalculus class. I wanted to start with a problem that felt playful at first but then might have some really cool deep math attached to it. So we did the “one cut” problem where you try to fold up a paper so that you can make ONE SINGLE SNIP and cut out a certain shape.

They just started working on some shapes and 10 minutes in I asked them… what’s going on here? What mathematical principles are happening? This was most of the conversation in the next 20 minutes or so. It was fairly quiet, but students progressed further and further.

- One student said “symmetry”. Great. What about symmetry?
- Another student “Well, this has to do with angle bisectors.” We talked about why that makes sense and then that led many students in the class to nail the scalene triangle.
- “Well if those are angle bisectors, that means that the point here that matters in the incenter.” Yup!
- The next piece was an irregular quadrilateral. “Well, does it make sense for a quadrilateral to have a single incenter?” The class decided, no, not really.
- Is this shape possible? I shared that there’s a theorem here, that EVERY shape is possible. This might have jumped the gun a bit and stolen some thought from them, but I wanted them to keep persisting with the shape.
- They worked for a while and a few students got an acceptable solution for the quadrilateral. One student’s idea was to draw the angle bisectors, find the pairs that connected, call these two “incenters” in a way, connect those and fold along all the lines (see the diagram above).
- One student: “Okay, so if an incenter is the center of a circle, do these have to do with that? Can you inscribe a circle in a quadrilateral so that it touches all four sides?” We decided no. A simple rectangle helped us with that.
- “Well if there are two incenters in a quadrilateral, I wonder if they are the foci of an inscribed ellipse.” Whoa. Probably not, but cool idea!
- Then, right at the end, a student noticed that the incenters themselves form a smaller quadrilateral inside the shape. The shape looks similar to the outside one. Is it!?? Could that help with the solution?

This had nothing to do with what we are going to do soon (i.e. tomorrow), but man – what an amazing progression of mathematical ideas that I didn’t see coming. It shows what happens when you start with a beautiful, rich problem that you might not even know how to solve.