Math 8 · Multiplying and dividing integers
Groups of negatives, and the signs they leave
Multiplying is repeated adding, and that alone explains most of the sign rules: three groups of −4 make −12. A pattern explains the rest, including why a negative times a negative is positive. Dividing runs multiplying backwards, so it follows the same rules, and the order of operations decides which step comes first.
- 1. The words, first
- 2. Multiplying integers
- 3. Dividing integers
- 4. Order of operations with integers
- 5. What costs marks
The words, first
The idea: Product and quotient are the two answers this unit is about: what multiplying gives, and what dividing gives.
| Word | What it means |
|---|---|
| Integer | A whole number that is positive, negative or zero: …, −2, −1, 0, 1, 2, … A number with no sign is positive. |
| Product, factor | The answer to a multiplication, and the numbers multiplied. In 3 × (−4) = −12, the product is −12. |
| Quotient | The answer to a division. In (−12) ÷ 3 = −4, the quotient is −4. |
| Integer tiles, zero pair | Counters for +1 and −1. One of each makes a zero pair, worth 0, so adding or removing zero pairs never changes a value. |
| Mean | Add the values, then divide by how many there are. |
| Order of operations | Brackets first. Then multiply and divide, left to right. Then add and subtract, left to right. |
Multiplying integers
The idea: Multiply the sizes as usual. Then count the negative factors: an even number of them gives a positive product, an odd number a negative one.
Repeated adding. 3 × (−4) means three groups of −4: (−4) + (−4) + (−4) = −12. On a number line that is three jumps of 4 to the left from 0; with tiles, three groups of four negatives. Order does not matter, so (−4) × 3 = −12 too.
Why a negative times a negative is positive. Follow a pattern. Each step multiplies −4 by one less, and the product goes up by 4:
| −4 × | 3 | 2 | 1 | 0 | −1 | −2 | −3 |
|---|---|---|---|---|---|---|---|
| Product | −12 | −8 | −4 | 0 | 4 | 8 | 12 |
Nothing makes the pattern turn round at zero, so (−4) × (−1) = 4 and (−4) × (−3) = 12. With tiles: start with zero pairs and take away three groups of four negatives, and twelve positives are left.
| Signs | Product | Example |
|---|---|---|
| positive × positive | positive | 3 × 4 = 12 |
| positive × negative | negative | 3 × (−4) = −12 |
| negative × positive | negative | (−3) × 4 = −12 |
| negative × negative | positive | (−3) × (−4) = 12 |
More than two factors. Count the negatives. (−2) × (−3) × (−5) has three, an odd number, so it is −30.
Worked example. The temperature is falling 3 °C an hour. Five hours from now, the change is 5 × (−3) = −15: 15 degrees colder. Five hours ago is −5 hours, and (−5) × (−3) = 15: it was 15 degrees warmer than now. Going back in time undoes the cooling — a negative times a negative, making sense.
Dividing integers
The idea: Every division is a multiplication run backwards, so the sign rules are the same: same signs give a positive quotient, different signs a negative one.
(−12) ÷ 3 = −4, because 3 × (−4) = −12 · (−12) ÷ (−3) = 4, because (−3) × 4 = −12 · 12 ÷ (−3) = −4, because (−3) × (−4) = 12
Picture it. (−12) ÷ 3 shares 12 negative tiles into 3 groups: −4 in each. (−12) ÷ (−3) asks how many groups of −3 make −12: 4.
Worked example. A weather station recorded these daily lows one week in January: −8, −3, 2, −7, −12, −1 and 1 °C. What was the mean low?
- Add: (−8) + (−3) + 2 + (−7) + (−12) + (−1) + 1 = −28.
- Divide by how many: (−28) ÷ 7 = −4 °C.
- Check the sign. Most days were below zero, so the mean should be negative. Writing 4 would say the week was mild.
Zero. 0 ÷ (−5) = 0, because (−5) × 0 = 0. But (−5) ÷ 0 has no answer: nothing times 0 makes −5.
Order of operations with integers
The idea: Brackets, then multiplying and dividing from left to right, then adding and subtracting from left to right. Keep every negative sign attached to its number.
Worked example. Evaluate −8 + 6 × (−3) ÷ 2 − (−5).
- Multiply and divide, left to right: 6 × (−3) = −18, then (−18) ÷ 2 = −9. The line is now −8 + (−9) − (−5). (The brackets round −3 and −5 only keep the signs clear.)
- Add and subtract, left to right: −8 + (−9) = −17, then −17 − (−5) = −17 + 5 = −12.
The classic trap. In −3 − 4 × (−2), multiply first: 4 × (−2) = −8, and −3 − (−8) = −3 + 8 = 5. Going straight from left to right gives (−7) × (−2) = 14.
Left to right matters. In (−4 − 6) ÷ (−2) × 3, the bracket gives −10, then (−10) ÷ (−2) = 5, and 5 × 3 = 15. Multiplying first gives (−10) ÷ (−6), not even a whole number.
Worked example: a quiz. A right answer scores +3, a wrong one −2 and a blank 0. Jo gets 7 right and 5 wrong, and leaves 3 blank: 7 × 3 + 5 × (−2) + 3 × 0 = 21 + (−10) + 0 = 11.
What costs marks
The idea: Nearly all of these are a sign that went missing or a step taken out of order.
- Using the multiplying rule for adding. Two negatives multiply to a positive, but −3 + (−5) = −8.
- The wrong sign with three negatives. An odd number of negative factors gives a negative product.
- Forgetting that dividing has sign rules too. (−20) ÷ (−4) = 5, not −5.
- Going left to right past a multiplication. −3 − 4 × (−2) = 5, not 14.
- Losing a negative between lines. Copy −8 + (−9) exactly, sign and all.