At a glance. One lesson of about 60 minutes. Students test answer boxes for gaps and overlaps, compare three samples of one group, and scale a sample up to a whole group.
Calculator: no calculator. Nothing here needs one. Every sample ratio is a friendly percentage of the whole.
What students will learn
Know
- Primary data is collected first-hand; secondary data was collected by someone else.
- Answer boxes should cover everyone exactly once, with no gaps or overlaps.
- A fair sample is spread across the whole group.
Understand
- Data can be wrong because of how it was collected, even when it is first-hand.
- A sample is used to estimate a whole group by scaling its percentage.
- Leading questions push the answers one way.
Do
- Judge a question or set of boxes as fair or flawed and say why.
- Find the number asked when every nth member is asked.
- Estimate a whole-group count from a sample.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: primary and secondary data, fair questions and option sets (questions 1 to 5).
- Criterion B. Investigating patterns: the answer-box figure. Ask students to find a rule for when a set of boxes has a gap and when it has an overlap, then test it on a set they design.
- Criterion C. Communicating: explaining in a full sentence why a question or a sample is unfair (the what-if on sample size, and the “When could every 3rd name be unfair?” prompt; questions 4, 6, 7 and 8 are auto-marked).
- Criterion D. Applying mathematics in real-life contexts: school travel, lunches and timetables (the hook and questions 4 and 9).
The one thing to do with the figure
Ask for a prediction before the first slide. “Is Rahul right that most of the school takes the bus?” Then set the second figure to the first 6 at the bus stop and ask what students notice before showing all 30.
In the first figure ask students to find a number of hours that fits two boxes, and one that fits none, before they read the verdict line.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, primary and secondary data, and the first figure on answer boxes. |
| 10 min | Key idea on fair questions; the second figure on who is asked. |
| 10 min | Worked example from sample to whole group; students finish the chess club example in pairs. |
| 5 min | Question 1, the check. If many choose A, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 24; 1, 2, 3, 4, 6, 12; discrete. |
|---|---|
| Worked example | About 105 students. |
| Faded example | About 60 members. |
| 1. Options with no gaps or overlaps | B. A has overlaps at 2 and 4; C leaves out 0; D leaves out 2. |
| 2. Secondary data | B. A is primary; C and D ignore who collected each set. |
| 3. Size of a systematic sample | 60. 480 ÷ 8 = 60. |
| 4. A fair question | B. A, C and D each push towards one opinion. |
| 5. Values that fit no box | 3. The hours 10, 11 and 12. |
| 6. What is wrong with the options | B. A misreads the wording; C and D invent faults. |
| 7. Estimate a whole group from a sample | 168. 40% of 420, or 14 × 12. |
| 8. Primary data is not automatically right | B. A treats first-hand as accurate; C and D confuse the source with the quality. |
| 9. A fair sample | C. A, B and D each draw from one corner of the school; C is spread across the register. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1 | Overlapping ranges: 2 and 4 fit two boxes. | Ask which box a student with exactly 2 pets ticks. |
| C or D in question 1 | Did not check the ends: 0 pets or 2 pets fit no box. | Test every whole number from 0 up to 6 in turn. |
| A in question 2 | Thought data a student measured must be secondary. | Ask who collected each set. First-hand is primary. |
| 8 in question 3 | Gave the step between students asked. | Ask how many groups of 8 fit into 480. |
| 472 in question 3 | Subtracted 8 from 480. | Ask: if one student is asked from each group of 8, how many are asked? |
| A, C or D in question 4 | Did not hear the push of a leading question. | Ask what answer each question seems to want. |
| 4 in question 5 | Read “8 to 9” as ending before 9. | Write the hours in a row and shade the covered ones. |
| 2 in question 5 | Found the distance rather than the count. | Ask to list the missing hours one by one. |
| A, C or D in question 6 | Blamed the wording or the number of options, not the missing “other”. | Ask whether every student has a box that is true for them. |
| 14, 490 or 40 in question 7 | Gave the sample count, multiplied by the sample size, or stopped at the percentage. | Ask how many students each person asked stands for. |
| A, C or D in question 8 | Treated first-hand as accurate, or secondary as better. | Ask what could still go wrong when you collect it yourself. |
| A, B or D in question 9 | Picked a convenient or self-chosen group. | Ask who could not be in this sample. |
Questions worth asking
- Can a question be fair and the sample still unfair? Yes. Listen for “the wording is fine but they all come from one place”.
