I’m going through Jim Propp’s piece on base 3/2 with my kids this week.
His essay is here:
Jim Propp’s How do you write one hundred in base 3/2?
And the our first project using that essay is here:
Sharing Jim Propp’s base 3/2 essay with kids – Part 1
Originally I wanted to have the kids read the essay and give some of their thoughts for part 2, but I changed my mind on the approach this morning. Instead I asked each of them to answer the question in the title of Propp’s essay -> How do you write 100 in base 3/2?
Propp points out in his essay that his approach to base 3/2 via chip firing / Engel machines / exploding dots is not what mathematicians would normally consider to be base 3/2. The boys are not aware of that statement, though, since they have not read the essay yet.
Here’s how my younger son approached writing 100 in base 3/2. The first video is an introduction to the problem and, from knowing how to write numbers like 100 (in base 10) in other integer bases.
I think the first 3 minutes of this video are interesting because you get to hear his ideas about why this approach seems like a good idea. The remainder of this video plus the next two videos are a long march down the road to discovering why this approach doesn’t work in the version of base 3/2 we are studying:
So, after finding that the path we were walking down led to a dead end, we started over. This time my son decided to try to write 100 as 10×10. This approach does work!
Next I introduced the problem to my older son. He also started by trying to solve the problem the same way that you would for integer bases, though his technique was slightly different. He realized fairly quickly (by the end of the video, I mean) that this approach didn’t work:
My older son needed to find a new approach, and he ended up finding an idea different from my younger son’s idea to find 100 in base 3/2. His idea was to use chip firing:
I thought that today’s project would be a quick reminder of how base 3/2 works (at least the version we are studying). That thought was way off base and was completely influenced by me knowing the answer! Instead we found – by accident – a great example of how to explore a challenging problem in math. Sometimes the first few things you try don’t work, and you have to keep trying new things.
Definitely a fun morning!