Growing rational patterns | Year 6

01 / LEARN

Understand the idea.

Measure the change

A constant-difference sequence changes by the same amount between neighbouring terms. Subtract each term from the next to check the change in more than one place. The repeated change can be a fraction or decimal as well as a whole number.

For $0.50, 0.75, 1.00, 1.25$ , every step adds $0.25$ . Writing the values with matching decimal places helps you see the repeated change. Reaching a whole number does not change the size of the step.

A constant quarter-step pattern
Term number	Value
1	0.50
2	0.75
3	1.00
4	1.25

Cross a whole without changing the rule

Fractions can continue past one whole. After three quarters, another quarter makes four quarters, or one. The next quarter makes five quarters, or $1 \frac{1}{4}$ . Improper fractions and mixed numbers can describe the same term.

A few terms alone may fit many rules. Here, use the stated constant difference and check that it reproduces every supplied step. Do not change the rule just because the notation crosses a whole number.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

A constant-difference sequence starts 1/3, 2/3, 1. Write its next two terms.

Worked solution

1
Express the whole in thirds.
$1 = \frac{3}{3}$
Using the same denominator makes the differences easy to compare.
2
Confirm the constant step.
$\frac{2}{3} - \frac{1}{3} = \frac{1}{3}, \frac{3}{3} - \frac{2}{3} = \frac{1}{3}$
Both supplied steps add one third.
3
Continue twice and rename the results.
$\begin{aligned} \frac{3}{3} + \frac{1}{3} & = \frac{4}{3} = 1 \frac{1}{3} \\ \frac{4}{3} + \frac{1}{3} & = \frac{5}{3} = 1 \frac{2}{3} \end{aligned}$
The same one-third increase gives the next two terms.

Answer

1 1/3 and 1 2/3.

EXAMPLE 02

Worked example 2

A sequence starts 2.4, 2.1, 1.8. It decreases by the same amount each time. Find the next two terms.

Worked solution

1
Find the repeated decrease.
$2.4 - 2.1 = 0.3, 2.1 - 1.8 = 0.3$
Both given steps decrease by three tenths.
2
Continue to the next term.
$1.8 - 0.3 = 1.5$
The constant decrease remains three tenths.
3
Continue once more.
$1.5 - 0.3 = 1.2$
A second equal decrease gives the next two terms, 1.5 and 1.2.

Answer

1.5 and 1.2.

03 / YOUR TURN

Try it yourself.

Check your understanding

A constant-step pattern begins 0.2, 0.6, 1.0. What comes next?

Show answer & reasoning

1.4.

$0.6 - 0.2 = 0.4$ and $1.0 - 0.6 = 0.4$ . Add the same step again: $1.0 + 0.4 = 1.4$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

Continue the constant-step pattern with two terms: 0.3, 0.6, 0.9, … .
2 marks
Continue by adding $\frac{1}{5}$ each time: $\frac{2}{5}$ , $\frac{3}{5}$ , $\frac{4}{5}$ , … .
2 marks
Find the missing value in this constant-difference pattern: 1.2, 1.6, __, 2.4.
1 mark
A pattern decreases by 0.25. Start at 2 and write the next three terms.
2 marks
A jogger increases a route by 0.5 km each week. Week 1 is 2 km. How far is week 4?
2 marks
A learner continues $\frac{1}{2}$ , $\frac{3}{4}$ , 1 with $\frac{1}{4}$ . Explain the error under the add- $\frac{1}{4}$ rule.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Continue the constant-step pattern with two terms: 0.3, 0.6, 0.9, … .
2 marks
Add 0.3 each time.
1.2, 1.5.
Continue by adding $\frac{1}{5}$ each time: $\frac{2}{5}$ , $\frac{3}{5}$ , $\frac{4}{5}$ , … .
2 marks
The next fractions are $\frac{5}{5}$ and $\frac{6}{5}$ .
1, $1 \frac{1}{5}$ .
Find the missing value in this constant-difference pattern: 1.2, 1.6, __, 2.4.
1 mark
Add 0.4 at each step.
2.0.
A pattern decreases by 0.25. Start at 2 and write the next three terms.
2 marks
Subtract one quarter at each step.
1.75, 1.50, 1.25.
A jogger increases a route by 0.5 km each week. Week 1 is 2 km. How far is week 4?
2 marks
The weekly distances are 2, 2.5, 3 and 3.5 km.
3.5 km.
A learner continues $\frac{1}{2}$ , $\frac{3}{4}$ , 1 with $\frac{1}{4}$ . Explain the error under the add- $\frac{1}{4}$ rule.
2 marks
The rule keeps adding $\frac{1}{4}$ ; it does not restart after 1.
The next term should be $1 \frac{1}{4}$ .

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Growing rational patterns; rules; brackets/order of operations; unknowns; algorithms.

Official curriculum source ↗

In this lesson

Understand the idea.

Measure the change

Cross a whole without changing the rule

See the method in action.

Worked example 1

Express the whole in thirds.

Confirm the constant step.

Continue twice and rename the results.

Worked example 2

Find the repeated decrease.

Continue to the next term.

Continue once more.

Try it yourself.

Check your understanding

Keep practising