- Why does asking every 3rd student spread the sample, while the first 6 do not? It takes students from all along the register. Listen for “from everywhere, not one spot”.
- Would a sample of 5 chosen well beat a sample of 50 chosen badly? Neither is good: 5 is too few to be steady, and 50 badly chosen is steadily wrong. Listen for size and fairness both.
- When could every 3rd name be unfair? When the list has a pattern, for example a register sorted by form or by bus route, so every 3rd name is always the same kind of student. Listen for “the list is not in a random order”.
- When is it better to use secondary data? When a large reliable table already exists and collecting your own is impossible. Listen for a reason, not a rule.
Differentiation
A way in for anyone stuck
- Give students the boxes as strips of paper and a row of numbered hour cards to place under them.
- For the sample question, say aloud: “each person asked stands for how many?” before any percentage.
- Use only friendly ratios first: 25%, 50%, 10%.
Stretch
- Question 5, then: design three boxes for minutes of homework (whole minutes, 0 to 180) that leave no gap and no overlap.
- A sample of 40 from 600 has 6 cyclists. Give a range you would expect for the whole school and say why not a single number.
Students learning in English as an additional language
“Survey” means a set of questions asked of many people. “Sample” means a small group taken from a larger one. Say “first-hand” with “I collected it myself”.
| Word | Plain meaning | Example |
|---|---|---|
| primary data | data you collect yourself, first-hand | a class survey |
| secondary data | data someone else collected | a table in a report |
| survey | a set of questions asked of a group of people | how students travel to school |
| sample | a smaller group chosen to stand for a whole group | every 20th name on the register |
| leading question | a question that pushes the answer one way | “Don't you agree it is too noisy?” |
| biased | leaning one way because of who was asked or how | only asking at the bus stop |
| random sample | every member has the same chance of being chosen | names drawn from a hat |
| gap and overlap | a value that fits no box, or a value that fits two | 2 hours fits “0 to 2” and “2 to 4” |
Sentence starters for explaining aloud or in writing:
- This question is leading because …
- These boxes have a gap/overlap at … because …
- This sample is not fair because …, so I would choose …
Two things not to say
“A bigger sample is always right.” Size helps only when the sample is spread fairly.
“Secondary data is worse.” It can be excellent; check who collected it and why.
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.
- Primary or secondaryA student counts the cars passing the school gate in 10 minutes. Is that primary or secondary data?
Primary: collected first-hand. - Every nthA register has 360 students. Every 6th student is asked. How many are asked?
60. 360 ÷ 6 = 60.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA café asks customers: “Our delicious cakes are fantastic. How many do you buy each week?” Name two problems.
Leading wording (“delicious”, “fantastic”) and no options given; it also only asks customers. - Earlier work (Unit 2, lesson 3)Find 25% of 480 and say what it would mean for a sample of 480 asked.
120; a quarter of those asked.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA sample of 50 from a school of 900 has 20 who walk. Estimate the number who walk in the whole school.
360. 20 ÷ 50 = 40%, and 40% of 900 = 360. - Fix the boxesBoxes: “0 to 5”, “5 to 10”, “10 to 15” minutes. Rewrite so every whole number of minutes from 0 to 20 fits exactly one box.
For example 0 to 4, 5 to 9, 10 to 14, 15 or more.
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? Cut a strip into answer boxes and let students lay hour cards beside it; use 30 cards in a row for the sample.