We had a busy day yesterday that ended with us getting home from a wedding about 2 hours past bedtime for the kids. As a result, I got started a little late with my younger son this morning and we were both a little tired. While we were erasing the board to get going with our number theory course he asked me a question about yesterday’s Family Math project.
We’d talked about the Koch Snowflake but had mainly focused on the fact that the perimeter gets larger and larger. At the end I showed that the figure has a finite area because you can fit the whole shape inside of a large circle. I did not, however, prove that the area was finite or actually go into any detail about the area at all.
The idea that the area is finite was bothering my son. He thought that since the perimeter was infinite, the shape itself might also grow really large as you add more and more triangles. I’m glad that he was thinking about ideas like this one rather than just taking what I tell him for granted. I thank that many kids would like thinking about these sorts of ideas, too.
While I didn’t want to go into the complete calculation of the area to answer his question, I thought there was a relatively easy way to help him see that you really could fit the shape inside of a circle. The idea I wanted to show him was that the (infinite) sum of all of the side lengths was finite. It turned out to be a really great conversation about some math ideas that I think many, many kids would love to see.
During the course of this conversation we touch on some basic arithmetic, fractions, a little bit of algebra, and even some ideas about infinite series – so it isn’t like the topic is that far away from normal school math. I love having these conversations: