Learning intention
We are learning to judge whether the evidence we have is strong enough to answer the question we are asking.
Success criteria
- I can say what a result would look like if a die were fair.
- I can explain why a small number of rolls cannot settle the question.
- I can design a test: how many trials, why that many, and what result would convince me.
- I can say "we cannot tell yet" when that is the honest answer.
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.
Lesson 5 established that more trials make the fraction settle. Today applies it under pressure, when there is something at stake in the answer.
Shape of the lesson
| Phase | Time | What happens |
|---|---|---|
| Warm-up | 6 min | Two students, two dice, two very different results |
| Launch | 12 min | The mystery pair. Twenty rolls each, then a class vote. |
| Explore | 24 min | Design a test that would actually settle it, then run it |
| Summarise | 10 min | Evidence proportionate to the claim. Reveal. |
| Review | 4 min | Exit ticket |
Withdrawal-safe — five-minute rejoin note
No new content is taught today. The lesson applies Lessons 4 and 5 rather than adding to them, so the extension withdrawal group loses nothing. When they return, reinstate three things:
- The headline. "Somebody rolled a die 20 times and got six 6s. Does that prove it is loaded?" Thirty seconds tells you whether they have it. The answer is no — that happens to a fair die about one time in ten.
- The phrase. "We cannot tell yet" is a legitimate mathematical answer, and it will be needed again in Lesson 14.
- The Games Day brief is coming in Lesson 7. Make sure they know their Family Group team is already assigned.
Then set them the Going Further items, which need no part of the Launch.
What we are learning today
We are learning to judge whether the evidence we have is strong enough to answer the question we are asking.
I will know I can do it when…
- I can say what a result would look like if a die were fair.
- I can explain why a small number of rolls cannot settle the question.
- I can design a test: how many trials, why that many, and what result would convince me.
- I can say "we cannot tell yet" when that is the honest answer.
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? Saying in advance what result would convince you is the hardest part today — and the most important.
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 · 6 minutes
Who has the fair die?
Ana counted her sixes: she got six 6s.
Ben counted his sixes: he got none at all.
One of them has a loaded die. Which one?
Hands up for Ana. Hands up for Ben. Hands up for "cannot tell". Record the three counts on the board — you will come back to them in the Summarise.
· expected number of 6s: 3.33
· chance of getting six or more 6s: 10.2% — about one roll-of-twenty in ten
· chance of getting no 6s at all: 2.6%
So in a class of thirteen pairs all rolling fair dice, you would expect roughly one pair to see Ana's result. Ben's is rarer but hardly impossible.
Do not give these numbers away now. Take the vote, write the three counts on the board, and move on. The numbers land properly in the Summarise, after students have felt the uncertainty themselves.
Launch · 12 minutes
The mystery pair
Open the Mystery dice tab of the Chance Simulator on the board. Two dice, A and B. One is fair. The other is weighted so that one face is three times as likely as any other. Nothing on the screen says which.
Set the roll count to 20, roll both, and put the two tables side by side. Give groups two minutes to argue, then take a vote: A, B, or cannot tell.
| Rolls each | Right | Wrong | Too close to call |
|---|---|---|---|
| 20 | 74% | 13% | 13% |
| 60 | 97% | 1% | 1% |
| 200 | 100% | 0% | 0% |
The weighted face is 3/8, or 37.5%. A fair face is 1/6, about 16.7%. The gap is large — and twenty rolls still is not enough to see it reliably. Students find that genuinely surprising.
If the class votes overwhelmingly one way, ask the confident students what result would have changed their minds. Most will not have an answer, and noticing that is the point.
Explore · 24 minutes
Design a test that would actually settle it
Groups of three. The task is not to find the answer. The task is to design the test.
1. How many rolls will you use?
2. Why that many?
3. What result would convince you?
Only when all three are written do groups run the test in the simulator.
REASONING CHECKPOINT — collect these
Every student writes their own justification on page 2 of the worksheet, even though the design was a group decision. This is the formative evidence for the whole probability block.
A student who can name what the fair result should look like has understood expected frequency. A student who can say how far apart the two predictions are has understood why sample size matters. Both are A-standard signals for item A7.
