The idea, step by step
Measure the change
A sequence is an ordered list. A constant-difference sequence changes by the same amount each time. The change can be a whole number, a fraction or a decimal. Compare more than one pair of neighbouring terms before deciding on a rule.
For 0.5, 0.75, 1, 1.25, every step adds 0.25. The notation changes when 1 is reached, but the size of the step does not. Write 1 as 1.00 if that helps you line up the place values.
| Term number | Value |
|---|---|
| 1 | 0.50 |
| 2 | 0.75 |
| 3 | 1.00 |
| 4 | 1.25 |
Cross a whole without changing the rule
A fraction pattern can continue beyond one whole. Adding one quarter to three quarters gives four quarters, or 1. Adding another quarter gives five quarters, or 1 and one quarter.
A short list can fit many rules. In this lesson the rule is stated to have a constant difference, or you are asked to describe one reasonable continuation. Check that your proposed rule reproduces every supplied term.
A common trap
A fractional pattern must restart at zero when it reaches one whole.
Instead: The rule continues across a whole. For example, 3/4, 1, 1 1/4 still increases by 1/4 each time.
See the thinking
Worked example 1
A constant-difference sequence starts 1/3, 2/3, 1. Write its next two terms.
- Each step adds 1/3. Rewrite 1 as 3/3.
- The next terms are 4/3 and 5/3.
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.
- 2.4 to 2.1 decreases by 0.3, and 2.1 to 1.8 does too.
- Subtract 0.3 twice: 1.8−0.3=1.5, then 1.5−0.3=1.2.
Check your understanding
A constant-step pattern begins 0.2, 0.6, 1.0. What comes next?
Show answer
1.4.
The increase is 0.4 each time.
Your turn
6 questions · 11 marks · Work at your own pace.
Read the questions online
- Continue the constant-step pattern with two terms: 0.3, 0.6, 0.9, … .2 marks
- Continue by adding 1/5 each time: 2/5, 3/5, 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 1/2, 3/4, 1 with 1/4. Explain the error under the add-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 1/5 each time: 2/5, 3/5, 4/5, … .2 marks
The next fractions are 5/5 and 6/5.
1, 1 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 1/2, 3/4, 1 with 1/4. Explain the error under the add-1/4 rule.2 marks
The rule keeps adding 1/4; it does not restart after 1.
The next term should be 1 1/4.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Growing rational patterns; rules; brackets/order of operations; unknowns; algorithms.
Official curriculum source ↗