How do you help your students make sense of the formula for the surface area of a sphere?

We started with an orange that was spherical. We talked about a great circle, and we used string to draw some great circles for the sphere on a piece of paper.

If we peel the orange, and fill the circles with peeling, then how many circles will we fill? I sent a Quick Poll to get their estimates. 20% were correct, which reminds me that I must find a way to work in more estimation. (Or I’ll admit I actually wondered whether someone who taught below me could work in more estimation. I don’t know what to replace!)

This year we didn’t actually peel the orange. I had been at #NCSM14 on the day we were going to do this in class, so after our conversation, we looked at a picture from another geometry class that their teacher had emailed me while I was out of town.

Four of the great circles were filled for this spherical grapefruit. At this point, I almost said the formula for the surface area of a sphere. My students would have nodded, and we would have moved on. But I stopped myself. Instead, I asked my students to write down on their paper what the formula was. I walked around, monitoring their work. I made myself wait on them. I thought the formula would be obvious, but it wasn’t obvious to many of my students. We talked more about the great circle, the dimensions of the great circle, and what those dimensions have to do with the sphere. A few more wrote down that the area of the surface area of the sphere was πr^{2}*4. One student wrote down that the surface area of the sphere was πr^{3}. I asked him to share what he had written with the rest of the class so that we could learn from this misconception. What would the units be if we calculated πr^{3}? Cubic. What do cubic units represent? Volume. We can often use units to help us make sense of a formula.

Finally, we settled on what we would see in a textbook or on a formula chart if we looked up the surface area of a sphere: 4πr^{2}. But it was hard won. Will my students remember it? The journey continues … with hopes that the image of the orange and 4 circles filled with peeling is imprinted in their memory.

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Travis

April 20, 2014 at 1:38 am

“Will they remember it?” I have thought about that. I am wondering if I even care if they remember the exact formula. I want them to remember the concept of removing the peel and that there were a certain number of great circles filled (it sounds like you used string, which is good if you want all to do it, but another way is to slice the orange in half and trace around it–kind of messy, but seeing that cross section is memorable.)

Formulas can easily be looked up. I think I am more interested in what they can do with the formula and what its graph looks like. I want to think more about what I want.

I think it is great that you held your tongue–great opportunity for a quick poll. I probably would have ruined the exploration by telling, but not now!

jwilson828

April 20, 2014 at 1:44 am

Thanks for making me think more about what is important for them to remember. Thanks, too, for suggesting we cut the orange in half … I honestly wouldn’t have thought of that since we’ve done it this other way for so long. This activity is definitely messy, but at least it makes the room smell good! I’ll look forward to next year thanks to your suggestions.