What to do with a group that says "1000 rolls". Ask whether they need that many. Then ask what the smallest number would be that they would still trust. Over-testing is not a mathematical error, but choosing a number without reasoning is.
Circulating question: "What would you expect to see if it were fair?" Groups that cannot answer this cannot design a test, because they have nothing to compare against. Send them back to their Lesson 4 expected-frequency table.
Complexity bands on the worksheet
SF Expected frequency for a fair die at a given roll count.
CF Two dice, two sets of results — which is more likely weighted, and how sure can you be?
CU Explain why more rolls give better evidence. "More is better" does not count.
Differentiation
Support
- Trial numbers supplied — 20, 60 and 200. The student judges the evidence rather than choosing the scale.
- Sentence stems from the shared pack, on the desk before they start.
- Expected-frequency table pre-calculated for a fair die at each scale.
- Group of three with a named recorder.
Core
- Choose their own roll count and justify it.
- State in advance what result would convince them.
- Individual written justification.
Extension
- Explain why more rolls reduce variation, not just that they do. (AC9M7P02)
- What is the smallest number of rolls you would trust? Defend it.
- What would you say to someone who insists their twenty rolls prove the die is loaded?
Part B · Design a test that would settle it
How many rolls will you use?
Why that many? What made you choose that number?
What result would convince you? Write it down BEFORE you roll.
A student who can name what the fair result should look like has understood expected frequency. A student who can say how far apart the two predictions are has understood why sample size matters. Both are A-standard signals for item A7.
Part C · Show your thinking
A fair six-sided die is rolled 60 times. How many 4s would you expect? Show your working.
Two dice were each rolled 30 times. Die A showed a six 5 times. Die B showed a six 11 times. Which is more likely weighted? How sure can you be?
Explain WHY more rolls give better evidence. "More is better" is not an explanation — say what more rolls actually do.
2 Die B looks more likely, since 11 is well above the 5 you would expect. But 30 rolls is still a small sample and you cannot be sure. "B, but I am not certain" is the strongest answer.
3 Each single roll has less power to move the fraction. One extra six out of 20 rolls shifts the result by 5%; out of 2000 it shifts it by 0.05%. An unlucky run gets swamped by all the ordinary results around it.
Talk it through with a partner
Show your partner your answer to question 3 of Part B. Your partner picks one of these to ask you:
- What would you expect to see if the die WERE fair?
- How far apart are your two predictions?
- Would you still believe your answer if you got the opposite result?
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.
| Rolls | Each face should be about | A weighted face would be about |
|---|---|---|
| 20 | 3 | 8 |
| 60 | 10 | 22 |
| 200 | 33 | 75 |
Roll both dice 20 times each. Write down which die you think is weighted.
Now roll both 200 times each. Did your answer change? Which set of rolls do you trust more, and why?
What is the smallest number of rolls you would trust? Defend it.
What would you say to someone who insists their twenty rolls prove the die is loaded?
★2 A fair die produces six or more 6s in twenty rolls about one time in ten. Their result is not surprising enough to prove anything. Strong answers offer a way to settle it rather than just disagreeing.
Summarise · 10 minutes
Now reveal it
Press Reveal on the simulator. Then, before any celebrating, go back to the vote you recorded in the Launch and count how many groups were right.
Then return to the warm-up and give the real numbers: a fair die produces six or more 6s in twenty rolls about 10% of the time, and no 6s at all about 2.6% of the time. Both Ana and Ben could easily have had fair dice. Nobody in the room could have known.
Three things to land
- A small sample cannot settle a question about fairness. Twenty rolls looks like data and behaves like noise.
- The evidence has to be proportionate to the claim. Saying a die is loaded is a big claim, and it needs more than a hunch and twenty rolls.
- "We cannot tell yet" is a real answer. It is what a mathematician says when the evidence is not there, and it is stronger than a confident guess.
Vocabulary
Add to the Frayer boards: variation. Students now have the best possible example for the Examples quadrant — the spread of results between groups at twenty rolls, compared with the spread at two hundred.
Review · 4 minutes
Exit ticket
How many reds would you expect in 12 spins?
