Lesson 11 · The chart that lies with true numbers
One idea, all lesson. A bar chart can be completely accurate about the numbers and still tell a lie about the comparison, and the way it does that is by starting its axis somewhere other than zero. Students meet it on a poster about data they already know by heart.
| Minutes | What happens |
|---|---|
| 0–6 | Warm-up. The same numbers drawn twice. Which one is lying? |
| 6–18 | Launch. The Spinner Stall’s poster: eight times more Preps loved our stall. Catch it. |
| 18–42 | Explore. Two more charts to check, then redraw one honestly and rewrite its headline. |
| 42–52 | Summarise. Why bars must start at zero, and the one case where a line need not. |
| 52–55 | Exit ticket. |
Why only one trick
There is a long list of ways to mislead with a graph, and a lesson that races through five of them leaves students able to name none. The truncated axis is the most common one in real media, it is purely visual so it needs no arithmetic to see, and it has a clean rule attached: bar length is the claim, so bars start at zero. Lesson 12 does the written version — misleading words and dodgy sampling — so there is no need to cram it in here.
The move that does the work
Cover the axis numbers with your thumb. Ask what the picture says. Then uncover and read the real values. The gap between those two answers is the deception, and students can feel it rather than be told about it.
What the achievement standard wants out of this
AC9M6ST02 asks students to critique — not just spot. A spot is “the axis doesn’t start at zero.” A critique says what the chart made you believe, what the numbers actually are, and what an honest version would look like. The worksheet asks for all three every time, in that order.
The hard question, saved for the end
In Lesson 10 the dot-plot number lines ran from 21 to 36, not from zero. So did we mislead them? No — and the reason is the whole point. See the Summarise tab before you teach this, because a sharp student will ask.
Differentiation
Support
- Two charts instead of three, and the true numbers printed beside each one.
- The three critique questions as sentence frames to complete.
- The redraw grid already labelled 0 to 20, so the axis decision is made for them.
- Thumb test modelled once with a partner before working alone.
Core
- All three charts, critiqued in the three-part structure.
- Redraw one honestly on a blank grid — they choose the axis.
- Write an honest headline for it.
Extension
- Question 8: when is it fair not to start at zero?
- Draw a chart that is misleading in the opposite direction — one that hides a difference that is real.
- Find a truncated axis in a real newspaper or website chart.
What we are learning
- That a bar chart can show true numbers and still give a false impression.
- To check where a chart’s axis starts before believing what it shows.
- To work out what a chart is really saying, using the numbers rather than the picture.
- To redraw a misleading chart honestly, and say what changed.
How you will know you have it
- You can say what the poster made you believe, and what the numbers actually show.
- You can explain why bars have to start at zero.
- Your redrawn chart has an axis that starts at zero and a headline that matches it.
Which of those is the biggest stretch for you? Write it down.
Not sure? The last one. Spotting the trick is easy; saying clearly what it did to you is the hard part.
How hard is each question?
Every question is labelled so you know what kind of thinking it needs.
SF — simple familiar
Read where an axis starts, and read the two true values.
CF — complex familiar
Explain how the chart misleads, and redraw it honestly.
CU — complex unfamiliar
Decide when not starting at zero is fair, and justify it.
Warm-up · same numbers, drawn twice
“Two charts. Both are about the same two stalls, and both use the same two numbers. Nobody has made anything up.
Which one is lying?”
Take answers. Most will pick the left one.
“Here is the problem. Neither of them is lying. Every number on both charts is correct. Check them against your book from Lesson 9 — sixteen Preps at the Spinner Stall gave it a 3 or a 4, and nine at the Treasure Bag.
So if neither is lying, why do they look nothing like each other?”
Do not rush to the answer
The useful discomfort is the thirty seconds where both charts are true and they still disagree. If you name the axis too early you get a rule to remember instead of a problem to solve. Let two or three students point at things that are not it — bar width, colour, the order of the bars — before anyone looks at the numbers up the side.
Expect this
“The left one is lying because the bars are the wrong size.” Push once: wrong compared to what? The bars are exactly the right size for that axis. The axis is the thing doing the work.
Launch · the Spinner Stall’s poster
Project page 2 of the stimulus. It is a poster the Spinner Stall team made for the school newsletter.
