I’ve just been reading the recently published e-book Nix the Tricks. I love the spirit of this book! I am 100% on board.
The first “trick” the authors tackle has to do with adding integers: “keep-change-change.” I remember hearing this for the first time when I was a brand-new teacher 11 years ago, and I was completely befuddled! Example: 2 – -3 = 2+ +3 via keep-change-change, or so they say.
Anyway, the authors suggested remedy for this rule is to use a number line. While that is certainly possible, I am going to take a stand here and suggest that “movement on a number line” — start here, go right or left some number of units, and end somewhere — is not the easiest approach to learning integers. For instance — can students explain which way to go when doing 2 – -3? Seems a bit confusing if you ask me.
I owe my preferred approach to James Tanton: just make piles of sand! Let’s take a trip to the sandbox to see how easy it is to work with positive and negative integers.
So “3” is just 3 piles of sand, and “-3” is just 3 holes in the sand. Now what happens when I push some piles of sand into the holes?
Looks like the piles got flattened! Apparently a hole is, in some sense, “the opposite of a pile.” So we would like to believe that 2 + -2 = 0. This makes adding integers a piece of cake!
I’m pretty confident that this is something that children of all ages can get their minds around! No need to worry about “moving this way or that way” on a number line.
Now for the sticky part: what about subtraction? Well, subtraction is all fine when the numbers behave nicely: “Johnny has a group of 8 apples. He takes away 3 of them. How many does he have left?” 8-5 = 3 is perfectly all right for this situation, but things get a bit out of hand when we try to understand what 3-8 must mean, to say nothing of -3 – (-8)! So what shall we do? Just do what mathematicians do at the highest levels of the profession: ignore subtraction entirely! From now on when we write 8 – 5, we are really saying something about *addition* — we take 8 and add the *opposite of 5,* so that 8 – 5 is — by definition — 8 + -5. With this simple understanding, we don’t even have to talk about the following four subtraction problems, as they are merely addition problems!
The only sticky point is to be careful about what we mean by -(-2). Piece of cake! -2 is 2 holes in the sandbox. -(-2) is the opposite of 2 holes in the sandbox, which is………………two piles of sand! So -(-2) really is 2 after all.
Another nice feature of this approach is that it allows us to handle algebraic expressions like the following: what is (9x+10) – (4x+3)? Well, we really just want to *add the opposite* of 4x+3, so we have 9x+10 + -(4x+3). What does that mean? I want the opposite of “4x piles and 3 piles,” which is “4x holes and 3 holes,” which is -4x + -3! So now we can write 9x + 10 + -4x + -3, and the rest is easy.
- Every addition problem can be handled by drawing piles and holes.
- Every subtraction problem can be construed as an addition problem. This is not a “trick” as in “keep-change-change,” it is a sound mathematical principle: a – b = a + -b, *by definition.*
- Thus every subtraction problem can be handled via piles and holes as well, by viewing it as an addition problem!
After drawing a few million diagrams with piles and holes, children will naturally internalize the principles and begin to do these problems in their heads like grown-ups!