Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Active learning works well for HCF and LCM because these concepts rely on visual grouping and repeated cycles, which are easier to grasp through hands-on manipulation than abstract rules. Students need to see how factors and multiples relate to real objects or events to move beyond procedural memorisation to true understanding.

• Secondary 1
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Calculate the HCF and LCM of two or more numbers using prime factorisation and the division algorithm.
2. Compare the efficiency of listing factors/multiples versus prime factorisation for finding HCF and LCM.
3. Analyze how HCF and LCM are applied to solve problems involving scheduling, grouping, or cyclical events.
4. Explain the relationship between the product of two numbers and the product of their HCF and LCM, providing examples.
5. Solve word problems requiring the identification of HCF for greatest common grouping or LCM for least common occurrence.

Collaborative Problem-Solving

⏱ 30 min·Pairs

Manipulative Sort: Tile Grouping for HCF

Provide sets of tiles in quantities matching the numbers, such as 12 and 18 tiles. Students group tiles into equal bundles to find the largest common group size, which is the HCF. Discuss and record findings on mini-whiteboards. Extend to three numbers.

Prepare & details

Analyze how HCF and LCM are used to solve problems involving common groupings or cycles.

Facilitation Tip: During Manipulative Sort, circulate and ask each group to explain why a particular grouping represents the HCF, not just the number.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: LCM Scheduling

Set up stations with calendars or number lines for problems like bus timetables (every 12 and 15 minutes). Groups solve for first common meeting time using listing or prime factors, then verify with drawings. Rotate stations and share solutions.

Prepare & details

Compare the efficiency of different methods for finding HCF and LCM for large numbers.

Facilitation Tip: For Station Rotation, set a timer for each station and encourage students to rotate only after verifying their LCM answer with the next station’s task.

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

Collaborative Problem-Solving

⏱ 25 min·Pairs

Prime Factor Race: Efficiency Challenge

Pairs race to factorise large numbers using division ladders or trees, then compute HCF and LCM. Compare times and accuracy across methods. Class discusses why prime factorisation wins for big numbers.

Prepare & details

Explain the relationship between the product of two numbers and the product of their HCF and LCM.

Facilitation Tip: In Prime Factor Race, provide calculators for checking prime factors but insist on written verification before moving to the next pair.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Real-World Relay: Problem Applications

Teams relay to solve chained problems: find HCF to simplify ratios, LCM for cycles. Pass baton with answer to next teammate. Debrief connections to the product rule.

Prepare & details

Analyze how HCF and LCM are used to solve problems involving common groupings or cycles.

Facilitation Tip: During Real-World Relay, assign roles such as recorder, presenter, and verifier to ensure every student contributes to the problem-solving process.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Teachers should introduce HCF and LCM using concrete examples before formal methods, as research shows this builds stronger mental models. Avoid rushing to the formula HCF × LCM = a × b, as students need time to see why it works through prime factorisation first. Encourage students to compare methods and justify their preferences to deepen understanding.

What to Expect

Successful learning looks like students confidently using multiple methods for HCF and LCM, explaining their choices, and applying these skills to solve problems without prompting. They should discuss their reasoning with peers and correct each other’s misconceptions during collaborative tasks.

Watch Out for These Misconceptions

Common Misconception

During Manipulative Sort, watch for students assuming the HCF is always 1 if numbers look different.

What to Teach Instead

Prompt groups to list all factors of each number using the tiles, then circle shared factors to identify the largest. Ask them to compare their lists with another group’s to confirm the HCF.

Common Misconception

During Station Rotation, watch for students calculating the LCM as the sum or average of the numbers.

What to Teach Instead

Have students draw number lines for each station’s numbers and mark multiples until they find the smallest common point. Ask them to explain why this method works better than addition.

Common Misconception

During Prime Factor Race, watch for students believing the product rule HCF × LCM = a × b only applies to coprime numbers.

What to Teach Instead

Provide varied pairs, including non-coprime numbers, and have students verify the rule with prime factorisation. Ask them to present one example where the rule holds true to the class.

Assessment Ideas

Exit Ticket

After Manipulative Sort, provide students with two numbers, e.g., 18 and 24. Ask them to find the HCF using prime factorisation and write one sentence explaining which method they found easier and why.

Quick Check

During Station Rotation, present a word problem: ‘Two friends jog around a track. One completes a lap every 4 minutes, the other every 6 minutes. If they start together, when will they meet again at the start?’ Students write down the calculation needed (LCM) and the answer on a sticky note.

Discussion Prompt

After Real-World Relay, pose the question: ‘If you have 15 pencils and 20 erasers, and you want to make identical sets with no leftovers, what mathematical concept would you use and why?’ Facilitate a brief class discussion on HCF and its application, noting students’ justifications.

Extensions & Scaffolding

• Challenge students who finish early to create their own word problems involving HCF or LCM and exchange them with peers for solving.
• For students who struggle, provide number lines and counters at Manipulative Sort to reinforce the concept of shared divisors visually.
• Deeper exploration: Ask students to investigate why the HCF of two consecutive numbers is always 1, using both examples and algebraic reasoning.

Key Vocabulary

Highest Common Factor (HCF)	The largest positive integer that divides two or more integers without leaving a remainder. It is also known as the Greatest Common Divisor (GCD).
Lowest Common Multiple (LCM)	The smallest positive integer that is a multiple of two or more integers. It is the smallest number that all the given integers divide into evenly.
Prime Factorisation	Expressing a composite number as a product of its prime factors. This method is efficient for finding HCF and LCM of larger numbers.
Division Algorithm	A systematic method, often using repeated division by common prime factors, to find the HCF and LCM of a set of numbers.