Learning intention
We are learning to run simulations with more and more trials, and to describe what improves when we do.
Success criteria
- I can set up a chance device in the simulator and check its sample space is right.
- I can predict before I run, then record what the simulator gave me.
- I can work out relative frequency: observed divided by trials.
- I can say which number gets closer to the probability, and which one does not.
Curriculum
AC9M6P02 — conduct repeated chance experiments and run simulations with an increasing number of trials using digital tools; compare observations with expected results and discuss the effect on variation of increasing the number of trials.
Shape of the lesson
| Phase | Time | What happens |
|---|---|---|
| Warm-up | 5 min | Three expected-frequency calculations on whiteboards |
| Launch | 12 min | Demonstrate the simulator on the board: 20, then 200, then 2000 |
| Explore | 25 min | Own device, four scales, recorded in the Excel workbook |
| Summarise | 10 min | Why more trials help — the actual mechanism |
| Review | 4 min | Exit ticket |
What we are learning today
We are learning to run simulations with more and more trials, and to describe what improves when we do.
I will know I can do it when…
- I can set up a chance device in the simulator and check its sample space is right.
- I can predict before I run, then record what the simulator gave me.
- I can work out relative frequency: observed divided by trials.
- I can say which number gets closer to the probability, and which one does not.
Set your goal for today
Read the success criteria above. Which one is the biggest stretch for you? Write one thing you are going to focus on. You will come back to it at the end.
Not sure? Watching the RIGHT column is what this lesson is about — two columns behave differently.
How hard is each question?
Every question in this unit is labelled so you know what kind of thinking it needs. The assessment has questions at all three levels.
SF — simple familiar
You have seen this kind of question before and it is clear what to do. One step, friendly numbers.
CF — complex familiar
You know the method, but there is more than one step, or you have to pick which method to use.
CU — complex unfamiliar
Nobody tells you the method. You work out what to do, and explain why you did it.
Warm-up · 5 minutes
Three on whiteboards
Quick retrieval from Lesson 4. Show one at a time, thirty seconds each, boards up together.
| Question | Answer | |
|---|---|---|
| 1 | A bag has 4 counters, 1 is red. You draw 40 times. How many reds do you expect? | 10 |
| 2 | A spinner has 5 equal sections, 3 shaded. You spin 100 times. How many shaded do you expect? | 60 |
| 3 | A coin. You toss it 2000 times. How many heads do you expect? | 1000 |
Launch · 12 minutes
Two thousand trials would take all term
Do the arithmetic out loud with the class: Lesson 4's forty draws took about ten minutes. Two thousand draws would take around eight hours. Twenty thousand would take four school weeks of solid drawing.
Demonstrate on the board, in this order
- Choose the device. Counters in a bag: 2 red, 1 blue, 1 green — the same bag as Lesson 4, so students recognise it.
- Check the picture. Point at the drawn bag. The picture is the sample space; if it looks wrong, the numbers will be wrong.
- Predict. Enter the class prediction for reds out of 20. Lock it in.
- Run 20. Read the observed count and the relative frequency aloud.
- Run 200. Then 2000. Each time, write the relative frequency on the board beneath the last one.
What the board should look like after the demonstration. Something close to:
20 trials → 0.55 200 trials → 0.48 2000 trials → 0.506
Ask what they notice. Somebody will say the numbers are getting closer to 0.5. Ask how they know 0.5 is the target — the answer is the bag, not the results, and that is Lesson 2 coming back.
Explore · 25 minutes
Your own device, four scales
Students work in pairs on one device, with the simulator open in one window and resources/l5-03-simulation-record.xlsx in another.
The steps are on the task-analysis strip
- Set up your chance device in the simulator. Check the picture matches what you meant.
- Type the same device into the workbook — outcome names and ways. The probability column fills itself in.
- Choose one outcome to follow. Type its name into the yellow cell.
- Predict, lock in, and run at 20. Type the observed count into the workbook.
- Repeat at 200, 2000 and 20 000.
- Look at the graph the workbook draws, then answer the four questions underneath.
