Mathematics • Year 10 • Unit 4 • Lesson 13
Line of Best Fit — Mixed Challenge
Combine every Lesson 13 idea: drawing the line of best fit, calculating slope and intercept, interpolation vs extrapolation, residuals, and the two big misconceptions ("interpolation = extrapolation" and "trends continue forever").
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 13. Decide whether the question is about drawing, equation-finding, prediction, or a misconception before you start writing. 3 marks each
1.1 A line of best fit is y = 2.5x + 10. The data x ranges from 0 to 20. (a) Predict y at x = 8. (b) Predict y at x = 25. (c) Classify each prediction.
1.2 Find the equation of a line of best fit that passes through (3, 7) with slope 1.5. Then predict y when x = 0 (the y-intercept).
1.3 A residual for a point (x, y) using the line of best fit y = mx + c is defined as y − (mx + c). For the line y = 2x + 1 and data point (3, 8), find the residual and state what a positive residual means.
1.4 Data ranges from x = 2 to x = 8. For each prediction, state interpolation or extrapolation and give a one-line reliability comment: (a) x = 5, (b) x = 1, (c) x = 9.5, (d) x = 7.
1.5 A line of best fit y = 4x + 12 is found for data ranging from x = 0 to x = 15. (a) What is the predicted y at x = 7.5? (b) If a real observed point is (7.5, 38), find the residual and comment on the fit at this point.
1.6 A linear model of CO₂ emissions from 1990–2020 fits the data well. Lesson 13 warns trends don't continue forever. (a) Why might extrapolating to 2050 still be useful, even if imperfect? (b) Name one external factor that could break the linear trend after 2020.
2. Find the mistake
Another Year 10 student has fitted a line of best fit and made predictions. Their reasoning is below. Exactly one line breaks a Lesson 13 rule. Spot it, explain why it's wrong, and re-do the work correctly. 3 marks
Student's reasoning — height vs age, data range 5–15 years:
Line 1: Mean point (x̄, ȳ) = (10, 140 cm). Line of best fit: y = 6x + 80.
Line 2: Predicted height at age 12: y = 6 × 12 + 80 = 152 cm (interpolation, reliable).
Line 3: Predicted height at age 40: y = 6 × 40 + 80 = 320 cm — "this is interpolation, so the prediction is reliable."
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that claim is wrong, quoting the lesson misconceptions card.
(c) Re-state Line 3 correctly, including the correct classification and a comment on reliability.
Stuck? The data range is 5–15. Age 40 is FAR outside it. And humans stop growing around 18.3. Open-ended challenge — design a prediction task
This question has many valid answers. Be creative but follow every rule. 4 marks
3.1 Design a bivariate scenario where a line of best fit is helpful for one prediction but misleading for another. Your write-up must include:
- a realistic scenario with at least 6 data points (made-up but plausible),
- the mean point (x̄, ȳ) and an estimated slope,
- the equation of the line of best fit,
- one interpolation prediction with a reliability comment,
- one extrapolation prediction with at least one specific external reason it might be wrong.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Predict and classify
(a) y = 2.5 × 8 + 10 = 30 (interpolation, within 0–20).
(b) y = 2.5 × 25 + 10 = 72.5 (extrapolation, outside 0–20).
(c) (a) is more reliable than (b) because (a) is within the observed data range.
1.2 — Build equation
y = 1.5x + c. Using (3, 7): 7 = 4.5 + c → c = 2.5. Equation: y = 1.5x + 2.5. At x = 0: y = 2.5 (the y-intercept).
1.3 — Residual
Predicted y at x = 3: y = 2 × 3 + 1 = 7. Residual = observed − predicted = 8 − 7 = +1. A positive residual means the actual point sits ABOVE the line of best fit (the model under-predicted at that x).
1.4 — I or E with reliability
(a) x = 5 → I (reliable). (b) x = 1 → E (less reliable; below data range). (c) x = 9.5 → E (less reliable; above data range). (d) x = 7 → I (reliable).
1.5 — Prediction + residual
(a) y = 4 × 7.5 + 12 = 30 + 12 = 42.
(b) Residual = 38 − 42 = −4. The actual value is 4 below the line; the line over-predicted at this point. The fit at this x is reasonable but not perfect.
1.6 — CO₂ extrapolation
(a) Even if imperfect, a 2050 extrapolation shows what would happen if current policies don't change — useful for planning and motivating action.
(b) Examples: large-scale renewable energy adoption, carbon tax policy, a global recession, or a technological breakthrough in carbon capture — any of these could bend the trend down (or up).
2 — Find the mistake
(a) The mistake is on Line 3.
(b) Age 40 is FAR OUTSIDE the data range (5–15 years), so this is extrapolation, NOT interpolation. The lesson misconception card warns "Interpolation and extrapolation are equally reliable" is wrong. Also, humans stop growing well before age 40, so the linear trend definitely fails.
(c) Corrected: "At age 40, this prediction is EXTRAPOLATION — far outside the data range. The model gives 320 cm, which is biologically impossible because humans stop growing in their late teens. The linear trend simply does not hold at age 40."
3 — Open-ended challenge (sample solution)
Scenario: Phone battery percentage left after x hours of continuous video streaming on a Year 10 student's phone.
Data: (0, 100), (1, 85), (2, 70), (3, 55), (4, 40), (5, 25).
Mean point: x̄ = 15/6 = 2.5; ȳ = 375/6 = 62.5. Slope: m = (25 − 100)/(5 − 0) = −15.
Equation: 62.5 = −15 × 2.5 + c → c = 62.5 + 37.5 = 100. So y = −15x + 100.
Interpolation: at x = 2.5 hr, y = −15 × 2.5 + 100 = 62.5 % — reliable, within data range.
Extrapolation: at x = 7 hr, y = −15 × 7 + 100 = −5 %. Impossible — battery cannot be negative. External factor breaking the trend: phones throttle once they reach low battery, and once at 0 % they simply shut down (the linear model doesn't know about this floor). Lesson 13: trends don't continue forever.
Marking: 1 mark for realistic data, 1 for correct mean/slope/equation, 1 for an interpolation prediction + comment, 1 for an extrapolation prediction with a specific reason it fails.