Exploring HCF & LCM

Overview

This 60-minute lesson is designed for Year 7 students (ages 11-12) and aligns closely with the National Curriculum for England, specifically the Programme of Study for Number – Multiplication and Division:
"Pupils should be taught to use common factors to simplify fractions; use common multiples to express fractions in the same denomination." (DfE National Curriculum for Mathematics, Key Stage 3)
The focus is on understanding and calculating the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) through engaging, interactive activities to develop numerical fluency and problem-solving skills.

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Learning Objectives

By the end of the lesson, students will be able to:

• Identify factors and multiples of given numbers (NC ref: KS3 Number)
• Calculate the Highest Common Factor (HCF) using listing and prime factorisation methods
• Calculate the Lowest Common Multiple (LCM) using listing and prime factorisation methods
• Apply HCF and LCM knowledge to solve real-life problems, such as scheduling or sharing items equally
• Explain reasoning and methods clearly, demonstrating understanding and fluency

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National Curriculum Coverage

• Number – Multiplication and Division (KS3):
  • Use common factors to simplify fractions
  • Use common multiples to express fractions in the same denomination
  • Solve problems involving calculations of HCF and LCM
  • Understand and use prime factorisation as a method for finding HCF and LCM

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Resources

• Whiteboard and markers
• Sets of coloured counters or small cubes (20 sets, 20 counters each)
• Worksheets with factor and multiple trees, problem-solving questions
• Mini whiteboards or exercise books for student responses
• Timer or stopwatch
• Projector (optional) to display visual aids

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Lesson Structure

1. Starter (10 minutes) – Factors and Multiples Warm-up

• Activity: “Factor & Multiple Snap”
  • Write a number on the board (e.g., 24).
  • Students work in pairs, with mini whiteboards, to write as many factors and multiples (under 100) of this number as they can in 3 minutes.
  • Share answers, discuss the difference between factors and multiples.
• Purpose: Activate prior knowledge and key vocabulary.

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2. Introduction (10 minutes) – Concept of HCF and LCM

• Use a visual approach:
  • Take two numbers, e.g., 12 and 18.
  • List their factors on the board:
    • 12 → 1, 2, 3, 4, 6, 12
    • 18 → 1, 2, 3, 6, 9, 18
  • Identify the common factors and highlight the HCF = 6.
  • Repeat with multiples to find LCM:
    • Multiples of 12: 12, 24, 36, 48, 60...
    • Multiples of 18: 18, 36, 54, 72...
  • LCM = 36 (first common multiple).
• Relate to real-life context, e.g., “If two buses come every 12 and 18 minutes respectively, when will they arrive together?”

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3. Main Activity (25 minutes) – Deep Dive with Prime Factorisation

• Prime Factorisation Trees:
  • Briefly explain prime numbers and prime factorisation.
  • Model how to break numbers (e.g., 24 and 36) into prime factors using factor trees.
• Group Task:
  • Students in pairs choose two numbers (between 10 and 50).
  • Build factor trees to find prime factors, then use these to calculate HCF and LCM.
  • Provide scaffolding worksheet to guide the process.
• Challenge Extension:
  • Provide word problems involving HCF and LCM, such as planning event schedules or dividing items equally.
• Circulate and support during group work, emphasising reasoning and clear written explanations.

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4. Plenary (10 minutes) – Applying and Explaining

• “Teach Back” Mini Presentations:
  • Pairs explain their chosen numbers and how they found the HCF and LCM to another pair or the class.
  • Emphasise proper mathematical vocabulary and clear reasoning.
• Quick Assessment:
  • Mini whiteboard quiz with 3 rapid-fire questions requiring HCF or LCM calculations.
  • Example question: Find the LCM of 15 and 20.
• Review common misconceptions and clarify.

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5. Independent Reflection (5 minutes)

• Students write a short reflection in their exercise books:
  • What is the difference between HCF and LCM?
  • Why would you need to find them in real-life situations?
• Collect reflections for formative assessment.

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Assessment & Differentiation

Formative Assessment

• Observation during paired/group work on the prime factorisation and problem-solving tasks
• Responses in the plenary mini whiteboard quiz
• Reflection exercise to check conceptual understanding and ability to articulate learning

Differentiation

• Support:
  • Use of concrete materials (counters)
  • Step-by-step worksheets with guided questions
  • Partner discussion to scaffold explanations
• Challenge:
  • Extension problem-solving tasks involving three or more numbers for HCF and LCM
  • Exploring the use of HCF and LCM to solve fraction problems per the National Curriculum focus on fractions

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Teacher Notes and Tips

• Stress the practical importance of HCF and LCM – they are not just abstract concepts but help solve real-life problems such as scheduling, sharing, or simplifying fractions.
• Encourage precise mathematical language (factor, multiple, highest, lowest, prime) during all explanations.
• Use questioning to prompt reasoning (“Why do you think 6 is the highest common factor?”, “Can there be a common multiple smaller than both numbers?”) to deepen understanding.
• For future lessons, link this understanding to simplifying fractions and working with ratio problems.

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This lesson plan aims to engage students with hands-on activities, promote conceptual understanding through visual and practical methods, and embed clear reasoning skills essential for mathematical development at Key Stage 3.