At a glance. One lesson of about 60 minutes. Tree diagrams for two and three stages, counting paths, and counting the outcomes that meet a condition.
Calculator: no calculator. The trees are small enough to draw and count; the lesson is about structure, not arithmetic.
What students will learn
Know
- A tree has a branch for every result at every stage.
- A complete path from left to right is one outcome.
- The number of paths is the product of the branches at each stage.
Understand
- HT and TH are different outcomes.
- A condition picks some of the paths, and they can be counted.
Do
- Draw a tree for two or three stages.
- Count all paths by multiplying.
- Count the paths that meet a condition, including “at least”.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: drawing and reading trees (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, count the paths with each number of heads for 3 tosses and describe the pattern (1, 3, 3, 1).
- Criterion C. Communicating: labelling branches and listing outcomes consistently, and the reasoning in the exit ticket.
- Criterion D. Applying mathematics in real-life contexts: families, codes and menus.
The one thing to do with the figure
Open on 2 tosses and 1 head only after they have said how many outcomes “one of each” is. Then go to 3 tosses and ask for the number of paths with each number of heads before showing them.
The counts 1, 3, 3, 1 are worth a pause: ask why 1 head and 2 heads have the same count (swap H and T). Do not name Pascal's triangle; it arrives later.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 10 min | Hook: Rafa's three outcomes. Ask for the evidence on both sides before the figure. |
| 15 min | The figure; draw the 3-toss tree by hand once. |
| 10 min | Worked examples; the A or B, 1 or 2 or 3 tree 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 | 10; 1/4; 12. |
|---|---|
| Worked examples | Meals: 12. Counter and spinner: 2 outcomes. |
| Faded example | 4 outcomes. |
| 1. Three stages | B, 8. A is a multiplication with the wrong factors; C swaps the number of results and the number of tosses. |
| 2. Paths on a tree | 12 = 4 × 3. |
| 3. At least one head | 3: HH, HT, TH. |
| 4. At least two girls | 4: GGB, GBG, BGG, GGG. |
| 5. A total on two spins | 3: (1, 3), (2, 2), (3, 1). |
| 6. Reading a tree | A. B is the adding slip; C reverses the stages. |
| 7. Working backwards | 4 toppings. |
| 8. A rule on the first stage | 12 = 1 × 3 × 4. |
| 9. Same twice | C. A is the total number of outcomes in the sample space. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 7 in question 2 | Added the branches instead of multiplying. | Draw the first 4 branches and ask how many leaves hang from each. |
| 2 in question 3 | Counted exactly one head and read “at least” as “exactly”. | Cross out only the outcomes with no head. |
| 3 or 1 in question 4 | Counted exactly two girls, or only the all-girl outcome. | List all eight in order and tick. |
| 2 or 1 in question 5 | Missed the double (2, 2). | List with the first spin fixed at 1, then 2, then 3. |
| 18, 8 or 12 in question 7; 24 or 8 in question 8 | Divided by only one of the known stages, mishandled a restricted stage, or used the wrong operation. | Divide by both known stages in turn, and draw a restricted stage with one branch. |
Questions worth asking
- Why do HT and TH count as two outcomes? Listen for “two different paths” or “the coins are separate”.
- What happens to the number of outcomes when you add one more toss? It doubles; ask why.
- How would the tree change if the first stage had only one result? One branch: it changes nothing but the labelling.
Differentiation
A way in for anyone stuck
- Supply a half-drawn tree with the first stage and some of the second stage filled in.
- Colour the first-stage branches differently so that each path's start is visible.
- Use two actual coins on a desk and record each throw as a path.
Stretch
- How many outcomes of 4 tosses have exactly 2 heads? (6.)
- A coin and a 6-sided die: how many outcomes have a head and an even number? (3.)
Students learning in English as an additional language
Pair each word with a gesture: two fingers splitting for “branch”, a hand tracing left to right for “path”.
| Word | Plain meaning | Example |
|---|---|---|
| tree diagram | a diagram that branches at each stage | the coin tree |
| branch | one result at one stage | H from a toss |
| path | one branch from every stage, joined, which is one outcome | H then T then H |
| stage | one step of a multi-step event | the second toss |
| at least | this many or more | at least one head |
Sentence starters for explaining aloud or in writing:
- The tree has … branches at the first stage and … at the second, so … paths.
- HT and TH are different outcomes because …
- I counted at least / exactly … so I included / did not include …
Two things not to say
“It's the same thing in a different order.” For two coins or two dice, HT and TH are different paths. Keeping them apart is what makes the outcomes equally likely, so the count depends on it. (For a team of two people, order would not matter; see the Lesson 2 exit.)
“Count the ends of the branches.” Students then miss the multiplication; say “count the paths, left to right”.
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.
- PathsA tree has 3 branches, each with 2 branches. How many paths?
6. - TossesHow many outcomes are there when a coin is tossed 4 times?
16.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ConditionTwo coins are tossed. In how many outcomes are both the same?
2: HH and TT. - Earlier work (listing)A cafe has 4 drinks and 6 snacks. How many drink-and-snack choices?
24.
3Challenge
Reasoning, generalizing and working backwards.
- Three stagesA menu has 2 starters, 3 mains and 2 puddings. How many meals?
12. - SpinnersA 3-section spinner is spun 3 times. How many outcomes?
27.
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 tree drawn by hand with highlighters does the same job as the figure.