Equivalence, comparing and ordering fractions
🎯 In this topic you will
- Recognise proper fractions as fractions that are less than one whole.
- Recognise when two or more fractions are equivalent.
- Compare and order fractions.
🧠 Key Words
- equivalent fraction
- proper fraction
Show Definitions
- equivalent fraction: Fractions that may look different but represent the same value or proportion of a whole.
- proper fraction: A fraction in which the numerator is smaller than the denominator, so the value is less than one whole.
📘 Building on What You Know
I n Stages 2 and 3, you worked with equivalent fractions for halves, quarters, fifths and tenths. In this unit, you will work with some other proper fractions.
🥧 Understanding Equivalent Fractions
F ractions such as $\dfrac{3}{6}$ and $\dfrac{1}{2}$ are equal in value. This means they represent the same fraction of a whole and are called equivalent fractions.
⚠️ Fractions and the Whole
B e careful when comparing fractions of different objects. Even if two fractions are equivalent, the actual amounts may not be the same if the wholes are different sizes.
💡 Quick Math Tip
Finding equivalent fractions: You can show that fractions are equivalent by dividing shapes into equal parts, placing fractions on a number line, or using a fraction wall to see which fractions line up at the same value.
❓ EXERCISES
1. These diagrams show equivalent fractions. Copy and complete the following:
👀 Show answer
2. Find four pairs of equivalent fractions in the table. Which fraction is not used?
| $\dfrac{8}{10}$ | $\dfrac{7}{10}$ | $\dfrac{3}{10}$ |
| $\dfrac{1}{2}$ | $\dfrac{4}{5}$ | $\dfrac{4}{10}$ |
| $\dfrac{5}{10}$ | $\dfrac{35}{50}$ | $\dfrac{30}{100}$ |
👀 Show answer
3. Which is the odd one out? Explain your answer.
$\dfrac{3}{4}\quad \dfrac{9}{12}\quad \dfrac{4}{6}$
👀 Show answer
4. Alana makes a fraction using two number cards. Alana says, “My fraction is equivalent to $\dfrac{1}{2}$.” One of the number cards is $6$. What fractions could Alana make?
👀 Show answer
5. Use the number line as a guide to help you order these fractions. Start with the smallest fraction.
$\dfrac{1}{2},\ \dfrac{3}{8},\ \dfrac{3}{4},\ \dfrac{5}{8},\ \dfrac{7}{8},\ \dfrac{1}{4}$
👀 Show answer
6. Here are three fraction cards. Use the cards to make this number sentence correct.
$\dfrac{3}{8},\ \dfrac{1}{4},\ \dfrac{5}{16}$
👀 Show answer
7. Raphael says that $\dfrac{3}{8} > \dfrac{3}{4}$ because $8 > 4$. Do you agree with him? Explain your decision.
👀 Show answer
🧠 Think like a Mathematician
Challenge: Make as many different pairs of equivalent fractions as you can using the numbers $1$ to $10$.
💡 Tip
Try using number cards. For example: $\dfrac{1}{2} = \dfrac{2}{4}$ and $\dfrac{1}{2} = \dfrac{3}{6}$.
Work systematically by starting with a simple fraction and multiplying both the numerator and denominator by the same whole number.
Examples you could find:
- $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6} = \dfrac{4}{8} = \dfrac{5}{10}$
- $\dfrac{1}{3} = \dfrac{2}{6} = \dfrac{3}{9}$
- $\dfrac{2}{5} = \dfrac{4}{10}$
- $\dfrac{3}{4} = \dfrac{6}{8}$
You are showing that you are specialising when you look for patterns and generate many correct examples based on the same idea.
Show Answer Ideas
- Equivalent fractions are formed by multiplying or dividing the numerator and denominator by the same non-zero whole number.
- Limiting numbers to $1$–$10$ means not all fractions can be extended indefinitely, so careful checking is needed.
- Organising your work by starting with unit fractions (like $\dfrac{1}{2}$ or $\dfrac{1}{3}$) helps you find more examples efficiently.