Why Does a Negative Times a Negative Make a Positive? | KS3 Maths

Start at zero, facing right. $3 \times 4$ means three jumps of four: $4$, $8$, $12$.

$3 \times 4 = +12$

In $3 \times -4$ it is still three jumps, but each jump is negative four. A negative sign tells you to do one thing: turn around. Now facing left, the same three jumps of four land on $-12$.

$3 \times -4 = -12$

Write out the times table for $-4$, starting with the rows you can check by counting lots of $-4$:

$3 \times -4 = -12,\quad 2 \times -4 = -8,\quad 1 \times -4 = -4,\quad 0 \times -4 = 0$

Every row, the answer goes up by four. That makes sense: each row has one fewer lot of $-4$ in it, so the answer climbs back up by four. Keep the pattern going past zero and it has to carry on climbing:

$-1 \times -4 = +4,\quad -2 \times -4 = +8,\quad -3 \times -4 = +12$

If $-3 \times -4$ were negative, the pattern would suddenly break for no reason.

The negative sign on the $3$ does exactly the same job as the one on the $4$: it turns you around. So "negative three jumps" just means three jumps, turned around. With one negative sign, $-3 \times 4$ is one turn and three jumps left: $-12$.

With two negative signs, turn around, then turn around again. You are facing right, the way you started, so the three jumps of four go right:

$-3 \times -4 = +12$

Two completely different reasons, the pattern and the turning, give the same answer.

Yes. Dividing undoes multiplying, so ask: what times $-4$ makes $12$? It is $-3$, because $-3 \times -4 = 12$. The signs are different, so the answer is negative:

$12 \div -4 = -3$

Sign first: count the negative signs. There are three, so that is three turns, which leaves you facing left: the answer is negative. Then the size: $2 \times 3 \times 1 = 6$.

$-2 \times -3 \times -1 = -6$

An even number of negatives gives a positive; an odd number gives a negative.