Sample Spaces and Events

Overview

Unit: Exploring Probability Concepts
Lesson: 2 of 20
Duration: 60 minutes
Class size: 30 students
Year level: 13 (NCEA Level 3, New Zealand)
Location: Waiheke Island Secondary High School

Learning Intentions

WALT

We Are Learning To create sample spaces for simple events.

Success Criteria

List all possible outcomes for given scenarios accurately.
Construct clear sample spaces using lists, tables, or tree diagrams.
Apply understanding by solving example probability problems (e.g., drawing cards, rolling dice).

Curriculum Alignment and Pedagogical Principles

Mathematics and Statistics Strands: Probability and Statistical Investigation
NCEA Level 3 Achievement Standards Focus: Develop understanding of probability distributions, sample spaces, and events.
Te Mātaiaho Framework: Emphasises logical reasoning, critical thinking, and statistical literacy through inquiry and digital tools.
Key Competencies:
- Thinking: logical and statistical reasoning.
- Using language, symbols, and texts: interpret and create representations of probability.
- Relating to others: collaborating to discuss outcomes and strategies.
- Managing self: working independently on tasks.
- Participating and contributing: group discussions and challenges.

References:

Te Mātaiaho Mathematics and Statistics Curriculum years 9-13 (2025 Draft) – Probability strand learning for Year 13 emphasises representing probability outcomes, constructing sample spaces, and critical evaluation of claims

Learning Objectives (Year 13, Level 3)

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

Identify all possible outcomes of simple probability experiments.
Construct clear sample spaces using appropriate representations (lists, tables, tree diagrams).
Explain the difference between independent and dependent events in sample spaces.
Use correct mathematical language and notation (outcomes, events, sample spaces).
Apply digital tools for representing probability where applicable.

Detailed Lesson Plan

Time	Activity	Aim / Description	Resources	Differentiation / Extension
0-10 mins	Starter Discussion	Recap outcomes from Lesson 1 (basic probability concepts). Engage students by asking: ‘What were some possible outcomes you identified in yesterday’s task?’ Introduce lesson focus: sample spaces and events.	Whiteboard, interactive quiz (Kahoot or Mentimeter)	Support: Provide dyslexia-friendly list of vocabulary (highlight key words like sample space, event, outcome).
10-20 mins	Direct Teaching and Demonstration	Teacher models constructing sample spaces with simple events. Use examples:

Draw a card from a deck (only suits or colours).
Rolling two dice (list possible sums or ordered pairs).
Use lists first, then tables and tree diagrams to visualize outcomes. Emphasize systematic listing to avoid missing outcomes. | Visual aids (spreadsheets or pre-made diagrams), projector | Support: Use colour coding and dyslexia-friendly fonts. Provide step-by-step written instruction on slides/handout. Extension: Challenge advanced students to find sample spaces for compound events (e.g., multiple draws with or without replacement). |
| 20-30 mins | Guided Practice | In pairs, students create sample spaces for given scenarios:

Tossing two coins
Drawing two cards with replacement
Teacher circulates to support students, prompt systematic listing, correct misconceptions. | Worksheets, digital tablets or laptops | Support: Sentence starters and graphic organisers to help structure their answers. Extension: Extend to three coins or conditional scenarios for advanced students. |
| 30-45 mins | Collaborative Activity: Probability Challenges | Small groups (4-5) given scenarios (e.g., rolling two dice and drawing a card). Each group lists all possible outcomes and describes events (e.g., sums > 7, drawing a heart). Share findings with class and discuss different sample space representation methods. | Scenario cards, large paper, markers. Digital tool suggestion: use an interactive whiteboard app. | Support: Pair dyslexic students with peers and provide printed scenario cards with pictorial cues. Extension: Groups create their own complicated event and challenge other groups to list outcomes. |
| 45-55 mins | Independent Application | Students complete a brief individual task: Given a scenario, write out the sample space and identify events. E.g., “Draw one card from a deck; list sample space for suits and describe the event ‘drawing a red card’.” | Individual worksheet, digital or paper | Support: Provide a template for listing outcomes. Extension: Encourage advanced students to start linking sample spaces to probabilities (finding the probability of defined events). |
| 55-60 mins | Exit Reflection and Plenary | Whole-class discussion reflecting on what was learned. Q&A to confirm success criteria. Students write one ‘ticket out the door’ response mentioning:

What are sample spaces?
Why is it important to list all outcomes?
Give an example of an event from today’s lesson. | Exit slips or short digital response | Support: Oral reflection option for students who prefer speaking over writing. Extension: Encourage students to suggest real-life situations where sample spaces are useful. |

Assessment for Learning (AfL)

Observation and questioning during activities to check understanding.
Peer and self-assessment during group work using success criteria checklist.
Exit slips to confirm individual learning achieved.

Differentiation Strategies

For Diverse Learners and Dyslexic Students:
- Provide dyslexia-friendly reading supports: use clear, sans-serif fonts, colour-coded outcomes, and visual organisers.
- Use oral explanations supported by diagrams.
- Provide sentence starters and graphic organisers for structuring answers.
- Pair students strategically for peer support.
For Advanced Learners:
- Challenge to create sample spaces for compound or dependent events.
- Use probability tree diagrams for extended tasks.
- Integrate digital tools for simulation and modelling of outcomes, encouraging exploration beyond simple cases.

Resources Needed

Whiteboard and markers
Printed worksheets and scenario cards
Digital devices (tablets or laptops) with spreadsheet or probability simulation software (optional)
Visual aids for modelling (cards, dice, diagrams)
Dyslexia-friendly vocabulary and instructions handouts

Teacher Reflection Prompts

Were all students able to list all possible outcomes accurately?
Did the differentiation strategies allow learners of different needs to engage effectively?
How effective were the digital tools and visual representations in aiding understanding?
What misconceptions about sample spaces and events emerged during activities?
How might this lesson sequence build towards the next lesson on probability calculations and event probabilities?

This lesson plan is designed with reference to the New Zealand Curriculum Refresh’s probability progression for Year 13 students, promoting logical reasoning and critical statistical thinking. By combining direct teaching, collaborative tasks, and differentiated supports, it ensures engagement across diverse learners, fostering deep understanding of sample spaces and events that are foundational to advanced probability study .

Teaching Instructions