I don’t start school for almost four weeks, but since other people are, here’s my favorite way to start the year.
On the first day of school, we play a game.
I project the numbers 1 to 9 on the board.
And I project the rules:
- Select a player to go first
- Take turns selecting a number from 1 to 9
- Each number can only be selected once.
- The first person to have exactly 3 number that sum to 15 wins.
I demonstrate with a student volunteer.
- I ask if they want to go first.
- I try to win.
- I always show what it looks like to win with 4 numbers to emphasize the last rule.
Here’s an example of how the game could go:
- Player 1 chooses 4
- Player 2 chooses 8
- Player 1 chooses 6
- Player 2 chooses 5
- Player 1 cannot choose 5 because it is already taken, so they choose 2.
- Player 2 chooses 1
- Player 1 chooses 7 and wins because 6 + 2 + 7 = 15. They do not need to use their 4.
I ask students play with a partner for while. I circulate, trying to encourage everyone to play. I jump in to play with a reluctant or left out student. I ask who’s winning or what their strategy is. I ask what they are noticing. If someone is undefeated, I ask if they want to play me, and I usually win.
When they start to seem bored, I pull the class back together and ask what they notice and wonder. I record their responses on a whiteboard. I try to get as much as possible from as many people as possible.
I hope to get all of the following, and I add the last one if they don’t ask it.
- Is it better to go first or second?
- What is the best number to choose first?
- Is there always a winner?
- What is your strategy?
- What is this game called?
I ask for a few initial ideas about each question, but I don’t confirm or deny anything.
I ask students to play a few more times with our questions in mind. I try to stoke some curiosity about the name of this game. If someone figures out the name of the game, I don’t tell them it’s wrong, but I try and get them to justify it. They’ll probably not realize they were right and act indignant later.
I pull the class together and try to build some consensus around the answers to some of our questions.
- It is better to go first.
- It’s not clear what number to choose first, but a common conjecture is 5.
- There isn’t always a winner. In fact, there are LOTS of ties.
- The simplest (best?) strategy is to play defense.
- But what is this game called?!
I also usually give a hint: you all know the name of this game because you’ve played it before.
Sometimes we get stuck here for a while. But we keep playing and keep talking about the game until they are convinced they know the name of the game. I’ve done this at least 20 times and my students have never not figured it out eventually (although sometimes right before the bell).
When we figure out the name of the game, we search for justifications and connections. We reorganize the numbers into a magic square and there is a class-wide aha! moment.
Finally, we talk about the goals of this activity.
- Math can and should be fun
- Math is about more than calculating; It’s about noticing, wondering, experimenting, …
- Math is surprise and wonder and the good kind of frustration
- Representations influence how we think