Multiplying and dividing signed numbers has its own sign rules. They are simple, but they belong to multiplying and dividing only.
Skill 0.01 reviewed adding and subtracting signed numbers. Multiplying and dividing follow different rules, and keeping the two sets apart is the main job.
The good news: for multiplying and dividing, the sign is decided by one short rule, and the sizes are multiplied or divided as usual.
Multiplying by a whole number means repeated addition. So 3 × (−2) is three jumps of −2.
Figure 1. Multiplying by 3 repeats the jump three times.
Starting at 0, three jumps of 2 to the left land on −6. So 3 × (−2) = −6, and (−2) × 3 gives the same, because order does not matter in multiplying.
Dividing undoes multiplying: −6 ÷ 3 = −2.
Multiplying or dividing: same signs give a positive answer, different signs give a negative answer.
A pattern shows why two negatives multiply to a positive.
Figure 2. The pattern forces negative times negative to be positive.
As the first number drops by 1, the product goes up by 2. Continuing the pattern past zero, −1 × (−2) must be 2, and −3 × (−2) must be 6.
Division follows the same rules, because each division is a multiplication run backwards: 6 ÷ (−2) = −3 because (−3)(−2) = 6.
The two sets of rules are easily mixed. The multiplication rule gets used for adding, and the adding idea gets used for multiplying.
Figure 3. Each sign rule belongs to its own operation.
On the left, −3 + (−4) is an addition. Both moves go left, so the answer is −7. “Two negatives make a positive” does not apply.
On the right, (−3)(−4) is a multiplication. Same signs give a positive, so the answer is 12, not −12.
For any multiplication or division, settle the sign first, then work with the sizes.
Figure 4. Decide the sign first, then multiply or divide the sizes.
With several factors, count the negatives. Each pair of negatives makes a positive, so an even count gives a positive answer and an odd count a negative one.
(−4) × 5
Different signs: negative. 4 × 5 = 20, so −20.
−36 ÷ (−9)
Same signs: positive. 36 ÷ 9 = 4, so 4.
× and ÷: SAME signs +, DIFFERENT signs −.
Skill 0.03 reviews the four operations with fractions.
Cover the right-hand column with your hand. Answer each one out loud, then slide your hand away and check.
| (−6)(−5) = ? | 30. |
| −48 ÷ 6 = ? | −8. |
| −3 + (−4) = ? | −7. |
| (−1)(−1)(−1)(−1) = ? | 1. |
| Why is (−3)(−4) not −12? | Same signs multiply to a positive: 12. |