Foundations of Mathematical Thinking · Junior Infants

Active learning ideas

Sample Space and Tree Diagrams

Active learning transforms abstract ideas like sample space into concrete experiences. When children physically spin, toss, or build, they see every possible outcome instead of guessing. These hands-on moments build confidence before moving to symbols or drawings.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Strand 3: Statistics and Probability - P.1.4

• 15–30 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Think-Pair-Share
• 20 min · Small Groups

Spinner Station: Single Events

Provide colour spinners and mats marked with outcomes. Students spin ten times, list results on charts to identify the full sample space. Discuss as a group why every colour belongs on the list.

Explain the purpose of a sample space in probability.

Facilitation Tip

During Spinner Station, remind pairs to spin at least five times to notice rare outcomes and add them to their shared list.

What to look for

Provide students with a spinner that has 3 different colors. Ask them to draw a picture showing all the possible outcomes when the spinner is spun once. Then, ask them to name one outcome.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 02

• Think-Pair-Share
• 25 min · Pairs

Pair Build: Fruit Tree Diagrams

Pairs get red/green apple cards and circle/square shape cards. First stage: draw branches for colours, second for shapes. Count total outcomes and test by picking cards randomly.

Construct a tree diagram to represent all possible outcomes of a two-stage experiment.

Facilitation Tip

For Pair Build, circulate and prompt pairs to ask, ‘Are we missing any color-shape pairs?’ before gluing their tree diagrams.

What to look for

Give each student a card with a simple two-stage experiment, such as 'toss a coin, then pick a colored block from a bag with red and blue blocks'. Ask them to draw a tree diagram showing all possible outcomes and list them.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 03

• Think-Pair-Share
• 30 min · Whole Class

Whole Class: Coin and Die Chain

Toss a coin for heads/tails, then roll a three-faced die. Class draws a large tree diagram on the board together, predicting and checking outcomes with real tosses and rolls.

Analyze how the number of outcomes changes with additional stages in an experiment.

Facilitation Tip

While running Coin and Die Chain, model how to label each branch with both results, like ‘heads and 1’ on one path.

What to look for

Pose the question: 'If you flip a coin once, there are two outcomes (heads, tails). If you flip it twice, how many outcomes are there? How do you know?' Facilitate a discussion where students explain their reasoning, potentially using drawings or examples.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Activity 04

• Think-Pair-Share
• 15 min · Individual

Individual Draw: Snack Choices

Students draw trees for biscuit (choc/plain) then drink (milk/juice). List all four outcomes, colour them, and share one path with a partner.

Explain the purpose of a sample space in probability.

Facilitation Tip

In Snack Choices, ask students to read their completed diagram aloud to a partner to practice naming all outcomes.

What to look for

Provide students with a spinner that has 3 different colors. Ask them to draw a picture showing all the possible outcomes when the spinner is spun once. Then, ask them to name one outcome.

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

A few notes on teaching this unit

Teachers start with real objects so children feel the randomness and see outcomes appear in real time. Avoid rushing to worksheets; instead, use repeated trials to reinforce that every branch matters. Research shows concrete materials build stronger mental models before moving to abstract representations like written lists or diagrams.

Successful learning shows when children can name or show all outcomes for single events and draw full tree diagrams for two-stage events. They explain their thinking using the materials and correct each other’s missing paths or outcomes with gentle peer feedback.

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Watch Out for These Misconceptions

• During Spinner Station, watch for children who list only the colors they see in their first few spins and ignore other sections.

  Prompt students to count all sections on the spinner first, then predict outcomes before spinning. Ask, ‘Did your list include every section? What did you miss?’

• During Pair Build, watch for pairs who stop branching after the first stage and only show one outcome per color.

  Hand pairs a checklist with ‘Have we shown all shapes for each color?’ while they build with colored blocks and sticky notes.

• During Coin and Die Chain, watch for students who say outcomes decrease when adding a second stage, like claiming ‘three outcomes’ instead of six.

  Use counters to model each branch doubling: place one counter for heads, then split into two for heads-1 and heads-2, repeating for tails.

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Methods used in this brief

• Think-Pair-Share