Mathematics · Secondary 1

Active learning ideas

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.

• MOE Syllabus Outcomes
• MOE: Primes, HCF and LCM - S1
• MOE: Numbers and Algebra - S1

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Placemat Activity
• 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.

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.

What to look for

Provide students with two numbers, e.g., 24 and 36. Ask them to: 1. Find the HCF using prime factorisation. 2. Find the LCM using the division algorithm. 3. Write one sentence explaining which method they found easier and why.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Relationship Skills

Activity 02

• 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.

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.

What to look for

Present a word problem: 'Two buses depart from a station. Bus A leaves every 15 minutes, and Bus B leaves every 20 minutes. If they both leave at 8:00 AM, when will they next leave at the same time?' Students write down the calculation needed (LCM) and the answer.

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

Activity 03

• Placemat Activity
• 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.

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.

What to look for

Pose the question: 'If you have 18 apples and 24 oranges, and you want to make identical fruit baskets with the greatest number of fruits possible in each, what mathematical concept would you use and why?' Facilitate a brief class discussion on HCF and its application.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Relationship Skills

Activity 04

• Placemat Activity
• 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.

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.

What to look for

Provide students with two numbers, e.g., 24 and 36. Ask them to: 1. Find the HCF using prime factorisation. 2. Find the LCM using the division algorithm. 3. Write one sentence explaining which method they found easier and why.

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Relationship Skills

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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

  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.

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Methods used in this brief

• Placemat Activity
• Stations Rotation