Mathematics · 3rd Year

Active learning ideas

Introduction to Factors and Multiples

Active learning builds concrete understanding of factors and multiples by letting students see and touch the concepts. When students arrange counters into arrays or race to extend sequences, they move from abstract rules to visual and kinesthetic memories that stick. These hands-on experiences correct misconceptions early and make abstract patterns feel real and memorable.

• 25–40 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Concept Mapping
• 35 min · Small Groups

Array Models: Factor Pairs

Provide counters and grid paper. Students select a number up to 20 and build rectangular arrays, recording side lengths as factor pairs. They test different arrangements and list all unique pairs. Groups share one array with the class for verification.

Explain the difference between a factor and a multiple.

Facilitation Tip

During Array Models, move between groups to prompt students to turn their counters into a rectangle and ask, 'Could you flip this rectangle sideways? What does that show about the pairs?'

What to look for

Present students with a number, such as 18. Ask them to write down: 1. Three factors of 18. 2. The first four multiples of 18. 3. One sentence explaining the difference between factors and multiples.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 02

• Concept Mapping
• 25 min · Small Groups

Multiples Relay: Sequence Race

Divide class into teams. Each student adds the next multiple of a given number (up to 20) on a whiteboard strip, passing to the next teammate. First team to reach a target multiple wins. Review sequences for errors as a class.

Construct a list of all factors for a given number like 12.

Facilitation Tip

In the Multiples Relay, stand near the sequence board to listen for repeated addition language and redirect any students counting by ones instead of by the starter number.

What to look for

Give each student a card with a number (e.g., 15). Ask them to list all factor pairs for that number and then write the next three multiples in the sequence starting from 15. They should also state if 30 is a factor or a multiple of 15 and why.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 03

• Concept Mapping
• 30 min · Whole Class

Factor Bingo: Number Hunt

Create bingo cards with numbers up to 20. Call out factors; students mark numbers with those factors. First to complete a line shouts 'Factors!' and explains one pair. Play multiple rounds with different caller numbers.

Predict the next three multiples in a given sequence.

Facilitation Tip

Set a timer during Factor Bingo so students must justify their marks with full factor pairs before calling 'Bingo!' to reinforce completeness.

What to look for

Pose the question: 'Can a number be both a factor and a multiple of another number?' Guide students to discuss examples, such as 4 being a factor of 8 and 8 being a multiple of 4. Encourage them to explain their reasoning using the precise vocabulary learned.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

Activity 04

• Concept Mapping
• 40 min · Pairs

Sharing Puzzle: Real-World Dividers

Give scenarios like dividing 18 cookies among friends. Students draw models or use objects to find factor groups. Pairs justify their divisions and predict multiples for buying more packs.

Explain the difference between a factor and a multiple.

Facilitation Tip

For the Sharing Puzzle, circulate with counters to watch how students physically distribute items; pause any group that starts by dividing one by one.

What to look for

Present students with a number, such as 18. Ask them to write down: 1. Three factors of 18. 2. The first four multiples of 18. 3. One sentence explaining the difference between factors and multiples.

• Understand
• Analyze
• Create
• Self-Awareness
• Self-Management

A few notes on teaching this unit

Start with real-world contexts like sharing sweets or tiling floors to anchor the concepts. Avoid rushing to rules; instead, let students discover patterns through arrays and sequences. Research shows that students who construct arrays themselves understand factor pairs faster than those who only see textbook lists. Always pair concrete models with precise vocabulary so students connect actions to terms like 'divisor' and 'product'.

Students will confidently list all factors of a number up to 20 and generate correct multiples in sequence. They will use precise language to explain that factors divide evenly while multiples come from repeated addition. Peer discussions and quick checks will show clear understanding, not just correct answers.

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Watch Out for These Misconceptions

• During Array Models, watch for students who do not include 1 and the number itself as factor pairs.

  Direct students to build a 1xN rectangle and a Nx1 rectangle for their number, then ask, 'What do these two arrays show about the number’s relationship with 1 and itself?' Have peers compare arrays to confirm completeness.

• During Multiples Relay, listen for students who assume multiples are only even numbers.

  Stop the relay after the first round and ask groups to share their sequences. Highlight odd starters like 3 and guide students to notice the pattern of adding the starter number each time, not just counting by twos.

• During Factor Bingo, observe students who skip marking 1 as a factor.

  Require players to say the factor pair aloud before marking; if they omit 1, ask, 'Does 1 divide evenly into every number? Show me with your counters how 1 fits.' Let peers verify with their own bingo cards.

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Methods used in this brief

• Concept Mapping