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.

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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?

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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!

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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!

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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.

Summing up:

  • 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!

For more information about James Tanton’s approach, check out his newsletter where he discusses this in more detail.  And be sure to visit his web site Thinking Mathematics!