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.

• 3rd Year
• Mathematical Foundations and Real World Reasoning
• 4 activities
• 25 min–40 min

Learning Objectives

1. Identify all factors for any whole number up to 20.
2. Calculate the first five multiples for any given number up to 20.
3. Compare and contrast the definitions of factors and multiples.
4. Construct a sequence of multiples for a given number, extending it by at least three terms.
5. Explain the relationship between a number and its factors and multiples.

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.

Prepare & details

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?'

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

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

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.

Prepare & details

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.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

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

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.

Prepare & details

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.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

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

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.

Prepare & details

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.

Setup: Tables with large paper, or wall space

Materials: Concept cards or sticky notes, Large paper, Markers, Example concept map

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

Teaching This Topic

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'.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Assessment Ideas

Quick Check

After Array Models and Multiples Relay, give students 18 on a slip of paper. Ask them to write: 1. Three factors of 18. 2. The first four multiples of 18. 3. One sentence explaining the difference between factors and multiples.

Exit Ticket

After Factor Bingo, hand each student a card with 15. Ask them to list all factor pairs for 15, write the next three multiples starting from 15, and state whether 30 is a factor or multiple of 15, explaining their reasoning.

Discussion Prompt

During Sharing Puzzle, pose the question: 'Can a number be both a factor and a multiple of another number?' Have students discuss examples like 4 and 8, then choose volunteers to explain using arrays or counters to justify their answers with precise vocabulary.

Extensions & Scaffolding

• Challenge early finishers to find a number that has exactly six factors and explain their array strategy to a peer.
• For students who struggle, provide a half-filled array template with some counters already placed to reduce cognitive load.
• Deeper exploration: Ask students to compare two numbers and find all common multiples below a target, then present their method to the class.

Key Vocabulary

Factor	A factor is a whole number that divides evenly into another number without leaving a remainder. For example, 3 is a factor of 12 because 12 divided by 3 is 4.
Multiple	A multiple is the result of multiplying a number by any whole number. For example, 24 is a multiple of 12 because 12 times 2 is 24.
Factor Pair	A factor pair consists of two numbers that multiply together to equal a given number. For 12, the factor pairs are (1, 12), (2, 6), and (3, 4).
Divisible	A number is divisible by another number if it can be divided evenly, with no remainder. This means the second number is a factor of the first.