Learning intention
We are learning to work backwards from probabilities to build a sample space, and to find more than one answer that works.
Success criteria
- I can build a deck that matches the probabilities I am given.
- I can find more than one deck that works.
- I can explain why some deck sizes are impossible.
- I can work out the probability of the outcome nobody mentioned.
Curriculum
AC9M6P01 — recognise that probabilities lie on numerical scales of 0–1 or 0%–100% and use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals.
It is also a Peter Sullivan-style open task. There is no single answer, and the number of cards is deliberately not given. The withheld information is the task.
Shape of the lesson
| Phase | Time | What happens |
|---|---|---|
| Warm-up | 5 min | An event with a probability of 1/4 |
| Launch | 8 min | Pose the deck problem. Refuse to say how many cards. |
| Explore | 27 min | Think alone, build in pairs, test with 20 draws, post to the class board |
| Summarise | 8 min | Why every deck size is a multiple of 4 — and what stayed the same |
| Review | 5 min | Exit ticket, collected |
Withdrawal-safe — five-minute rejoin note
Nothing new is taught today. This lesson consolidates Lessons 1 and 2, so students in the extension withdrawal group lose no content. When they return, reinstate three things:
- The deck-size rule. "A deck had a 1/4 chance of a picture card. Could it have had 10 cards? Why not?" Thirty seconds tells you whether they have it.
- The class norm. Point at the board of valid decks and say the task had many correct answers. This norm carries into Lesson 7.
- The open question from Lesson 2 is still on the wall and still unanswered. Make sure they know it is coming.
Then give them the Going Further items from the worksheet to do independently. They do not need the Launch to attempt them.
What we are learning today
We are learning to work backwards from probabilities to build a sample space, and to find more than one answer that works.
I will know I can do it when…
- I can build a deck that matches the probabilities I am given.
- I can find more than one deck that works.
- I can explain why some deck sizes are impossible.
- I can work out the probability of the outcome nobody mentioned.
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? Finding a SECOND deck that works is the part most people stop short of.
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
A one-in-four chance of what?
Individual, on whiteboards. Collect four or five that use different equipment — a spinner, a bag, a die, something from real life.
Watch for the student who writes 1/4 on the number line but cannot invent an event. That student needs the four-card entry point in Explore, not the eight.
Launch · 8 minutes
The mystery deck
If I draw one card without looking:
— the probability of a picture card is 1/4
— the probability of an even number is 1/2
What might my deck be?
One deck that works. There are many others.
Set the rules of the deck
Agree these as a class before students start, and write them on the board:
- Picture cards are J, Q, K. They have no number.
- Numbered cards run from 1 to 10. Odd numbers are 1, 3, 5, 7, 9. Even numbers are 2, 4, 6, 8, 10.
- Cards may repeat. You can have three 4s if you want.
- Every card is either a picture card or a numbered card. Nothing is both.
Better: use the blank card templates. Students write their own values, which removes the Ace problem entirely and means you do not need fifteen real decks.
Explore · 27 minutes
Think alone first
Four steps, in this order
- Alone, five minutes. Each student writes down one deck that they think works. No talking. This matters — pairs that start talking immediately produce one idea between two people.
- In pairs. Compare. Are they the same? Are both right? Can both be right?
- Build it. Make the deck physically with the card blanks.
- Test it. Draw a card, record whether it is a picture card, put it back. Twenty times.
The class board
As pairs finish, they write their deck on the board as three numbers: picture / even / odd, and the total. Do not sort them. Let the list build up messily.
| Picture | Even | Odd | Total | Valid? |
|---|---|---|---|---|
| 1 | 2 | 1 | 4 | yes — the smallest one |
| 2 | 4 | 2 | 8 | yes |
| 3 | 6 | 3 | 12 | yes |
| 5 | 10 | 5 | 20 | yes |
| 2 | 4 | 0 | 6 | no — 2 out of 6 is not 1/4 |
If nobody produces it, put it on the board yourself as "a deck someone tried in another class" and ask whether it works.
Circulating. Ask every pair the same question: "could your deck have been bigger?" Students who say yes and can produce the next size up have the generalisation. Students who say "no, this is the answer" have solved an arithmetic problem rather than an open one.
Complexity bands on the worksheet
SF Build one deck that satisfies both probabilities and record it.
CF Find the probability of drawing an odd-numbered card — the outcome the problem never mentioned. Then test whether a given deck size is possible.
CU Explain why every deck that works has a total that is a multiple of 4.
Differentiation
Support
- One constraint at a time: build a deck with P(picture) = 1/4 first, then add the even-number rule.
- Start with exactly 4 cards, then double it.
- Card blanks pre-cut.
- Conversion mat on the desk.
Core
- Both constraints from the start.
- Find at least two decks of different sizes.
- Full worksheet including the multiple-of-4 explanation.
Extension
- A deck has exactly 20 cards. How many of each kind? Is there more than one answer?
- If 60 Prep students each drew one card and put it back, how many picture cards would you expect? (AC9M7P01)
- A different deck has P(picture) = 1/3 and P(even) = 1/2. Which deck sizes work now, and what is P(odd)?
Part D · Show your thinking
Nobody told you about the odd-numbered cards. What is the probability of drawing an odd-numbered card? How do you know?
Two students both made decks that work. One used 8 cards, the other used 20. Are they both right? How do you know?
Look at every deck that works. Why is the total always a multiple of 4?
2 Yes, both. 2 picture of 8 is 1/4, and 5 picture of 20 is 1/4. Different sizes, identical probabilities.
3 A quarter of the deck has to be a whole number of picture cards. You cannot have 2.5 kings, so the total must divide by 4.
Talk it through with a partner
Show your partner your answer to question 2. Your partner picks one of these to ask you:
- Could there be a deck bigger than 20 that also works?