8 TIMES MORE PREPS LOVED OUR STALL!
The Spinner Stall — Buddy Games Day
“Their chart is on the poster. Thumb over the numbers up the side. What does the picture tell you?”
“Loads more. About eight times as many.”
“Now take your thumb off. What are the two real numbers?”
“Sixteen and nine.”
“Is sixteen eight times nine?”
The thumb test
| 1 | Thumb over the numbers. What does the picture make you believe? |
|---|---|
| 2 | Thumb off. What are the real values? |
| 3 | Where does the axis start? If it is not zero, the bar lengths are not the values. |
Why the bars read as eight to one
The axis starts at 8. The Spinner Stall’s bar covers 8 to 16, so it is eight units tall. The Treasure Bag’s covers 8 to 9, so it is one unit tall. Eight units against one unit, and the eye divides bar lengths. The headline is not an exaggeration of the data — it is an accurate reading of the picture.
Say the rule once, in these words
On a bar chart, the length of the bar is the claim. A bar twice as long is saying “twice as many”. That is only true if the bars start at zero, so bars start at zero.
If someone says “but the Spinner Stall really was more popular”
Agree, and keep going. It was — 80% gave it a 3 or 4 against 45%, which is a solid, defensible finding and it is what Lesson 9 concluded. The poster did not need to cheat. Misleading charts usually dress up a difference that was real but not impressive enough, and that is worth saying out loud.
Explore · three charts to check
Pages 3 and 4 of the stimulus, also on the worksheet. For each one: what did it make you believe, what do the numbers say, and where does the axis start?
Choose your path
Answer questions 1 to 4 about chart 2, then do the redraw on the grid that is already labelled 0 to 20.
Chart 2: where does the axis start? Chart 3: where does that axis start?
Chart 2 shows 18 Preps and 13 Preps. How many more Preps won a prize at the Spinner Stall?
The poster for chart 2 says six times more. Thumb over the numbers — why does the picture agree with the poster?
Is 18 six times 13? Write one sentence saying what the chart made you believe and what the numbers really say.
Chart 3 says nine times more. The real numbers are 16 and 8. What is the truth about those two numbers?
Chart 3 is about a difference that is real — twice as many. Why do you think the team exaggerated it anyway?
Redraw one of the three charts on the blank grid with the axis starting at zero, and write a headline that matches your chart.
A weather chart of temperatures between 24° and 31° starts its axis at 20, not 0. Is that misleading? Explain what makes it different from chart 2.
1. Chart 2 starts at 12. Chart 3 starts at 7.
2. Five more. 18 − 13 = 5.
3. From 12, the bars are 6 units and 1 unit tall. The eye compares the lengths, so it reads six to one — and that is exactly what the headline claims.
4. No. 18 is about 1.4 times 13, not 6 times. The chart made it look like six times as many; really it is five extra Preps out of twenty.
5. 16 is exactly double 8. Twice as many, not nine times. From an axis at 7 the bars are 9 units and 1 unit, which is where “nine times” comes from.
6. Open. Good answers: twice as many did not sound exciting enough; they wanted to win the argument with the other team; they may not have understood what their own chart was doing. Accept any answer that separates the finding was real from the picture was dishonest.
7. Axis from 0. Chart 1 redrawn: bars at 16 and 9. Chart 2: 18 and 13. Chart 3: 16 and 8. Honest headlines — “Nearly twice as many Preps loved the Spinner Stall”, “Five more Preps won a prize at the Spinner Stall”, “Twice as many Preps got a quick turn at the Treasure Bag”.
8. Not necessarily misleading, and the reason is the key idea. Nobody is claiming 31° is “a lot more times” than 24° — temperature is read as a position on a scale, not as a length, and a chart of it is drawn as a line or dots rather than bars. On chart 2 the bar length was the claim, and a length that does not start at zero makes a false claim. A careful chart also marks the break in its axis so the reader knows.
Then redraw one honestly
Pick any of the three charts. Draw it again on the blank grid with the axis starting at zero, then write a headline that matches your chart.
Marking the redraw quickly
Three things only: does the axis start at zero, are the two bar heights right, and does the headline match the bars? A headline that still says “8 times” over an honest chart is the common error and it is worth a conversation — the student has fixed the picture without noticing the picture was the evidence for the words.
