A negative multiplier takes tokens out

So far the multiplier has been positive, and we PLACED tokens into the bag. What could a NEGATIVE multiplier mean? The natural opposite: we REMOVE tokens from the bag.

Take (−4) × 2. That means removing 2 positives — 2 green tokens — from the bag, 4 times. (Green, not red, because the multiplicand is +2.)

But the bag starts EMPTY, so there is nothing to remove. This is exactly the difficulty we met in subtraction, and it has the same solution: first place zero pairs in, which changes nothing, and then remove what we need.

So place 2 zero pairs in and remove the 2 positives; and do that whole move 4 times. After removing the positives, 8 negatives are left in the bag. That is −8, so (−4) × 2 = −8.

Now the case everyone finds strangest: (−4) × (−2). Read it the same way — remove 2 NEGATIVES from the bag, 4 times. Again there are no red tokens to remove, so place 2 zero pairs in and remove the 2 negatives, four times over.

What is left? Eight POSITIVES. So (−4) × (−2) = 8. Nothing was assumed; the rule 'negative multiplier means remove' produced it.

There is a shortcut worth seeing. Removing 2 positive tokens leaves the bag exactly as adding 2 negative tokens would. So (−4) × 2 can be got by ADDING 2 negatives to the empty bag 4 times, with no zero pairs needed at all. So the four results stand: 4 × 2 = 8, 4 × (−2) = −8, (−4) × 2 = −8, and (−4) × (−2) = 8.