Sharing an e surprise with kids

Yesterday I saw a neat request from Sam Shah on twitter asking for ideas about how to “stumble upon” e with kids in Algebra 2 (other than compound interest). I shared an old project we did (and am doing again below) which I think is a terrific way to share a fun and surprising idea about e with kids.

Later in the thread, though, there was a tweet that surprised me:

Strogatz has done more math for the public that just about anyone, and he’s also taught a college course that shared beautiful and advanced ideas in math with students not intending to be math majors, so I was really caught off guard by his thoughts about e.

But rather than getting into an academic discussion about whether or not ideas about e can be shared with Algebra 2 students, I decided to revisit our old project with the boys today.

The idea we’ll take a look at today is this -> Take an NxN set of squares and place a random integer from 1 to N^2 in each of the squares. How many of the integers from 1 to $N^2$ do you expect to not appear in any of the boxes?

I introduced the idea with a 2×2 square and selecting random integers from 1 to 4 by rolling a 4-sided die:

Next we moved on to a 5×5 grid and talked about what we’d expect to happen:

Now we moved to a computer to help us look at the grids more quickly. In this video I explain the program using a few simple examples. The program itself is picking random numbers and counting how often each integer from 1 to N^2 appears in the list of numbers selected.

Although I struggled a little bit with the output of the program (the joy of filming these things live . . . ) we eventually found our way and the kids noticed some potentially interesting patterns in the number counts:

Now we moved up to some larger grids and the kids began to notice more and more patterns in the number counts – :

Finally, we looked at a few very large grids – starting with a 50×50 grid – and the boys began to notice the pattern emerging in the number counts that allowed you to take a guess at each number in the list. It was fun to see them begin to understand these patterns more and more throughout this project:

I guess I’ll conclude by saying that my view differs from Strogatz’s view. I think this project would be appropriate for Algebra 2 kids. It shows them a pretty advanced idea but also gives them a chance to explore that idea using things they’ve learned in K-12 math ranging from simple arithmetic, to a bit of geometry and algebra, and also elementary statistics. I’m happy that we were able to go through this project again today.