I did a bad job (in my opinion) of teaching infinite geometric series in precalculus in my previous class. I told them I did a bad job. I was rushing. They were confused. (One of them said: “you did a fine job, Mr. Shah” which made me feel better, but I still felt like they were super confused.)

At the start of the lesson, I gave each group one colored piece of paper. (I got this idea last year from my friend Bowen Kerins on Facebook! He is not only a math genius but he’s also a 5 time world pinball player champion. Seriously.) I don’t know why but it was nice to give each group a different color piece of paper. Then I had them designate one person to be the “paper master” and two people to be the friends of the paper master. Any group with a fourth person simply had to have the fourth person be the observer.

I did not document this, so I have made photographs to illustrate ex post facto.

I started, “Paper master, you have a whole sheet of paper! One whole sheet of paper! And you have two friends. You feel like being kind, sharing is caring, so why don’t you give them each a third of your paper.”

The paper master divided the paper in thirds, tore it, and shared their paper.

Then I said: “Your friends loveeeed their paper gift. They want just a little bit more. Why don’t you give them each some more… Maybe divide what you have left into thirds so you can keep some too.”

And the paper master took what they had, divided it into thirds, and shared it.

To the friends, I said: “Hey, friends, how many of you LOOOOOVE all these presents you’re getting? WHO WANTS MORE?” and the friends replied “MEEEEEEEEEEEEEEE!”

“Paper master, your friends are getting greedy. And they demand more paper. They said you must give them more or they won’t be your friends. And you are peer pressured into giving them more. So divide what little you have left and hand it to them.”

They do.

“Now do it again. Because your greedy friends are greedy and evil, but they’re still your friends.”

“Again.”

“Again.”

Here we stop. The friends have a lot of slips of paper of varying sizes. The paper master has a tiny speck.

I ask the class: “If we continue this, how much paper is the paper master going to eventually end up with?”

(Discussion ensues about whether the answer is 0 or super duper super close to 0.)

I ask the class: “If we continue this, how much paper are each of the friends going to have?”

(A more lively short discussion ensues… Eventually they agree… each friend will have about 1/2 the paper, since there was a whole piece of paper to start, each friend gets the same amount, and the paper master has essentially no paper left.)

I then go to the board.

I write

and then I say: “How much paper did you get in your initial gift, friends?”

I write

and then we continue, until I have:

Ooohs and aahs.

Next year I am going to task each student to do this with two friends or people from their family, and have them write down their friends/family member’s reactions…

I love this.