Mathematics • Year 8 • Unit 1 • Lesson 13
Unitary Method in the Real World
Use “find 1 first, then scale” in real situations: tutoring fees, supermarket prices, road-trip planning, paint coverage. Then explain why the unitary method is your most reliable maths habit.
1. Word problems
Each problem uses the “find 1 first” idea. Show your two-step working — a single final answer with no working only earns half marks.
1.1 — Tutoring fees. Lara's first 3 hours of tutoring cost her $\$45$ in total.
(a) Find the per-hour rate.
(b) What does a 5.5-hour week cost at the same rate?
(c) Lara has a budget of $\$120$ for tutoring this month. How many hours can she afford? 3 marks
1.2 — Tomatoes by the kilo. Jay's mum pays $\$24$ for 4 kg of tomatoes at the market.
(a) Find the price per kg.
(b) What does 7 kg cost at the same rate?
(c) For $\$15$, how many whole kg can she buy (round DOWN, you can't go over)? 3 marks
1.3 — Road trip planning. On a test drive, a car covers 270 km in 3 hours of steady motorway driving. The driver wants to plan a 450 km trip at the same speed.
(a) Find the car's average speed in km/h.
(b) How long would 450 km take at the same speed?
(c) If the driver leaves at 9:00 am, what time will they arrive? 3 marks
1.4 — Painting the classroom. A tin of paint covers 24 m² and costs $\$36$. The classroom needs to be repainted: 4 walls, each 5 m wide and 3 m high, plus a ceiling 5 m by 4 m. (Ignore doors and windows.)
(a) Find the cost per m² covered.
(b) Find the total area to be painted.
(c) Find the total cost of paint at the unit rate from (a). 3 marks
1.5 — Shared pizza bill. A group of 6 friends share a pizza order that costs $\$48$ in total. Two more friends arrive and they reorder the same kind of pizzas, this time for 8 people.
(a) Find the per-person cost of the first order.
(b) At the same per-person rate, what will the bill be for 8 people?
(c) If the new arrivals only have $\$5$ each, can they afford the same per-person amount? Explain. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A friend looks at this problem — “3 pens cost $\$4.50$, find the cost of 7 pens” — and just writes “$7 \times \$4.50 = \$31.50$”. In your own words, explain (i) what mistake they have made, (ii) what the unitary method would do instead, (iii) what the correct answer is, and (iv) why the unitary method “always works” for proportional situations. Use the phrase “find 1 first” somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Tutoring
(a) Per hour: $\$45 \div 3 = \textbf{\$15/h}$.
(b) 5.5 hours: $5.5 \times \$15 = \textbf{\$82.50}$.
(c) $\$120 \div \$15/h = \textbf{8 hours}$ exactly.
1.2 — Tomatoes
(a) Per kg: $\$24 \div 4 = \textbf{\$6/kg}$.
(b) 7 kg: $7 \times \$6 = \textbf{\$42}$.
(c) $\$15 \div \$6 = 2.5$ kg theoretically, but whole kg only → 2 kg (costing $\$12$, leaving $\$3$).
1.3 — Road trip
(a) Speed: $270 \div 3 = \textbf{90 km/h}$.
(b) Time for 450 km: $450 \div 90 = \textbf{5 hours}$.
(c) Leaving 9:00 am → arrive at 2:00 pm.
1.4 — Painting
(a) Cost per m²: $\$36 \div 24 = \textbf{\$1.50/m²}$.
(b) Total area = 4 walls (4 × 5 × 3 = 60 m²) + ceiling (5 × 4 = 20 m²) = 80 m².
(c) Total cost: $80 \times \$1.50 = \textbf{\$120}$.
1.5 — Pizza bill
(a) Per person: $\$48 \div 6 = \textbf{\$8/person}$.
(b) 8 people at $\$8$ each: $8 \times \$8 = \textbf{\$64}$.
(c) $\$5 \lt \$8$ — so no, the new arrivals can't cover their share at the same per-person rate; they'd be $\$3$ short each.
2.1 — Explain your thinking (sample response)
The friend multiplied the total 3-pen price by 7 instead of multiplying the per-pen price by 7. The unitary method would find 1 first: $\$4.50 \div 3 = \$1.50$ per pen, then multiply by 7 to get $\$10.50$ — that's the correct answer. The friend's $\$31.50$ is actually the cost of 21 pens (3 pens × 7), not 7 pens, which you can spot if you sanity-check: 7 pens should cost just over twice as much as 3 pens (about $\$9$), not seven times. The unitary method always works for proportional situations because the per-1 rate captures the whole relationship in one number, and multiplying by any new quantity gives the matching total — no matter whether the new number is bigger, smaller, or a fraction.
Marking: 1 mark for identifying the “multiplied the wrong thing” mistake; 1 mark for showing the “find 1 first” step ($\$4.50 \div 3 = \$1.50$); 1 mark for the correct answer $\$10.50$; 1 mark for explaining why the method works in general (one number captures the proportional relationship).