Simple Probability Models

Overview

This second lesson in the unit builds from identifying chance outcomes to calculating simple theoretical probabilities. Students use familiar objects and visual models to connect favourable outcomes with the total number of equally likely outcomes, then communicate probabilities as fractions and decimals.

Learning intentions

• WALT calculate simple theoretical probabilities.
• WALT identify favourable outcomes and the total number of equally likely outcomes.
• WALT represent probabilities as simplified fractions and decimals.
• WALT explain how the numerator and denominator relate to the context.

Success criteria

• I can list or count all equally likely outcomes.
• I can identify the favourable outcomes for an event.
• I can write and simplify a probability as a fraction and convert it to a decimal where appropriate.
• I can explain what my numerator and denominator represent.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making informed decisions from probability representations.
• Demonstrate mathematical reasoning by using appropriate methods, mathematical language, and representations.
• Use mathematical methods to explore problems related to life in Aotearoa New Zealand or the Pacific, including accurate calculations and communication.
• New Zealand Curriculum Refresh: develop mathematical and statistical thinking, make connections between representations, and communicate reasoning clearly.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Teacher displays a bag containing different-coloured balls and asks, “Which colour is most likely to be selected, and how could we prove it?” Open with the probability hook and retrieval questions. Students make a prediction, recall the meanings of outcome and event, and explain what information would be needed.

2. 8–25 min · Teacher modelling. Teacher explicitly models the structure (P(\text{event})=\frac{\text{number of favourable outcomes{\text{number of equally likely outcomes). Use a six-sided die, a set of coloured pencils, playing cards, LOTTO balls, and a spinner to show that the sample space must include every possible outcome. Record examples such as (P(4)=\frac16), (P(\text{even})=\frac36=\frac12=0.5), and (P(\text{red})=\frac{3}{8}). Refer to the worked examples and numerator-denominator visual. Students annotate the guided probability examples by circling favourable outcomes and labelling the numerator and denominator.

3. 25–43 min · Guided practical practice. Teacher places students in groups of three with roles of equipment manager, recorder, and explainer. Groups rotate through four short stations: balls, pencils, cards, and dice; they identify the sample space and calculate one or two event probabilities at each station. Use colour-coding on the worksheet to separate favourable outcomes from all outcomes, and prompt students to justify whether outcomes are equally likely. Students record fractions, simplify where possible, convert selected answers to decimals, and explain one answer to a partner. Display instructions through the station routine and discussion prompts.

4. 43–50 min · Checkpoint and misconception clinic. Teacher pauses for mini-whiteboard questions: “What is the denominator if there are five balls?”, “What does the numerator count?”, and “Can a probability be greater than 1?” Address common errors, including counting only favourable outcomes, using an incorrect sample space, and failing to simplify. Students show answers simultaneously and improve one response on their worksheet.

5. 50–77 min · Independent problem solving. Teacher distributes the independent probability problem set and conferences with students. Problems progress from single-object situations to contexts involving LOTTO balls and unequal-looking but equally divided spinners; each requires an answer, simplification where appropriate, and a written explanation. Students solve independently, then use a structured “compare, check, explain” routine with a peer for selected questions. The teacher checks that students distinguish an outcome, an event, and the complete sample space.

6. 77–88 min · Extension and challenge. Teacher invites confident students to create two probability questions: one involving LOTTO outcomes and one involving a spinner. Each question must state the complete sample space, identify the favourable outcomes, and include a worked answer. Students swap questions with a partner, solve them, and check whether the wording makes the outcomes equally likely. Other students complete unfinished worksheet questions or a teacher-selected support example. Refer to the question-design challenge.

7. 88–95 min · Review and exit ticket. Teacher leads a brief review: “What does the denominator tell us?” and “When should we simplify?” Students complete the final two questions on the exit reflection: (a) A bag has 2 red, 3 blue, and 5 green balls. Find (P(\text{blue})) and write it as a decimal; (b) explain the numerator and denominator. Students hand in the response before leaving.

Resources

• the simple theoretical probability teaching deck
• the guided, independent, and exit-ticket worksheet
• Six-sided dice
• Bags or containers with coloured balls
• Sets of coloured pencils
• Standard playing cards
• LOTTO-style numbered balls or labelled counters
• Spinners or paper spinner equipment
• Mini-whiteboards and pens
• Uncluttered, dyslexia-friendly printed copies and optional audio instructions

Assessment

• Use mini-whiteboard responses and questioning to check understanding of favourable outcomes, sample spaces, and probability notation.
• During stations and independent work, listen for explanations linking the numerator and denominator to the context; provide immediate feedback.
• Collect the exit ticket to identify students needing further support with counting outcomes, simplifying fractions, or converting to decimals.

Differentiation

• Support students with concrete manipulatives, colour-coded outcome lists, partially completed examples, a word bank, and sentence frames: “The numerator is ___ because…” and “The denominator is ___ because…”.
• Provide structured peer roles and pair students strategically; allow students to explain orally before recording written reasoning.
• Offer audio instructions, clear sans-serif fonts, increased spacing, uncluttered worksheets, and teacher-read questions for students with dyslexia or reading needs.
• Extend advanced learners by requiring them to design and solve LOTTO and spinner questions, compare two different representations, and critique whether the stated outcomes are equally likely.

Extension

• Investigate how changing the number or colour of LOTTO balls changes the probability of an event.
• Design a spinner that gives a chosen probability, such as (0.25), and justify the number and size of its equal sections.