Jesse’s math homework last night included the following problem:

Jesse was stumped. So was I. How can it be that knowing the length of only one side of a shape, you can know its area? So Jesse and I talked, and it was a complicated affair for a 9-year-old. Unless I’m missing something, you have to make significant assumptions about the number of sides and the angles involved for Aiden’s assertion to be true. But if you’re working on the principle of right-angle four-sided arrays, which is what the kids have been doing for two months as they learn multiplication, Aiden doesn’t seem to have enough information at all with only the length of one side. I think he needs to know a width as well as a length. And there’s something goofy about answering “it can be true if it’s a square,” because then of course Aiden “knows” the length of all four sides of his garden. Or I suppose you could argue that if the adjoining sides of the rectangle are some factor of the one known side, like they’re exactly twice as long, then Aiden can use the one known length to measure the other sides and so on. It doesn’t feel right.

I asked Jesse to stretch her thinking by leaving straight lines in the dust. Boring. What if there were a shape with only one side… The only thing you have to assume then is the shape: it’s a circle garden. Then Aiden knows the “length” of the one “side” – the circumference – and then he can calculate the radius, and from that he can calculate the area. I think I remember these basic equations right, so I wrote them down for Jesse and suggested she stretch her third-grade teacher’s thinking.

Jesse officially thinks I’m crazy now, just a complete lunatic. I feel like I must be missing something really obvious, and it’s making me feel stupid today. What do you think?

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