Mathematics • Year 10 • Unit 4 • Lesson 13
Line of Best Fit — Skill Drill
Build fluency with Lesson 13's three-step routine: (1) draw a line of best fit by eye so the line passes through the mean point (x̄, ȳ) with roughly equal points above and below; (2) write the equation y = mx + c; (3) use the line to predict — and decide whether the prediction is interpolation (reliable) or extrapolation (risky).
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Data shows the height of a plant (y, cm) after x weeks of growth:
(1, 5), (2, 8), (3, 12), (4, 14), (5, 18).
Draw a line of best fit, write its equation, then predict the height at x = 3.5 weeks and x = 10 weeks. Classify each prediction.
Step 1 — Find the mean point (x̄, ȳ).
x̄ = (1+2+3+4+5)/5 = 3. ȳ = (5+8+12+14+18)/5 = 57/5 = 11.4.
Reason: Lesson 13 says a good line of best fit passes through (x̄, ȳ).
Step 2 — Pick the slope from two clear points along the trend.
Using (1, 5) and (5, 18): m = (18 − 5)/(5 − 1) = 13/4 = 3.25.
Reason: a line of best fit is straight, so use rise/run.
Step 3 — Write y = mx + c using (3, 11.4).
11.4 = 3.25 × 3 + c → c = 11.4 − 9.75 = 1.65.
Equation: y = 3.25x + 1.65.
Step 4 — Predict, then classify each prediction.
At x = 3.5: y = 3.25 × 3.5 + 1.65 = 11.375 + 1.65 = 13.025 cm.
At x = 10: y = 3.25 × 10 + 1.65 = 32.5 + 1.65 = 34.15 cm.
x = 3.5 is INSIDE [1, 5] → interpolation (reliable). x = 10 is OUTSIDE [1, 5] → extrapolation (less reliable).
Answer: Equation y = 3.25x + 1.65. Predicted heights: 13.0 cm (interpolation) and 34.2 cm (extrapolation — the plant may not keep growing at 3.25 cm/week).
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. The data below pairs ice cream sales (y, in $1000s) with maximum daily temperature (x, °C):
(20, 4), (22, 5), (24, 6), (26, 8), (28, 9).
Step 1 — Mean point. x̄ = (20+22+24+26+28)/5 = ________. ȳ = (4+5+6+8+9)/5 = ________.
Step 2 — Slope. Using (20, 4) and (28, 9): m = (9 − 4)/(28 − 20) = _______ / _______ = ________.
Step 3 — Equation. Use the mean point to find c. y = mx + c, so ȳ = m × x̄ + c → c = ________.
Equation: y = __________ x + __________.
Step 4 — Predict y at x = 23 °C and x = 35 °C. Classify each.
At x = 23 → y ≈ ________ ($'000); type: ____________________.
At x = 35 → y ≈ ________ ($'000); type: ____________________.
3. You do — independent practice
For each task, do the calculation shown. The first four are foundation (one step). The middle two are standard (two ideas). The last two are extension (lesson misconception traps).
Foundation — one-step calculations
3.1 Find the mean point (x̄, ȳ) of (1, 3), (2, 5), (3, 7), (4, 9), (5, 11). 1 mark
3.2 Find the slope of the line passing through (2, 10) and (8, 28). 1 mark
3.3 The line of best fit is y = 2x + 5. Predict y at x = 4. 1 mark
3.4 A data set has x values from 5 to 15. State whether each prediction is interpolation (I) or extrapolation (E):
(a) x = 8, (b) x = 20, (c) x = 12, (d) x = 2. 1 mark
Standard — equation + prediction
3.5 A line of best fit has slope 3 and passes through the mean point (4, 14). (a) Write the equation. (b) Use it to predict y when x = 6. 2 marks
3.6 A scatter plot has data x = 0 to x = 50 with line of best fit y = 1.5x + 4. (a) Predict y at x = 25 (interpolation). (b) Predict y at x = 80 (extrapolation). (c) Which prediction is more reliable and why? 2 marks
Extension — the lesson's misconceptions
3.7 A model fits data from 2010–2020 perfectly. Lesson 13 misconception card: "A trend will always continue in the same direction." Explain in one sentence why this is wrong, and give one example where a real-world trend changed direction. 3 marks
3.8 Two students draw a line of best fit for the same scatter plot. Student J's line passes through every single point but zigzags. Student K's line is straight and passes through the mean point with roughly equal points above and below. Whose line is correct, and what specific Lesson 13 rule did the other student break? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (ice cream vs temperature)
Step 1: x̄ = 120/5 = 24; ȳ = 32/5 = 6.4.
Step 2: m = 5/8 = 0.625.
Step 3: 6.4 = 0.625 × 24 + c → c = 6.4 − 15 = −8.6. Equation: y = 0.625x − 8.6.
Step 4: x = 23 → y ≈ 0.625 × 23 − 8.6 = 14.375 − 8.6 = 5.775 ≈ $5.78k (interpolation).
x = 35 → y ≈ 0.625 × 35 − 8.6 = 21.875 − 8.6 = 13.275 ≈ $13.28k (extrapolation — less reliable; at extreme heat sales may level off).
3.1 — Mean point
x̄ = 15/5 = 3, ȳ = 35/5 = 7. Mean point = (3, 7).
3.2 — Slope
m = (28 − 10)/(8 − 2) = 18/6 = 3.
3.3 — Predict
y = 2 × 4 + 5 = 13.
3.4 — I or E
(a) I, (b) E, (c) I, (d) E.
3.5 — Equation through mean
(a) y = 3x + c. Using (4, 14): 14 = 12 + c → c = 2. Equation: y = 3x + 2.
(b) y = 3 × 6 + 2 = 20.
3.6 — Predictions
(a) y = 1.5 × 25 + 4 = 41.5 (interpolation).
(b) y = 1.5 × 80 + 4 = 124 (extrapolation).
(c) The x = 25 prediction is more reliable because it is within the observed data range. Lesson 13: interpolation uses observed patterns; extrapolation assumes the trend continues.
3.7 — Trends don't continue forever
Real-world trends often change direction due to outside factors. Example: Australian house price growth, which trended strongly upward through the 2010s, slowed and locally reversed during 2022's interest-rate rises. Other valid examples: COVID-era population change, smartphone sales after market saturation, climate-policy emissions reductions.
3.8 — Zigzag vs straight
Student K is correct. Student J broke the rule that a line of best fit must be a STRAIGHT line that summarises the trend — it does NOT pass through every point. Lesson 13: the line should pass through (x̄, ȳ) with roughly equal numbers of points above and below.