Replacement Tree Models

Overview

This is lesson 10 of 12 in Probability: From Chance to Models. Students build on prior work with single-event probability by using tree diagrams to model successive events, distinguishing between replacement and no replacement, and combining probabilities along appropriate paths.

Learning intentions

• WALT use tree diagrams to represent successive events.
• WALT calculate probabilities with and without replacement.
• WALT multiply probabilities along a path and add probabilities for suitable outcomes.
• WALT explain how the context changes the probabilities at each stage.

Success criteria

• I can label every branch with a probability.
• I can update probabilities when an item is not replaced.
• I can multiply probabilities along a complete path.
• I can add appropriate paths to find a combined probability.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making an informed judgement from a probability model.
• Demonstrate mathematical reasoning by using appropriate methods, mathematical statements and representations.
• Use mathematical methods to explore a problem related to life in Aotearoa New Zealand or the Pacific, communicating accurate probabilities.
• New Zealand Curriculum Refresh: develop mathematical and statistical thinking, communicate reasoning, and use representations to make sense of uncertainty in a meaningful context.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Open with the hook and retrieval slides showing a bag containing 4 red and 6 blue marbles and ask, “Is the chance of blue always 6 out of 10?” Students complete a quick individual retrieval task on single-event probability, then share what they think changes after one marble is drawn. Teacher establishes that the question depends on whether the marble is replaced and introduces the WALT and success criteria.

2. 8–23 min · Concrete model. Teacher uses 4 red and 6 blue physical marbles to draw one marble, record the result, and model replacing it before a second draw. Students predict the probabilities for red and blue on both draws and explain why the second set of probabilities stays the same with replacement. Repeat without replacement, physically removing the first marble, and invite students to describe what changes in the numerator and denominator.

3. 23–40 min · Branch-by-branch modelling. Display the worked tree-diagram slides and model a tree for “red then blue” without replacement. Students use the probability tree diagram template cards as a visual scaffold, copying the tree into their books or onto the replacement and tree-diagram practice sheet. Teacher explicitly models: label branches, multiply along one complete path, and check that branches from each point total 1. For red then blue, students calculate ( \frac{4}{10}\times\frac{6}{9}=\frac{4}{15} ). Briefly compare this with replacement: ( \frac{4}{10}\times\frac{6}{10}=\frac{6}{25} ).

4. 40–62 min · Guided practice in pairs. Distribute the replacement and tree-diagram practice sheet and assign pairs a sequence of questions involving two draws, including “blue then red”, “same colour”, and “different colours”, with and without replacement. Students take turns being the “branch checker” and “calculation checker”, explaining each decision aloud. Teacher circulates, asking: “What is the total now?”, “What has changed?”, “Which paths match the event?” and checks that students do not add probabilities along a single path.

5. 62–79 min · Independent application. Students complete the final worksheet problems independently: construct both trees for the marble experiment and write a short comparison explaining why the probabilities differ. They must use correct fraction notation and state their answer in context. Teacher conferences with students who need support, while confident students justify why the probabilities of all four final outcomes add to 1.

6. 79–90 min · Extension and mathematical discussion. Open the extension and discussion slides. Students investigate two selections with replacement from a box of hard-centre and soft-centre caramel chocolates, using the given proportions on the slide, and calculate the probability of exactly one hard-centre chocolate. Students explain why there are two successful paths—hard then soft, or soft then hard—and compare their answer with the probability of two hard-centre chocolates. Invite selected pairs to present a clear chain of reasoning.

7. 90–95 min · Plenary and exit check. Return to the plenary slides and ask students to complete a short exit response: “A red marble is drawn from 4 red and 6 blue marbles and is not replaced. What is the probability that the next marble is blue? Explain.” Students also identify one difference between replacement and no replacement. Collect responses to plan the final two lessons.

Resources

• the complete probability teaching deck
• the replacement and tree-diagram practice sheet
• the probability tree diagram template cards
• Physical bag or container
• 4 red and 6 blue marbles
• Hard-centre and soft-centre chocolate probability information
• Whiteboard and markers
• Calculators, if normally used by the class
• High-contrast copies and dark pens

Assessment

• Listen during concrete modelling and pair rehearsal for accurate use of “with replacement”, “without replacement”, “path”, “multiply” and “add”.
• Check worksheet trees for labelled branches, changing denominators, correct multiplication and appropriate addition of paths.
• Use the exit response to identify whether students understand why the second probability changes without replacement and can communicate reasoning in context.

Differentiation

• Provide pre-drawn, high-contrast tree templates, one branch at a time, and physical marbles for students who need reduced cognitive load. Keep diagrams uncluttered and use consistent colours or patterns for red and blue.
• Use sentence starters: “The total changes from ___ to ___ because…”, “This path represents…”, and “I multiply because…”. Pair students for oral rehearsal before written responses.
• Offer dyslexia-friendly worksheet versions with a clear sans-serif font, increased line spacing, short instructions, uncluttered diagrams, and key words highlighted consistently. Read questions aloud and allow students to explain reasoning verbally before recording it.
• Extension students investigate the caramel-chocolate problem, compare alternative paths, verify that all outcomes sum to 1, and explain the result using relational and extended reasoning rather than only giving a calculation.

Extension

• Calculate the probability of exactly one hard-centre caramel chocolate in two selections with replacement.
• Explain why both orders must be included and compare the result with the probability of two hard-centre chocolates.
• Generalise: describe how the tree would change if the probability of a hard-centre chocolate changed.