Tiny Game Re: Euler’s Number

I’m teaching Algebra 2 this year and the other teacher and I decided that we should introduce e to our kids. The reason it’s challenging is that it’s hard to motivate in any real way. You can do compound interest, but that doesn’t do much for you in terms of highlighting how important the number is. [1] I asked on Twitter for some help, and I got a ton of amazing responses (read them all here). My mind was blown. This year, though, I didn’t have time to execute my plan that I outlined at the bottom of that post. So here’s what I did:

1. The core part of what I did to get the number to pop up was to use @lukeselfwalker’s Desmos activity. I like it for so many reasons, but I’ll list a few here. It starts by “building up” a more and more complicated polynomial of the form $(1+\frac{x}{n})^n$, but in a super concrete way so kids can see the polynomial for different n-values. It shows why the x-intercept travels more and more left as you increase n, so when you finally (in the class discussion) talk about what happens when n goes to infinity, you can have kids understand this is how to “build” a horizontal asymptote. It gets kid saying trying to articulate sentences like “this number is increasing, but slower and slower” (when talking about the value of the polynomial when $x=1$. And they see how this polynomial gets to look more and more like an exponential function as you increase the value of n. If you want to introduce e, this is one fantastic way to do it.
2. A few days later, I had everyone put their stuff down and take only a calculator with them. They paired up. (If someone didn’t have a pair, it would be fine… they just sit out the first round.) On the count of three, both people say a number between 0 and 5. (I reinforce the number doesn’t have to be an integer, so it can be 4.5 or something.)Then using their calculators, they calculate their score: they take their number and raise it to their competitor’s number. The winner has the higher number. (If it’s a tie, they go again until there is a winner.)

Then the loser is done. They “tag” along with the winner and cheer them on as they find another winner to play. This goes on. By the end, you have the class divided into two groups each cheering on one person. (I learned this game this year as an ice breaker for a large group… it’s awesome. This is the best youtube video I could find showing it.)

Finally there is a class winner.

So I then went up against them.

And when we both said our numbers, I said: e.

The class groans, realizing it was all a trick and I was going to win. We did the calculations. I obviously won.

We sit down and I show them on my laptop how this works:

The red graph is my score, for any student number chosen ($e^x$).
The blue graph is the student score, for any student number chosen ($x^e$).

Clearly I will always win, except for if my opponent picks e.

I tell kids they can win money off of their parents by playing this game for quarters, losing a few times, and then doing a triple or nothing contest where they then play 2.718. WINNER WINNER CHICKEN DINNER!

3. After this, I show kids these additionally cool things (from the blogpost), saying I just learned them and don’t know why they work (yet), but that’s what makes them so intriguing to me! And more importantly, they all seem to have nothing to do with one another, but e pops up in all of them!

I re-emphasize e is a number like $\pi$ and I showed them this to explain that it pops up in all these places in math that seem to have nothing to do with that polynomial we saw. And that even though we don’t have time to explore e in depth, that I wanted them to get a glimpse of why it was important enough to have a mathematical constant for it, and why their calculators have built in e and ln.

That is all. I honestly really just wrote this just because I was excited by the “game” I made out of one of the properties of and wanted to archive it so I would remember it. (And in case someone out there in the blogoversesphere might want to try it.)

UPDATE: Coconspirator in math teaching at my school, Tom James (blogs here) created the checkerboard experiment using some code. You can access the code/alter the code here. The darker the square, the more times the number for the square has been called by the random number generator. And with some updates, you can make more squares! In the future, we can give this to kids and have them figure out an approximation for e.

[1] And introducing it with compound interest means you have to assume 100% interest compounded continuously. Where are you going to get 100% interest?!?!