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Home/ Mathematics/ Year 7/ Unit 1/ Lesson 6

Year 7 Maths Unit 1 Lesson 6 ~25 min

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.

Choose how you work, type your answers below or write in your book.

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

NumeratorThe top number of a fraction, how many parts you have.

DenominatorThe bottom number of a fraction, how many equal parts the whole is divided into.

Mixed numberA number made of a whole number and a fraction together.

Core Content

Formula

Types of Fractions

Proper Fraction

The numerator is smaller than the denominator. The value is less than 1.

Examples: $ \frac{1}{2}, \frac{3}{4}, \frac{7}{10} $

Improper Fraction

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} $

Mixed Number

A whole number combined with a proper fraction.

Examples: $ 2\frac{1}{3}, 5\frac{3}{4} $

Fraction of a Quantity

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:

$ \frac{3}{8} \quad \longleftarrow \quad $ The top number is the numerator (how many parts we have)
$ \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

Example 1

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} $.

Example 2

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 $.

Example 3

Convert $ \frac{17}{5} $ to a mixed number.

Answer: $ 17 \div 5 = 3 $ remainder $ 2 $. So $ \frac{17}{5} = 3\frac{2}{5} $.

Example 4

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} $

Answers:

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 questions from the lesson bank.

Short Answer

Show Your Working

Easy2 marks

1. Shade 3/4 of a rectangle divided into 8 equal parts. How many parts did you shade?

Medium3 marks

2. Convert 17/4 into a mixed number and explain what it means.

Hard4 marks

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.]

Mark lesson as complete

Tick when you have finished all activities and checked your answers.

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