At a glance. One lesson of about 60 minutes. Sample spaces for two stages, organised lists and two-way tables, and counting outcomes by multiplying.
Calculator: no calculator. The numbers are small so that the work is in listing systematically and noticing when order matters.
What students will learn
Know
- The sample space is the list of every possible outcome.
- Each outcome is one complete result of every stage.
- A table or an organised list keeps outcomes from being missed or repeated.
Understand
- Every first result pairs with every second result, so the number of outcomes is a product.
- If order matters, (a, b) and (b, a) are different outcomes.
Do
- List the sample space for two stages systematically.
- Count outcomes by multiplying.
- Find the outcomes that meet a condition from a table.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: listing and counting outcomes for two-stage events (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, predict the count before each move and describe how it changes when one spinner gains a result. Look for a rule in words.
- Criterion C. Communicating: using the notation (2, 3) or 2C for an outcome consistently, and the reasoning in the exit ticket.
- Criterion D. Applying mathematics in real-life contexts: menus, codes and choosing roles, and judging whether order matters.
The one thing to do with the figure
Ask for the count before each move, and for the number of cells added when a spinner gains a result. Going from 3 letters to 4 with 4 numbers adds a whole column of 4 cells, not 1.
Then ask: what would the grid look like if the order mattered, as with president and secretary, and the diagonal were not allowed? That is the exit ticket and the bridge to tree diagrams next lesson.
A possible order
| 5 min | Warm-up, then answers; dwell on the seats example (rows and columns). |
|---|---|
| 10 min | Hook: shirts and caps. Take 7 seriously and ask what they were counting. |
| 15 min | The figure with predictions; organised lists and the table. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 0.3; 0; 24. |
|---|---|
| Worked examples | Coin and die: 8 outcomes. Spinners: 3 outcomes with total 5. |
| Faded example | 6 outcomes. |
| 1. Counting outcomes | B, 12. A is the adding slip; D treats the coin as a die. |
| 2. Soup and bread | 15 = 5 × 3. |
| 3. Two spinners | 20 = 5 × 4. |
| 4. A total from the table | 5 outcomes. |
| 5. Different numbers | 12 = 16 − 4. |
| 6. Organised list | B. Look for a list that fixes the coin result and runs through the colours. |
| 7. Working backwards | 5 drinks. |
| 8. No repeats | 20 numbers. |
| 9. Order matters | B. A treats the pair as a team and ignores who holds which job; C allows repeats. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 8 in question 2; 9 in question 3 | Added the number of results instead of multiplying. | Draw the grid and ask what each cell stands for. |
| 3 or 4 in question 4 | Missed that (6, 2) and (2, 6) differ, or missed the double (4, 4). | List the first die fixed at 2, then 3, 4, 5, 6 in a column. |
| 4 or 6 in question 5 | Counted doubles, or treated (1, 2) and (2, 1) as one. | Ask: if the dice were two different colours, would these be the same outcome? |
| 16, 24 or 80 in question 7 | Used the wrong operation to work backwards. | Ask which operation was used to make 20, then undo it. |
| 25, 10 or 9 in question 8 | Allowed repeats, ignored order, or added 5 + 4. | Write the first few numbers: 12, 13, 14, 15, 21, … |
Questions worth asking
- How do you know your list is complete? Listen for a system: the first result held fixed, then moved on.
- When does (a, b) differ from (b, a)? When the two stages are different or have different roles.
- Why do the outcomes multiply rather than add? Listen for “each one of the first goes with all of the second”.
Differentiation
A way in for anyone stuck
- Give a ready-made grid to fill in before they build their own.
- Use real objects: three T-shirts and four caps drawn on cards that can be paired.
- Let students colour all outcomes with the same first result the same colour.
Stretch
- How many outcomes have a sum that is a multiple of 3 when two 6-sided dice are rolled? (12.)
- Three spinners each have 3 results. How many outcomes? Explain why it is a product of three numbers. (27.)
Students learning in English as an additional language
Pair each word with a gesture: a finger running along a row for “list”, two hands joining for “pair”.
| Word | Plain meaning | Example |
|---|---|---|
| sample space | the list of every possible outcome | H1, H2, H3, H4, T1, T2, T3, T4 |
| outcome | one possible result, taking every stage together | tails and a 3 |
| organised list | a list with a system, so none is missed | H1, H2, H3, then T1, T2, T3 |
| two-way table | a grid with one stage down the side and one along the top | the dice grid |
| combination | one particular pairing of choices | a soup with a bread |
Sentence starters for explaining aloud or in writing:
- There are … outcomes because … × …
- I know I have all of them because …
- (2, 5) and (5, 2) are / are not the same outcome because …
Two things not to say
“Just add them.” Students then add the number of results at each stage; draw the grid first.
“Order doesn't matter in probability.” It matters whenever the stages are different things, such as two different dice or two jobs.
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.
- CountingA quiz has 6 questions in round 1 and 4 in round 2, and each team picks one of each. How many picks?
24. - SpinnersOne spinner has red, green and blue. Another has 1 and 2. How many outcomes?
6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- TotalTwo 4-sided dice are rolled. In how many outcomes is the total 5?
4: (1, 4), (2, 3), (3, 2), (4, 1). - Earlier work (probability scale)A bag has 3 green counters in 12. What is P(green) as a percentage?
25%.
3Challenge
Reasoning, generalizing and working backwards.
- RolesFour friends choose a leader and a deputy, who must be different. How many ways?
12. - Working backwardsA menu has 3 starters and some mains, giving 21 starter-and-main choices. How many mains?
7.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The sliders work with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: a grid drawn by hand does the same job as the figure.