Patterns in integer multiplication

The token model gave us four results. Here is a second route to the same place, and it may convince you more, because it uses nothing but a pattern you can see.

Write a sequence, keeping the multiplicand fixed at 3 and stepping the multiplier down by one each time. 4 × 3 = 12, 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0.

What do you notice? When the multiplicand is positive, for every unit DECREASE in the multiplier, the product decreases by the multiplicand — here it drops by 3 each line.

Now the important question: will the pattern continue when the multiplier goes below zero? If it does, the next lines must be −1 × 3 = −3, then −2 × 3 = −6, then −3 × 3 = −9. And they do. The pattern continues exactly, and it agrees with what the tokens told us.

Now make the multiplicand negative and do the same. 4 × (−3) = −12, 3 × (−3) = −9, 2 × (−3) = −6, 1 × (−3) = −3, 0 × (−3) = 0.

This is the inverse of the first pattern. When the multiplicand is negative, for every unit decrease in the multiplier, the product INCREASES by the multiplicand — here it rises by 3 each line.

Continue below zero and the same steps carry on: (−1) × (−3) = 3, (−2) × (−3) = 6, (−3) × (−3) = 9. So what is true of multiplication when the integers are positive stays true when they are negative — and the two negatives giving a positive is not a special rule bolted on, but the only way the pattern can continue.