Revisiting an old James Tanton / James Key pyramid project

Several months ago this tweet –

inspired this project:

A neat geometry project inspired by a James Tanton / James Key tweet

Fast forward a few months today and I’m actually studying pyramids with my older son and it seemed like a great time to revisit the project. Because my younger son is also starting to look at some basic geometry, I asked him to join in today as well.

We started off by building and talking about some of the shapes from the previous project. The main ideas I wanted to hear the boys talk about here were (i) that the volume of these pyramids was 1/3 of the volume of a cube, and (ii) the 3d printed pyramids looked like a stack of squares.

For the next part of the project we went to the living room to build some larger – but similar – pyramids to see what the kids would notice:

and we kept building. With the double-sized pyramid in the last video, the kids had a hard time seeing how to divide the pyramid into boxes. With the larger pyramid in this video, they have a slightly easier time.

Now we built a giant pyramid. The bottom layer is a 10×10 square. We had to add some support struts just to keep it standing, but all of the extra building was worth it because the kids can really see the box approximations now. The boys are also able to begin to see the connection between the sum of squares and the volume of the pyramid. They also are starting to see that the more layers you have, the better the approximation. Fun!

At the end of this video there is one open question, though – if we are adding up a bunch of perfect squares to find the volume, where does the 1/3 come from?

For the last part of the project we turn to some algebra / arithmetic to answer the open question from the last video. I’m not yet interested in deriving the formula here, though I do think that it is interesting for them to see this formula. For my younger son testing a few of the simple sums is a nice way to get in some simple algebra practice.

I asked my older son what this formula would start to look like as n got large and he is able to notice that the “+1″‘s wouldn’t matter so much and the whole formula would start to look like n^3 / 3. So we get to see a fun and surprising connection between arithmetic, algebra, and geometry here.

Fun little project, and we end with a little speculation about what the formula for the volume of a pyramid would be in dimensions higher than 3 🙂