UK K12 GCSE/A-Level: Year 5 KS2 Mathematics - Fractions, Multiplying and Dividing

Learning Objectives

By the end of this topic, students will be able to:

• Understand the concept of multiplying and dividing fractions
• Multiply and divide fractions using various methods (e.g., equivalent fractions, diagrams)
• Apply multiplication and division of fractions to solve problems in real-world contexts
• Identify and explain the relationship between multiplication and division of fractions and whole numbers
• Recognize and correct common misconceptions related to multiplying and dividing fractions

Core Concepts

What are Fractions?

A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). The denominator tells us how many equal parts the whole is divided into, and the numerator tells us how many of those parts we have.

Multiplying Fractions

When we multiply fractions, we are essentially finding a part of a part. To multiply fractions, we multiply the numerators together and the denominators together:

Example: 1/2 × 3/4

To solve this problem, we multiply the numerators (1 × 3) and the denominators (2 × 4):

1 × 3 = 3 2 × 4 = 8

So, 1/2 × 3/4 = 3/8

Dividing Fractions

When we divide fractions, we are essentially finding how many times one part fits into another part. To divide fractions, we invert the second fraction (i.e., flip the numerator and denominator) and then multiply:

Example: 1/2 ÷ 3/4

To solve this problem, we invert the second fraction (3/4 becomes 4/3) and then multiply:

1 × 4 = 4 2 × 3 = 6

So, 1/2 ÷ 3/4 = 4/6, which can be simplified to 2/3

Equivalent Fractions

Equivalent fractions are fractions that represent the same value, but with different numerators and denominators. We can use equivalent fractions to help us multiply and divide fractions more easily.

Example: 1/2 and 2/4 are equivalent fractions because they represent the same value (half of a whole)

Worked Examples

Example 1: Multiplying Fractions

Tom has 1/4 of a pizza and his friend has 3/4 of a pizza. If they combine their pizzas, what fraction of a pizza do they have together?

To solve this problem, we multiply the fractions:

1/4 × 3/4 = 3/16

Example 2: Dividing Fractions

A recipe calls for 1/2 cup of flour to make 2 cakes. If you want to make 4 cakes, how much flour will you need?

To solve this problem, we divide the fraction:

1/2 ÷ 2 = 1/4

So, you will need 1/4 cup of flour to make 4 cakes.

Example 3: Real-World Application

A water tank can hold 3/4 of a tank of water. If 1/2 of the tank is already filled, what fraction of the tank is still empty?

To solve this problem, we subtract the filled fraction from the total fraction:

3/4 - 1/2 = 1/4

So, 1/4 of the tank is still empty.

Common Misconceptions

• Many students think that multiplying fractions is the same as adding fractions. However, multiplying fractions involves finding a part of a part, whereas adding fractions involves combining two or more parts.
• Some students think that dividing fractions is the same as multiplying fractions. However, dividing fractions involves finding how many times one part fits into another part, whereas multiplying fractions involves finding a part of a part.
• Students may also struggle with equivalent fractions, thinking that they are different values rather than the same value with different numerators and denominators.

Exam Tips

• When multiplying fractions, make sure to multiply the numerators together and the denominators together.
• When dividing fractions, invert the second fraction and then multiply.
• Use equivalent fractions to help you simplify your answers.
• Make sure to read the question carefully and understand what is being asked.

MCQs with Explanations

MCQ 1: [F]

What is the result of multiplying 1/2 and 3/4?

A) 3/8 B) 1/2 C) 3/4 D) 2/3

Correct answer: A) 3/8 Why the distractors fail: * B) 1/2 is the original fraction, not the result of multiplying. * C) 3/4 is one of the original fractions, not the result of multiplying. * D) 2/3 is not the result of multiplying 1/2 and 3/4.

MCQ 2: [H]

What is the result of dividing 1/2 by 3/4?

A) 4/6 B) 2/3 C) 1/4 D) 3/2

Correct answer: B) 2/3 Why the distractors fail: * A) 4/6 is not the result of dividing 1/2 by 3/4. * C) 1/4 is not the result of dividing 1/2 by 3/4. * D) 3/2 is not the result of dividing 1/2 by 3/4.

MCQ 3: [F]

What is the result of multiplying 1/4 and 2/3?

A) 1/12 B) 2/12 C) 3/12 D) 4/12

Correct answer: A) 1/12 Why the distractors fail: * B) 2/12 is not the result of multiplying 1/4 and 2/3. * C) 3/12 is not the result of multiplying 1/4 and 2/3. * D) 4/12 is not the result of multiplying 1/4 and 2/3.

MCQ 4: [H]

What is the result of dividing 3/4 by 1/2?

A) 6/8 B) 3/4 C) 2/3 D) 4/6

Correct answer: A) 6/8 Why the distractors fail: * B) 3/4 is one of the original fractions, not the result of dividing. * C) 2/3 is not the result of dividing 3/4 by 1/2. * D) 4/6 is not the result of dividing 3/4 by 1/2.

MCQ 5: [F]

What is the result of multiplying 1/2 and 1/4?

A) 1/8 B) 2/8 C) 3/8 D) 4/8

Correct answer: A) 1/8 Why the distractors fail: * B) 2/8 is not the result of multiplying 1/2 and 1/4. * C) 3/8 is not the result of multiplying 1/2 and 1/4. * D) 4/8 is not the result of multiplying 1/2 and 1/4.

Short-Answer Questions

Question 1

Tom has 1/4 of a pizza and his friend has 3/4 of a pizza. If they combine their pizzas, what fraction of a pizza do they have together? Show your working.

Question 2

A recipe calls for 1/2 cup of flour to make 2 cakes. If you want to make 4 cakes, how much flour will you need? Show your working.

Question 3

A water tank can hold 3/4 of a tank of water. If 1/2 of the tank is already filled, what fraction of the tank is still empty? Show your working.

Question 4

What is the result of multiplying 2/3 and 3/4? Show your working.

Question 5

What is the result of dividing 1/2 by 2? Show your working.

Note: These short-answer questions are designed to assess students' understanding of the concepts and their ability to apply them to solve problems.