“But I thought a minus and a minus makes a plus. We learned that last year.”
One problem with rules is that they depend on the user’s memory. One little jurisdictional detail forgotten or misremebered and the whole shebang can go sour. Students may have memorized rules for sign management when it comes to doing arithmetic with signed numbers and variables, but chances are pretty good that time away from the game, like, say, Summer, has eroded the fidelity of those memories. By “and,” do you mean addition, in which case (–)+(–)=(–), or by “and,” do you mean multiplication, in which case (–)•(–)=(+)? Can we offer students an understanding here that is more than a rerun of those earlier rules of engagement?
Maybe. There are several ways to illustrate the outcomes of the addition of signed objects, and multiplying differently signed objects is just a few steps away. In these cases, intuition, to the extent that my number rockets will have math-flavored intuition, is fairly reliable. The 18 ft. Great White in the pool, however, is the multiplication of two negatives. Why oh why is negative 2 times negative 3 a positive 6? Who was on watch when they rolled that one out? Middle school is not the time to break out number theory proofs, so what kind of model can we ask students to handle that will feed their intuition and not overwhelm their budding abilities for formal, abstract thinking?
Here’s one idea towards helping to correct this signage misconception. It’s a work in progress, but it basically breaks down like this. Let’s say we’re playing games in an unusual casino, Casino Zero. Some of this is imaginary, of course, but some can certainly be acted out in small groups in class.
- You get your playing chips from the cashier. Everybody starts out with zero credits in the form of, say 20 positive credits and 20 negative credits. Let’s say all the chips have a value of either +5 or –5, so you’d have this set of 8 chips:(+ + + +,– – – –), four +5 credit chips and four –5 credit chips. And you certainly understand that the net value in your hands is zero credits. If you were to cash in these 8 chips, the cashier would give you nothing in return.
- Say you win your bet of 10 credits. You have two ways to act that will register this change in value. You can GRAB two +5 chips from the pot, or you can TOSS two –5 chips from your hand. In either case, if you go straight to the cashier and cash in your chips, you have clearly increased your wealth by 10 credits, which the cashier will happily pay you. Students see this because they are comfortable with the addition of differently signed numbers and because they are holding the chips in their hands.
- Of course, that’s one action that would register that change in value. Another action would be to GRAB one +5 chip and TOSS one –5 chip. Same outcome. Same change in wealth. Same huge haul at the cashier’s window. And students are becoming fluent with these kinds of equivalences.
- We need to be explicit about these actions. Every action has a multiplier, which is a positive number if you GRAB and a negative number if you TOSS. (I think that’s reasonably intuitive.) And you can GRAB and TOSS both kinds of chips, + chips and –chips. For example, you lose 15 credits: TOSS three +5 chips, written as –3•5. Or, you could GRAB three –5 chips, written as 3•–5. In both cases, your wealth has decreased by 15 credits and you can see that in your hands. You know exactly what will happen at the cashier’s window.
- Let’s say a player wins 25 credits but there are no +chips to GRAB. The player is clever and so she knows that there is another action besides GRABBING from the pot; she can TOSS from her hand. She TOSSES five –5 chips from her hand, getting rid of (– – – – –). She writes this as –5•–5. When she looks in her hands, she sees that her wealth has increased by 25 credits. She skips happily towards the cashier’s cage, secure in her understanding that a negative times a negative is a positive. And so rich, to boot.