Understand the idea.
Make instructions precise
An algorithm gives precise steps that someone can follow without guessing. A pattern algorithm needs a start, an instruction for changing the value, a point at which to record it, and a condition or count that ends the repetition.
The recording order matters. Starting at one and writing it before adding two includes one as the first output. The phrase at most nine includes nine; less than nine would exclude it. Test the boundary words carefully.
Trace, then improve
Trace an algorithm by writing the changing value after each step. Check the first output, several repeats and the stopping point. This can reveal a skipped term or an equality condition the pattern jumps past and never reaches.
A decision also needs a clear action for each outcome. For example, a process that labels numbers even or odd must say what to record in both cases. Testing instructions by hand helps find ambiguity before a computer follows them repeatedly.
| Step | Recorded number |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 5 |
| 4 | 7 |
| 5 | 9 |
See the method in action.
Worked example 1
Follow: start at 0.5; write the number; add 0.25; write again if the new number is at most 1.5; repeat.
Worked solution
- 1
Record the starting value and add the step.
The start 0.50 is written first, and the new value is still at most 1.50.
- 2
Continue with the same increment.
Every new value satisfies the boundary, including 1.50 itself.
- 3
Test the next candidate before writing it.
The next value fails the condition, so it is not recorded.
- 4
Collect the recorded output.
These are exactly the values written before the stopping boundary is exceeded.
0.5, 0.75, 1, 1.25, 1.5.
Worked example 2
An algorithm says “start at 2, add 3 and continue until the value equals 10”. Explain the problem and repair it.
Worked solution
- 1
Trace the values around the proposed stop.
Adding three moves from eight straight past ten to eleven.
- 2
Explain why equality will never occur.
Every later value is two plus a whole number of threes, so ten is unreachable; once above ten the increasing sequence cannot return.
- 3
Replace equality with a boundary test.
Write a value only while this condition is true. It detects passing ten even when ten is not a term.
- 4
Trace the repaired output.
Record 2 first, then add three and test before each next recording; stop before writing 11.
The equality stop is never reached. Use a boundary the sequence can pass, such as stopping before a value exceeds 10.
Try it yourself.
Check your understanding
Start at 4. Record four terms, adding 3 each time. What is the output?
Show answer & reasoning
4, 7, 10, 13.
Record as term one, then add three for each next term: . Four recorded terms require three additions, not four.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- Start at 3. Write five terms, adding 4 each time.2 marks
- Start at 2. Write terms by doubling until you have written five terms.2 marks
- Start at 0.2 and record it. Keep adding 0.2 and recording the new value while it is at most 1. Write the output.2 marks
- What changes if the preceding algorithm records only values less than 1?2 marks
- Write precise steps that generate 5,10,15,20 and then stop.3 marks
- An algorithm starts at 1 and adds 2 until it equals 8. Why will it not stop? Give a repair.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Start at 3. Write five terms, adding 4 each time.2 marks
Record the start, then make four increases.
3,7,11,15,19. - Start at 2. Write terms by doubling until you have written five terms.2 marks
Each new term is twice the previous term.
2,4,8,16,32. - Start at 0.2 and record it. Keep adding 0.2 and recording the new value while it is at most 1. Write the output.2 marks
Include 1 because at most 1 allows equality.
0.2,0.4,0.6,0.8,1.0. - What changes if the preceding algorithm records only values less than 1?2 marks
Less than excludes equality.
The final 1.0 is omitted. - Write precise steps that generate 5,10,15,20 and then stop.3 marks
A correct algorithm specifies start, change, recording and a finite stop. Equivalent correct steps are acceptable.
For example: start at 5; record it; add 5; repeat recording and adding until four values have been recorded. - An algorithm starts at 1 and adds 2 until it equals 8. Why will it not stop? Give a repair.3 marks
Check the boundary values rather than assuming the target is reached.
It stays odd and never equals 8. Stop before the next value exceeds 8, giving 1,3,5,7.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Growing rational patterns; rules; brackets/order of operations; unknowns; algorithms.
Official curriculum source ↗