At a glance. One lesson of about 60 minutes. It assumes sample and population from Lesson 1. Students build a sampling frame, take a simple random sample from random numbers, take a systematic sample from an interval and a start, and judge when a pattern in the list spoils it.
Calculator: GDC optional. A GDC can generate random integers for a simple random sample or a random start, but a printed list of random digits does the same job, so the GDC is a convenience and nothing more.
What students will learn
Know
- The sampling frame is the numbered list used.
- A simple random sample uses random numbers and skips repeats and numbers off the list.
- A systematic sample uses an interval equal to population divided by sample size, and a random start.
Understand
- Each member needs the same chance of being chosen.
- A systematic sample can be biased when the list has a pattern whose length matches the interval or shares a factor greater than 1 with it.
- The random start is part of the method.
Do
- Find the interval, list the chosen positions, and find the last.
- Pick a simple random sample from a random-number list.
- Say why a given interval or list could mislead.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the frame, the interval, the start and the random-number rules (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask for the intervals that always give the same nozzle before trying them.
- Criterion C. Communicating: writing the chosen list in full, and explaining a skipped number (the second worked example).
- Criterion D. Applying mathematics in real-life contexts: production lines, registers, markets and timetables.
The one thing to do with the figure
Make the pattern visible first. In the first figure, set the interval to 5 and ask what every chosen bottle has in common. Then ask for an interval that is not affected, and test it.
In the second figure, ask for the interval before moving the slider, and for the third position, so the start-plus-two-intervals idea is spoken.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the five-nozzle machine. Collect predictions about the start. |
| 10 min | First figure and key idea: frame, random, systematic. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask what is still fixed after the random start. |
| 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 | 48; 4, 8, 12, 16, 20, 24; 7. |
|---|---|
| Hook | Yes, with an interval of 5 the start decides everything: every chosen bottle comes from one nozzle, so the sample mean is that nozzle's fill. With an interval of 3 the sample covers all five nozzles evenly. |
| Worked examples | Interval 8, the 25th position is 195; the random-number sample is 17, 08, 39, 24. |
| Faded example | Interval 15; first three 4, 19, 34; last 139. |
| 1. Simple random sample | D. A is systematic; B has no chance in it; C chooses itself. |
| 2. Interval of a systematic sample | 8. 360 ÷ 45. |
| 3. A later position in a systematic sample | 55. 7 + 4 × 12. |
| 4. How many members a systematic sample picks | 23. The last position is 226, and (226 − 6) ÷ 10 + 1 = 23. |
| 5. When a systematic sample misleads | C. A blames size; B is about divisibility, not bias; D says days cannot be sampled. |
| 6. Using a random-number list | 31. Chosen in order: 47, 12, 05, 31. |
| 7. The last member of a systematic sample | 237. 5 + 29 × 8. |
| 8. Writing out a systematic sample | A. B uses the sample size as the interval; C ignores the start; D is a block. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Calls a systematic sample random, picks the first names, or lets people choose themselves. | Ask: before any chance was used, who was already ruled out? |
| 405, 16 200 or 0.125 in question 2 | Adds, multiplies, or divides the wrong way up. | Ask: if 45 are chosen from 360, how many members does one chosen member stand for? |
| 67, 60 or 19 in question 3 | Adds five intervals, leaves out the start, or gives the second member. | List the first few positions in a row before using a rule. |
| 22, 24 or 226 in question 4 | Counts intervals not members, adds one too many, or gives the last position. | Write the list out for a shorter case, such as 6, 16, 26, 36, to count the members. |
| A, B or D in question 5 | Blames the size, the divisibility, or the item type. | Ask: what day of the week is every 7th day? |
| 47, 5 or 96 in question 6 | Skips nothing, skips only the numbers off the list, or skips only the repeats. | Cross out each rejected number with a reason beside it. |
| 245, 240 or 232 in question 7 | Adds 30 intervals to the start, uses 30 intervals with no start, or uses the right 29 intervals but no start. | Count the gaps between 30 chosen members: how many? |
| B, C or D in question 8 | Uses the sample size as the interval, ignores the start, or lists a block. | Test each list: is the gap constant, and is it 9, and does it begin at 3? |
Questions worth asking
- Why must a systematic sample have a random start? So every member has the same chance of being first in the pattern.
- When does an interval of 5 suit a production line, and when does it not? It is fine if the line has no cycle of 5. It fails when a 5-cycle repeats.
- What would you do if the random-number list repeats a number? Skip it, as the member is already in the sample.
- Is the first 30 names on a register random? Why not? No: the rest had no chance, and the first names may differ from the rest.
Differentiation
A way in for anyone stuck
- Give a table with the first six positions written, and ask students to continue it.
- Provide a checklist for random numbers: on the list? repeat? then chosen.
- Use a strip of 30 squares and a counter for the systematic sample.
Stretch
- A line has a cycle of 6. Which intervals from 2 to 12 always pick the same position in the cycle? Which pick every position?
- Explain why a systematic sample with a random start gives every member the same chance of being chosen, even though the sample is not a simple random sample.
Students learning in English as an additional language
“Frame” here is the list that a sample is taken from, not a picture frame. “Interval” is the gap between chosen members. “Random” means decided by chance. “Systematic” means following a fixed rule.
| Word | Plain meaning | Example |
|---|---|---|
| sampling frame | the numbered list of all the members the sample is taken from | a class register |
| simple random sample | a sample in which every member has the same chance and each choice is by chance | names drawn from a hat |
| systematic sample | a sample taken by choosing a random start then every interval-th member | every 6th seat in a hall |
| interval | the gap between one chosen member and the next | population size divided by sample size |
| random start | a starting number chosen by chance between 1 and the interval | a number drawn from a bag of slips |
| bias | a lean in the method that pushes results the same way every time | asking only the first people through the door |
Sentence starters for explaining aloud or in writing:
- The interval is … because ….
- I skipped … because ….
- This could be biased because the list repeats every ….
Two things not to say
“Random means without any plan.” A random method has a precise rule and a source of chance.
“A systematic sample is never random.” With a random start each member has the same chance of being chosen.
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.
- IntervalA list of 480 members, a sample of 60. Find the interval.
480 ÷ 60 = 8. - PositionStart 2, interval 15. What is the 4th position?
2 + 3 × 15 = 47.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA hotel has 300 rooms numbered 1 to 300. A systematic sample of 20 rooms is checked, with a random start. Find the interval and say what the start can be.
Interval 15; the start is any number from 1 to 15. - Earlier work (Lesson 1)A sample of 25 from a batch of 1 000 has 2 faulty. Estimate the faulty items in the batch.
2 ÷ 25 = 0.08 and 0.08 × 1 000 = 80.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA list is sorted by height from shortest to tallest. Is a systematic sample with interval 10 more or less likely to cover the range of heights than a simple random sample of the same size? Explain.
More likely to cover the range, because the chosen members are spread evenly along the sorted list. - Working backwardsA systematic sample has interval 12 and its 6th member is at position 70. What was the start?
70 − 5 × 12 = 10.
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? Give each pair a printed row of 60 numbered squares, and have them mark every 5th and every 7th from a start of their choice.