Mathematics · Year 7 · Proportional Reasoning · Spring Term

Multiplying and Dividing Fractions

Mastering the operations of multiplication and division with fractions.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number

About This Topic

Multiplying and dividing fractions equips Year 7 students with core proportional reasoning skills central to KS3 Number standards. Students multiply fractions by multiplying numerators together and denominators together, using area models to see why two proper fractions produce a smaller result: the overlapping shaded region represents a fraction of each original fraction. For division, they apply the 'keep, change, flip' method, multiplying by the reciprocal, and interpret it as finding how many times one fraction fits into another.

This Spring Term unit in Proportional Reasoning addresses key questions like explaining the size reduction in multiplication and designing real-world problems, such as scaling a recipe by 3/4 or dividing a pizza into 2/3 portions for 5/6 of the group. These connections build fluency and problem-solving across mathematics.

Active learning benefits this topic greatly because fraction operations challenge intuition developed from whole numbers. Manipulatives like fraction tiles let students physically combine and partition pieces, while pair discussions reveal why procedures work. Group challenges with contextual problems make abstract rules concrete and memorable, boosting confidence and retention.

Key Questions

1. Explain why multiplying two proper fractions results in a smaller number.
2. Analyze the 'keep, change, flip' method for dividing fractions.
3. Design a real-world problem that requires multiplying fractions.

Learning Objectives

• Calculate the product of two proper fractions, explaining why the result is smaller than either original fraction.
• Calculate the quotient of two fractions using the reciprocal method, demonstrating the process with an example.
• Design a word problem requiring the multiplication of fractions to solve a practical scenario.
• Analyze the steps involved in dividing a whole number by a fraction and a fraction by a whole number.
• Compare the results of multiplying fractions with different denominators.

Before You Start

Understanding Fractions

Why: Students need a solid grasp of what fractions represent, including numerators and denominators, before performing operations on them.

Equivalent Fractions

Why: Understanding how to create equivalent fractions is helpful for conceptualizing multiplication and division, though not strictly required for the standard algorithms.

Key Vocabulary

Numerator	The top number in a fraction, representing the number of parts being considered.
Denominator	The bottom number in a fraction, representing the total number of equal parts in a whole.
Reciprocal	A number that, when multiplied by a given number, results in 1. For a fraction, it is found by flipping the numerator and denominator.
Proper Fraction	A fraction where the numerator is smaller than the denominator, representing a value less than one whole.

Watch Out for These Misconceptions

Common Misconception

Multiplying fractions always makes a larger number, like whole numbers.

What to Teach Instead

Visual area models show the product as a fraction of each factor, explaining the smaller result. Pair sharing of models prompts students to compare and revise ideas, building correct intuition through evidence.

Common Misconception

Dividing fractions means dividing numerators by denominators separately.

What to Teach Instead

The reciprocal method clarifies division as 'how many fit into'; hands-on partitioning with fraction bars lets students see and correct this. Group verification reinforces the 'keep, change, flip' steps.

Common Misconception

Cancel terms before multiplying without reason.

What to Teach Instead

Discuss equivalence first; collaborative equation building shows valid cancellation preserves value. Active manipulation of tiles helps students internalize rules over blind procedures.

Active Learning Ideas

Area Model: Fraction Multiplication

Provide grid paper; students draw rectangles for each fraction, shade the first fully then overlay the second's fraction to show product. Pairs compare results and explain size reduction. Extend to multiply three fractions.

35 min·Pairs

Keep Change Flip Stations

Set up stations with division problems on cards; students solve using reciprocal method, record steps, and rotate. Small groups verify each other's work with visual checks like number lines. Include word problems at final station.

45 min·Small Groups

Recipe Scaling Relay

Teams get recipe cards with fractions; one student multiplies or divides an ingredient then tags next teammate. Whole class debriefs errors and real-world accuracy. Adjust fractions for challenge.

30 min·Small Groups

Fraction Wall Builder

Students cut and assemble fraction strips to model multiplication as combining lengths and division as splitting. Individuals build then pair to test 'keep change flip' on walls. Share findings class-wide.

40 min·Pairs

Real-World Connections

• Bakers frequently multiply fractions when scaling recipes. For example, if a recipe calls for 2/3 cup of flour and they need to make 1/2 of the batch, they calculate (1/2) * (2/3) to find the new amount of flour needed.
• Home improvement projects often involve dividing lengths or areas by fractions. A carpenter might need to cut a 5/6 meter plank into sections that are each 1/3 meter long, requiring the calculation (5/6) ÷ (1/3) to determine how many sections can be made.

Assessment Ideas

Quick Check

Present students with the problem: 'A recipe requires 3/4 cup of sugar. You only want to make 1/3 of the recipe. How much sugar do you need?' Ask students to write down their calculation and the final answer.

Discussion Prompt

Pose the question: 'Imagine you have 5/8 of a chocolate bar and you want to share it equally among 3 friends. How much of the original chocolate bar does each friend receive?' Ask students to explain their method for solving this division problem.

Exit Ticket

Give each student a card with two fractions, e.g., 2/5 and 3/4. Ask them to perform one multiplication and one division calculation using these fractions. They should show their working for both.

Frequently Asked Questions

Why does multiplying two proper fractions give a smaller product?

Each fraction represents part of a whole, so the product is part of a part, visually smaller via area or length models. Students grasp this by shading rectangles: for 2/3 x 3/4, shade 2/3 then 3/4 of that shaded area. Practice with varied examples solidifies the rule across contexts like scaling down areas.

How to teach the keep change flip method for dividing fractions?

Introduce as multiplying by reciprocal: keep first fraction, change division to multiplication, flip second. Use visuals like partitioning circles to show 3/4 divided by 1/2 equals 1.5. Scaffold with steps on cards, then fade for independence; real problems like sharing costs build fluency.

What real-world problems use multiplying and dividing fractions?

Examples include scaling recipes (multiply ingredients by 2/3), dividing pizzas (2/3 of pie into 1/4 slices), or speeds (3/5 hour for 2/3 distance). Students design their own, like mixing paint colours or sharing travel costs, connecting maths to daily life and deepening proportional understanding.

How can active learning help teach multiplying and dividing fractions?

Active approaches like fraction strips or relay games make abstract operations tangible: students physically combine for multiplication and partition for division. Collaborative stations encourage explaining 'why' during peer checks, addressing misconceptions instantly. These methods outperform worksheets by engaging multiple senses, improving retention by 30-50% in similar studies, and fostering problem-solving confidence.