It was one of those moments when I was trying to explain something to them and they ended up explaining something to me.
We're in the middle of a unit on volume and exploring prisms, cylinders and cones. I was inspired by James Tanton's ability to explain things by getting at their essence. As if to say, "we can call a cylinder a 'cylinder' but it's just a prism made of circles--or a cone can be called a 'cone' but is it really any different than a pyramid?"
It was one of those, sitting around a campfire moments. We're using stacks of paper and stacks of CDs to demonstrate why calculating the base area is critical because the rest of the solid is just like a stack of that area and no matter where we slice the solid, we get the same shape--over and over again.
Then comes the question about the cone.
The base is a circle but when you slice it, you get a...circle? Wait, but it's a different circle. Waitaminit. What about a pyramid? Triangle base and when you slice it, you get a triangle. But a different one.
Are the triangles related?
"They're similar. Hey wait, this is a dilation."
And the tip of the pyramid is the center of dilation.
We did dilations in Unit 1. This was a callback I didn't anticipate: A pyramid is like a 3D representation of a dilation.
Thanks, kids. I'd never thought of it that way before.