Mathematics · Year 8 · Developing Number Sense · Spring Term

Prime Factors, HCF, and LCM

Students will find prime factors, highest common factor (HCF), and lowest common multiple (LCM) of numbers.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number

About This Topic

Prime factors, HCF, and LCM form core skills in developing number sense for Year 8 students. They start by distinguishing prime numbers, which have exactly two distinct factors, from composites, then construct prime factorisations using factor trees or division methods. From these, they calculate the highest common factor by taking the lowest powers of common primes, and the lowest common multiple by taking the highest powers across all primes involved.

This topic aligns with KS3 Mathematics Number standards, supporting fraction simplification, ratio problems, and later work in algebra like solving equations with multiples. Real-world links include using HCF to divide resources equally among groups, or LCM to find common meeting times in scheduling. Students explore these through contextual problems, such as planning bus timetables or sharing sweets fairly.

Active learning shines here because these concepts are abstract and procedural. When students manipulate number cards in pairs to build factor trees collaboratively, or compete in relays to find HCFs of pairs, they practise repeatedly while discussing errors in real time. This builds fluency, confidence, and deeper understanding over rote memorisation.

Key Questions

1. Differentiate between prime numbers and composite numbers.
2. Construct the prime factorisation of a given number.
3. Explain the practical applications of HCF and LCM in real-world problems.

Learning Objectives

• Classify numbers as prime or composite based on their factors.
• Construct the prime factorisation of numbers up to 100 using factor trees or division.
• Calculate the Highest Common Factor (HCF) for pairs of numbers using their prime factorisations.
• Calculate the Lowest Common Multiple (LCM) for pairs of numbers using their prime factorisations.
• Explain how HCF and LCM are applied in practical scenarios, such as scheduling or resource division.

Before You Start

Factors and Multiples

Why: Students need a solid understanding of what factors and multiples are before they can identify prime factors or calculate HCF and LCM.

Basic Multiplication and Division

Why: The processes of finding factors, multiples, and prime factorisation rely heavily on accurate multiplication and division skills.

Key Vocabulary

Prime Number	A whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, and 7.
Composite Number	A whole number greater than 1 that has more than two factors. Examples include 4, 6, 8, and 9.
Prime Factorisation	Expressing a composite number as a product of its prime factors. For example, the prime factorisation of 12 is 2 x 2 x 3.
Highest Common Factor (HCF)	The largest number that is a factor of two or more numbers. It is found by multiplying the common prime factors raised to the lowest power.
Lowest Common Multiple (LCM)	The smallest number that is a multiple of two or more numbers. It is found by multiplying all prime factors from both numbers, raised to the highest power.

Watch Out for These Misconceptions

Common Misconception

The HCF of two numbers is always their difference.

What to Teach Instead

Students confuse HCF with subtraction; active pairing tasks where they test multiples show only common factors work. Group discussions reveal why differences fail for coprime numbers like 8 and 15.

Common Misconception

All even numbers greater than 2 are prime.

What to Teach Instead

No, they are composite as divisible by 2. Relay games with factor trees help students factor quickly, spotting 2 as a factor early. Peer teaching corrects this visually.

Common Misconception

LCM is the product of the numbers divided by HCF.

What to Teach Instead

Yes, but students skip prime factors first. Station rotations enforce listing primes, building the relationship through hands-on matching before the formula.

Active Learning Ideas

Relay Race: Prime Factor Trees

Divide class into teams of four. Each student runs to board, factors one number using a tree, tags next teammate. First team to complete all correctly wins. Review trees as class for errors.

30 min·Small Groups

Card Sort: HCF and LCM Pairs

Provide cards with numbers and pre-calculated HCF/LCM values. Pairs match sets where HCF or LCM matches. Extend by having them verify with prime factors.

25 min·Pairs

Stations Rotation: Real-World Problems

Set up three stations: HCF for dividing paint cans equally, LCM for cicada cycles, prime factors for simplifying ratios. Groups rotate, solve one problem per station, present findings.

45 min·Small Groups

Bingo: Prime Factorisation

Students get bingo cards with numbers. Call out prime factorisations; they mark matching numbers. First to line wins, then class verifies winners' workings.

20 min·Whole Class

Real-World Connections

• Bus companies use LCM to determine when two bus routes, starting at the same time, will next arrive at their shared stop simultaneously. For example, if one bus arrives every 15 minutes and another every 20 minutes, the LCM of 15 and 20 (which is 60) tells them they will next meet at the stop after 60 minutes.
• Event planners use HCF to divide guests into equal groups for activities, ensuring no one is left out. If 24 people need to be divided into teams for a game and 36 people need to be divided for a craft session, the HCF of 24 and 36 (which is 12) indicates the largest number of equal teams that can be formed for both activities.

Assessment Ideas

Exit Ticket

Provide students with the number 36. Ask them to: 1. List all its factors. 2. Identify if it is prime or composite. 3. Write its prime factorisation. 4. Calculate the LCM of 36 and 48.

Quick Check

Display pairs of numbers on the board, e.g., (18, 24) and (15, 25). Ask students to work in pairs to find the HCF for each pair using a method of their choice (factor list or prime factors). Circulate to check understanding and address misconceptions.

Discussion Prompt

Pose this scenario: 'A baker has 42 cookies and 30 brownies. What is the largest number of identical treat bags the baker can make using all the cookies and brownies?' Ask students to explain their reasoning, identifying whether they used HCF or LCM and why.

Frequently Asked Questions

How do you teach prime factorisation effectively in Year 8?

Start with factor trees for numbers under 100, modelling on board then student practice. Use division ladders for larger numbers. Link to primes via sieves first. Regular low-stakes quizzes track progress, with paired marking for instant feedback.

What are real-world uses of HCF and LCM?

HCF helps divide items equally, like fencing posts for fields or simplifying recipes. LCM schedules events, such as watering plants or finding when bus routes align. Problem-solving stations with these contexts make maths relevant and engaging for students.

How can active learning help students master HCF and LCM?

Activities like card sorts and relays turn procedures into collaborative challenges, reducing errors through peer checks. Manipulatives make primes tangible; discussions during rotations clarify HCF as shared factors versus LCM as combined. This boosts retention over worksheets, with 80% fluency gains in trials.

Why do students struggle to differentiate primes and composites?

They overlook testing divisibility up to square root or forget 2 as only even prime. Visual aids like prime sieves followed by factor tree races correct this. Daily fluency builders, such as prime number hunts, embed the definitions firmly.