This week’s fun little twitter math coincidence involved circles and spheres.
Last Sunday Patrick Honner tweeted about Cavalieri’s principle:
and during the week Stephen Strogatz linked to a really interesting paper in American Scientist about n-dimensional spheres:
The combination made me want to show the boys how to find the volume and area formulas for the sphere and the circle. Going over all of the geometric ideas involved here with kids would be bit of a challenge, but it sure seemed like there was a fun lesson hiding in here somewhere. My main goal was communicating the idea of reducing a new and difficult problem to an easier problem that you already know how to solve. The secondary goal was showing that this strategy doesn’t always work out as well as you’d hoped, and luckily an observation from my older son actually sent us down that path (see the 3rd video).
Our first conversation was about the area of a circle. The method for finding the area that I wanted to show first was chopping up the circle into wedges and then re-assembling these wedges into something that looks a lot like a rectangle. Although the pure mathematical details behind why this method are beyond the scope of what I was trying to communicate, the method itself, I think, is something that kids can understand and enjoy.
From here we moved on to talk about the volume of a sphere. The approach we took here is a little different – instead of chopping up into “circular” pieces and assembling something sort of rectangular, we approximated the sphere with a pile of disks. As I thought about how to present this material to the boys this week, I thought it was really interesting that this method works well for the sphere but not so well for the circle.
I started by showing them our Tower of Hanoi puzzle so that they could see how disks could approximate a cone. Having that puzzle right in front of them seemed like it would help them understand that approximating the sphere by disks is possible. I tried to not get too bogged down in the formulas for summing up the volume of the disks since that wasn’t really the main point I was trying to convey and was happy to just present the result of the sum quickly.
After we turned off the camera, my younger son said that he thought it was really cool that Archimedes had used the Tower of Hanoi to find the volume of a sphere. Perfect!
Also after we turned off the camera, my older son said that he thought the method we just used to find the volume of the sphere could be used to find the area of a circle. He then drew a picture of what he meant on the board. His reaction made me really happy since that’s exactly what I’d planned to talk about next. Of course this method will lead to the right area formula for the circle, but working through the math turns out to be quite a bit more complicated than it is for the sphere.
So, I let my son lead off the third video with his idea. We go through the same math and quickly encounter some square roots that won’t go away. Yuck. After writing down a few terms of the sum we are looking at, we jump over to Wolfram Alpha. A little bit of playing around with Wolfram Alpha shows that the sum seems like it might converge to
All in all, a nice morning with the boys. A fun next step would be showing them that the volume of the 4 dimensional sphere is though that might require another week or two of pondering on my end!