Pingpong ball

In my multivariable calculus class today, I gave the kids a simply stated problem: you have a pingpong ball in the corner of the room, touching the floor and two walls. Then you have a beach ball roll over and it happens to be just the right size that it not only touches the pingpong ball but also floor and two walls.

What can you tell me about the relative size of the beachball to the pingpong ball?

I paired up the kids, and they came up with some great ideas. Initially, all the groups were stuck, but with some prodding by me or some perseverance by them, they got really close to the answer. All three groups got to the 2D answer, but thought it was the answer to the 3D question. WHICH WAS AWESOME because we had spent the entire previous few days talking about extending problems. So they then had to rethink their work, but systematically by extending their existing work into 3D.

Watching them work together, ask each other questions, go down dead ends and come back… this made me supremely happy. Last year I felt like most of my multivariable calculus students were working individually. Getting them to hunker down together was almost impossible, at least when I wanted to see true collaboration. But this year, they are acting as a team (at least, so far they are). And I’m liking it!