Is Kai right that the spinner is rigged? Explain your answer.
What would you do to find out for sure? Say how many spins you would use and why.
Answers.
1. 3. Method 1: 12 ÷ 4 = 3, then × 1 = 3. Method 2: 1 ÷ 4 = 0.25, then × 12 = 3.
2. No — or more precisely, we cannot tell. Five is more than three, but twelve spins is a very small sample and a fair spinner would produce five reds fairly often. Accept "we cannot tell yet" as the strongest answer. Accept "no, because 12 spins is not enough" as fully correct. Do not accept "no, because 5 is close to 3" without the sample-size reasoning — that is the right answer for the wrong reason.
3. Any number with reasoning attached. Look for a student who says what the fair result would look like at their chosen scale — "if I spun it 200 times I would expect about 50 reds, so if I got 90 I would believe Kai". The number matters far less than whether a prediction comes with it.
The discriminator is question 3, not question 2. Most students will get question 2 right by now. Question 3 asks them to design, which is what item A7 asks. A student who writes a number with no reasoning has not met the success criteria, even if the number is sensible.
Support version. Same three questions, but the spinner is spun 16 times and lands on red 6 times, so the expected number is 4. The excess is smaller relative to the sample, which makes "we cannot tell" easier to reach.
Sorting for Lesson 7. Students who write a prediction into question 3 are ready to lead their Family Group team's game design. Put one in each team.
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 — the Mystery dice tab. One per group plus the board copy. The weighted face is chosen fresh each time, so the answer cannot travel between groups or between your two classes.
- resources/l6-01-worksheet-core.pdf — 3 pages
- resources/l6-02-worksheet-support.pdf — 3 pages, roll counts supplied and the fair-die table pre-calculated
- shared/sentence-stems.pdf — the Judging fairness block. Put it out before students start, not after they struggle.
- shared/frayer-boards.pdf — print the variation page
Optional
- A physical weighted die, if you have one. Not required — the simulator does the job and gives you a fresh pair every time.
Lesson-specific adjustments
Standing adjustments are on the linked Personalised Adjustments document and are not repeated here.
- WITHDRAWAL-SAFE lesson. No new content. The rejoin note is on the Overview tab.
- Private advance notice for students who find being wrong aversive. The Launch is designed so that most of the class votes without enough evidence, and about a quarter will be wrong. Tell those students beforehand that being wrong is the plan, and that nobody will be singled out.
- Roll counts supplied — 20, 60 and 200 — so the student judges the evidence rather than designing the scale. The reasoning is still theirs.
- Fair-die expected frequencies pre-calculated on the Support worksheet, so the comparison is available without redoing Lesson 4.
- Sentence stems on the desk from the start. This lesson asks for more written justification than any lesson so far.
- Named recorder in each group of three, but every student writes their own justification — the checkpoint is individual.
Misconceptions to expect today
"Six 6s in twenty rolls proves it's loaded"
The headline misconception. A run that feels extreme is treated as proof.
Response. The number: a fair die does that about 10% of the time. Roll twenty on the simulator ten times over and let the class watch how often something surprising turns up.
"I was right, so my method was good"
Groups that guessed correctly at twenty rolls conclude their reasoning was sound. It was not — they were lucky.
Response. Ask them to predict again with a fresh mystery pair, still at twenty rolls. A second round separates method from luck quickly.
"More rolls is better" with nothing behind it
Students repeat Lesson 5's conclusion as a slogan without the mechanism.
Response. "Better how?" Then point at the one-extra-roll table from Lesson 5. At twenty rolls, one extra six moves the fraction by 5%.
Refusing to commit to any answer
The opposite failure, and it appears after the Summarise. Students learn that "cannot tell" is safe and apply it to everything.
Response. Run 2000 rolls on the mystery pair. The evidence is overwhelming and "cannot tell" is now the wrong answer. Enough evidence means you should commit.
What comes next
Lesson 7 — Designing for our buddies. Family Group teams design a chance game for their Prep buddies, list the sample space, assign probabilities, and simulate it. The design brief constraints are non-negotiable, and peer critique uses the protocol cards.