Mathematics · Secondary 2

Active learning ideas

Combined Events and Sample Space

Students need to move beyond abstract counting when learning about combined events and sample spaces. Active learning through structured group work and hands-on trials helps them visualize how outcomes combine, corrects misconceptions about independence, and builds confidence in systematic enumeration. Kinesthetic tasks like relay races and spinner trials make abstract probability concepts concrete and memorable.

• MOE Syllabus Outcomes
• MOE: Probability - S2
• MOE: Statistics and Probability - S2

• 20–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Plan-Do-Review
• 30 min · Pairs

Pairs: Tree Diagram Relay

Pairs take turns adding branches to a tree diagram for two-stage events like coin toss and die roll. One student draws while the partner calls outcomes; they switch after each stage and calculate probabilities. End with sharing complete diagrams.

How does the sample space size affect the probability of a specific event occurring?

Facilitation Tip

For the Tree Diagram Relay, provide each pair with a different scenario and have them rotate stations to check each other’s diagrams for completeness before calculating probabilities.

What to look for

Present students with a scenario: 'A bag has 3 red marbles and 2 blue marbles. You draw one marble, note its color, and replace it. Then you draw a second marble.' Ask: 'Draw a tree diagram to show all possible outcomes and calculate the probability of drawing two red marbles.'

• Remember
• Apply
• Analyze
• Self-Management
• Decision-Making
• Self-Awareness

Activity 02

• Plan-Do-Review
• 45 min · Small Groups

Small Groups: Spinner Trials

Groups create two spinners for colors and shapes, conduct 50 combined trials, and record outcomes on tables. They draw tree diagrams from results and compute theoretical probabilities. Compare group data class-wide.

Construct a tree diagram to represent the outcomes of combined events.

Facilitation Tip

During Spinner Trials, require students to record at least 30 spins per trial to ensure the frequency distribution stabilizes and reflects the theoretical probabilities.

What to look for

Give students two scenarios: Scenario A: Rolling a die and flipping a coin. Scenario B: Drawing two cards from a deck without replacement. Ask them to identify which scenario involves independent events and explain why, referencing the definition of independent events.

• Remember
• Apply
• Analyze
• Self-Management
• Decision-Making
• Self-Awareness

Activity 03

• Plan-Do-Review
• 35 min · Whole Class

Whole Class: Systematic Listing Board Race

Divide class into teams; project a three-event scenario like weather choices. Teams race to list sample spaces on mini-whiteboards, vote on completeness, then calculate event probabilities together.

Analyze the independence of events in a probability experiment.

Facilitation Tip

Use the Systematic Listing Board Race by having students write outcomes on sticky notes and race to post them on a shared board, then immediately discuss gaps or duplicates as a class.

What to look for

Pose this question: 'Imagine you are playing a board game where you roll two dice to move. How does the size of the sample space (the total number of outcomes when rolling two dice) affect your chances of landing on a specific square?' Facilitate a discussion on how larger sample spaces can mean smaller probabilities for individual outcomes.

• Remember
• Apply
• Analyze
• Self-Management
• Decision-Making
• Self-Awareness

Activity 04

• Plan-Do-Review
• 20 min · Individual

Individual: Real-Life Probability Cards

Students draw cards with combined event problems, like bus delays and rain. They list outcomes or sketch trees individually, then pair to check work before class discussion.

How does the sample space size affect the probability of a specific event occurring?

What to look for

Present students with a scenario: 'A bag has 3 red marbles and 2 blue marbles. You draw one marble, note its color, and replace it. Then you draw a second marble.' Ask: 'Draw a tree diagram to show all possible outcomes and calculate the probability of drawing two red marbles.'

• Remember
• Apply
• Analyze
• Self-Management
• Decision-Making
• Self-Awareness

A few notes on teaching this unit

Teachers should start with small, manageable sample spaces like coin tosses or single dice rolls to establish the habit of systematic listing. Move quickly to tree diagrams for two-stage experiments, but avoid rushing the counting process. Research shows that students grasp independence better when they experience both replacement and non-replacement scenarios side by side. Emphasize the language of ‘and’ for multiplication and ‘or’ for addition to reduce confusion between union and intersection events.

Students will confidently list all possible outcomes for small sample spaces and use tree diagrams to represent larger ones. They will calculate probabilities correctly by counting favorable outcomes or multiplying branch probabilities, and articulate whether events are independent based on trial results. Peer discussions and teacher checks will confirm understanding before moving to independent tasks.

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Watch Out for These Misconceptions

• During the Systematic Listing Board Race, watch for students who only list favorable outcomes or assume some outcomes are impossible without justification.

  Circulate with a checklist of all possible outcomes for their scenario and ask pairs to compare their list against it, adding missing outcomes and explaining why each item belongs. Use a timer to keep the race brisk but thorough.

• During Spinner Trials, watch for students who assume all branch probabilities on a tree must add up to 1, regardless of whether they represent ‘and’ or ‘or’ events.

  Have students tally the actual frequencies of outcomes from their spins and compare these to the tree probabilities. Ask them to explain why the probabilities of two red outcomes multiply to 0.25, but the probabilities of red or blue outcomes add to 1.

• During the Tree Diagram Relay, watch for students who assume all combined events are independent without checking the scenario details.

  Provide some scenarios with replacement and others without (e.g., drawing cards vs. rolling dice). After they build the tree, ask them to state whether the events are independent and justify with the probability calculations from their diagram.

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Methods used in this brief

• Plan-Do-Review