B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional ReasoningMultiplying Integers
B2.7 A1About 45 minutes
Learning goals- I can multiply a positive and a negative integer and explain the answer as repeated addition.
- I can use a pattern to explain why a negative times a negative is positive.
- I can find the sign of a product with many factors by counting the negative signs.
- I can evaluate powers with negative bases, like (−2)³.
Words to know
product
The answer to a multiplication.
factor
A number being multiplied. In (−3) × 5, the factors are −3 and 5.
sign
Whether a number is positive (+) or negative (−). Zero has no sign.
negative base
A power whose base is negative, written in brackets, like (−2)⁴.
New in Grade 8
In Grade 7 you added and subtracted integers. Multiplying and dividing integers is new this year. The good news: you already know all the multiplication facts. The only new thing is deciding the sign of the answer.
Let's start with a cold front
A cold front moves in, and the temperature falls 3 degrees every hour for 4 hours. What is the total change?
Each hour the change is −3. Four hours of that is (−3) + (−3) + (−3) + (−3) = −12. As a multiplication: 4 × (−3) = −12. The temperature dropped 12 degrees.
Positive × negative
Multiplication is repeated addition, so 3 × (−4) means 3 groups of −4, which is −12. Because multiplication is commutative, (−4) × 3 = −12 too.
A positive times a negative is negative.
| Multiplication | Product |
|---|
| (−3) × (+3) | −9 |
| (−3) × (+2) | −6 |
| (−3) × (+1) | −3 |
| (−3) × 0 | 0 |
| (−3) × (−1) | +3 |
| (−3) × (−2) | +6 |
| (−3) × (−3) | +9 |
Negative × negative
The pattern in the table shows that a negative times a negative is positive.
You can also see it with chips. (−2) × (−3) means "remove 2 groups of (−3)". Start with zero: put out 6 zero pairs. Remove 2 groups of 3 negative chips. 6 positive chips are left, so (−2) × (−3) = +6.
The sign rules
- Same signs (+ × + or − × −) → positive product.
- Different signs (+ × − or − × +) → negative product.
- Many factors: count the negative factors. An even number of negatives gives a positive product; an odd number gives a negative product. (If any factor is 0, the product is 0.)
Powers with negative bases
(−2)² = (−2) × (−2) = +4, and (−2)³ = (−2) × (−2) × (−2) = −8. Brackets matter: −2² means −(2 × 2) = −4, because the exponent belongs only to the 2.
Worked exampleExample 1: Different signs
- Different signs → negative.
- 6 × 7 = 42.
Worked exampleExample 2: Same signs
- Same signs → positive.
- 8 × 5 = 40.
Worked exampleExample 3: Many factors
Find (−2) × (+3) × (−5) × (−1).
- Count the negative factors: −2, −5 and −1. That is 3, an odd number, so the product is negative.
- Size: 2 × 3 × 5 × 1 = 30.
Worked exampleExample 4: In context
Every week, $15 is taken from Owen's savings for a bus pass. What is the change in his savings after 6 weeks?
- Each week the change is −15.
- 6 × (−15) = −90.
−$90 (his savings go down by $90)
Worked exampleExample 5: Brackets and exponents
- (−3)² = (−3) × (−3) = +9.
- −3² = −(3 × 3) = −9.
Sign first, then size
Decide the sign of the answer before you multiply. Then multiply the numbers as if they were positive. This two-step habit prevents most mistakes.
Hands-on activityCoin chips
You need: about 20 coins of two kinds (for example nickels and dimes), or two kinds of buttons. Decide that one kind is +1 and the other is −1.
- Show 3 × (−2): put out 3 groups of 2 negative coins. Count: −6.
- Show (−3) × (+2): start with 6 zero pairs (a + coin and a − coin side by side, worth 0). Remove 3 groups of 2 positive coins. What is left?
- Show (−3) × (−2): start with 6 zero pairs. Remove 3 groups of 2 negative coins. What is left?
- Write a sentence that explains why removing negatives leaves a positive.
Model answers: 1. −6. 2. Six negative coins are left: −6. 3. Six positive coins are left: +6. 4. Taking away debts (negatives) leaves you better off, so the result is positive.
Just for funEnglish grammar sometimes agrees with integer rules: two negatives make a positive. So if you say "I don't have no homework," you just admitted you do have homework.
Your turn
−1. Three negatives (odd), and 1 × 1 × 1 = 1.
−27. (−3) × (−3) = +9, then (+9) × (−3) = −27.
+60. Two negatives (even) → positive. 10 × 2 × 3 = 60.
7The temperature drops 2°C each hour for 5 hours. What is the total change?
−10 degrees. 5 × (−2) = −10.
More practice: warm-up
0. Anything times 0 is 0.
−12. Multiplying by −1 gives the opposite.
More practice: on your own
+60. Same signs; 12 × 5 = 60.
+16. Four negative factors (even) → positive. 2⁴ = 16.
+300. Two negatives → positive. 4 × 25 = 100, and 100 × 3 = 300.
−5² = −25 and (−5)² = +25. Without brackets, only 5 is squared, and then the negative is applied.
5A submarine dives at a steady 12 m per minute. What is its change in depth after 7 minutes?
−84 m. 7 × (−12) = −84. It is 84 m deeper.
Challenge
1Find four different integers whose product is −24.
Many answers. For example (−1) × 2 × 3 × 4 = −24, or (−1) × (−2) × (−3) × 4 = −24. The product is negative when there is an odd number (1 or 3) of negative factors.
2Use the distributive property to find (−6) × 98 in your head.
−588. (−6) × (100 − 2) = (−6) × 100 − (−6) × 2 = (−600) − (−12) = −600 + 12 = −588. Check: 6 × 98 = 588, and different signs make it negative. ✓
3Is (−1)¹⁰⁰ positive or negative? What about (−1)⁹⁹?
(−1)¹⁰⁰ = +1 because 100 negative factors is an even number. (−1)⁹⁹ = −1 because 99 is odd.
Quiz
3. What is the sign of (−2) × (−3) × (−4) × (+5)?
5. A tank loses 3 L of water each minute for 8 minutes. What is the change in the amount of water?
6. Which product equals +20?
Big idea: Ontario's Grade 8 expectation B2.7 asks students to multiply and divide integers, using appropriate strategies, in various contexts. This is new in Grade 8. Students should be able to explain the sign rules (using repeated addition, patterns or chips), not just memorize them.
Common difficulties:
- Mixing up the addition and multiplication rules. (−3) + (−4) = −7, but (−3) × (−4) = +12. Ask "Are we adding or multiplying?"
- Brackets with exponents: −3² = −9, but (−3)² = +9.
- Many factors: encourage counting the negative signs first.
How to help: Use real situations: "You spend $4 a day for 5 days; what is the change in your money?" (5 × (−4) = −20.) Ask your child to explain the pattern table in their own words.