Mathematics • Year 8 • Unit 4 • Lesson 13
Misleading Graphs — Mixed Challenge
Pull together truncated axes, inconsistent intervals, 3D/pictograph distortion, cherry-picking, and accurate percentage-change calculations. Six mixed problems, one "find the mistake", and one open-ended challenge where YOU design a misleading graph.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 13. Show working. 3 marks each
1.1 Identify the misleading technique in each scenario:
(a) A bar chart of profits with y-axis from 96 to 102.
(b) A line graph showing only the months when sales were strong.
(c) A pictograph that triples both the height and width of a coin to show triple revenue.
1.2 A graph shows population values 50 000, 52 000, 54 000 with the y-axis starting at 48 000. (a) Calculate the actual % growth from 50 000 to 54 000 (1 d.p.). (b) On the truncated graph, the third bar appears 3× taller than the first. Calculate the visible height ratio (using axis start 48 000) to confirm.
1.3 A pictograph doubles the width of an icon but keeps the height the same. The icon now represents twice the original value. Is this honest or misleading? Justify by calculating the area change.
1.4 A graph shows axis labels: 0, 10, 50, 100, 500, 1 000. Identify the misleading technique and explain in one sentence the visual effect.
1.5 A graph of monthly rainfall shows ONLY February-April (the wettest months) of a single year, claiming "rainfall is at record highs". List TWO questions you would ask before trusting this claim.
1.6 A truncated graph shows test scores 75, 78, 81. The y-axis starts at 72 and ends at 84. (a) Calculate the actual % change from 75 to 81 (1 d.p.). (b) Calculate the visible bar height ratio (third vs first) using the truncated axis. (c) By what factor does the graph exaggerate the change?
2. Find the mistake
A student analysed a misleading chart. Exactly one line below contains a mistake. Spot it, explain why, and correct it. 3 marks
Problem: A graph shows sales rising from 200 units to 250 units. The y-axis starts at 180 (not 0). Calculate the actual % change.
Line 1: The y-axis starts at 180, not 0 — this is a truncated y-axis.
Line 2: % change formula: (new − old) ÷ new × 100.
Line 3: = (250 − 200) ÷ 250 × 100 = 50 ÷ 250 × 100 = 20%.
Line 4: Conclusion: sales rose by 20%.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working and the correct % change.
Stuck? The % change formula divides by the ORIGINAL (old) value, not the new value. The student got the denominator wrong.3. Open-ended challenge — design a deliberately misleading graph
This question has many valid answers. 4 marks
3.1 Your job: design a misleading graph that overstates a small change, then design the honest version that tells the true story.
(i) Choose a real-feeling scenario (e.g. school fundraising, gaming sales, study hours, weather, sports stats) and invent THREE data points where the actual percentage change between the first and last is between 2% and 6%.
(ii) Write the actual percentage change (calculation shown).
(iii) Describe (in words OR rough sketch) the MISLEADING version: state the trick you used, the y-axis range, and the visual impression it gives (e.g. "looks like it doubled").
(iv) Describe the HONEST version: state the correct y-axis range starting at 0 and what the bars actually look like.
(v) Write a one-line caption that would be honest for the data, AND a one-line caption that would be misleading.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Name the tricks
(a) Truncated y-axis. (b) Cherry-picking. (c) Pictograph area trick — tripling both dimensions makes the area 3 × 3 = 9 times larger, not 3 times.
1.2 — Population growth
(a) % change = (54 000 − 50 000) ÷ 50 000 × 100 = 4 000 ÷ 50 000 × 100 = 8.0%.
(b) Visible heights (axis at 48 000): first bar = 50 000 − 48 000 = 2 000; third bar = 54 000 − 48 000 = 6 000. Ratio = 6 000 ÷ 2 000 = 3 times. The graph shows a 3× visual change for an actual 8% rise — a major exaggeration.
1.3 — Doubling only width
If width doubles and height stays the same, new area = 2 × 1 = 2 times the original. This matches the doubled value being represented, so it is honest. (Caution: if your eye reads the icon as 2D and intuits both dimensions, it can still be slightly misleading — best practice is to use TWO icons rather than one stretched icon.)
1.4 — Uneven intervals
Inconsistent intervals on the axis. The visual distance between 500 and 1 000 looks the same as between 0 and 10, so massive differences appear equal — trends can be hidden or invented depending on which jump is exaggerated.
1.5 — Cherry-picked rainfall
(1) "Can I see ALL 12 months of the year, not just Feb-Apr?" — to check whether the wet months are unusually wet or just the normal wet season.
(2) "What does the long-term annual rainfall look like over the past 10-20 years?" — single-year data can't establish a "record high" trend.
1.6 — Test scores
(a) Actual % change = (81 − 75) ÷ 75 × 100 = 6 ÷ 75 × 100 = 8.0%.
(b) Visible heights (axis from 72): first = 3, third = 9. Ratio = 9 ÷ 3 = 3×.
(c) The graph exaggerates the change by a factor of 3 ÷ 1.08 ≈ 2.8 (an 8% change looks like a 200% change visually — about 25 times more dramatic than reality).
2 — Find the mistake
(a) The mistake is on Line 2 (and propagated to Line 3).
(b) The % change formula divides by the OLD value, not the new value. So it should be (new − old) ÷ old × 100, not (new − old) ÷ new × 100. The student got the denominator wrong.
(c) Corrected: % change = (250 − 200) ÷ 200 × 100 = 50 ÷ 200 × 100 = 25%. Sales rose by 25%, not 20%.
3 — Open-ended challenge (sample solution)
(i) Scenario: School fundraising over three years. Data: 2021 = $980, 2022 = $1 000, 2023 = $1 020.
(ii) Actual % change: (1 020 − 980) ÷ 980 × 100 ≈ 4.1%.
(iii) Misleading version: Truncated y-axis from $970 to $1 030. Visible bar heights = 10, 30, 50 — third bar looks 5× the first. Visual impression: "fundraising has skyrocketed!"
(iv) Honest version: y-axis from $0 to $1 100 with ticks every $200. Bars appear almost identical in height, accurately showing the modest 4% rise.
(v) Captions: Honest — "Fundraising up 4% over three years — a steady but small increase." Misleading — "Fundraising has soared — donations through the roof!"
Marking: 1 mark for a sensible scenario with three data points and actual % change between 2-6%. 1 mark for showing the correct % change calculation. 1 mark for a clear, specific misleading version (named trick + axis range). 1 mark for a clear honest version AND two contrasting captions.