Mathematics · Year 9

Active learning ideas

Tree Diagrams for Dependent Events

Active learning helps Year 9 students grasp how probabilities shift without replacement by making abstract dependencies concrete. When students physically draw marbles or cards, they see totals shrink and probabilities adjust, turning symbolic notation into tangible evidence.

• National Curriculum Attainment Targets
• KS3: Mathematics - Probability

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Simulation Game
• 35 min · Pairs

Pairs Practice: Card Draw Simulations

Pairs share a standard deck and predict probabilities for two draws without replacement using tree diagrams. They perform 20 trials, recording outcomes on a class chart. Discuss how actual frequencies match tree calculations and adjust diagrams based on results.

Explain how the probability of an event changes if the previous outcome is not replaced.

Facilitation Tip

For Custom Scenario Trees, give students blank templates with placeholders for initial and adjusted probabilities to guide correct structure.

What to look for

Present students with a scenario: 'A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?' Ask students to draw a tree diagram and calculate the final probability, showing their adjusted probabilities on the second branches.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Activity 02

• Simulation Game
• 45 min · Small Groups

Small Groups: Marble Bag Trees

Provide bags with 20 mixed colored marbles per group. Groups construct trees for sequences like two reds, draw without replacement, and update branches after each draw. Tally class results to compare theoretical versus experimental probabilities.

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

What to look for

Pose the question: 'Imagine you are playing a game where you draw colored balls from a box. How does the probability of drawing a blue ball change if you draw a red ball first and do not put it back?' Facilitate a class discussion where students explain the concept of conditional probability and how the total number of balls affects subsequent draws.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Activity 03

• Simulation Game
• 40 min · Whole Class

Whole Class: Dependency Relay

Divide class into teams with shared bags of counters. Teams relay to the board, adding branches to a large tree diagram after each draw without replacement. Calculate path probabilities as a group and vote on most likely outcomes.

Assess the impact of sampling without replacement on subsequent probabilities.

What to look for

Give students two scenarios: Scenario A (independent events, e.g., flipping a coin twice) and Scenario B (dependent events, e.g., drawing socks from a drawer without replacement). Ask them to write one sentence describing the key difference in how probabilities are represented on a tree diagram for each scenario.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Activity 04

• Simulation Game
• 30 min · Individual

Individual: Custom Scenario Trees

Students create their own bag scenarios with 15-20 items, draw trees for dependent events, and solve three probability questions. Swap with a partner to verify calculations and simulate draws for validation.

Explain how the probability of an event changes if the previous outcome is not replaced.

What to look for

Present students with a scenario: 'A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?' Ask students to draw a tree diagram and calculate the final probability, showing their adjusted probabilities on the second branches.

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

A few notes on teaching this unit

Teachers should pair concrete manipulations with symbolic notation right away. Students benefit from seeing both the physical removal of items and the parallel calculation on the board. Avoid rushing to the formula—instead, let students articulate why the second draw’s denominator is smaller through repeated trials.

Students will correctly draw tree diagrams that show adjusted probabilities after each dependent draw and explain why second branches differ from the first. They will multiply path probabilities accurately and justify their calculations with reference to changing totals.

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Watch Out for These Misconceptions

• During Pairs Practice: Card Draw Simulations, watch for students who keep the same denominator for both draws.

  Have partners pause after the first draw to recount the remaining cards and recalculate the second draw’s probability together before continuing.

• During Small Groups: Marble Bag Trees, watch for students who treat the second draw as independent.

  Ask groups to physically remove the first marble and place it on the table so the reduced total is visible, then recalculate the second branches as a group.

• During Whole Class: Dependency Relay, watch for students who add instead of multiply along paths.

  Pause the relay and ask the student to trace one path aloud, saying, ‘I drew red, then blue, so I multiply 3/10 by 7/9’ to reinforce the multiplication rule.

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Methods used in this brief

• Simulation Game