Mathematics • Year 10 • Unit 4 • Lesson 13
Line of Best Fit in the Real World
Apply Lesson 13's line of best fit + interpolation/extrapolation skills to real Australian contexts: house prices, exam study, surf forecasting and fuel economy. Then justify when a prediction is trustworthy and when it isn't.
1. Word problems
For each scenario, do the calculation and then judge whether the prediction is reliable. A "yes" or "no" alone earns half marks — quote the data range.
1.1 — House prices vs distance from CBD. Sydney median house prices (in $M) at various distances (km) from the CBD:
(5, 1.8), (10, 1.5), (15, 1.2), (20, 1.0), (25, 0.85), (30, 0.75).
(a) Find the mean point (x̄, ȳ) and the slope using (5, 1.8) and (30, 0.75).
(b) Write the equation of the line of best fit.
(c) Use it to predict the price at 12 km (interpolation) and at 100 km (extrapolation). Comment on the reliability of each. 4 marks
1.2 — Study hours vs exam score. A Year 10 student records weekly:
(study hr/week, test score %) = (2, 55), (4, 62), (6, 68), (8, 75), (10, 82), (12, 88).
(a) Estimate the slope using the first and last points.
(b) Use the mean point to write the equation y = mx + c.
(c) Predict the score for 7 hr/week and 25 hr/week. Which is interpolation? Why is the other one unrealistic — quote the misconception card. 4 marks
1.3 — Fuel economy. Average fuel use (L/100 km) of a sedan increases with cruising speed:
(60, 6.5), (80, 7.0), (100, 8.2), (110, 9.0), (120, 10.5).
(a) Compute the mean point.
(b) Estimate the slope from (60, 6.5) and (120, 10.5).
(c) Predict fuel use at 90 km/h and 150 km/h. State which is interpolation, and explain why the 150 km/h prediction is risky. 3 marks
1.4 — Surf wave height. A coastal surf model predicts wave height (m) from wind speed (km/h) on the day:
Line of best fit: y = 0.05x + 0.4, with data range x = 10 to x = 60 km/h.
(a) Predict the wave height for 35 km/h winds.
(b) Predict the wave height for 100 km/h winds.
(c) For each, state whether it is interpolation or extrapolation, and which prediction you would trust. 3 marks
1.5 — Pandemic projection. A 2019 model used data from 2010–2019 to predict café sales in 2025. The line of best fit gave a confident upward projection. By 2025, café sales had actually dropped sharply due to a global event no one had modelled.
(a) Is the 2025 prediction interpolation or extrapolation? Justify with the data range.
(b) Using Lesson 13's misconception card, write one sentence explaining why this prediction failed. 3 marks
2. Explain your thinking
This question is about justifying a prediction, not just calculating. Use full sentences. 4 marks
2.1 A marketing analyst tells the boss: "Our line of best fit shows monthly online sales rising by $2,000 per month — by 2035 we'll be selling $1 million a month, problem solved!" Using Lesson 13's misconceptions card and the interpolation/extrapolation distinction, write a four-sentence reply that (i) identifies whether the 2035 figure is interpolation or extrapolation, (ii) explains why the linear trend may not hold, (iii) names ONE external factor that could break the trend, and (iv) suggests a more honest way to report the prediction.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — House prices vs CBD distance
(a) x̄ = (5+10+15+20+25+30)/6 = 105/6 = 17.5. ȳ = (1.8+1.5+1.2+1.0+0.85+0.75)/6 = 7.1/6 ≈ 1.183.
Slope: m = (0.75 − 1.8)/(30 − 5) = −1.05/25 = −0.042.
(b) 1.183 = −0.042 × 17.5 + c → c = 1.183 + 0.735 = 1.918. Equation: y = −0.042x + 1.918.
(c) At x = 12: y = −0.042 × 12 + 1.918 ≈ 1.42 ($M) — interpolation, reliable. At x = 100: y = −0.042 × 100 + 1.918 ≈ −2.28 ($M) — extrapolation, NEGATIVE price means the linear trend has broken down; 100 km is rural and the relationship is no longer linear.
1.2 — Study hours vs score
(a) Slope from (2, 55) and (12, 88): m = (88 − 55)/(12 − 2) = 33/10 = 3.3.
(b) Mean point: x̄ = 42/6 = 7; ȳ = 430/6 ≈ 71.67. So 71.67 = 3.3 × 7 + c → c = 71.67 − 23.1 = 48.57. Equation: y = 3.3x + 48.57.
(c) At x = 7: y = 3.3 × 7 + 48.57 ≈ 71.7 % (interpolation, within 2–12 hr). At x = 25: y = 3.3 × 25 + 48.57 ≈ 131 % — impossible (over 100 %). Extrapolation. Lesson misconception: a trend will not continue forever; at high study hours, returns diminish.
1.3 — Fuel economy
(a) x̄ = 470/5 = 94, ȳ = 41.2/5 = 8.24.
(b) m = (10.5 − 6.5)/(120 − 60) = 4/60 ≈ 0.067.
(c) At 90 km/h: y ≈ 0.067 × 90 + c. c = 8.24 − 0.067 × 94 ≈ 8.24 − 6.298 ≈ 1.942. So y(90) ≈ 0.067 × 90 + 1.942 ≈ 7.97 L/100 km (interpolation). At 150 km/h: y ≈ 0.067 × 150 + 1.942 ≈ 11.99 L/100 km (extrapolation). Risky because air resistance grows non-linearly at high speeds, so the linear model underestimates fuel use.
1.4 — Surf wave height
(a) y = 0.05 × 35 + 0.4 = 1.75 + 0.4 = 2.15 m. Interpolation (35 is within 10–60).
(b) y = 0.05 × 100 + 0.4 = 5.4 m. Extrapolation (100 is far outside 10–60).
(c) Trust the 35 km/h prediction. At 100 km/h winds (cyclone-strength), the linear model has not been calibrated and waves behave very differently.
1.5 — 2025 café sales prediction
(a) Extrapolation: 2025 is OUTSIDE the data range (2010–2019). Lesson 13: predictions outside the data range are less reliable.
(b) The misconception card warns "a trend will always continue in the same direction" is wrong. Real-world trends change when external factors (like a pandemic) intervene — the linear fit cannot capture this.
2.1 — Explain your thinking (sample)
The 2035 figure is an extrapolation well outside the company's actual sales data range, so by Lesson 13 it is much less reliable than a within-range prediction. The linear trend may NOT hold because real-world markets saturate, competitors enter, or buying habits shift; the lesson warns "never assume a linear trend continues indefinitely." One external factor that could break the trend is a recession or a major shift in consumer behaviour (e.g. a competitor undercutting price). A more honest report would describe a range of plausible outcomes (best/likely/worst case) rather than a single $1M projection, and would explicitly flag that 2035 is far beyond the observed data.
Marking: 1 mark for naming extrapolation, 1 for explaining why linear may fail, 1 for a plausible external factor, 1 for a more honest reporting suggestion.