When I asked my younger son to tell me about a math idea that he’s see but that he doesn’t believe to be true, he brought up the area and perimeter of the Koch snowflake. The perimeter is infinite while the area is finite, and he does not believe that these two facts can go together.

Today I thought it would be fun to talk about the perimeter of the Koch snowflake – no need to tackle both ideas at once. Here’s the introduction to the Koch snowflake and some thoughts from my younger son on what he finds confusing about the shape:

After that introduction we began to tackle the problem of finding the perimeter. We began by looking at the first couple of iterations in the construction of the snowflake to try to find a pattern. At this point in the project the boys didn’t quite see the pattern:

As a way to help the boys see the pattern in the perimeter, I asked my younger son to calculate the perimeter of the 4th iteration. My older son had been doing most of the calculating up to this point, and I hoped that my younger son working though the details here would shed a bit more light on what was going on as you move from one step to the next.

The counting project we reference at the end of this video is here:

Finally, we looked at how we could use math to describe the pattern that we found in the last video. We also discuss what it means mathematically for the perimeter to be infinite.

We need fairly precise language to describe the situation here, so this part of the project also gives the kids a nice way to learn the language of math.