Prime Factors, HCF, and LCM

Active learning transforms abstract concepts like prime factors, HCF, and LCM into concrete, visual processes. Students manipulate numbers directly, which builds number sense and reveals patterns they might miss with only textbook exercises.

• Year 8
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

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

Stations Rotation

⏱ 30 min·Small Groups

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.

Prepare & details

Differentiate between prime numbers and composite numbers.

Facilitation Tip: During the Relay Race, circulate and ask each group to explain one step of their factor tree aloud before passing the marker to keep everyone engaged.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 25 min·Pairs

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.

Prepare & details

Construct the prime factorisation of a given number.

Facilitation Tip: In the Card Sort, listen for students to verbalize why a pair belongs together, such as ‘Both numbers share 2 and 3, so the HCF is 6.’

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 45 min·Small Groups

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.

Prepare & details

Explain the practical applications of HCF and LCM in real-world problems.

Facilitation Tip: At the Stations, provide calculators only after students attempt the prime factorisation by hand to prevent skipping the core skill.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Stations Rotation

⏱ 20 min·Whole Class

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.

Prepare & details

Differentiate between prime numbers and composite numbers.

Facilitation Tip: For Bingo, have students call out their prime factors as they mark squares to reinforce the connection between factors and multiples.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach prime factorisation first through factor trees or repeated division, as these methods build a visual and procedural foundation. Avoid rushing to the formula for LCM = (a×b)/HCF before students understand why it works. Research shows that students who construct factorisations themselves retain the concept longer than those who memorize shortcuts too early.

What to Expect

Students will confidently break down composite numbers into primes, pair common factors correctly for HCF, and combine primes appropriately for LCM. They will explain their method and justify their answers using clear reasoning.

Watch Out for These Misconceptions

Common Misconception

During the Card Sort: HCF and LCM Pairs, watch for students who pair numbers based on their difference rather than common factors.

What to Teach Instead

Have students write the prime factors of each number on the cards first, then physically group cards with matching primes to find the HCF.

Common Misconception

During the Relay Race: Prime Factor Trees, watch for students who mark even numbers greater than 2 as prime.

What to Teach Instead

Pause the race and ask the group to factor out 2 immediately for any even number, reinforcing that all evens over 2 are composite.

Common Misconception

During the Stations: Real-World Problems, watch for students who confuse HCF with LCM when solving for identical groupings.

What to Teach Instead

Ask them to list the prime factors of both quantities first, then decide whether to take the lowest or highest powers before calculating.

Assessment Ideas

Exit Ticket

After the Relay Race: Prime Factor Trees, give students the number 84. Ask them to write its prime factorisation and find the HCF of 84 and 60 using their factor tree.

Quick Check

During the Card Sort: HCF and LCM Pairs, circulate and ask pairs to explain their chosen HCF for one card set. Note who can justify using prime factors versus those relying on trial and error.

Discussion Prompt

After the Stations: Real-World Problems, pose the baker scenario and ask students to share whether they used HCF or LCM and why. Listen for references to prime factors or common groupings.

Extensions & Scaffolding

• Challenge early finishers to find the smallest number with a prime factorisation of 2³×3² and explain their steps.
• For students who struggle, provide partially completed factor trees with one branch missing for them to finish.
• Deeper exploration: Ask students to investigate why the HCF of two consecutive numbers is always 1, using factor trees as evidence.

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.