Mathematics · Year 10

Active learning ideas

Tree Diagrams for Dependent Events

Active tasks let students see how probabilities shift when items are removed, turning abstract fractions into concrete evidence. Moving from cards to marbles makes the shrinking pool visible, so students trust the updated branch fractions on their diagrams.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Probability

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Decision Matrix
• 35 min · Small Groups

Hands-On: Card Simulation Stations

Provide decks of cards at stations. Groups draw two or three cards without replacement, record outcomes on mini whiteboards, and construct tree diagrams for probabilities like both red or first ace then king. Compare group results to class predictions. Rotate stations for variety.

Explain how the concept of 'without replacement' alters probabilities in subsequent events.

Facilitation Tip

During Card Simulation Stations, have each pair record the remaining cards after every draw so they can immediately see why the next branch fraction must be smaller.

What to look for

Present students with a scenario: 'A bag contains 3 red marbles and 2 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?' Ask students to draw the tree diagram and calculate the final probability, checking their branch probabilities and final multiplication.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 02

• Decision Matrix
• 25 min · Pairs

Pairs: Custom Problem Swap

Pairs design a 'without replacement' scenario, such as coloured marbles from a bag, and draw its tree diagram with probabilities. Swap problems with another pair, solve using trees, then discuss solutions and check calculations together.

Compare the structure of tree diagrams for independent versus dependent events.

Facilitation Tip

While pairs swap problems in Custom Problem Swap, circulate and ask one partner to explain the updated branch fractions to the other before they calculate.

What to look for

Pose the question: 'How does the calculation of probabilities change when drawing marbles from a bag with replacement versus without replacement? Use a specific example to illustrate your explanation.' Encourage students to refer to their tree diagrams and discuss the concept of conditional probability.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 03

• Decision Matrix
• 30 min · Whole Class

Whole Class: Tree Diagram Relay

Divide class into teams. Project a multi-stage dependent event. One student per team adds a branch to the shared board tree with updated probability, next teammate multiplies or adds as needed. First accurate tree wins.

Design a problem where a tree diagram is essential for understanding dependent probabilities.

Facilitation Tip

In Tree Diagram Relay, stop each group after the first branch and ask them to justify their updated fraction with the cards or marbles they have left in front of them.

What to look for

Give students a partially completed tree diagram for a dependent event scenario (e.g., selecting two students from a group for different roles). Ask them to fill in the missing probabilities on the branches and calculate the probability of a specific outcome, such as 'Student A is chosen first and Student B is chosen second'.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Activity 04

• Decision Matrix
• 20 min · Individual

Individual: Marble Bag Trials

Each student gets a bag of coloured marbles, performs 20 draws without replacement for pairs of colours, tallies results, and builds a tree diagram to predict theoretical probabilities. Share findings in plenary.

Explain how the concept of 'without replacement' alters probabilities in subsequent events.

Facilitation Tip

For Marble Bag Trials, ask students to sketch the first two branches on mini whiteboards before they pick any marbles, forcing them to predict the changing totals.

What to look for

Present students with a scenario: 'A bag contains 3 red marbles and 2 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?' Ask students to draw the tree diagram and calculate the final probability, checking their branch probabilities and final multiplication.

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start with a physical model—cards or marbles—so students feel the pool shrink before they draw the tree. Ask them to predict what the next branch fraction should be, then confirm it with the remaining items. Avoid rushing to the abstract diagram; let the evidence from the removal shape the fractions on the branches.

You will hear students explain why probabilities change after each draw, watch them label branches correctly with updated fractions, and see them multiply along paths then add across paths without mixing the two steps.

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Watch Out for These Misconceptions

• During Card Simulation Stations, watch for students who keep the same branch fractions across all draws because they assume the deck never changes.

  Have them stop after the first draw, count the cards left in their hand, and recalculate the next branch fraction before they continue drawing. Ask them to write the updated fraction on the board for the group to see.

• During Custom Problem Swap, watch for partners who multiply probabilities across different branches rather than along a single path.

  Tell them to trace one colored path with their finger from start to end and label each branch fraction before they multiply; if they try to multiply from different branches, hand them a red pen and have them cross it out.

• During Tree Diagram Relay, watch for groups who skip updating the second branch because they believe the event is still independent.

  Freeze the relay and ask each group to hold up the remaining cards or marbles; then require them to write the new denominator on the board before they proceed to the next step.

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Methods used in this brief

• Decision Matrix