Mathematics • Year 9 • Unit 4 • Lesson 20
Unit 4 Synthesis in the Real World
Apply integrated problem solving across Lessons 16-19 in authentic STEM contexts: engineering a bridge, designing a tank model, planning a hike, surveying a school, and quality-testing a factory line. Then explain how to check the reasonableness of a multi-step answer.
1. Word problems
Each scenario combines TWO or more topics from Unit 4. Use the lesson's 5-step routine and show working.
1.1 — Civil engineering a bridge. An engineer is designing a steel arch bridge. The arch is approximately triangular with base 60 m and height 20 m at the peak (above road level).
(a) Find the angle the arch makes with the road at one end (use right triangle, half-base 30 m).
(b) Find the length of each diagonal steel beam from road to peak (the hypotenuse).
(c) A 1 : 200 scale model of the bridge is built. Find the model's beam length in cm. 5 marks
1.2 — Designing a water tank model. A real water tank is a cylinder of radius 1 m and height 2.5 m, made from steel that weighs 7.85 g/cm³. The architect builds a 1 : 20 scale model from the same steel.
(a) Find the real tank's capacity in litres (1 m³ = 1 000 L).
(b) Find the model's capacity in millilitres (1 cm³ = 1 mL).
(c) If the real tank weighs 580 kg, find the model's weight in grams. (Hint: weight scales like volume.) 5 marks
1.3 — Planning a hike via probability. Forecasting suggests P(rain on any given day in Sydney's Royal National Park during May) = 0.25, based on 20 years of weather data.
(a) Find P(NO rain on a particular day).
(b) A hiker plans a 3-day hike. Assuming rain on each day is independent, find P(no rain on all 3 days).
(c) Find P(at least one rainy day) using the complement rule. 4 marks
1.4 — School survey on screen time. A high school of 900 students wants to know the median daily screen time. A stratified random sample of 60 students is asked, with results: mean = 4.2 hours, median = 3.6 hours, IQR = 2.4 hours.
(a) Why is the sample stratified rather than convenience? (Reference Lesson 19's checklist.)
(b) Mean > median tells you what about skew?
(c) Estimate the total daily screen time across all 900 students, in hours. (Use the mean — the median doesn't give totals.) 4 marks
1.5 — Factory quality control + scale up. A bottling plant tests a sample of 800 bottles and finds 12 are leaking.
(a) Find the experimental P(leaking).
(b) If the plant produces 50 000 bottles per day, how many leaking bottles are expected per day?
(c) A second factory uses bottles that are double the size (k = 2 for linear dimensions). Assume the leak rate per bottle stays at the same 1.5%. How does the total volume of wasted liquid per day compare with the first plant, assuming the same number of bottles? 5 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate, working through a multi-step problem, ends up with a "building height" of 12 000 m. Without doing any new calculations, explain in your own words:
(i) Why this answer must be wrong (give a real-world reference point).
(ii) Which of the lesson's reasonableness checks it fails (refer to the "Does it match a rough estimate?" idea).
(iii) Which specific kind of error in a multi-step problem most often causes wildly wrong answers (refer to the lesson's misconceptions).
(iv) What concrete first step you'd recommend they take before redoing the problem from scratch.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Bridge arch
(a) tan θ = 20 ÷ 30 = 0.667 → θ = tan⁻¹(0.667) ≈ 33.7°.
(b) Beam length = √(20² + 30²) = √(400 + 900) = √1 300 ≈ 36.06 m. (Or: 30 ÷ cos 33.7° ≈ 36.06 m.)
(c) Model beam = 36.06 ÷ 200 = 0.180 m = 18.03 cm.
1.2 — Scale model of water tank
(a) V_real = π × 1² × 2.5 = 2.5π m³ ≈ 7.854 m³ = 7 854 L.
(b) Volume scale = (1/20)³ = 1/8 000. V_model = 7.854 ÷ 8 000 ≈ 0.000 982 m³ = 982 cm³ = 982 mL.
(c) Weight scales by k³ = 1/8 000. Model weight = 580 000 g ÷ 8 000 = 72.5 g.
1.3 — Rain on a 3-day hike
(a) P(no rain on a day) = 1 − 0.25 = 0.75.
(b) P(no rain on all 3 days) = 0.75³ = 0.4219 ≈ 0.42.
(c) P(at least one rainy day) = 1 − 0.4219 = 0.5781 ≈ 0.58. So there's about a 58% chance the hike gets at least one wet day.
1.4 — School survey on screen time
(a) Stratified is better because it ensures all year groups are proportionally represented; a convenience sample (e.g., just the students near the cafeteria) would over-represent some groups. This satisfies the lesson's "was the sample representative?" check.
(b) Mean (4.2) > median (3.6) → right skew — a few students with very high screen time pull the mean above the median.
(c) Estimated total daily screen time across 900 students = 4.2 × 900 = 3 780 student-hours per day. (The sample mean is the best estimate of the population mean; multiplying by total students gives the total.)
1.5 — Factory leaks
(a) P(leaking) = 12 ÷ 800 = 0.015 = 1.5%.
(b) Leaks per day = 50 000 × 0.015 = 750 bottles/day.
(c) Same number of leaking bottles per day (750), but each holds 2³ = 8× the volume. Total wasted liquid is 8× the first plant (for the same number of bottles produced).
2.1 — Explain your thinking (sample response)
(i) 12 000 m is absurd as a building height — the tallest building on Earth (Burj Khalifa) is 828 m, so a Year 9 problem would never legitimately give a number 14× larger. Even Australia's tallest building (Q1 in the Gold Coast) is only 322 m. (ii) This fails the lesson's "Does it match a rough estimate?" check: a quick mental estimate should give a number in the tens or hundreds of metres for any plausible building, not the kilometres. (iii) The most common cause is a unit error — e.g., mixing cm with m, or forgetting to divide by 100 — which the lesson lists as the second misconception of multi-step problems. A second common cause is using the wrong trig ratio (cos instead of sin) which puts the variable in the wrong place. (iv) Tell them to re-check the units at each step — write down the units beside each number — and to compute an order-of-magnitude estimate (e.g., "for a building, I expect 10-300 m") before redoing the problem.
Marking: 1 per part (i)–(iv). Award full marks for any clear, lesson-grounded answer that names a real-world reference and a unit-error/check-each-step concept.