Chance, Products and Fractions

Overview

Students investigate whether outcomes are equally likely by rolling two standard dice and recording products. They use multiplication facts, frequency and fractions to compare experimental results with predictions, then consider how an uneven spinner changes likelihood.

Learning intentions

Students will:

• list possible outcomes in a chance experiment and identify whether they are equally likely
• use multiplication facts to calculate products from two dice
• conduct repeated trials and record frequencies accurately
• represent frequencies as fractions and use them to compare likelihoods

Success criteria

• I can list possible outcomes and explain whether they are equally likely.
• I can use times-table knowledge to calculate products efficiently.
• I can record results using tally marks and frequency totals.
• I can compare results using fractions and explain what the data suggests.

Curriculum links

• Probability — listing possible outcomes and comparing equally likely and unequally likely events.
• Probability — conducting repeated chance experiments and using frequency to estimate likelihood.
• Number — applying multiplication and division strategies, including interpreting fractions.
• Statistics — acquiring, representing and discussing data from a chance experiment.

Lesson structure (50 minutes)

1. 0–5 min · Hook: Is it fair? Teacher opens the hook and prediction slides and displays two questions: “Is every product made by rolling two dice equally likely?” and “Which product do you predict will occur most often?” Students make an individual prediction, then justify it to a partner using a multiplication fact.

2. 5–12 min · Model outcomes and likelihood. Teacher rolls two dice, models recording an ordered outcome such as (3 \times 4 = 12), and explains that each die has six equally likely face outcomes, but products are not equally likely because some products can be made in more ways than others. Teacher demonstrates that 6 can be made by (1 \times 6, 2 \times 3, 3 \times 2) and (6 \times 1), while 1 has only one combination. Students help list possible products from 1 to 36 and identify products that may be more or less likely.

3. 12–27 min · Hands-on repeated experiment. Teacher places students in 13 pairs and distributes the two-dice probability investigation sheet and two dice to each pair. Using the experiment instruction slides, students roll both dice 30 times, calculate the product, and record each result with a tally and frequency. Partners take turns rolling, calculating, checking the multiplication fact and recording. Pause after 10 and 20 rolls for pairs to compare emerging patterns without changing their results.

4. 27–35 min · Analyse frequency as fractions. Teacher asks pairs to total their frequencies and write selected results as fractions out of 30, such as “product 12 occurred 5 out of 30 times”. Students use the fraction wall and strip cards to compare fractions such as (2/30, 4/30) and (6/30), then identify their most frequent and least frequent products. Discuss why experimental results will not be identical across pairs.

5. 35–43 min · Compare equal and unequal likelihood. Teacher displays an uneven six-section spinner on the comparison and discussion slides with sections representing 1, 1, 1, 2, 3 and 4. Students predict which number is most likely, explain why the outcomes are not equally likely, and conduct 20 spins in pairs using a paperclip and pencil or a digital spinner. They record frequencies and compare the spinner data with the dice-product data.

6. 43–50 min · Plenary and exit check. Teacher returns to the opening predictions and leads a brief discussion: “Does the most frequent result always prove it is most likely?” Students complete the final questions on the reflection and exit questions: list two ways to make a product of 12, explain why products are not equally likely, and write one observed frequency as a fraction. Invite two students to share contrasting results and reasoning.

Resources

• the complete probability investigation slide deck
• the two-dice probability investigation sheet
• the fraction wall and strip cards
• 26 standard six-sided dice
• 13 paperclips and pencils, or one digital spinner per pair
• Whiteboard and markers
• Counters or mini-whiteboards for predictions
• Timer

Assessment

• Listen during prediction and modelling for correct use of “equally likely”, “not equally likely”, “outcome”, “frequency” and “product”.
• Check pair recording for 30 completed trials, accurate multiplication facts, tally marks and frequency totals.
• Use the exit questions to assess whether students can list outcomes, compare likelihoods and express frequency as a fraction.

Differentiation

• Support: provide a multiplication grid, allow students to use the dice as arrays, and offer sentence starters such as “The outcomes are not equally likely because…” and “The fraction ___ is greater than ___ because…”.
• Support: pair students strategically, assign one student as roller and one as recorder initially, and reduce the product range to products from 1–6 times tables if needed.
• Enabling prompts: “How many ordered pairs make this product?”, “Can you find a different way?”, “What does the denominator 30 represent?”, and “Are your results proof or evidence from a sample?”
• Extending prompts: “Calculate and compare the number of combinations for products 6, 12, 18 and 24”, “Predict the product most likely to occur before testing”, and “Design an uneven spinner whose most likely outcome is 3.”