Mow a huge lawn. Earn $75.
Knock down clothes line; shred grandmother’s robe. Pay $30.
For a short while this summer, I’ve been working with a rising ninth-grader, Hat (not his real name), establishing and repairing his foundational math concepts and skills. For one of our episodes, I taught operations with negative numbers using an idea that I’d left undeveloped since I’d thought about it almost a year ago (during the New Blogger Initiation festival!). At the time, I wasn’t sure about Casino Zero’s working mechanism, mostly because I didn’t need one. Now I needed one.
Hat is an entrepreneurial number rocket with access to a riding lawn mower, making semi-mad stacks of cash this season. A simple story and board-game model provided all the control I needed. Here’s the set-up:
- The Game is played for pretend money but not with pretend money; it’s played with value chips.
- Chips are in two places: the World has chips (in the center of the game board) and the Player has chips.
- The Game starts with the World and the Player both having no money, but in the form of zero value, that is, as equal amounts of positive and negative value chips.
- Chip denominations are 5 and 25. (I would have also used 10 if I’d had a fourth color.) Those denominations come in both positive and negative, too. So, chips are +5, -5, +25, and -25, with positive and negative denoting increasing or decreasing wealth. That kind of “direction” of monetary value is not so strange.
- Positive and negative were used to denote one other direction: the flow of value. I suppressed the urge to talk about magnitude and vectors, but I did define positive flow as meaning “in the direction from World to Player” and negative flow as “in the direction from Player to World.” Hat agrees that this language captures the idea of chips either coming to him (getting paid) or going away from him (having to pay).
- The Game is played by rolling a die, moving a marker, and either earning money (a wealth-increasing event) or paying money (a wealth decreasing event), with plenty of ridiculousness along the way. Obviously, this game is easily adaptable to any story you want to make.
- Hat keeps a ledger with columns labeled Direction (In/Out), Number of Chips, Chip Value, Change in Wealth, and Total Wealth. He writes an expression to describe what happens for each play, then simplifies it to see the result. For example, you earn $75, so your ledger looks like +, 3, +25, +75, (new sum): . If you had to buy gas, your ledger might look like -, 8, +5, -40, (new sum): . An important supporting piece here is that the result of the operations must match the change in wealth, which is obvious because the game tile says what that change is and also because you can see your rising and falling fortunes for yourself in your pile of chips. (Hat sees that having equal stacks of opposite chips means having zero value and zero dollars.) With each event, Hat and I look at the outcome of the operations: (+) • (+) = (+), (–) • (+) = (–), and so on, building some sense about the “sign rules” for multiplication.
- Finally, the big mathematical moment happens! The World owes you $30 but it has none to give because there are no more +chips in the middle of the board. What can you do? If you go to the cashier right now and cash in your chips (for the pretend money), what will your wealth be? What about the wealth that the World owes you? What can you do with the chips in your hand that will increase the wealth you have in your hand, and so, the money you get from the cashier?
- Give away negative chips? Really? What would that look like? So, -, 6, -5, +30, or . Now, when you go to the cashier with fewer negative chips, you’ll receive more money; your wealth has increased: (–) • (–) = (+).
We only had time for one game and never got into some of the more interesting combinations of operations that might be required to create the necessary change in wealth. And for sure, not only would this have been more fun and more kindly chaotic with 3 or 4 players playing at 5 or 6 tables, but Hat and his game-mates would also have had some conversation, argument (I hope), and other social-learning energy to help drive the lesson forward. Having concrete experiences that illustrate abstract ideas helps secure those ideas. Even semi-authentic, not-exactly-real-world experiences can help students create cognitive pegs to hang math concepts on.
To add more support to the idea of negative factors making positive products, I drew Hat into Chris Adams’s great response to Dan Meyer’s call-for-tweets about A Negative Times a Negative. Hat’s gears audibly clicked into place when he played that movie in his head.
If you try a Casino Zero type game with your students, ping back how it went and add your ideas for making it better.