Mathematics • Year 7 • Unit 1 • Lesson 4
Multiplying and Dividing Integers — Mixed Challenge
Combine sign rules for × and ÷ with brackets and the add/subtract rules from Lesson 3. Choose the right tool for each problem, spot a sign-rule mistake, and tackle an open-ended factor puzzle.
1. Mixed problems — choose the right rule
Each question uses a different combination of × , ÷ , + and − with integers. Decide the sign of each step before you calculate. Show your working. 2 marks each
1.1 Calculate (−7) × (−6).
1.2 Calculate 56 ÷ (−8).
1.3 Calculate (−5) × 4 × (−2).
1.4 Calculate (−45) ÷ (−9) × 2.
1.5 Calculate (−6) × 3 + (−10) ÷ (−2). (Hint: × and ÷ are done before + .)
1.6 Without doing the full calculation, predict whether (−1) × (−2) × (−3) × (−4) × (−5) is positive or negative. Then evaluate it.
2. Find the mistake
Another Year 7 student has tried to calculate (−4) × (−3) × 2. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — calculate (−4) × (−3) × 2:
Line 1: First: (−4) × (−3). Different signs → answer is negative.
Line 2: 4 × 3 = 12, so (−4) × (−3) = −12.
Line 3: Now: −12 × 2. Different signs → negative.
Line 4: 12 × 2 = 24, so final answer = −24.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Look carefully at Line 1: are −4 and −3 the SAME sign or DIFFERENT signs? That decides whether (−4) × (−3) is positive or negative.3. Open-ended challenge — different ways to multiply to −24
This question has more than one correct answer. Show two and explain. 4 marks
3.1 Find three different multiplications of the form (integer) × (integer) that equal −24. Each multiplication must use exactly two integers, and at least one of the two integers must be negative.
For each multiplication you find: (a) write it down; (b) check that the magnitudes multiply to 24; (c) explain why the answer comes out negative.
Bonus: Now find a multiplication of three integers whose product is −24, where all three integers are different.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — (−7) × (−6)
Same signs → positive. 7 × 6 = 42. Answer: +42.
1.2 — 56 ÷ (−8)
Different signs → negative. 56 ÷ 8 = 7. Answer: −7.
1.3 — (−5) × 4 × (−2)
Magnitudes: 5 × 4 × 2 = 40. Count negatives: 2 (even) → positive. Answer: +40.
Check pair by pair: (−5) × 4 = −20; −20 × (−2) = +40. ✓
1.4 — (−45) ÷ (−9) × 2
(−45) ÷ (−9): same signs → +5.
+5 × 2 = +10.
1.5 — (−6) × 3 + (−10) ÷ (−2)
Do × and ÷ before +:
(−6) × 3 = −18 (different signs).
(−10) ÷ (−2) = +5 (same signs).
Now add: −18 + 5 = −13.
1.6 — (−1) × (−2) × (−3) × (−4) × (−5)
Count negatives: 5 (odd) → answer negative.
Magnitudes: 1 × 2 × 3 × 4 × 5 = 120.
Answer: −120.
2 — Find the mistake
(a) The mistake is on Line 1 (which then makes Line 2's sign wrong).
(b) The student called −4 and −3 "different signs", but they're both negative — that's same signs, which gives a POSITIVE answer. The student confused the multiplication rule with the addition rule for "two minus signs".
(c) Corrected working:
Line 1 (fixed): (−4) × (−3) — same signs → positive.
Line 2: 4 × 3 = 12, so (−4) × (−3) = +12.
Line 3: Now +12 × 2 — same signs → positive.
Line 4: 12 × 2 = 24, so final answer = +24.
"Count the negatives" check: 2 negatives (even) → positive answer. ✓
3 — Open-ended challenge (sample solutions)
Factor pairs of 24: 1 × 24, 2 × 12, 3 × 8, 4 × 6.
For the product to equal −24, exactly one of the two integers must be negative (different signs → negative).
Three sample multiplications:
(i) (−4) × 6 = −24. Magnitudes 4 × 6 = 24 ✓. Different signs → negative ✓.
(ii) 8 × (−3) = −24. Magnitudes 8 × 3 = 24 ✓. Different signs → negative ✓.
(iii) (−12) × 2 = −24. Magnitudes 12 × 2 = 24 ✓. Different signs → negative ✓.
Other valid pairs: (−1) × 24, (−24) × 1, (−2) × 12, (−6) × 4, (−8) × 3 — any pair where one is negative and the magnitudes multiply to 24.
Bonus — three different integers: for example (−2) × 3 × 4 = (−6) × 4 = −24. ✓ (One negative + two positives = odd count of negatives = negative answer.)
Other examples: (−1) × 4 × 6, (−3) × 2 × 4, 1 × (−4) × 6.
Marking: 3 marks for three valid distinct two-integer products with sign reasoning shown; 1 mark for a valid three-integer product.