*“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*and a**GRAB**. (I think that’s reasonably intuitive.) And you can**negative**number if you**TOSS****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.

I just had this conversation about signed numbers with some of my 8th graders this week. We’ve been doing exponent rules, and here those misconceptions that a negative plus a negative equals a positive. Love your idea of using Casino Zero to teach the concepts.

thanks for a new analogy – equivalent but much more personally satisfying to the one with an old west town with good guys and bad guys coming to and leaving town …

Never heard the old west town version, but I do remember hearing that a negative times a negative is a positive because “when bad things happen to bad people, that’s a good thing.” Grimaced then, as now.

I’m glad I’m not the only one who has trouble with students and “memory aids” when it comes to adding/subtracting vs multiplying/dividing integers. I think it’s a perfect example of how easy it can be to teach based on memorization instead of teaching based on the idea of having the students internalize the ideas being used so that they’re really in there on an unconscious level.