- What is the smallest deck that works, and how do you know there is nothing smaller?
- If both are right, what is actually the same about them?
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.
What is the smallest deck that works?
Draw a deck of 8 cards that works. Write how many of each kind you used.
What do the totals of all the decks that work have in common?
A deck has exactly 20 cards. How many of each kind? Is there more than one answer?
If 60 Prep students each drew one card and put it back, how many picture cards would you expect?
A DIFFERENT deck has P(picture) = 1/3 and P(even) = 1/2. Which deck sizes work now? What is P(odd) for that deck?
★2 60 × 1/4 = 15. Expect some students to say "15 exactly" — press them, because Lesson 4 makes this distinction the whole point.
★3 Multiples of 6 now: 6, 12, 18, 24. And P(odd) is 1/6, not 1/4. The invariant changed with the constraints — this separates students who found a rule from students who found the rule.
Summarise · 8 minutes
What stayed the same
Now sort the class board by total size. Put the invalid ones to one side rather than rubbing them out.
Three things to land
- There were many correct answers. Count them on the board. This is not a lesson where one pair won.
- Every deck that works is a multiple of 4. Because a quarter of the deck has to be a whole number of picture cards. You cannot have 2.5 kings.
- Every valid deck has the same three probabilities: 1/4, 1/2, 1/4. The decks are different sizes and the probabilities are identical. That is the point of the lesson.
The route to the odd cards is the complement, which is on the conversion mat: the two given probabilities are 1/4 and 1/2, which total 3/4, and everything must total 1, so the odd cards are the remaining 1/4. Some students will instead count the leftovers in their own deck. Both are correct. Ask the counters whether their method would still work for a deck of 400 cards.
Vocabulary
No new Frayer words today. Revisit sample space from Lesson 2 — students now have a second, harder example to add to the Examples quadrant.
Review · 5 minutes
Exit ticket
Give two DIFFERENT bags that would work. They must have different numbers of counters altogether.
Could the bag hold 8 counters? Explain your answer.
In one of your bags, what is the probability of drawing a counter that is NOT red?
Answers.
1. Any bag whose total is a multiple of 3 with a third red: 1 red of 3, 2 of 6, 3 of 9, 4 of 12. Two different sizes are required — two bags of 3 is not two answers.
2. No. 8 is not a multiple of 3, so a third of 8 is 2.67 counters, and you cannot have part of a counter. Accept any correct reasoning; the number 2.67 does not have to appear.
3. 2/3, whichever bag they chose. A student who gives a different answer for a different bag has missed the invariant and should be paired with a confident student in Lesson 4.
Going Further answers (worksheet page 2).
a. 5 picture, 10 even, 5 odd. There is only one answer for the counts — though the specific cards can vary, which is worth asking about.
b. 60 × 1/4 = 15 picture cards. Expect some students to say "15 exactly" — press them, because Lesson 4 is about to make this distinction the whole point.
c. Now the sizes are multiples of 6: 6, 12, 18, 24. And P(odd) is 1/6, not 1/4. This is the item that separates students who found a rule from students who found the rule — the invariant depends on the constraints, and it changed.
Sorting for Lesson 4. Question 2 is the discriminator. Students who explain it cleanly are ready for the non-whole expected frequencies in Lesson 4's extending prompt.
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
- resources/l3-01-worksheet-core.pdf — 3 pages, most of the class
- resources/l3-02-worksheet-support.pdf — 3 pages, one constraint at a time
- resources/l3-03-card-blanks.pdf — blank cards to cut, plus a reference page showing picture, even and odd cards. One sheet per pair; pre-cut for students who need it.
- shared/conversion-mat.pdf — needed for the complement in the Summarise
Equipment
- Scissors, one pair per pair of students
- Envelopes or small bags to hold each deck
- Board space for the class list of decks — leave it up for the Summarise
- Real playing cards are optional. The blanks are easier and avoid the Ace argument.
Lesson-specific adjustments
Standing adjustments are on the linked Personalised Adjustments document and are not repeated here.
- WITHDRAWAL-SAFE lesson. No new content. The five-minute rejoin note is on the Overview tab.
- Four-card deck as the entry point, extended to eight. A student who builds only the smallest valid deck has still met the success criteria.
- Card blanks pre-cut for students for whom scissor work would consume the thinking time.
- Reference page showing what counts as a picture card, an even card and an odd card — for students unfamiliar with a standard deck, which includes more of the class than you expect.
- Silent thinking time is protected. For students who find open tasks anxiety-provoking, hand them the four-card version at the start rather than after they have sat blank for five minutes.
Misconceptions to expect today
"There must be one right answer"
Students look for the deck rather than a deck, and stop as soon as they find one. Some will assume the pair next to them is wrong.
Response. Ask both pairs to test their decks. Two different decks that both pass is more convincing than anything you can say.
Forgetting the odd-numbered cards
The problem mentions picture cards and even numbers, so students build a deck of only those two kinds. The probabilities then come out wrong and they cannot see why.
Response. "Add up your two probabilities. What do you get? What do all the probabilities have to add to?" Then let them find the missing quarter.
Choosing a deck size that satisfies one constraint but not both
Six cards works for the even-number half. It fails for the picture-card quarter. Students check one and stop.
Response. Make them state both checks explicitly before posting to the board. The worksheet has a check column for exactly this.
Drawing without replacement
Students deal the cards out rather than drawing and returning. The probabilities shift with every draw.
Response. Catch it while circulating, not afterwards. Ask what the probability of a picture card is once they have already taken it out.
What comes next
Lesson 4 — Expected or observed? The first lesson of the AC9M6P02 block, and where the open question from Lesson 2 starts to be answered. Students meet expected frequency: probability multiplied by the number of trials.