Mathematics · Grade 9

Active learning ideas

Compound Events and Tree Diagrams

Active learning works because compound events involve visualizing multiple steps, and hands-on tasks let students map outcomes physically. When students manipulate spinners, coins, or cards, probabilities become tangible rather than abstract symbols on paper.

• Ontario Curriculum Expectations
• CCSS.MATH.CONTENT.7.SP.C.8.A
• CCSS.MATH.CONTENT.7.SP.C.8.B

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Pairs

Pairs: Dual Spinner Trees

Pairs create two spinners with unequal sections, draw tree diagrams for two spins, and calculate probabilities for specific outcomes. They spin 50 times, tally results, and compare to predictions. Discuss discrepancies and refine diagrams.

Explain how tree diagrams help visualize all possible outcomes of a compound event.

Facilitation Tip

During Dual Spinner Trees, have pairs take turns spinning spinners and recording outcomes before building the tree, linking the activity to their data.

What to look for

Present students with a scenario involving two independent events, such as spinning a spinner twice. Ask them to draw a tree diagram showing all possible outcomes and label the probability of each path. Check if the diagram is correctly structured and probabilities are accurate.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 40 min · Small Groups

Small Groups: Coin Sequence Challenges

Groups build tree diagrams for three coin flips, predict heads-tails-heads probability using multiplication. Flip coins 30 times per group, pool data class-wide. Analyze total outcomes against diagram.

Justify the use of the multiplication rule for independent compound events.

Facilitation Tip

For Coin Sequence Challenges, insist students write each sequence twice: once as letters (e.g., HH, HT) and once with probabilities to reinforce labeling.

What to look for

Give students a problem: 'A baker makes cookies and cakes. 80% of cookies are chocolate chip, and 90% of cakes are chocolate. If the baker randomly selects one cookie and one cake, what is the probability that both are chocolate?' Ask students to show their work using the multiplication rule or a tree diagram.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 30 min · Whole Class

Whole Class: Weather Decision Trees

Project a tree for two-day weather (sunny/rainy, each 50%). Class votes on paths, calculates probabilities. Simulate with random draws, track class results on board, discuss real Canadian weather patterns.

Construct a tree diagram to represent a sequence of two or more events.

Facilitation Tip

In Weather Decision Trees, assign each group a different city’s weather data so they see diverse applications of the same method.

What to look for

Pose the question: 'When might a tree diagram be more helpful than just using the multiplication rule for calculating compound probabilities?' Facilitate a discussion where students explain how tree diagrams help visualize the entire sample space and identify specific outcomes.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 25 min · Individual

Individual: Card Draw Diagrams

Students draw tree diagrams for drawing two cards from a deck without replacement. Calculate probabilities for both red. Verify with 20 simulated draws using a deck model.

Explain how tree diagrams help visualize all possible outcomes of a compound event.

Facilitation Tip

With Card Draw Diagrams, provide limited decks (e.g., only red suits) to force students to adjust probabilities rather than assuming equal likelihoods.

What to look for

Present students with a scenario involving two independent events, such as spinning a spinner twice. Ask them to draw a tree diagram showing all possible outcomes and label the probability of each path. Check if the diagram is correctly structured and probabilities are accurate.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Teachers should start with concrete experiments before formalizing rules, as students need to see why multiplication works for sequential events. Avoid rushing to the multiplication rule; instead, let students derive it from their tree diagrams. Research shows that students who construct diagrams themselves retain probability concepts longer than those who only observe finished examples.

Students will correctly draw tree diagrams for independent events, label branches with accurate probabilities, and calculate compound probabilities using the multiplication rule. They will explain how each branch represents a unique outcome and why paths multiply rather than add.

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Watch Out for These Misconceptions

• During Dual Spinner Trees, watch for students adding probabilities across branches instead of multiplying along paths.

  Have pairs calculate both the sum of all branch probabilities and the product along one path, then compare their totals to the actual frequency from 50 trials to correct the misconception.

• During Coin Sequence Challenges, watch for students assuming all outcomes are equally likely even when the coin is biased.

  Provide biased coins and ask students to adjust branch labels based on observed frequencies before calculating compound probabilities.

• During Weather Decision Trees, watch for students drawing incomplete trees that omit rare weather outcomes.

  Require groups to list all possible weather combinations from the provided data set before constructing the tree, then cross-check each other’s diagrams for omissions.

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Methods used in this brief

• Collaborative Problem-Solving