Mathematics • Year 8 • Unit 4 • Lesson 6
Line Graphs — Mixed Challenge
Pull together everything from Lesson 6: trends, interpolation, extrapolation, chart-type choice, and reading multi-line graphs.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 6. Show your working. 3 marks each
1.1 Classify each estimate as interpolation or extrapolation:
(a) Data for years 2010–2020; estimating 2015.
(b) Data for years 2010–2020; estimating 2025.
(c) Temperature taken at 6am, 12pm, 6pm; estimating 3pm.
1.2 A line graph shows the daily temperature falling steadily for a week: 28°C, 26, 24, 22, 20, 18, 16. (a) Describe the trend. (b) Estimate the temperature on day 4.5 by interpolation. (c) Estimate the temperature on day 9 by extrapolation.
1.3 Choose the best chart for each scenario (line graph / bar chart / pie chart) and justify in one sentence:
(a) Showing the gender split of the school: 52% female, 48% male.
(b) Showing daily rainfall (mm) for 30 days.
(c) Comparing the number of students in each year level (Years 7–12).
1.4 A line graph plots two cities' temperatures on the same axes. City A: Jan 5°C, Jul 28°C. City B: Jan 18°C, Jul 32°C. The lines cross in April at approximately 20°C. (a) What does the crossing point mean in context? (b) Which city likely has more extreme seasonal variation?
1.5 A school's enrolment over 5 years: 480, 510, 535, 550, 570. (a) Describe the trend with reference to whether it is accelerating, steady, or slowing. (b) Estimate next year's enrolment using a simple extrapolation. (c) Comment briefly on reliability.
1.6 Identify FOUR features that must appear on every correctly drawn line graph.
2. Find the mistake
Another student attempted this line-graph estimation problem. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: A line graph plots a city's population: 2000 = 50 000; 2010 = 65 000; 2020 = 80 000. Estimate the population in 2015 by interpolation, and in 2030 by extrapolation, and state the reliability of each.
Line 1: 2015 is halfway between 2010 (65 000) and 2020 (80 000).
Line 2: 2015 estimate ≈ (65 000 + 80 000) ÷ 2 = 72 500.
Line 3: Pattern: +15 000 each decade.
Line 4: 2030 estimate = 80 000 + 15 000 = 95 000.
Line 5: 2015 is INTERPOLATION (within range) → less reliable than 2030 estimate.
Line 6: 2030 is EXTRAPOLATION (beyond range) → more reliable.
(a) Which line(s) contain a mistake?
(b) Explain in one or two sentences why those lines are wrong.
(c) Write out the corrected lines.
Stuck? The student has reversed which is more reliable. Interpolation is the reliable one.3. Open-ended challenge — design and analyse a time-series
This question has many valid answers. 4 marks
3.1 Your job: pick ONE topic and invent a realistic time series of 6–8 values that would be displayed on a line graph. Possible topics:
• Your daily screen-time (minutes) for one week.
• The closing price of a stock for 6 trading days.
• Weekly weight (kg) of a puppy for 8 weeks.
• Hours of sleep per night for 7 nights.
Write up your analysis with the following:
(i) State your topic and list your time points + values.
(ii) Sketch a labelled line graph in your workbook (axes, title, units, even scale).
(iii) Describe the trend in 1–2 sentences.
(iv) Estimate ONE value within the data range (interpolation) — show working.
(v) Estimate ONE value beyond the data range (extrapolation) — show working AND comment on reliability.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Classify estimates
(a) 2015 is within 2010–2020 → interpolation. (b) 2025 is beyond 2020 → extrapolation. (c) 3pm is between 12pm and 6pm → interpolation.
1.2 — Falling temperature
(a) Decreasing trend, falling by 2°C per day (constant linear decrease).
(b) Day 4.5: between day 4 (22°C) and day 5 (20°C) → (22 + 20) ÷ 2 = 21°C.
(c) Day 9: extrapolating the pattern → 16 − 2 − 2 = 12°C. (Less reliable — the trend may not continue.)
1.3 — Choose the best chart
(a) Pie chart — parts of a whole, 2 categories.
(b) Line graph — continuous numeric data over time (30 days).
(c) Bar chart — comparing 6 discrete year-level categories.
1.4 — Two cities
(a) The crossing point means both cities had the same temperature in April (~20°C); before April City B was warmer, after April City A becomes warmer.
(b) City A — range 5 to 28 = 23°C; City B range 18 to 32 = 14°C. So City A has the larger seasonal variation.
1.5 — School enrolment
(a) Increasing trend that is slowing — annual gains of +30, +25, +15, +20 are getting smaller on average.
(b) Next year ≈ 570 + 20 = 590 students.
(c) Less reliable than interpolation — the trend could continue slowing, stay flat, or reverse if local conditions change.
1.6 — Line graph features
Any four of: descriptive title; horizontal axis label + units (usually time); vertical axis label + units; even scale on each axis; dot/marker at every data point; lines connecting consecutive points; ideally axis starting at 0 (or with a clearly marked break).
2 — Find the mistake
(a) The mistake is on Lines 5 and 6 (the reliability labels are swapped).
(b) Interpolation (within the data range) is always more reliable than extrapolation (beyond the data range). The student has reversed the two.
(c) Corrected Line 5: "2015 is INTERPOLATION (within range) → more reliable." Corrected Line 6: "2030 is EXTRAPOLATION (beyond range) → less reliable." Numbers (72 500 and 95 000) remain correct.
3 — Open-ended (sample solution — puppy weight)
(i) Topic: Puppy weight (kg) over 8 weeks. Values: W1 = 2.5, W2 = 3.2, W3 = 3.8, W4 = 4.5, W5 = 5.1, W6 = 5.8, W7 = 6.4, W8 = 7.0.
(ii) Graph: sketched in workbook with x = "Week (1–8)", y = "Weight (kg)", title "Puppy weight over 8 weeks", scale 0–8 in 1-kg steps.
(iii) Trend: Increasing — the puppy gains roughly 0.6–0.7 kg per week, growth is fairly steady.
(iv) Interpolation: Week 4.5 weight ≈ (4.5 + 5.1) ÷ 2 = 4.8 kg.
(v) Extrapolation: Week 10 ≈ 7.0 + 2 × 0.6 = 8.2 kg. Reliability: less reliable — puppies' growth typically slows as they near adult weight, so the linear extrapolation likely overestimates Week 10 weight.
Marking: 1 mark for plausible time-series data; 1 mark for a clear trend description; 1 mark for correct interpolation with working; 1 mark for extrapolation + sensible reliability comment.