The token model

Besides the number line, we used the token model for integers in Grade 6. A green token stands for +1 and a red token for −1. Put one of each together and they cancel: that is a zero pair, worth nothing at all.

That cancelling is what makes the model powerful, because you can add as many zero pairs as you like without changing the value.

Find (+7) − (+18). To subtract 18 from 7 we must REMOVE 18 positives from 7 positives. But there are not enough tokens to remove — only 7 positives are there.

So put in enough zero pairs to make the removal possible. Think of it as borrowing: every green token you bring in comes with a red one, so the total stays the same. We have 7 positives and need 11 more, so put in 11 zero pairs. The value has not changed, but now there are 18 positives present (and 11 negatives).

Now remove the 18 positives. What is left? Eleven negatives — meaning −11. So 7 − 18 = −11.

The model also shows something more general. Removing 18 positives left exactly what ADDING 18 negatives would have left. So 7 − 18 = 7 + (−18). Likewise, 4 − (−12) = 4 + 12: removing 12 negatives leaves what adding 12 positives leaves.

That is the rule: Subtracting a number is the same as adding its additive inverse. The additive inverse of an integer a is written −a, so the additive inverse of 18 is −(18) = −18, and the additive inverse of −18 is −(−18) = 18.