# Volume, Scaling, and a surprising relationship to the Fibonacci numbers.

We used our Zome Geometry book for another Family Math project today. The project we stumbled on used the blue struts to talk about volume and scaling. It starts by asking a seemingly simple question about the volume of two boxes you can make with the blue struts. Here’s the problem:

and the constructed boxes – I liked hearing their ideas about the volume of the larger box:

Next they re-built the “golden box” using only medium and short blue struts. Building the box this way gave them a little bit of extra insight into the volume. Their first guess about the volume was that it would be the same as a cube with side length equal to 2x the small blue struts.

We took a few minutes to build the cube they wondered about in the last video and looked to see if the volume of that cube matched the volume of the golden box:

Once we saw that the 2x2x2 cube was too large, the kids thought to build a cube out of the medium struts. It seems possible that these two structures have the same volume – but we need to find a relationship between the mediums and the small blue struts in order to calculate the volume.

While we had the camera off, the boys played with the medium and small struts to find an aproximate relationship. They used this approximation to estimate the volume of the golden box. It was really interesting to me to hear and see their approach here:

After we finished the project, my younger son was playing around with the struts a little more. I thought it would be fun to see if they could see the interesting relationship between the medium and small blue struts. We did a little postscript. Here they see the numbers 5, 8, 13 for the long struts and 3, 5, 8 for the short ones and guess that the next numbers will be 13 and 20.

We checked if 20 shorts matched 13 mediums and found a fun surprise!

So, a fun project with a surprise postscript. Who would have thought you’d see the Fibonacci numbers pop up just by building Zometool boxes 🙂