Multiplying & Dividing Fractions

Overview

Duration: 90 minutes
Class size: 30 Year 10 students
National Curriculum links:

• Number - Fractions (KS3 Programme of Study, continuing into GCSE Foundation content)
• Probability (KS4)
• Developing fluency, reasoning and problem solving (Mathematics Programme of Study for KS4, English National Curriculum 2014, GCSE Mathematics 1MA1)

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Learning Objectives

By the end of this lesson, students will be able to:
a) Perform multiplication of fractions and mixed numbers by converting to improper fractions where necessary (National Curriculum: Number - fractions, decimals and percentages, solve problems involving multiplication of fractions).
b) Divide fractions by multiplying by the reciprocal and apply this to real-life and mathematical problems (National Curriculum: Number - fractions, division of fractions including by whole numbers and unit fractions).
c) Link fractions to probability, solving problems involving likelihood of independent and dependent events using multiplication and division (National Curriculum: Probability - calculate the probability of independent and dependent events using fractions including combined events).

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Resources

• Whiteboard and markers
• Individual mini-whiteboards and pens (for students)
• Probability spinners or dice (physical or images)
• Worksheets with fraction multiplication/division and probability problems
• Fraction and probability visual aids (fraction strips, probability trees)
• Calculator (optional, for checking work)

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Lesson Structure

1. Starter – 5 minutes: Maths Retrieval

Purpose: Activate prior knowledge and prepare students for the new concepts.

Activity:

• Quick quiz on simple fraction operations and probabilities (e.g., add/subtract fractions, find fraction of an amount, basic probability words).
• Sample questions on mini whiteboards:
  • Simplify 3/6.
  • What is 1/2 + 1/4?
  • If a spinner has 4 equal parts, what is the probability of landing on one specific part (write as fraction)?

Assessment: Teacher circulates, checks answers quickly to identify misconceptions.

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2. Introduction & Explanation – 15 minutes

Aim: Introduce multiplying and dividing fractions with visual and formal methods.

• Recap the concept of improper fractions vs mixed numbers. Show conversion both ways.
• Demonstrate fraction multiplication:
  • Multiply two proper fractions (e.g., 2/3 × 3/4).
  • Multiply a mixed number by a fraction (convert mixed number to improper fraction first).
• Demonstrate dividing fractions by multiplying by their reciprocal (e.g., 2/5 ÷ 3/7 = 2/5 × 7/3).
• Use diagrams (fraction bars) to visually illustrate multiplication and division.

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3. Guided Practice – 20 minutes

Aim: Reinforce procedures and improve fluency.

Tasks:

• Students solve differentiated questions on mini-whiteboards:
  • Multiply proper fractions (e.g., 4/7 × 2/5).
  • Multiply mixed numbers (e.g., 1 3/5 × 2/3).
  • Divide fractions by reciprocal (e.g., 5/8 ÷ 3/4).
• Teacher models problem-solving with step-by-step verbalisation, pupils follow and ask questions.
• Peer discussion: In pairs, students explain the steps to each other.

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4. Linking to Probability – 25 minutes

Aim: Apply fraction multiplication/division to probability of independent and dependent events.
National Curriculum Reference: Use the language of probability and calculate combined probabilities.

Explanation:

• Explain independent events with examples (e.g., flipping coins, rolling dice) – probability of both happening = multiply fractions.
• Introduce dependent events with examples (drawing cards without replacement) – show how probabilities change and division or fractions arise.

Activities:

• Use probability spinners or dice in pairs: calculate probability of sequences of independent events (e.g., landing on red then blue).
• Worksheet: word problems involving probability of combined events, requiring multiplication/division of fractions.
• Introduce simple tree diagrams showing probabilities, demonstrate multiplication along branches.

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5. Challenge & Problem Solving – 15 minutes

Aim: Deepen understanding by solving multi-step problems combining fractions and probability.

Tasks:

• Multi-step context problems:
  • Example: A bag has 5 red and 3 blue balls. If one ball is taken and not replaced, what is the probability the next ball is red?
  • Example: Calculate the probability of rolling an even number and then drawing a card from a deck with some fraction-based probabilities.

Extension: Create own probability problems involving fraction multiplication/division, present in pairs to class.

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6. Plenary & Assessment – 10 minutes

Purpose: Review learning, self-assess and consolidate understanding.

Activity:

• Exit ticket questions (student writes answers on post-its or mini whiteboards):
  1. Multiply and simplify: 3/4 × 2 1/3
  2. Divide: 5/6 ÷ 2/9
  3. Probability: Probability of independent events, flipping heads on a coin and rolling 4 on a dice. Write answer as fraction.
• Teacher collects responses and gives instant feedback.

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Assessment Opportunities

• Continuous formative assessment during starter, guided practice, and plenary.
• Peer assessment during paired work and discussion.
• Exit tickets provide summative check of objectives.
• Use mini-whiteboard answers to target misconceptions immediately.

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Differentiation

Support:

• Use visual aids and concrete manipulatives (fraction strips, counters).
• Scaffolded worksheets with step-by-step guidance for fraction operations.

Challenge:

• Include problems with mixed numbers and improper fractions.
• Multi-step probability problems involving conditional probabilities and tree diagrams.

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Curriculum Links Justification

This lesson plan directly supports the Mathematics Programmes of Study for KS3 and KS4 in England:

• Number - fractions (Programme of Study, Number section): "Perform operations with fractions including multiplication and division" and "Convert between improper fractions and mixed numbers" (GCSE content).
• Probability (Programme of Study): "Calculate the probability of combined events for independent and dependent events" using fractions.
• Emphasises fluency, reasoning, problem solving (cross-cutting themes in the National Curriculum).

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Teacher Reflection Notes

• Monitor students' understanding of turning mixed numbers into improper fractions as this is a common barrier.
• Check that students can confidently multiply by reciprocals rather than attempt complicated division.
• Use frequent formative assessment to tailor complexity of probability problems for confidence and challenge.
• Allow creativity during problem-creation to increase engagement and provide ownership.

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This lesson plan provides a clear structure combining core fraction skills with practical probability applications to deepen students’ mathematical understanding in line with the National Curriculum.