Mathematics • Year 7 • Unit 1 • Lesson 4
Multiplying and Dividing Integers
Build the sign rule for × and ÷: same signs → positive answer; different signs → negative answer. Multiply or divide the magnitudes as usual, then attach the sign.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Calculate (−8) × (−5).
Step 1 — Check the signs.
−8 and −5 → both negative → same signs.
Reason: the sign rule is the FIRST thing to settle before you multiply. Same sign means a positive answer.
Step 2 — Multiply the magnitudes.
8 × 5 = 40
Reason: ignore the signs and just multiply like normal whole numbers.
Step 3 — Apply the sign.
Same signs → positive answer → +40
Reason: positive × positive = positive, and negative × negative = positive. Two negatives "cancel" each other in multiplication.
Step 4 — Sanity check (mini-pattern).
(−1) × (−1) = +1. So adding more negative factors must keep giving positives — confirms our sign.
Reason: count the negative signs. Even number → positive. Odd number → negative. Here we had 2 negatives → positive.
Answer: (−8) × (−5) = +40.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Calculate (−6) × 7.
Step 1 — Check the signs: −6 and +7 → __________ (same / different) signs.
Step 2 — Multiply the magnitudes:
6 × 7 = _______
Step 3 — Apply the sign: different signs → __________ (positive / negative) answer.
(−6) × 7 = _______
Step 4 — Pattern check: there is ____ negative sign in the question, which is an ____ (even / odd) number, so the answer must be ____ (positive / negative). This matches Step 3.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Calculate (−4) × 3. 1 mark
3.2 Calculate (−9) × (−2). 1 mark
3.3 Calculate 20 ÷ (−4). 1 mark
3.4 Calculate (−18) ÷ (−6). 1 mark
Standard — combine two ideas
3.5 Calculate (−3) × (−4) × (−1). Use the "count the negatives" pattern to check your sign. 2 marks
3.6 Calculate (−36) ÷ 9 + 2. 2 marks
Extension — push your thinking
3.7 Evaluate [24 ÷ (−3)] × (−2) + (−4). 3 marks
3.8 Without doing the full multiplication, decide whether each answer is positive or negative. Explain how you decided.
(a) (−2) × (−3) × (−5) × (−7) (b) (−1) × 4 × (−6) × 2 × (−1). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do ((−6) × 7)
Step 1: different signs (−6 negative, +7 positive).
Step 2: 6 × 7 = 42.
Step 3: different signs → negative; (−6) × 7 = −42.
Step 4: 1 negative sign, an odd number, so the answer must be negative — matches Step 3.
3.1 — (−4) × 3
Different signs: 4 × 3 = 12, sign negative. Answer: −12.
3.2 — (−9) × (−2)
Same signs: 9 × 2 = 18, sign positive. Answer: +18.
3.3 — 20 ÷ (−4)
Different signs: 20 ÷ 4 = 5, sign negative. Answer: −5.
3.4 — (−18) ÷ (−6)
Same signs: 18 ÷ 6 = 3, sign positive. Answer: +3.
3.5 — (−3) × (−4) × (−1)
Magnitudes: 3 × 4 × 1 = 12. Count negatives: 3 (odd) → answer negative. Answer: −12. Check by pairing: (−3) × (−4) = +12, then +12 × (−1) = −12. ✓
3.6 — (−36) ÷ 9 + 2
Division first: (−36) ÷ 9 = −4 (different signs).
Then add: −4 + 2 = −2.
3.7 — [24 ÷ (−3)] × (−2) + (−4)
Brackets first: 24 ÷ (−3) = −8 (different signs, 24 ÷ 3 = 8, sign negative).
Multiply: (−8) × (−2) = +16 (same signs).
Add: 16 + (−4) = 12.
3.8 — Count the negatives
(a) (−2) × (−3) × (−5) × (−7) has 4 negative factors → even → answer is positive.
(b) (−1) × 4 × (−6) × 2 × (−1) has 3 negative factors → odd → answer is negative.
Rule: count only the negative factors; whether or not there are positives doesn't change the sign.