Students take many random samples of a simulated school, see their results vary, and find that bigger samples vary less, for proportions and means (AC9M8ST03, nodes 1 and 2).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min
Do the Run board yourself first. Students need sampling methods and describing a sample (last two lessons), and relative frequency from Chance.
The one idea: random samples differ by chance (sampling variation), and bigger samples differ less. The common mistake is deciding one of two different samples must be wrong (Mia and Jack).
The board plots each sample's result under the school, with the school's value dashed. Changing n clears the results so each spread belongs to one size. With 200 in the school, samples of 150 hardly vary (they hold most of the school).
Wider: polls' margin of error (about ±3% for 1000 people), and ensemble weather forecasts.
Seeing variation as normal, not an error. Ask: could both samples have been chosen fairly?
| What you will see | A question to ask | The fix |
|---|---|---|
| A student thinks a different result means a mistake. | Did the two samples hold the same students? | Different students, different results: sampling variation. |
| A student thinks bigger samples are always closer. | Is every big sample closer, or are they closer on the whole? | Bigger samples vary less on the whole; any one can still be a little off. |
| A student confuses variation with bias. | Do the results scatter around the dashed line, or sit to one side? | Variation scatters around the truth; bias pushes to one side. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Stretch: Record the range of 10 sample results for n = 10, 20, 40, 80 and 150. Plot range against n. What pattern do you see?
Support: A table to record n and the lowest and highest of 10 results, for three sizes.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 8 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "How samples vary". The ideas: - Sampling variation — same population, different samples: I can Students recognise that samples from the same population produce different distributions due to chance, and describe the range of variation. - Effect of sample size on variation: I can Students investigate and describe how increasing sample size reduces variation in sample statistics, and apply this to making inferences. What understanding looks like: - Explains that samples from the same population will naturally vary. - Describes the range of variation observed across multiple samples. - Does not conclude that sampling variation means the data is wrong. - States that larger samples produce less variation in statistics. - Demonstrates the relationship using simulation data. - Applies the principle to justify a recommendation about sample size in a practical context. Mistakes students often make here: - A student thinks a different result means a mistake. - A student thinks bigger samples are always closer. - A student confuses variation with bias. How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "How samples vary".
Two random samples of 10 students give 0.4 and 0.7 for the proportion who walk to school. Is one wrong? What would you do?
Not necessarily: small samples vary a lot. Take bigger samples (or more of them) for a more reliable estimate.
This covers AC9M8ST03.
Where it leads: investigations with samples, where the uncertainty is reported (next lesson), and Year 9 and 10 statistics.