Fractions and equivalence

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Lesson notes

What is a Fraction?

A fraction shows a part of a whole or a group. For example, \frac{3}{4} means 3 parts out of 4 equal parts.
The numerator is the top number. It tells how many equal parts you have. In \frac{3}{4}, the numerator is 3.
The denominator is the bottom number. It tells how many equal parts make a whole. In \frac{3}{4}, the denominator is 4.
The line between them is called the fraction bar. It means 'divide' or 'out of'.
A fraction can also mean division. \frac{3}{4} is the same as 3 ÷ 4.

Equivalent Fractions

Equivalent fractions are different fractions that name the same amount. For example, \frac{1}{2} and \frac{2}{4} are equivalent.
You can make an equivalent fraction by multiplying the numerator and denominator by the same number. For example, \frac{1}{2} \times \frac{2}{2} = \frac{2}{4}.
You can also make an equivalent fraction by dividing the numerator and denominator by the same number. For example, \frac{4}{8} \div \frac{4}{4} = \frac{1}{2}.
When you multiply or divide the top and bottom by the same number, the fraction's value does not change.
Use fraction strips or pie charts to see that \frac{1}{2}, \frac{2}{4}, and \frac{3}{6} all cover the same area.

Simplifying Fractions

Simplifying a fraction means making it as simple as possible. You divide the numerator and denominator by their greatest common factor (the biggest number that divides both evenly).
For example, to simplify \frac{6}{8}, divide both by 2: \frac{6}{8} \div \frac{2}{2} = \frac{3}{4}.
A fraction is in simplest form when the only common factor of the numerator and denominator is 1.
Simplifying is also called reducing a fraction.
Always check if you can simplify your answer after adding or subtracting fractions.

Comparing Fractions

When two fractions have the same denominator, the one with the bigger numerator is larger. For example, \frac{3}{5} > \frac{2}{5}.
When fractions have different denominators, you can compare them by finding equivalent fractions with a common denominator.
For example, to compare \frac{1}{2} and \frac{3}{8}, change \frac{1}{2} to \frac{4}{8}. Now \frac{4}{8} > \frac{3}{8}, so \frac{1}{2} > \frac{3}{8}.
You can also compare fractions by using cross multiplication. For \frac{a}{b} and \frac{c}{d}, compare a \times d and c \times b.
Use fraction bars or number lines to help you see which fraction is bigger.

Adding and Subtracting Fractions with the Same Denominator

To add fractions with the same denominator, just add the numerators and keep the denominator the same. For example, \frac{2}{7} + \frac{3}{7} = \frac{5}{7}.
To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same. For example, \frac{5}{8} - \frac{2}{8} = \frac{3}{8}.
The denominator does not change when you add or subtract fractions with the same denominator.
Always check if your answer can be simplified. For example, \frac{2}{4} + \frac{1}{4} = \frac{3}{4} (already simple), but \frac{1}{4} + \frac{1}{4} = \frac{2}{4} which simplifies to \frac{1}{2}.
When adding or subtracting mixed numbers, you can add the whole parts and the fractional parts separately, or convert to improper fractions first.

Improper Fractions and Mixed Numbers

An improper fraction has a numerator that is bigger than or equal to the denominator. For example, \frac{7}{4} is improper.
A mixed number has a whole number and a proper fraction. For example, 1\frac{3}{4} is a mixed number.
To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, and the remainder becomes the numerator of the fraction part. For example, \frac{7}{4} = 1\frac{3}{4}.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and put that over the original denominator. For example, 1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}.
Improper fractions and mixed numbers both represent the same amount. You can choose which one to use.

Finding a Fraction of a Quantity

To find a fraction of a quantity, divide the quantity by the denominator, then multiply by the numerator.
For example, to find \frac{3}{4} of 20: first divide 20 by 4 to get 5, then multiply 5 by 3 to get 15. So \frac{3}{4} of 20 is 15.
You can also multiply the quantity by the fraction: \frac{3}{4} \times 20 = 15.
This works for any quantity, like money, length, or number of objects.
Practice with real-life examples: 'What is \frac{2}{3} of 12 apples?' Answer: 8 apples.

Fractions on a Number Line

You can show fractions on a number line by dividing the space between 0 and 1 into equal parts.
For example, to show \frac{3}{4}, divide the line from 0 to 1 into 4 equal parts and count 3 parts from 0.
Equivalent fractions appear at the same point on the number line. For example, \frac{1}{2} and \frac{2}{4} are at the same spot.
Number lines help you compare fractions and see which is larger or smaller.
Improper fractions are placed beyond 1 on the number line. For example, \frac{5}{4} is between 1 and 2.

Key Vocabulary

Fraction: a number that shows part of a whole.
Numerator: the top number in a fraction; it tells how many parts you have.
Denominator: the bottom number in a fraction; it tells how many equal parts make a whole.
Equivalent fractions: fractions that name the same amount.
Simplify: to make a fraction as simple as possible by dividing the top and bottom by the same number.
Improper fraction: a fraction where the numerator is greater than or equal to the denominator.
Mixed number: a number made of a whole number and a proper fraction.

