Mathematics • Year 7 • Unit 1 • Lesson 4
Multiplying and Dividing Integers — Real World
Apply integer multiplication and division to repeated temperature drops, weekly debt repayments, scoring penalties in a game, and steady descents of a submarine. The sign rules stay the same — only the story changes.
1. Word problems
Each problem uses integer multiplication or division. Show your working — a single answer with no working only earns half marks.
1.1 — Steady temperature fall. A weather station records that the temperature in Bathurst falls by 3 °C every hour over 4 hours one winter night. (A fall of 3 °C per hour = a change of −3 °C per hour.)
(a) Write the total temperature change as a multiplication of integers.
(b) Evaluate it.
(c) If the temperature started at 5 °C, what is the temperature 4 hours later? 3 marks
1.2 — Repaying a debt. Mira owes her friend $48. She pays back $6 each week. We can write each weekly repayment as a change of +$6 to her balance. (Or, alternatively, "subtract a debt of $6 each week", which is −(−6) = +6.)
(a) Write a division of integers to find how many weeks it takes to clear the debt (use −48 ÷ −6 to keep the integer style).
(b) Evaluate it.
(c) After 5 weeks, what is Mira's balance? 3 marks
1.3 — Quiz game penalties. In a class quiz, every wrong answer takes 2 points off your score (a change of −2 per wrong answer). Lin gets 7 wrong answers in a row.
(a) Write the total score change as a multiplication of integers.
(b) Evaluate it.
(c) If Lin started on +10, what is Lin's score now? 3 marks
1.4 — Submarine descent. A submarine descends at a steady rate of 5 m every minute (a change of −5 m per minute). After some time, it has descended a total of 60 m (a change of −60 m from sea level).
(a) Write a division of integers to find how many minutes have passed.
(b) Evaluate it.
(c) If the submarine started at sea level (0 m), what is its depth (as an integer) at this point? 3 marks
1.5 — Sharing a loss. A small lemonade-stall partnership made a loss of $36 last weekend. The four friends running the stall agreed to share the loss equally — meaning each friend's balance changes by the same negative integer.
(a) Write a division of integers to find each friend's share of the loss.
(b) Evaluate it.
(c) If one friend already had $10 in their stall account, what is their balance after taking their share of the loss? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate sees (−4) × (−7) and writes "negative times negative — that's two negatives, so the answer is −28." In your own words, explain (i) which rule from Lesson 4 they have applied incorrectly, (ii) what (−4) × (−7) actually equals, and (iii) why "negative × negative" gives a POSITIVE answer in multiplication. Use either the "count the negatives" pattern or a real-life situation in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Temperature falls
(a) 4 × (−3).
(b) Different signs: 4 × 3 = 12, sign negative. Total change = −12 °C.
(c) Start at 5, add (−12): 5 + (−12) = −7 °C.
1.2 — Repaying a debt
(a) (−48) ÷ (−6) (alternatively 48 ÷ 6).
(b) Same signs: 48 ÷ 6 = 8 weeks (sign positive).
(c) After 5 weeks Mira has paid 5 × $6 = $30. Balance: −48 + 30 = −$18. She still owes $18.
1.3 — Quiz game penalties
(a) 7 × (−2).
(b) Different signs: 7 × 2 = 14, sign negative. Total change = −14 points.
(c) Start at +10, add (−14): 10 + (−14) = −4 points.
1.4 — Submarine descent
(a) (−60) ÷ (−5).
(b) Same signs: 60 ÷ 5 = 12 minutes (sign positive — time is always positive).
(c) After descending 60 m from 0 m, depth is −60 m.
1.5 — Sharing a loss
(a) (−36) ÷ 4.
(b) Different signs: 36 ÷ 4 = 9, sign negative. Each share = −$9.
(c) Friend's balance: $10 + (−$9) = +$1. They are still just above zero.
2.1 — Explain your thinking (sample response)
My classmate has used the wrong rule. Counting "two negatives" works for addition/subtraction ("two minus signs side by side become a plus"), but for multiplication and division the rule is different: same signs → positive, different signs → negative. With (−4) × (−7) both signs are negative (same), so the answer is positive. Multiply the magnitudes: 4 × 7 = 28. Answer: (−4) × (−7) = +28. A nice way to picture it: imagine a debt of $4 being taken away from you 7 times (you LOSE 7 debts of $4). Losing a debt is the same as gaining money, so you end up $28 better off — a positive change. The "count the negatives" pattern also confirms it: 2 negative factors is an EVEN number, so the answer is positive.
Marking: 1 for naming the addition vs multiplication rule confusion; 1 for stating "same signs → positive"; 1 for the correct answer +28; 1 for the real-life example or even-count justification.