What the workbook computes
| Trials | Observed | Expected | Difference | Relative frequency |
|---|---|---|---|---|
| 20 | 11 | 10 | +1 | 0.550 |
| 200 | 96 | 100 | −4 | 0.480 |
| 2000 | 1013 | 1000 | +13 | 0.506 |
| 20 000 | 10024 | 10000 | +24 | 0.501 |
One real set of results. The difference column climbs. The relative frequency column tightens. Both at once.
Circulate and ask one question: "which column is getting closer to the probability?" Not "is it getting closer" — that invites a yes from a student looking at the wrong column.
Complexity bands on the worksheet
SF Calculate relative frequency from an observed count and a trial count.
CF Compare two runs at different scales and say which is better evidence, and why.
CU Two students disagree about whether more trials help. Both are looking at real numbers. Explain what each is looking at and who is answering the right question.
Differentiation
Support
- Trial ladder reduced to 20, 200 and 2000 — three scales rather than four.
- Device supplied: use the Lesson 4 bag so no set-up is needed.
- Task-analysis strip, ticked as they go.
- School Officer support — this is a data-entry heavy session.
Core
- Own device, all four scales.
- Prediction locked in before each run.
- All four reflection questions in the workbook.
Extension
- Before running 20 000, predict the relative frequency to three decimal places. How confident are you, and why?
- Run 2000 three separate times. How much do the three answers differ? Now do the same at 20.
- Explain why more trials help, not just that they do. (AC9M7P02)
Part B · Show your thinking
Out of 250 spins, a spinner landed on red 137 times. What is the relative frequency? Give it as a decimal.
A simulator ran 2000 trials. The relative frequency came out at 0.336. What is the probability most likely to be? Explain how you decided.
Ali says more trials made it worse, because the difference got bigger. Bo says more trials made it better. They are both looking at real numbers. What is each of them looking at? Who is answering the right question?
2 About 1/3. 0.336 is very close to 0.333, and with 2000 trials the relative frequency has settled enough to trust. A device with 3 equal outcomes, or 2 of 6.
3 Both are right about their own column. Ali is watching the raw difference, which usually GROWS with more trials. Bo is watching the relative frequency, which settles towards the probability. Bo is answering the question that matters, because the fraction is what you compare with a probability.
Talk it through with a partner
Show your partner your answer to question 3. Your partner picks one of these to ask you:
- Which column would you show someone to prove more trials help?
- Can both columns be right at the same time?
- What would you say to Ali to change his mind?
Write down the question you were asked and what you said back.
Choose your path
Pick the one that matches where you are right now. You can change your mind.
Out of 50 spins, a spinner landed on red 21 times. What is the relative frequency?
The probability was 0.5. Is 0.42 close to 0.5? Write yes or no, and say why.
Which run gives you better evidence: 20 trials or 2000 trials? Explain your answer.
Before you run 20 000, predict the relative frequency to three decimal places. How confident are you? Say why.
Run 2000 three separate times. How different are the three answers? Now do the same at 20 trials. What do you notice?
Explain WHY more trials help — not just that they do.
★2 At 2000 the three answers agree to about two decimal places. At 20 they can differ wildly. That collapsing spread is variation.
★3 Because each single trial has less power to move the fraction. At 20 trials one extra red shifts it by 1/20, which is 5%. At 2000 it shifts by 1/2000, which is 0.05%.
Summarise · 10 minutes
Why more trials actually help
This is the explanation, and it is sayable to Year 6. Put it on the board as a question first.
| Trials | One extra red changes the fraction by |
|---|---|
| 20 | 1/20 = 0.05, which is 5% |
| 200 | 1/200 = 0.005, which is 0.5% |
| 2000 | 1/2000 = 0.0005, which is 0.05% |
| 20 000 | 1/20 000 = 0.00005, which is 0.005% |
An unlucky run at 20 trials swings the answer wildly because each trial is worth 5% of the total. The same unlucky run inside 2000 trials is swamped by all the ordinary results around it.
The variation point. If you ran this lesson with 200 classes, the relative frequencies at 20 trials would spread from about 0.20 to 0.80. At 20 000 they would spread from about 0.489 to 0.510. That collapsing spread is what "variation" means, and it is the word Lesson 6 needs. Introduce it here if the class is with you.
Three things to land
- Relative frequency is observed divided by trials. It turns a count into something you can compare with a probability.
