Mathematics · Year 7

Active learning ideas

Highest Common Factor (HCF)

Active learning helps students grasp the HCF because it turns abstract prime factors into visible, manipulable structures. When students build towers or sort cards, they see how numbers break down and share common parts. This hands-on work builds intuition before moving to symbolic calculations.

• ACARA Content Descriptions
• AC9M7N01

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

Activity 01

• Problem-Based Learning
• 30 min · Pairs

Prime Factor Tower Race: Build Factor Towers

Pairs build towers with linking cubes to represent numbers up to 100, labeling each layer with prime factors. They compare towers to find common factors and calculate HCF. Discuss which primes are shared and multiply lowest powers.

Differentiate between a factor and a multiple.

Facilitation Tip

During Prime Factor Tower Race, ensure teams label each prime layer on their tower so peers can verify shared factors.

What to look for

Provide students with two numbers, e.g., 24 and 36. Ask them to find the HCF using prime factorization and show their work. On the back, ask them to write one sentence explaining why the HCF is useful.

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

Activity 02

• Problem-Based Learning
• 35 min · Small Groups

HCF Card Sort: Real-World Scenarios

Small groups sort cards with number pairs and contexts like fencing lengths or recipe scaling. They find HCF for each, justify with prime factors, and match to correct groupings. Share one solution as a class.

Explain how prime factorization helps in finding the HCF.

Facilitation Tip

For HCF Card Sort, circulate to prompt students to explain their sorting choices using mathematical language.

What to look for

Display three numbers on the board, such as 18, 30, and 42. Ask students to work in pairs to find the HCF. Circulate to observe their methods and ask clarifying questions like, 'Which prime factors are common to all three numbers?'

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

Activity 03

• Problem-Based Learning
• 25 min · Whole Class

Factor Hunt Relay: Whole Class Chain

Divide class into teams. One student per team factorizes a number on board, passes to next for HCF with previous. First team to chain five correct HCFs wins. Review errors together.

Construct a real-world problem where finding the HCF is necessary.

Facilitation Tip

In Factor Hunt Relay, set a time limit to keep energy high and prevent rushing; emphasize accuracy over speed.

What to look for

Pose the following scenario: 'Sarah has 20 stickers and 28 pencils. She wants to make identical packs for her friends. What is the largest number of packs she can make, and how many stickers and pencils will be in each pack?' Facilitate a class discussion on how to solve this problem using the HCF.

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

Activity 04

• Problem-Based Learning
• 20 min · Individual

Individual Puzzle: HCF Problem Creator

Students create and solve their own HCF problems from everyday items, like tiles or books. Swap with a partner to check using prime factorization. Class votes on most creative real-world example.

Differentiate between a factor and a multiple.

Facilitation Tip

With Individual Puzzle, model how to use factor trees before independent work to reduce frustration.

What to look for

Provide students with two numbers, e.g., 24 and 36. Ask them to find the HCF using prime factorization and show their work. On the back, ask them to write one sentence explaining why the HCF is useful.

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

A few notes on teaching this unit

Teach HCF by starting with concrete examples students can break apart, such as grouping classroom supplies. Move to visual models like towers or cards before formal notation to avoid rote memorization. Avoid rushing to the algorithm; let students discover patterns through structured play. Research shows that students retain concepts better when they construct meaning through activity rather than receiving it passively.

Successful learning looks like students confidently explaining why some factors are common and others are not. They should articulate the process of finding the HCF using prime factors and justify their answers in real-world contexts. Look for clear links between theory, method, and application.

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Watch Out for These Misconceptions

• During Prime Factor Tower Race, watch for students who assume the HCF is always 1 because their towers look different.

  Have teams compare towers side by side and circle matching prime layers to highlight shared factors, reinforcing that common primes exist even if towers appear distinct.

• During HCF Card Sort, watch for students who confuse HCF and LCM when matching real-world scenarios.

  Ask students to explain their matches aloud, focusing on keywords like 'largest equal group' for HCF or 'smallest total' for LCM to clarify the difference.

• During Factor Hunt Relay, watch for students who omit 1 or the number itself when listing factors.

  Ask teams to check their lists against a shared example on the board and discuss why 1 and the number are always factors.

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Methods used in this brief

• Problem-Based Learning