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Understanding Fractions
Numerators, denominators, proper and improper fractions, mixed numbers, and finding fractions of quantities.
You and three friends order a pizza. You eat half, your friend eats a quarter, and someone claims they ate three-eighths. Who ate the most? Fractions settle the argument.
Think First
Before we begin, have a go at this:
If you cut a pizza into 8 equal slices and eat 3, what fraction did you eat? What fraction is left?
Learning Intentions
I Can...
- Identify numerators and denominators
- Recognise proper, improper fractions and mixed numbers
- Find fractions of quantities
Key Terms
Core Content
Formula
Types of Fractions
The numerator is smaller than the denominator. The value is less than 1.
Examples: $ \frac{1}{2}, \frac{3}{4}, \frac{7}{10} $
The numerator is larger than or equal to the denominator. The value is 1 or more.
Examples: $ \frac{5}{4}, \frac{7}{7}, \frac{11}{6} $
A whole number combined with a proper fraction.
Examples: $ 2\frac{1}{3}, 5\frac{3}{4} $
To find a fraction of a number, divide by the denominator, then multiply by the numerator.
$ \frac{2}{5} $ of $ 30 = (30 \div 5) \times 2 = 6 \times 2 = 12 $
What is a Fraction?
A fraction represents a part of a whole. It has two numbers separated by a line:
$ \quad \quad \quad \longleftarrow \quad $ The bottom number is the denominator (how many equal parts the whole is divided into)
In our pizza example, eating 3 out of 8 slices means you ate $ \frac{3}{8} $ of the pizza. The fraction left is $ \frac{5}{8} $.
$ \frac{3}{8} $ eaten
$ \frac{5}{8} $ left
Worked Examples
What fraction of a pizza is left if you eat 5 out of 8 slices?
Answer: $ 8 - 5 = 3 $ slices left. The fraction left is $ \frac{3}{8} $.
Find $ \frac{2}{5} $ of $ 30 $.
Answer: Divide 30 by 5 to get 6. Then multiply by 2: $ 6 \times 2 = 12 $. So $ \frac{2}{5} $ of $ 30 = 12 $.
Convert $ \frac{17}{5} $ to a mixed number.
Answer: $ 17 \div 5 = 3 $ remainder $ 2 $. So $ \frac{17}{5} = 3\frac{2}{5} $.
Convert $ 4\frac{1}{3} $ to an improper fraction.
Answer: $ 4 \times 3 + 1 = 13 $. So $ 4\frac{1}{3} = \frac{13}{3} $.
Practice Questions
Watch Out For These
Common Mistakes
Mistake: "The larger the denominator, the larger the fraction."
Correct: The larger the denominator, the smaller each part. $ \frac{1}{8} $ is smaller than $ \frac{1}{4} $ because eighths are smaller pieces than quarters.
Mistake: Thinking $ \frac{3}{3} $ is less than 1, or that $ \frac{5}{4} $ is impossible.
Correct: $ \frac{3}{3} = 1 $ (a whole). $ \frac{5}{4} = 1\frac{1}{4} $ (more than a whole). Improper fractions are perfectly valid.
Your Turn
Your Turn, Fraction Explorer
Part A: Write each situation as a fraction.
- 2 out of 7 days
- 5 out of 12 months
- A whole pizza cut into 6 slices, all eaten
Part B: Find the fraction of each quantity.
- $ \frac{3}{4} $ of $ 24 $
- $ \frac{2}{5} $ of $ 40 $
- $ \frac{5}{6} $ of $ 18 $
Part C: Classify each as proper, improper, or mixed.
- $ \frac{7}{8} $
- $ \frac{9}{4} $
- $ 2\frac{1}{3} $
- $ \frac{11}{11} $
Part A:
1. 2/7
2. 5/12
3. 6/6 = 1
Part B:
1. 18
2. 16
3. 15
Part C:
1. Proper
2. Improper
3. Mixed
4. Improper (equals 1)
Revisit Your Thinking
Look back at your Think First answer. How close was your first idea to what you learned? What changed?
Quick Summary
Copy these points into your notes:
- The main idea from this lesson is...
- The key formula or rule is...
- A common mistake to avoid is...
- I can check my answer by...
Interactive: Fraction Area Model
Use the sliders to build fractions and see the area model update. Find equivalent fractions and simplify!
Multiple Choice
Test your understanding with multiple choice questions.
Short Answer
Show Your Working
1. Shade 3/4 of a rectangle divided into 8 equal parts. How many parts did you shade?
2. Convert 17/4 into a mixed number and explain what it means.
3. A pizza is cut into 8 slices. You eat 3/8 and your friend eats 1/4. Who ate more? Show your working.
Model Answers
+Short Answer 1 (2 marks)
Model answer:
$\frac{3}{4}=\frac{6}{8}$, so shade 6 of the 8 equal parts. [1 mark for an equivalent fraction or calculation; 1 mark for 6 correctly shaded parts.]
Short Answer 2 (3 marks)
Model answer:
$17 \div 4=4$ remainder $1$, so $\frac{17}{4}=4\frac{1}{4}$. It represents four whole units and one quarter of another unit. [1 mark for the division; 1 mark for $4\frac{1}{4}$; 1 mark for the explanation.]
Short Answer 3 (4 marks)
Model answer:
Convert to eighths: $\frac{1}{4}=\frac{2}{8}$. Since $\frac{3}{8}>\frac{2}{8}$, you ate more, by $\frac{1}{8}$ of the pizza. [2 marks for the conversion and comparison; 1 mark for the conclusion; 1 mark for the difference.]