Multiplying and dividing fractions
🎯 In this topic you will
- Multiply a unit fraction by a whole number
- Divide a unit fraction by a whole number
🧠 Key Words
- repeated addition
- unit fraction
Show Definitions
- repeated addition: Adding the same number multiple times; a way to understand multiplication.
- unit fraction: A fraction with a numerator of 1, representing one part of a whole divided into equal pieces.
Understanding the Pictures
L ook at the pictures. They show how multiplying a unit fraction by a whole number can make a complete whole.
Sharing a Fraction
I magine you have one half of an apple and you share it with four friends. This helps you think about dividing a unit fraction among several people.
Using Calculations
T his idea helps you write a calculation to show how much of the fruit each person gets when a unit fraction is shared.
Learning the Goal
I n this section you will learn how to multiply and divide unit fractions, using simple examples to build understanding.
❓ EXERCISES
1. Calculate $\dfrac{1}{4} \times 3$. Draw a diagram to show your answer.
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2. Calculate $\dfrac{1}{5} \times 6$. Draw a diagram to show your answer.
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3. Amy, Kiki and Magda work out $\dfrac{1}{6} \times 4$. Here are their methods.
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4. Draw a diagram to help you calculate $\dfrac{1}{3} \times 4$.
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5. Arun multiplies a unit fraction by a whole number. He writes:
$\dfrac{1}{5} \times 5 = \dfrac{5}{25}$
Explain what Arun has done wrong.
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6. Draw diagrams to help you calculate
a. $\dfrac{1}{5} \div 2$
b. $\dfrac{1}{6} \div 3$
c. $\dfrac{1}{4} \div 5$
Check your answers with your partner.
👀 Show answer
a. $\dfrac{1}{5} \div 2 = \dfrac{1}{10}$
b. $\dfrac{1}{6} \div 3 = \dfrac{1}{18}$
c. $\dfrac{1}{4} \div 5 = \dfrac{1}{20}$
7. Zara has $\dfrac{1}{3}$ of a bottle of fruit juice. She divides it equally between two glasses. What fraction of the bottle is in each glass?
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🧠 Think like a Mathematician
Look at these pairs of calculations.
$\dfrac{1}{2} \times 7 = \dfrac{7}{2}$ $7 \div 2 = \dfrac{7}{2}$
$\dfrac{1}{2} \times 6 = \dfrac{6}{2}$ $6 \div 2 = \dfrac{6}{2}$
$\dfrac{1}{2} \times 5 = \dfrac{5}{2}$ $5 \div 2 = \dfrac{5}{2}$
Write the next three rows of the pattern.
What do you notice about multiplying by $\dfrac{1}{2}$ and dividing by 2?
What happens if you multiply by $\dfrac{1}{3}$ and divide by 3?
You will show that you are generalising when you explain the pattern and find examples that satisfy the pattern.
You will show that you are convincing when you explain the relationship between multiplying by $\dfrac{1}{3}$ and dividing by 3.
Show Answers
- Next three rows of the pattern:
$\dfrac{1}{2} \times 4 = \dfrac{4}{2}$ $4 \div 2 = \dfrac{4}{2}$
$\dfrac{1}{2} \times 3 = \dfrac{3}{2}$ $3 \div 2 = \dfrac{3}{2}$
$\dfrac{1}{2} \times 2 = \dfrac{2}{2}$ $2 \div 2 = \dfrac{2}{2}$ - What do you notice?
Multiplying any number by $\dfrac{1}{2}$ gives exactly the same result as dividing that number by 2. - What happens with $\dfrac{1}{3}$ and dividing by 3?
Multiplying a number by $\dfrac{1}{3}$ gives the same result as dividing the number by 3. In both cases you are finding one third of the number. - Generalising the pattern:
Multiplying by the unit fraction $\dfrac{1}{n}$ is equivalent to dividing by $n$.