- As trials increase, relative frequency settles towards the probability. The raw difference does not — it usually grows.
- The reason is that each trial matters less. One result out of 20 000 cannot move the fraction the way one result out of 20 can.
Vocabulary
Add to the Frayer boards: relative frequency. Use the blank page from the Frayer pack. Also revisit expected and observed from Lesson 4 — students now have far better examples for their quadrants.
Review · 4 minutes
Exit ticket
What is the relative frequency of heads? Give it as a decimal.
Is 262 heads close to what you would expect? Explain.
Jo ran the same coin 40 times and got 24 heads. Whose result is better evidence that the coin is fair? Explain your answer.
1. 262 ÷ 500 = 0.524. Accept 0.52.
2. Expected is 250. Observed is 262, so it is 12 out of. As a fraction, 0.524 against 0.5 — close. Both readings are acceptable; a student who only says "no, it should be 250" has missed Lesson 4.
3. The 500-trial run. Jo's 24 out of 40 is a relative frequency of 0.6, which looks further from 0.5 — but the real reason is the sample size, not the number. With only 40 trials, one lucky streak moves the answer a long way. The 500-trial run had less room to be misled.
Question 3 is the discriminator and it feeds straight into Lesson 6. A student who says "Jo's, because 24 is closer to half of 40 than 262 is to half of 500" is comparing raw differences and will be defeated by the mystery dice tomorrow. Anyone giving that answer should be paired deliberately in Lesson 6.
Note that 0.6 is further from 0.5 than 0.524 is, so the numbers happen to agree with the correct reasoning here. Do not let that pass as understanding — ask which sample you would trust if Jo's 40 trials had given exactly 20.
Think back
Your goal
Go back to the Intentions tab and read the goal you set. Did you meet it?
Nothing you type here is saved. Copy it into your book before you close the page.
What you need
Digital
- shared/chance-simulator.html — one per pair, plus the board copy. Test on a student device first.
- resources/l5-03-simulation-record.xlsx — one per pair. Formulas and the graph are already built; only the yellow cells are editable.
- Board or projector for the Launch demonstration
- resources/l5-01-worksheet-core.pdf — 3 pages. Also the paper pathway if devices fail.
- resources/l5-02-worksheet-support.pdf — 3 pages, three scales rather than four
- shared/task-analysis-strips.pdf — the Lesson 5 strip is pre-filled
- shared/conversion-mat.pdf and the Frayer blank for relative frequency
Lesson-specific adjustments
Standing adjustments are on the linked Personalised Adjustments document and are not repeated here.
- School Officer support flagged — this is the most data-entry heavy session in the cycle.
- Workbook supplied with formulas and trial counts already entered. Students type observed counts only.
- Trial ladder reduced to 20 / 200 / 2000 where four scales is too many. The pattern is visible in three.
- Device supplied rather than built for pairs who would spend the lesson on set-up: use the Lesson 4 bag, which they already know.
- Task-analysis strip — six steps, ticked as completed.
Misconceptions to expect today
"More trials made it worse — look, the difference got bigger"
Entirely reasonable, and based on a real column. The student is watching the difference rather than the relative frequency.
Response. Do not correct it as an error. "You are right about that column. Now look at the one next to it." Both facts are true at once, and holding both is the actual learning.
"The simulator is rigged — it gave me exactly 1000"
Or the reverse: a student gets an unusual result and concludes the tool is broken.
Response. Have them run the same scale three times and compare. Three different answers from the same set-up settles it faster than any explanation.
Typing the relative frequency in as the observed count
The simulator shows both. Students enter 0.55 where the workbook wants 11, and everything downstream breaks.
Response. The workbook's observed column is formatted as a whole number, so 0.55 appears as 1. Point at the column header: it says observed, and observed is a count of things that happened.
"20 000 trials means the answer is exact now"
Students swing from "results are random" to "big samples are certain". The relative frequency at 20 000 is close, not exact, and it is still different every run.
Response. Run 20 000 twice on the board. The two answers differ in the third decimal place. Close is not the same as certain.
What comes next
Lesson 6 — Is this die fair? Withdrawal-safe, and the reasoning checkpoint of the whole probability block. Students get twenty rolls to identify a weighted die, discover twenty is not enough, and then have to design a test that would actually settle it.