Topics
4. Fractions
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Stage 3

4.08 Fraction patterns

Lesson
Practice
Lesson

Are you ready?

To be able to identify patterns with fractions, it will help us if we can  add and subtract fractions  . Let's try this problem to review.

Examples

Example 1

What is \dfrac{1}{12} + \dfrac{1}{12}?

Worked Solution
Create a strategy

Use area models.

Apply the idea
Two circles with plus sign in between. Each circle is divided into 12 equal parts with 1 shaded part.

\dfrac{1}{12} means means one twelfth or 1 part out of 12 parts should be shaded in an area model.

Here is the area model of \dfrac{1}{12} + \dfrac{1}{12}.

A circle divided into 12 equal parts with 2 shaded parts.

We can see that adding the two shaded parts means we get 2 shaded parts out of 12 parts.

This can be written as the fraction \dfrac{2}{12}.

\dfrac{1}{12}+\dfrac{1}{12}= \dfrac{2}{12}

Idea summary

We can add or subtract fractions by using area models.

Patterns with tenths and hundredths

Just like we did with whole numbers, we can explore, continue and create patterns with fractions. Let's start by looking at tenths and hundredths.

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Examples

Example 2

Create a pattern by adding \dfrac{1}{10} each time.

\dfrac{3}{10}, \,\dfrac{4}{10}, \,⬚, \, ⬚, \,⬚

Worked Solution
Create a strategy

Add \dfrac{1}{10} to the last given value to complete the pattern.

Apply the idea
A rectangle divided into 10 parts and 4 parts are shaded.

This picture shows \dfrac{4}{10} blocks shaded blue. If another 1 block were shaded, 5 out of 10 would be shaded in total. So the next number is \dfrac{5}{10}.

\dfrac{4}{10} + \dfrac{1}{10} = \dfrac{5}{10}

We can see that the numerator increased by 1 and the denominator stayed the same. Similarly doing this for the following numbers we can complete the pattern:\dfrac{3}{10}, \,\dfrac{4}{10}, \,\dfrac{5}{10}, \, \dfrac{6}{10}, \,\dfrac{7}{10}

Idea summary

We can create patterns with fractions by adding or subtracting the same fraction each time.

Patterns with area models

We can also use the area models to represent fraction patterns, this video shows us how.

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Examples

Example 3

Which fraction comes next in the pattern?

A table where the first column has numbers 1 to 4. The second column has 3 triangles. Ask your teacher for more information.
A
A triangle divided into 3 parts. 1 part is shaded.
B
A triangle divided into 3 parts. 3 parts are shaded.
C
A triangle divided into 3 parts. No part is shaded.
D
A triangle divided into 3 parts. 2 parts are shaded.
Worked Solution
Create a strategy

Notice how the number of shaded parts changes each time.

Apply the idea

In the first shape, 3 out of 3 parts are shaded.

In the second shape, 2 out of 3 parts are shaded.

In the third shape, 1 out of 3 parts are shaded.

The number of shaded parts are decreasing by 1 each time. So the next shape should have \\ 1-1=0 parts shaded.

So option C is the answer.

Idea summary

Area models can be used to represent fractions in a pattern.

Create and continue fraction patterns

This video shows an example of what to look for when trying to determine the pattern in a sequence of fractions.

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Examples

Example 4

Complete the pattern:

\dfrac{1}{7},\,\dfrac{2}{7},\,\dfrac{3}{7},\,⬚,\,⬚,\,⬚

Worked Solution
Create a strategy

Find how much the numbers are increasing by each time and add that to the last given value to complete the pattern.

Apply the idea

To work out how much the numbers are increasing by each time, you can count up or find the difference between two fractions.

\displaystyle \dfrac{2}{7} - \dfrac{1}{7}\displaystyle =\displaystyle \dfrac{1}{7}Subtract the numerators

So we need to add \dfrac{1}{7} to each number to complete the pattern.

\displaystyle \dfrac{3}{7} + \dfrac{1}{7}\displaystyle =\displaystyle \dfrac{4}{7}Add \dfrac{1}{7} to \dfrac{3}{7}
\displaystyle \dfrac{4}{7} + \dfrac{1}{7}\displaystyle =\displaystyle \dfrac{5}{7}Add \dfrac{1}{7} to \dfrac{4}{7}
\displaystyle \dfrac{5}{7} + \dfrac{1}{7}\displaystyle =\displaystyle \dfrac{6}{7}Add \dfrac{1}{7} to \dfrac{5}{7}

The complete pattern is \dfrac{1}{7},\,\dfrac{2}{7},\,\dfrac{3}{7},\,\dfrac{4}{7},\,\dfrac{5}{7},\,\dfrac{6}{7}

Idea summary

So to complete a pattern with fractions, we need to:

  • Find the pattern - which will either be given or which will need to be found by looking at the list of numbers.

  • Continue the pattern - using the pattern that you found.

Outcomes

MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

MA3-8NA

analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane

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