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.

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## 2 comments:

Wow! This is a great story. Thank you so much for sharing it. I've never thought to talk about the similarity of those cross sections, much less how we know they are similar. We just spent some time on the Mathematics Assessment Project formative assessment lesson "2D Representations of 3D Objects", and even then the conversation didn't just happen. But I will make sure it does moving forward.

We did an activity like that with kids ages 5-10 in our Inspired by Calculus math circle. We made islands out of foam by taking our nested substitution fractals (Droste effect fractals) to the third dimension. Then we talked about 4th etc. dimensions. This is Week 5 of that circle, with other weeks linked from the blog post: http://www.moebiusnoodles.com/2014/04/inspired-by-calculus-math-circle-week-5/

We did not use the word "dilation" - but next time we will! Thanks!

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