While they work
The three charts get steadily more interesting. Chart 2 is a difference of 18 against 13 inflated to look like six to one — a small real difference. Chart 3 is 16 against 8, which genuinely is double, stretched to look like nine times. Chart 3 is the one to pull the class back for, because it shows a true and useful finding being spoiled by a dishonest picture.
Summarise · bars start at zero
“On a bar chart, the length of the bar is the claim. Twice as long means twice as many. So if the bars do not start at zero, the lengths are not the numbers, and the picture is making a claim the data does not support.
Three questions, every chart, for the rest of your life: where does the axis start, do the steps go up evenly, and does the headline match the numbers?”
The question someone will ask
“Good catch. What was each dot on that number line doing?”
“Showing where one Prep’s time sat.”
“Right. A dot shows a position. It never says one time is three times another — you read it by looking at where the dots sit. A bar is different: a bar has a length, and length is what we compare. Length is the claim, so length has to start at zero.
Starting at 21 wasted no space and claimed nothing. Starting a bar at 8 claimed something false.”
The honest version of not starting at zero
Grown-up charts do sometimes need a cut-off axis — temperatures around 37°, a share price, a long run of values all bunched together. When they do it properly, they mark the break in the axis and the chart is a line rather than bars. The bar-length problem simply does not arise for a line or a dot, and that is the distinction, not “zero always”. Year 6 does not need to draw one; they need to know the rule has a reason, so they can tell a cut-off axis from a cheat.
The stimulus pack’s last page is the Spinner Stall poster redrawn honestly — the same two numbers, the axis from zero, and a headline the chart actually supports. Put it next to the original and the argument is over.
Exit ticket
A fourth chart, unseen: 7 Preps against 3 gave a 4, with the axis starting at 2, so it reads as five to one when it is really about 2.3 times. One sentence on what is wrong, and one honest headline.
Check what stuck
Marking the exit ticket. Three bands, and you only need to sort them.
| Where they are | What the ticket looks like | Next move |
|---|---|---|
| Not yet | Says the chart is wrong but cannot say why, or says the numbers are made up. | Thumb test again, out loud, with one chart. The numbers are true — that is the thing to settle. |
| There | Names the axis, gives the two real values, writes a headline that matches. | Ready for Lesson 12. |
| Beyond | Works out the real factor, or notices that the difference was real and did not need exaggerating. | Ask them to find a truncated axis in a real chart this week. |
Evidence this lesson contributes
Part B of the folio has a critique item. This lesson is where that is taught and where you find out who can do it before the assessment.
| Dimension | What to look for |
|---|---|
| Understanding | Knows that bar length carries the comparison, so the axis must start at zero. |
| Reasoning | Explains the gap between what the picture says and what the numbers say. |
| Problem-solving | Redraws the chart honestly and writes a headline the new chart supports. |
Keep the redraws
The redrawn chart plus its honest headline is the single best piece of pre-assessment evidence in this half of the unit. Photograph or file them — it is the same task as Part B item 5, on different data.
What to print
| File | Who gets it |
|---|---|
| l11-01-stimulus-pack.pdf | Project it, or print one set for the wall. Five pages: the warm-up pair, the poster, charts 2 and 3, and the honest redraw. |
| l11-02-worksheet-core.pdf | One per student. All three charts, the critique structure, a blank grid and the exit ticket. |
| l11-03-worksheet-support.pdf | Two charts, true numbers printed beside them, sentence frames, grid pre-labelled 0 to 20. |
Where this sits in the achievement standard
AC9M6ST02 — identify statistically informed arguments presented in traditional and digital media; discuss and critique methods, data representations and conclusions.
Today is the data representation half of that descriptor, and the conclusion that is drawn from it: the chart and its headline. Lesson 12 does methods — who was asked, how many, and what was left out.
Why this is not a soft lesson
“Critique the media” can drift into opinion. It does not here, because every judgement is checkable: the axis starts at a number you can read, the true values are counts the students already have, and the factor the chart implies is arithmetic. A student either can or cannot say that a bar eight units long against one unit long is claiming eight times. That is mathematics, not a point of view.
Australian Curriculum content descriptions and achievement standard extracts © ACARA 2010–present, licensed under CC BY 4.0. ACARA does not endorse this material.