Analysing Real Choices

Overview

This engaging 90-minute mathematics lesson is specifically designed for Australian Year 10 students and aligns closely with the Australian Curriculum: Mathematics – Level 10, Chance and Data strand. The focus is on Probability using tree diagrams to explore compound events, deepen conceptual understanding, and enhance real-world problem-solving.

The lesson will balance conceptual explanation, collaborative work, individual practice, and real-life applications to build mathematical confidence and deepen statistical literacy.

---

🧭 Curriculum Alignment

Australian Curriculum - Mathematics – Year 10

Strand: Chance
Sub-strand: Describe the results of two- and three-step chance experiments, both with and without replacement, using tables, tree diagrams and Venn diagrams. Assign probabilities to outcomes and determine probabilities of events.
Code: ACMSP225

---

🎯 Learning Intentions

By the end of the lesson, students will be able to:

• Construct and interpret probability tree diagrams.
• Differentiate between independent and dependent events.
• Use probability trees to calculate the likelihood of compound outcomes.
• Apply tree diagrams to real-world contexts, making meaningful connections.

---

✅ Success Criteria

Students will demonstrate success by:

• Correctly representing scenarios using accurate tree diagrams.
• Clearly explaining their reasoning behind calculated probabilities.
• Successfully solving both procedural and application-based probability questions.

---

⏰ Lesson Duration

Total Length: 90 minutes
Class Size: 30 Year 10 Students

---

🧠 Prior Knowledge

Before this lesson, students should be familiar with:

• Basic probability calculations for single events.
• Concepts like ‘mutually exclusive’, ‘independent’, and ‘dependent’ events (at an introductory level).

---

🛠 Materials Required

• Mini-whiteboards or scrap paper for collaborative drawing
• Printed or digital worksheet
• Coloured pens or markers
• Access to dice, coins, coloured balls (for tactile probability experiment)

---

📋 Lesson Breakdown

0–10 mins | Real-world Hook & Introduction

Activity:
Show students an engaging video snippet or recount a real-world example such as:

"You’re applying for a job. You have a 50% chance of getting an interview. If you get the interview, there’s a 70% chance of being hired. What’s the overall chance that you get the job?"

Discussion Prompt:

• “How would we calculate that compound probability?”
• “What’s changing or staying the same in each part of the process?”

Outcome: Establish the need for clear, structured thinking – introducing the probability tree diagram as a visual model.

---

10–25 mins | Explicit Teaching: How Tree Diagrams Work

Teacher-led Explanation:

• Define tree diagrams.
• Step-by-step breakdown:
  1. Start with the first event and branch out possible outcomes.
  2. From each branch, branch again with the second event’s outcomes.
  3. Assign probabilities to each branch.
  4. Multiply along the branches to get the final compound probability of each outcome path.

Example 1 (Independent Events):

• Tossing a coin twice.
• Draw the tree diagram.
• Calculate P(Heads then Tails), P(Two Heads), etc.

Example 2 (Dependent Events):

• Drawing two balls from a bag without replacement.
• Tree branches change as probabilities shift.

Visual Modelling: Use whiteboard or projector. Take student volunteers to help draw.

---

25–35 mins | Think-Pair-Share: Draw Your Own

Task: In pairs, students receive different real-life scenarios (randomised between pairs):

• A restaurant offers 3 meal options and 2 dessert choices.
• A student rolls a die then draws a card from a deck.
• A team spins a wheel then flips a coin.

Instruction:

• Diagram the process using tree diagrams.
• Assign appropriate probabilities.
• Identify total outcome paths and calculate the probability of a chosen outcome (e.g., steak and ice cream).

Sharing: Pairs present their diagrams on mini-whiteboards. Class walkthrough of selected examples.

---

35–60 mins | Application Task: Mystery Bag

Hands-on Experiment:

Use tactile manipulatives for real-world engagement.

Setup:
Each group gets:

• A mystery bag with a mix of coloured balls (e.g., 3 red, 2 blue, 1 green).
• A random spinner with sections marked A, B, C.

Task Instructions:

1. Draw one ball from the bag with replacement.
2. Spin the spinner.
3. Use a tree diagram to model the event.
4. Calculate probability of red ball and section A selected.

Extension Challenge:

• Repeat with without replacement.
• Compare how the tree diagram structure and probabilities change.

---

60–80 mins | Individual Practice: Real-World Problems

Students complete a differentiated worksheet featuring:

• Independent and dependent event scenarios
• Worded problems (themed: weather predictions, sports win/loss, card games)
• Diagram construction + compound probability questions

Support Extension:

• Struggling students get scaffolded templates.
• Extension students try 3-step tree diagrams and inverse probability problems.

---

80–90 mins | Reflection & Exit Ticket

Class Discussion (5 mins): What surprised you about tree diagrams?
How could you use this outside the classroom?

Exit Ticket Prompts (individually on cards):

1. Draw and label one tree diagram with your own scenario.
2. Identify whether it's dependent or independent.
3. Calculate a compound probability.

---

🔁 Assessment Strategy

• Formative: Observation during group tasks and think-pair-share.
• Summative: Final worksheet and exit ticket assessed for accuracy and understanding of tree diagrams, correct multiplication of probabilities, and clarity of construction.

---

🔎 Differentiation

Support:

• Use of sentence starters and structured templates.
• Teacher-led mini-workshops during activity.

Extension:

• More complex problems with 3-step experiments.
• Questions involving event probabilities “at least” or “no more than”.

---

🧩 Cross-Curricular and Real-Life Links

• Business Studies: Risk and decision-making in investment.
• Science: Genetic trait predictions.
• PE: Probability in tactics and game theory.
• Everyday Life: Insurance risks, hiring practices, weather forecasting.

---

🌱 Future Follow-Up Lessons

• Using tree diagrams for expected value.
• Comparing tree diagrams to Venn diagrams and two-way tables.
• Exploring Bayesian conditional probability.

---

🎉 Extension Idea: Class-Wide Game

“Tree Trekker” Simulation Game
Students run a decision-making game where different path choices lead them to outcomes with ‘rewards’ based on probability. They build a collaborative tree on the wall across the week and update class stats.

---

This lesson blends visual, kinaesthetic, collaborative and abstract learning to give students an engaging and deep experience with probability tree diagrams. With hands-on activity, scaffolded instruction, and real-life relevance, this lesson supports diverse learners while aligning neatly with the Australian Curriculum.

Ready to bring compound probability to life in your classroom!