I’ve written before about making sense of the surface area of a sphere. The lesson this year unfolded (unpeeled?) a bit differently.
I’m not sure how students might guess that the surface area of a sphere has something to do with the area of a great circle of the sphere. We talked about what a great circle must be, we used fishing wire to measure the circumference of a great circle of the sphere (orange), and I asked them to estimate how many great circles would cover the orange. You can see the huge variety of responses.
We cut the orange in half. I showed them the surface of the great circle and the act of “stamping” it onto the orange peeling. Do you want to keep your estimate? Or change it?
We went from 20 responses below π, 9 infinites, and 3 correct to 13 responses below π, 5 infinites and 7 correct.
I hesitated about what to ask next. We were ready to peel the orange to see how many great circles we could cover and figure out what the surface area formula would be, but I was curious about whether students could make sense of the formula before we did that if I told them what the formula was. So we waited on peeling and I sent another poll.
68% understood that the sphere’s surface area formula, 4πr^2, meant that 4 great circles could cover the sphere. We peeled it to be sure.
And so the journey to figure out what questions to ask when continues …