Topics
5. Fractions
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AustraliaNSW
Stage 3

5.09 Add and subtract fractions with models

Lesson
Practice
Lesson

Are you ready?

Let's review how to  identify a fraction  using a model.

Examples

Example 1

What fraction of the square is shaded blue?

A square divided into 16 equal parts with 9 shaded parts.
Worked Solution
Create a strategy

Write the fraction as: \,\, \dfrac{\text{Number of shaded parts}}{\text{Total number of parts}}.

Apply the idea

There are 9 squares shaded blue and 16 squares in total.

So, the fraction shaded blue is \,\dfrac{9}{16}.

Idea summary

When a fraction is represented in an area model:

  • To find the numerator of the fraction, count the number of parts shaded.

  • To find the denominator, count the total number of parts.

Adding fractions using area models

This video shows how to add fractions using an area model.

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Examples

Example 2

In the image below, \dfrac{1}{2} has been shaded red and \dfrac{1}{4} has been shaded blue.

A rectangle divided into 4 equal squares where 2 squares are shaded red and 1 is shaded blue.
a

Write the addition that describes the image.

Worked Solution
Create a strategy

Add the two fractions to find the total fraction shaded.

Apply the idea

\text{Red}+\text{Blue}=\dfrac{1}{2}+\dfrac{1}{4}

b

What is the total fraction shaded?

Worked Solution
Create a strategy

Count how many parts are shaded and how many parts there are in total.

Apply the idea

There are 2 red squares shaded and 1 blue square shaded.

\displaystyle \text{Total squares shaded}\displaystyle =\displaystyle 2+1Add the shaded squares
\displaystyle =\displaystyle 3

The total number of squares in the image is 4.

So the total fraction shaded is \dfrac{3}{4}.

Idea summary

Area models can help us see the parts of fractions that are being added or subtracted.

Subtract fractions using area models

This video shows how to subtract fractions using an area model.

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Examples

Example 3

The image below shows \dfrac{1}{3} of the rectangle shaded.

This image shows a rectangle divided into 15 equal squares with 5 shaded squares.
a

What is the equivalent fraction in fifteenths?

\dfrac{1}{3}=\dfrac{⬚}{15}

Worked Solution
Create a strategy

To find the numerator of the fraction, count the number of parts shaded.

Apply the idea

There are 15 squares in total and 5 of them are shaded.

So the images also shows \dfrac{5}{15} of a rectangle shaded.

\dfrac{1}{3}=\dfrac{5}{15}

b

We now want to take away \dfrac{4}{15}.

This image shows a rectangle divided into 15 equal squares with 5 shaded squares. Ask your teacher for more information.

What is the answer to \dfrac{1}{3}-\dfrac{4}{15}?

Worked Solution
Create a strategy

Use the equivalent fraction we found in part (a).

Apply the idea
\displaystyle \dfrac{1}{3}-\dfrac{4}{15}\displaystyle =\displaystyle \dfrac{5}{15}-\dfrac{4}{15}Replace \dfrac{1}{3} with \dfrac{5}{15}
\displaystyle =\displaystyle \dfrac{5-4}{15}Subtract the numerators
\displaystyle =\displaystyle \dfrac{1}{15}
Reflect and check

We can also count the number of shaded squared left after 4 are taken away.

In the image, 4 squares are shaded lighter to suggest that they are being taken away. Then we would only have 1 shaded square left out of 15.

So again: \dfrac{1}{3}-\dfrac{4}{15}=\dfrac{1}{15}

Idea summary

Area models can help us see the parts of fractions that are being added or subtracted.

Add and subtract fractions on number lines

We can also use number lines to add and subtract fractions.

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Examples

Example 4

We want to calculate the sum \dfrac{1}{2}+\dfrac{3}{8}.

a

Complete the statement to make an equivalent fraction for \dfrac{1}{2}.

\dfrac{1}{2}=\dfrac{⬚}{8}

Worked Solution
Create a strategy

Use this a fraction wall showing halves and eighths.

A fraction wall showing a whole, halves and eighths. Ask your teacher for more information.
Apply the idea

By looking on the fraction wall, we can see that there are 4 eighths in 1 half.

So the complete statement is:

\dfrac{1}{2}=\dfrac{4}{8}

b

Plot \dfrac{4}{8} on the number line.

A number line from 0 to 1 with each tick going up by one eighth.
Worked Solution
Apply the idea

To plot \dfrac{4}{8}, we want to start at zero and jump right four spaces. So we have:

A number line from 0 to 1 with each tick going up by one eighth. A point is plotted on four eighths.
c

Choose the image below that represents the addition \dfrac{1}{2}+\dfrac{3}{8}.

A
A number line from 0 to 1 with each tick going up by one eighth. Ask the teacher for more information.
B
A number line from 0 to 1 with each tick going up by one eighth. Ask the teacher for more information.
C
A number line from 0 to 1 with each tick going up by one eighth. Ask the teacher for more information.
D
A number line from 0 to 1 with each tick going up by one eighth. Ask the teacher for more information.
Worked Solution
Create a strategy

Use the equivalent fraction found in part (a).

Remember when adding an amount on a number line we move to the right.

Apply the idea

We have found that \dfrac{1}{2}+\dfrac{3}{8} is the same as \dfrac{4}{8}+\dfrac{3}{8}.

So we should start at \dfrac{4}{8} then jump right 3 eighths.

The correct answer is option B.

Idea summary

We can use area models to change fractions to their equivalent fraction so the denominators are the same.

We can add and subtract fractions by moving right and left on the number line.

Outcomes

MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

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