Slides

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Practice questions

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1.In the fraction 3/4, what does the denominator 4 tell us?
Easy
- AHow many equal parts make up a whole
- BHow many parts are being described
- CThe total number of parts altogether
- DThe number of wholes
2.Which fraction is equivalent to 1/2?
Easy
- A2/3
- B2/4
- C3/4
- D1/3
3.The fraction 6/8 is equivalent to 3/4.
Easy
True or false?
4.Which of these fractions is the largest?
Medium
- A3/4
- B2/3
- C5/8
- D1/2
5.Which of the following are equivalent to 2/3? (Select all that apply)
Medium
- A4/6
- B6/9
- C3/4
- D8/12
- E5/7
6.Order these fractions from smallest to largest.
Medium
- 1/2
- 3/4
- 2/3
- 5/8
7.Match each fraction to its equivalent fraction.
Medium
- 1/2
- 2/3
- 3/4
- 6/8
- 4/6
- 3/6
8.A pizza is cut into 8 equal slices. Tom eats 3 slices and Jerry eats 2 slices. What fraction of the pizza is left?
Medium
- A3/8
- B5/8
- C2/8
- D1/8

Revision notes

## Fractions and Equivalence A **fraction** shows a part of a whole. It has two numbers: - **Numerator** (top number): how many parts you have. - **Denominator** (bottom number): how many equal parts make a whole. For example, in \(\frac{3}{4}\), the numerator is 3 and the denominator is 4. It means 3 parts out of 4 equal parts. ### Equivalent Fractions Equivalent fractions are different fractions that name the same amount. You can make them by **multiplying** or **dividing** the top and bottom by the same number (not zero). Example: \(\frac{1}{2} = \frac{2}{4} = \frac{4}{8}\) because you multiply top and bottom by 2. ### Comparing Fractions - If fractions have the **same denominator**, the one with the bigger numerator is bigger. - If denominators are different, change them to equivalent fractions with a common denominator, then compare. ### Adding and Subtracting Fractions Only add or subtract the numerators when the denominators are the same. Keep the denominator the same. Example: \(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\) ### Improper Fractions and Mixed Numbers - **Improper fraction**: numerator is bigger than or equal to the denominator, like \(\frac{7}{4}\). - **Mixed number**: a whole number and a fraction, like \(1\frac{3}{4}\). To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same. Example: \(\frac{7}{4} = 1\frac{3}{4}\) because 7 ÷ 4 = 1 remainder 3. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and put that over the denominator. Example: \(1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}\) ### Finding a Fraction of a Quantity To find a fraction of a number, divide by the denominator and multiply by the numerator. Example: Find \(\frac{2}{3}\) of 12. First, 12 ÷ 3 = 4, then 4 × 2 = 8. So \(\frac{2}{3}\) of 12 is 8.

fraction: A number that shows a part of a whole. It has a numerator and a denominator.
numerator: The top number in a fraction. It tells how many equal parts you have.
denominator: The bottom number in a fraction. It tells how many equal parts make a whole.
equivalent fractions: Fractions that name the same amount, like 1/2 and 2/4.
improper fraction: A fraction where the numerator is bigger than or equal to the denominator, like 7/4.
mixed number: A number made of a whole number and a fraction, like 1 3/4.

Flashcards

What does the numerator in a fraction tell you?

The numerator is the top number. It tells you how many equal parts you have.

What does the denominator in a fraction tell you?

The denominator is the bottom number. It tells you how many equal parts make up one whole.

In the fraction 3/4, what does the 3 mean?

The 3 is the numerator. It means you have 3 equal parts.

In the fraction 3/4, what does the 4 mean?

The 4 is the denominator. It means 4 equal parts make one whole.

What is an equivalent fraction?

Equivalent fractions are different fractions that name the same amount. For example, 1/2 and 2/4 are equivalent.

How can you make an equivalent fraction?

Multiply or divide the top and bottom numbers by the same number. For example, 1/2 = 2/4 because you multiply top and bottom by 2.

What is 1/2 equivalent to?

1/2 is equivalent to 2/4, 3/6, 4/8, and so on.

What is 2/3 equivalent to?

2/3 is equivalent to 4/6, 6/9, 8/12, and so on.

How do you compare fractions with the same denominator?

Look at the numerators. The fraction with the bigger numerator is larger. For example, 3/5 is bigger than 2/5.

Which is bigger: 3/4 or 1/4?

3/4 is bigger because 3 is more than 1.

How do you compare fractions like 1/2 and 3/8?

Change them to have the same denominator. 1/2 = 4/8, so 1/2 is bigger than 3/8.

What is an improper fraction?

An improper fraction has a numerator that is bigger than or equal to the denominator. For example, 5/3 is improper.

What is a mixed number?

A mixed number has a whole number and a fraction together. For example, 1 1/2 is a mixed number.

How do you change 5/3 to a mixed number?

Divide 5 by 3. You get 1 with 2 left over. So 5/3 = 1 2/3.

How do you change 2 1/4 to an improper fraction?

Multiply the whole number by the denominator: 2 × 4 = 8. Add the numerator: 8 + 1 = 9. Keep the denominator: 9/4.

How do you add fractions with the same denominator?

Add the numerators and keep the denominator the same. For example, 1/5 + 2/5 = 3/5.

What is 2/7 + 3/7?

2/7 + 3/7 = 5/7.

How do you subtract fractions with the same denominator?

Subtract the numerators and keep the denominator the same. For example, 4/6 - 1/6 = 3/6.

What is 5/8 - 2/8?

5/8 - 2/8 = 3/8.

How do you find 1/4 of 20?

Divide 20 by the denominator: 20 ÷ 4 = 5. So 1/4 of 20 is 5.

What is 2/3 of 12?

First find 1/3 of 12: 12 ÷ 3 = 4. Then multiply by 2: 4 × 2 = 8. So 2/3 of 12 is 8.

What does the fraction bar mean?

The fraction bar means division. So 3/4 means 3 ÷ 4.

What is a rational number?

A rational number is a number that can be written as a fraction a/b, where a and b are whole numbers and b is not zero.

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