Mathematics · Year 4 · Multiplicative Thinking · Term 1

Factors of Whole Numbers

Identifying factors of whole numbers and exploring their relationships through arrays and divisibility rules.

• ACARA Content Descriptions
• AC9M4N03

About This Topic

Factors of whole numbers are pairs of positive integers that multiply to produce the given number. Year 4 students identify these pairs for numbers up to 100, often using arrays to visualise multiplication as grids of counters or squares. They apply divisibility rules for 2, 3, 5, and 10 to check possibilities quickly, which connects to multiplication facts from earlier years. This work addresses AC9M4N03 by emphasising efficient strategies over rote listing.

Within the Multiplicative Thinking unit, students distinguish factors from multiples, develop methods like starting from 1 and pairing up to the square root, and link concepts to grouping in everyday scenarios such as sharing food or arranging seats. These skills build number sense and problem-solving, revealing patterns like square numbers having an odd count of factors.

Active learning benefits this topic greatly since physical manipulatives make invisible relationships visible and interactive. When students build and rearrange arrays collaboratively, they test ideas hands-on, correct errors in real time, and explain reasoning to peers, which strengthens retention and confidence in abstract mathematics.

Key Questions

1. Differentiate between a factor and a multiple of a number.
2. Construct a method to find all factors of a given number.
3. Explain how factors are used in real-world situations like grouping.

Learning Objectives

• Identify all factor pairs for whole numbers up to 100.
• Classify numbers as prime or composite based on their factors.
• Explain the relationship between factors and the construction of arrays.
• Apply divisibility rules for 2, 3, 5, and 10 to determine factors.
• Compare the number of factors for square numbers versus non-square numbers.

Before You Start

Multiplication Facts

Why: Students need to recall multiplication facts accurately to identify pairs of numbers that multiply to a given number.

Division Concepts

Why: Understanding division as the inverse of multiplication is foundational for identifying numbers that divide evenly into another number.

Key Vocabulary

Factor	A factor is a whole number that divides exactly into another whole number without leaving a remainder. For example, 3 and 4 are factors of 12.
Multiple	A multiple is the result of multiplying a whole number by another whole number. For example, 12 is a multiple of 3 and 4.
Array	An array is a visual arrangement of objects in rows and columns, representing multiplication. For example, 3 rows of 4 counters form an array for 12.
Divisibility Rule	A divisibility rule is a shortcut to determine if a number can be divided by another number without a remainder. Rules exist for numbers like 2, 3, 5, and 10.
Prime Number	A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, and 7.
Composite Number	A composite number is a whole number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, and 10.

Watch Out for These Misconceptions

Common Misconception

Factors and multiples are interchangeable terms.

What to Teach Instead

Factors divide evenly into the number, while multiples result from multiplying it by integers. Array activities clarify this: rows show factors of the total, columns show multiples. Peer teaching during group builds reinforces the distinction through shared explanations.

Common Misconception

All numbers have exactly two factors.

What to Teach Instead

Prime numbers do, but composites have more. Systematic pair-hunting games reveal this pattern, as students list and count pairs. Hands-on correction prevents overgeneralisation by letting them discover through exploration.

Common Misconception

1 does not count as a factor.

What to Teach Instead

1 pairs with every number to make itself. Manipulative arrays starting with 1xN make this obvious and non-negotiable. Discussions during activities normalise it as essential.

Active Learning Ideas

Array Builder: Factor Grids

Provide counters and grid paper. Students select a number like 24 and build rectangular arrays, recording factor pairs such as 3x8 or 4x6. They test if arrays fit perfectly and discuss why some dimensions fail. Extend by finding all pairs systematically.

35 min·Small Groups

Divisibility Dash: Rule Relay

Divide class into teams. Place number cards around the room. Students run to a card, apply one divisibility rule to identify a factor, and tag the next teammate. Teams compare lists at the end and verify with multiplication.

25 min·Small Groups

Factor Pair Match-Up: Card Game

Create cards with numbers and possible pairs. In pairs, students match factor pairs to products, then justify using arrays or rules. Shuffle for multiple rounds, timing for speed and accuracy.

20 min·Pairs

Grouping Challenge: Real-World Scenarios

Present problems like dividing 36 cookies among friends. Students draw arrays or list factors to find sharing options. Groups present solutions and vote on the fairest method.

30 min·Pairs

Real-World Connections

• Event planners use factors to arrange seating for guests at parties or conferences, ensuring equal numbers of people at each table for a balanced setup. For instance, if 48 guests attend, they might arrange tables in 6 rows of 8.
• Bakers use factors when dividing cakes or pizzas into equal slices. A baker might cut a cake into 12 equal pieces, using factors of 12 to decide on 3 rows of 4 or 2 rows of 6 slices.
• Teachers use factors to group students for collaborative activities. If 24 students need to work in groups, a teacher might form 4 groups of 6 or 3 groups of 8, depending on the task.

Assessment Ideas

Exit Ticket

Provide students with the number 36. Ask them to: 1. List all factor pairs of 36. 2. Draw an array representing 36. 3. State one divisibility rule they used to find factors.

Quick Check

Write the numbers 7, 15, and 25 on the board. Ask students to hold up fingers to indicate the number of factors each number has. Then, ask them to write down the factors for one of the numbers and explain why it is prime or composite.

Discussion Prompt

Pose the question: 'How are factors and multiples related?' Guide students to explain that multiples are built from factors. Ask: 'If a number is a multiple of 6, what do you know about its factors?'

Frequently Asked Questions

How do you teach students to find all factors of a number?

Guide students to list pairs starting from 1, increasing one factor while decreasing the other, up to the square root. Use divisibility rules to skip tests. Arrays visualise pairs concretely. Practice with numbers like 36: 1x36, 2x18, 3x12, 4x9, 6x6. This systematic approach ensures completeness and builds efficiency over time.

What is the difference between a factor and a multiple?

A factor of 12 divides it evenly, like 3 because 12 ÷ 3 = 4. A multiple of 12 is 12 × integer, like 36. Arrays for 12 show factor pairs in dimensions, while skip-counting lists multiples. Real-world: factors for grouping 12 items, multiples for batches of 12.

How can active learning help students understand factors?

Active approaches like building arrays with counters let students manipulate and see factor pairs form visually, turning abstract division into tangible shapes. Collaborative games encourage explaining rules to peers, correcting misconceptions on the spot. These methods boost engagement, as students discover patterns through trial, leading to deeper retention than worksheets alone.

What are real-world examples of factors for Year 4?

Factors appear in grouping: factors of 20 help divide runners into lanes (1 group of 20, 2 of 10, 4 of 5). Sharing 24 pizzas uses pairs like 3x8 slices per person. Tiling a floor with 18 tiles fits 2x9 or 3x6 patterns. These contexts make factors relevant and practical.