Jim Propp published a terrific essay last week:
Here’s a direct link in case the Twitter link has problems:
Yesterday we did a fun project about card shuffling using the ideas from Propp’s post:
Today we did a second project for kids based on some ideas from Propp’s post. The topic today was “primitive roots”. Unfortunately this isn’t a topic that I know well and I messed up one explanation in the first video below. Oh well . . . still a really neat idea to share with kids.
So, I started by introducing the concept of primitive roots by reminding them of the 8 card and 52 card shuffles we looked at yesterday (pay no attention to my explanation about powers and mods at the end. It will become clear in the next video that I goofed up that explanation . . . . ):
Now we looked at some examples of primitive roots with small numbers. These simple examples give a nice way for kids to get a little bit of arithmetic practice and also help them see the main ideas in the problem that we are studying.
After working through these smaller examples, we moved to the computer to continue studying the problem. My older son noticed that the examples that seemed take the longest time to work were primes, but not all primes took a long time. That’s exactly the math idea we are looking at here.
Next we made a small change to the program to study all of the odd numbers up to 1,000 all at once. After correcting a little bug we found that the numbers we were looking for were indeed all primes.
We wrapped up be talking about what was known and what wasn’t known about these primitive roots. I was happy that my older son seemed to be particularly interested in this problem.
Definitely a fun project. It is always fun to find unsolved problems that are accessible to kids (and lots of them seem to come from number theory!). We will definitely have to do some follow up projects to explore the ideas here in a bit more detail.