Mathematics · Year 5 · Additive and Multiplicative Structures · Autumn Term

Common Factors and Multiples

Students will identify common factors of two numbers and common multiples of two numbers, applying this to problem-solving.

• National Curriculum Attainment Targets
• KS2: Mathematics - Multiplication and Division

About This Topic

Common factors and multiples build essential multiplicative reasoning in Year 5. Students identify factors of numbers, such as 1, 2, 3, 4, 6, 12 for 12 and 1, 2, 3, 6, 9, 18 for 18, then spot shared ones: 1, 2, 3, 6. For multiples, they list the first five common ones for 3 and 4: 12, 24, 36, 48, 60. These skills apply to problems like scheduling events that align or sharing items evenly.

This topic connects to multiplication tables from earlier years and lays groundwork for highest common factor, lowest common multiple, fractions, and ratios. Students develop pattern recognition, logical listing, and problem-solving, key for algebraic thinking. Real-world contexts, such as dividing resources or timing cycles, show mathematics in action.

Active learning suits this topic perfectly. Collaborative sorting of factor tiles or racing to generate multiples turns abstract lists into dynamic challenges. Group problem-solving reveals strategies peers use, while hands-on models like bead strings for multiples make concepts visible and memorable, increasing confidence and retention.

Key Questions

1. Explain how to find the common factors of 12 and 18.
2. Construct a list of the first five common multiples of 3 and 4.
3. Analyze a problem where finding a common multiple helps to solve it.

Learning Objectives

• Calculate the common factors of two given numbers, listing all possibilities.
• Generate the first five common multiples of two given numbers.
• Analyze a word problem to determine if finding a common multiple is the most efficient solution strategy.
• Explain the process for finding common factors and common multiples using mathematical vocabulary.

Before You Start

Multiplication Facts

Why: Students need fluency with multiplication tables to efficiently identify multiples and factors.

Identifying Factors

Why: Understanding how to find all factors of a single number is a direct precursor to finding common factors.

Key Vocabulary

Factor	A number that divides exactly into another number without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiple	A number that can be divided exactly by another number. Multiples are found by multiplying a number by a whole number. For example, multiples of 3 are 3, 6, 9, 12...
Common Factor	A number that is a factor of two or more different numbers. For example, 3 is a common factor of 12 and 18.
Common Multiple	A number that is a multiple of two or more different numbers. For example, 24 is a common multiple of 4 and 6.

Watch Out for These Misconceptions

Common Misconception

Common factors are only prime numbers.

What to Teach Instead

All divisors count as factors, including 1 and composites. Hands-on sorting activities with physical tiles let students build complete lists visually, comparing with peers to spot overlooked factors like 6 for 12 and 18.

Common Misconception

Multiples start from zero or only even numbers.

What to Teach Instead

Multiples begin with the number itself and include all products. Relay games where students race to list accurately encourage checking work collaboratively, correcting the idea that multiples must be even through examples like 3 and 4.

Common Misconception

Factors and multiples are the same thing.

What to Teach Instead

Factors divide the number; multiples are products. Venn diagram tasks clarify the inverse relationship, with group discussions helping students articulate differences and connections.

Active Learning Ideas

Small Groups: Factor Tile Sort

Provide number tiles (1-20) and two numbers like 12 and 18. Groups sort tiles into factors for each number using Venn diagrams on large paper. Discuss and list common factors. Extend by predicting for new pairs.

35 min·Small Groups

Pairs: Multiples Relay Race

Pairs generate multiples of two numbers on mini-whiteboards, passing to partner after five each. First pair to list 10 common multiples wins. Review lists as a class, circling shared ones.

25 min·Pairs

Whole Class: Problem-Solving Carousel

Set up stations with problems needing common factors or multiples, like 'Share 24 cakes between 3 and 4 friends equally.' Groups rotate, solve, and justify. Debrief key strategies.

45 min·Small Groups

Individual: Common Multiple Hunt

Students list first eight multiples of given pairs, highlight commons, then solve a word problem using the smallest common multiple. Share one solution with the class.

20 min·Individual

Real-World Connections

• Event planners use common multiples to schedule recurring events, like booking a conference room that is available every 3 days and a catering service that is available every 4 days, to find when both are available simultaneously.
• Teachers use common factors when dividing students into groups for projects, ensuring that the number of students in each group divides evenly into the total number of students for fair distribution.

Assessment Ideas

Exit Ticket

Provide students with two numbers, for example, 15 and 20. Ask them to list all common factors of these two numbers and then list the first three common multiples.

Quick Check

Present a word problem: 'Sarah is making party bags. She has 24 sweets and 30 stickers. What is the largest number of party bags she can make so that each bag has the same number of sweets and the same number of stickers?' Ask students to explain how finding a common factor helps solve this problem.

Discussion Prompt

Pose the question: 'When might it be useful to find common multiples in real life?' Encourage students to share examples and explain their reasoning, focusing on scenarios involving cycles or regular intervals.

Frequently Asked Questions

How do you teach common factors in Year 5 maths?

Start with factor pairs using arrays or multiplication facts. Model listing factors for 12 and 18 on the board, then guide students to identify commons like 1, 2, 3, 6. Use real contexts like dividing sweets. Reinforce with pair shares to build fluency before independent problem-solving.

What activities work for common multiples Year 5?

Try multiples races or matching cards where students pair multiples of 3 and 4. Extend to finding the least common multiple in scheduling problems, like bus times. These keep pace brisk and link to practical uses, solidifying lists like 12, 24, 36.

How do common factors link to HCF and LCM?

Common factors lead to highest common factor (largest shared); common multiples to lowest common multiple (smallest shared). Teach by listing then selecting extremes. This prepares for fractions and problem-solving, with visuals like factor rainbows aiding recognition.

How can active learning help students master common factors and multiples?

Active methods like tile sorts, Venn diagrams, and relay races make listing interactive and social. Students physically manipulate items, discuss strategies, and compete lightly, which embeds patterns deeply. Real problems, such as sharing or timing, show relevance, boosting motivation and reducing errors from rote memorisation.