Infinite descent with kids inspired by Jim Propp

I saw another neat post from Jim Propp a few days ago:

Fermat’s Last Theorem: The Curious Incident of the Boasting Frenchman

The part that caught my attention as something that kids might find fun was the discussion of “infinite descent.” It reminded me a little of a Zometool project that we did last year:

Fibonacci Spirals and Pentagons with our Zometool set

We sure had a lot of Zome fun in the new house before we got furniture 🙂

To get going with the “infinite descent” discussion today I had the boys build another Fibonacci spiral shape from the Zometool set. We talked about the shape and my younger son remembered the connection to the Golden Rectangle, which was a nice surprise.

This shape is actually really fun to build because you don’t really get a sense for how quickly the shape grows until you try to fit it in your living room 🙂

 

My son remembering a connection with the Fibonacci Spiral and the Golden Rectangle made for a nice transition to the next part of the project. The first thing we needed to do was talk about what it means to be “similar” in geometry.

 

For the last part of the project we stepped through the infinite descent argument showing that the Golden Ratio cannot be the ratio of two integers.

The argument requires a little bit of algebra so I went through it slowly to make sure that my younger son could follow it. Hopefully he was able to follow it – and I think he did because at the end he asked about non-integer values for the ratio.

Pretty fun to start the project with a spiral that grows larger and larger and end with a spiral that shrinks and shrinks!

 

Another fun project inspired by Jim Propp’s blog. I originally wanted to look at the pentagon question mentioned in Propp’s post, but I think we’ll explore that question next weekend.