Mathematical Explorers: Building Number and Space · 3rd Class

Active learning ideas

Highest Common Factor and Lowest Common Multiple

Active learning helps students grasp HCF and LCM because these concepts rely on visualizing factors and multiples, which are easier to understand when manipulated physically. By breaking numbers into primes and rebuilding them, students move from abstract ideas to concrete reasoning.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Number - N.9
• NCCA: Junior Cycle - Problem Solving - PS.1

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Problem-Based Learning
• 25 min · Pairs

Pairs: Factor Tree Race

Pairs draw prime factor trees for two numbers on mini-whiteboards, then compute HCF or LCM. Switch roles after 2 minutes; first accurate pair wins a point. Debrief as a class on common patterns.

Differentiate between HCF and LCM and their applications.

Facilitation Tip

During the Factor Tree Race, circulate to ensure pairs are combining exponents correctly, not just listing factors.

What to look for

Present students with two numbers, for example, 24 and 30. Ask them to find the HCF using prime factorization and write down the steps they followed. Then, ask them to find the LCM of the same two numbers using prime factorization and show their work.

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

Activity 02

• Problem-Based Learning
• 35 min · Small Groups

Small Groups: Real-World Scenario Sort

Provide cards with problems needing HCF or LCM, like dividing pizzas or cicada cycles. Groups sort, solve using lists or factorization, and justify choices. Share one solution per group.

Explain how prime factorization can be used to find HCF and LCM.

Facilitation Tip

In the Real-World Scenario Sort, ask guiding questions to push students beyond surface-level connections.

What to look for

Pose the following scenario: 'Imagine you have 18 red balloons and 24 blue balloons. You want to make as many identical bouquets as possible, with the same number of red and blue balloons in each. What is the maximum number of bouquets you can make?' Ask students to identify whether they need to find the HCF or LCM and explain their reasoning.

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

Activity 03

• Problem-Based Learning
• 30 min · Whole Class

Whole Class: HCF/LCM Relay

Divide class into teams. One student per team runs to board, factors a number pair, next computes HCF/LCM. First team correct across 6 rounds wins. Review methods collectively.

Design a real-world problem that requires finding either the HCF or LCM to solve.

Facilitation Tip

For the HCF/LCM Relay, assign roles so every student contributes, such as recorder, calculator, or presenter.

What to look for

Give each student a card with a different scenario. For example: 'Bus A arrives every 10 minutes, and Bus B arrives every 15 minutes. When will they next arrive at the same time?' or 'You have 36 cookies and 48 brownies. You want to make identical treat bags with the same number of cookies and brownies in each. What is the largest number of bags you can make?' Students write down the HCF or LCM calculation needed and the answer.

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

Activity 04

• Problem-Based Learning
• 20 min · Individual

Individual: Problem Designer

Students create one HCF and one LCM problem from daily life, like sports schedules or recipe scaling. Swap with a partner to solve, then discuss solutions in pairs.

Differentiate between HCF and LCM and their applications.

Facilitation Tip

When students design problems, require them to include a solution key to encourage self-checking.

What to look for

Present students with two numbers, for example, 24 and 30. Ask them to find the HCF using prime factorization and write down the steps they followed. Then, ask them to find the LCM of the same two numbers using prime factorization and show their work.

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

A few notes on teaching this unit

Teach prime factorization as a tool, not a rote procedure. Start with small numbers to build intuition, then introduce larger numbers to highlight efficiency. Avoid overemphasizing shortcuts like the listing method, which becomes impractical with bigger numbers. Research shows hands-on factorization and peer discussion strengthen retention and transfer to problem-solving.

Successful learning looks like students confidently using prime factorization to find HCF and LCM, explaining their steps aloud, and justifying their choices in real-world contexts. They should connect the mechanics to practical applications without hesitation.

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Watch Out for These Misconceptions

• During the Factor Tree Race, watch for students assuming that the larger number is always the LCM or the smaller the HCF.

  Have peers compare their factor trees side by side and circle the shared primes for HCF and all primes for LCM, reinforcing the concept through visual overlap and difference.

• During the Real-World Scenario Sort, students may think prime factorization only works for two numbers.

  Ask groups to add a third number to their scenarios and adapt their method, then compare notes to see how combining exponents remains effective.

• During the HCF/LCM Relay, students might treat HCF and LCM as interchangeable in problems.

  Require each relay team to justify their choice in a one-sentence explanation before moving to the next station, using context clues from the problem.

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Methods used in this brief

• Problem-Based Learning