At a glance. One lesson of about 60 minutes. It assumes the summaries from Lessons 1 to 5 and percentage change. Students check axes, scaling, grouping and averages, and write a short report structure.
Calculator: no calculator. No calculator is needed: the divisions and percentages are chosen to be short, and the checking habit is the reasoning.
What students will learn
Know
- A vertical axis that starts above zero exaggerates differences between bars.
- Area grows as the square of a scale factor.
- A mean of group means must be weighted by group sizes.
Understand
- A picture can mislead even when every number is correct.
- A count is not a rate: compare shares, not totals, for groups of different sizes.
- A report needs a limitation as well as a conclusion.
Do
- Work out the true ratio or percentage from a chart's values.
- Combine group means properly.
- Write a short report with centre, spread, comparison and a limitation.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: ratios, percentages, weighted means (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to predict how the number of bars and the tallest bar change with the class width.
- Criterion C. Communicating: stating the impression and the truth with numbers, and adding a limitation (the worked example and questions 3 and 8).
- Criterion D. Applying mathematics in real-life contexts: newspaper charts, sales, population shares; judging data presented to them.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “How many times as tall will B look when the axis starts at 40?” and take a number before moving the slider.
In the second figure, ask whether the tallest class will stay the same when the width changes, then check at width 5 and width 20.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the two headlines. Collect what students would check on a graph. |
| 10 min | First figure and key idea: the four checks. |
| 10 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose B or C, return to the first figure and read the bar heights. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | A; 25%; 15. |
|---|---|
| Hook | The same figures can be drawn with different axis starts, so one graph shows a big difference and another a small one. Check where the axis starts, the scale, the grouping and the measure. For the first figure, with the axis starting at 50, B looks three times as tall as A although its value is only 1.08 times as large. |
| Worked example | Looks 1.3 times as tall (30%); really 5%. |
| Faded example | Mean of all 25 students: 64. |
| 1. What a truncated axis does | A. B says the values change; C says nothing is wrong; D confuses width with height. |
| 2. How tall a bar looks | 2. 20 ÷ 10. |
| 3. Choosing a limitation | B. A names a method, not a limitation; C thinks a clear result needs none; D treats a difference as an error. |
| 4. The true percentage change | 25. 20 ÷ 80. |
| 5. A mean of two groups | 55. 2200 ÷ 40. |
| 6. Scaling a pictogram | 4. 2 × 2. |
| 7. A rate against a count | 1. 7.5% − 6.5%. |
| 8. Impression against truth | C. A reads the drawn heights as the values; B gives the difference as a percentage; D uses the true ratio for the drawn heights. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Treats the printed values as the only information, or confuses height with width. | Hold up two strips cut to the drawn heights. |
| 1.25, 10 or 0.5 in question 2 | Gave the true ratio, a difference, or the inverted ratio. | Ask for the drawn heights first. |
| A, C or D in question 3 | Named a method, said no limitation was needed, or treated a difference as an error. | Ask: if the same study were done again next week, could the answer change? |
| 100, 20 or 80 in question 4 | Gave the impression, the difference, or the first value as a percentage of the second. | Ask: change divided by what? |
| 60, 2200 or 1500 in question 5 | Averaged the means, did not divide, or used one group's total. | Make a total for each group in a table. |
| 2, 8 or 60 in question 6 | Used the factor on one side, three dimensions, or the value. | Draw the two icons on squared paper and count squares. |
| 430, 6.5 or 7.5 in question 7 | Compared counts, or gave one share only. | Ask: which town is bigger, and does that explain the counts? |
| A, B or D in question 8 | Read the drawn heights as the values, gave the difference as a percentage, or ignored the cut-off. | Ask for the drawn heights and the values on two separate lines. |
Questions worth asking
- When is a truncated axis acceptable? On a line graph meant to show small changes, if it is labelled and stated.
- Why does doubling both sides of an icon quadruple the area? Area is width times height, so both factors of 2 multiply.
- Why is a rate often fairer than a count? It removes the effect of group size.
- What belongs in a limitation? Sample size, how data were collected, or what the data cannot show.
Differentiation
A way in for anyone stuck
- Give a pair of charts of the same data already drawn, one with a zero axis and one without.
- Use a checklist: axis start? scale? grouping? count or rate? weighted?
- Provide a report frame with the headings already written.
Stretch
- Find a set of two groups where the mean of the group means is exactly right, and say what must be true of the group sizes.
- Redraw one of your own charts with a misleading axis, then redraw it fairly, and describe how the headline would change.
Students learning in English as an additional language
“Misleading” means giving a wrong idea. “Exaggerate” means to make something seem bigger than it is. “Limitation” means a reason the conclusion may not be fully reliable. “Weighted” means counted according to size.
| Word | Plain meaning | Example |
|---|---|---|
| truncated axis | an axis that starts above zero, cutting off the bottom of the bars | a bar chart whose axis starts at 40 |
| pictogram | a chart that uses pictures to stand for values | icons of buses for numbers of passengers |
| weighted mean | a mean in which each group counts according to its size | the mean of all the plants, not of two group means |
| rate | a count compared with the size of the group | goals per match |
| limitation | a reason a conclusion may not be fully reliable | the data came from only one week |
| impression | the idea a reader gets from looking at a graph | a rise that looks large |
Sentence starters for explaining aloud or in writing:
- The graph gives the impression that … but the numbers show ….
- A fairer way to compare is … because ….
- A limitation of this data is ….
Two things not to say
“The graph is lying.” The numbers can be correct; it is the impression that misleads.
“Just average the two means.” Groups of different sizes need weighting.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- RatioBars of 30 and 45 on an axis that starts at 0. How many times as tall is the second?
1.5. - PercentageA value rises from 50 to 60. What is the percentage increase?
20%.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA chart of rainfall in Kampala has bars for 90 mm and 100 mm on an axis starting at 80. How many times as tall does the second bar look?
20 ÷ 10 = 2. - Earlier work (Lesson 5)Group P has median 18 and IQR 4; group Q has median 18 and IQR 9. Compare them.
Same typical value; Q varies more.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA report says “crime doubled” because a count went from 4 to 8 in a small town. What would you ask?
The population and the period: 4 to 8 is a small count and may be random variation. - Working backwardsTwo groups of 10 and 40 have an overall mean of 52. The group of 10 has mean 40. Find the mean of the group of 40.
2600 − 400 = 2200; 2200 ÷ 40 = 55.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Bring two printed charts of the same data with different axes and have groups rank them for how honest they are.