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.

• Year 7
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate the Highest Common Factor (HCF) of two or more numbers using prime factorization.
2. Compare and contrast the HCF of a set of numbers with their Least Common Multiple (LCM).
3. Explain the steps involved in finding the HCF using a factor tree diagram.
4. Construct a word problem that requires the calculation of the HCF for its solution.

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

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.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

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.

Prepare & details

Differentiate between a factor and a multiple.

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

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

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

Teaching This Topic

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.

What to Expect

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.

Watch Out for These Misconceptions

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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.

Common Misconception

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

What to Teach Instead

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

Assessment Ideas

Exit Ticket

After Prime Factor Tower Race, give students two numbers, e.g., 30 and 45. Ask them to draw towers and find the HCF, then write one sentence explaining how the towers helped them.

Quick Check

During HCF Card Sort, display three numbers on the board, such as 12, 18, and 24. Ask pairs to find the HCF and hold up their answer on mini whiteboards for immediate feedback.

Discussion Prompt

After Factor Hunt Relay, pose the scenario: 'A baker has 36 cookies and 54 brownies. He wants to make identical trays with no leftovers. What is the largest number of trays he can make?' Facilitate a class discussion on how the relay method applies to this problem.

Extensions & Scaffolding

• Challenge students to find the HCF of three numbers during the Prime Factor Tower Race.
• Provide scaffolding for the HCF Card Sort by including partially completed factor trees for students who struggle.
• Extend the Individual Puzzle by asking students to create a problem where the HCF is not immediately obvious, such as 48 and 72.

Key Vocabulary

Factor	A number that divides another number exactly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Highest Common Factor (HCF)	The largest factor that two or more numbers share. It is also known as the Greatest Common Divisor (GCD).
Prime Number	A whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, 7, and 11.
Prime Factorization	Expressing a number as a product of its prime factors. For example, the prime factorization of 12 is 2 x 2 x 3.