The idea, step by step
Separate fixed pieces from growth
A growing pattern follows the same construction rule at each stage. Look for what stays fixed and what gets added. Shared pieces must not be counted twice.
The first square in the train needs 4 matchsticks. Each extra square shares one side and therefore adds only 3 matchsticks.
Use a rule to jump ahead
The first counts are 4, 7 and 10. Adding 3 finds the next stage. A stage-number rule finds a distant stage without drawing all the ones before it.
For n squares, count 3 sticks per square plus one end stick: . Test the rule at more than one pictured stage before relying on it.
A common trap
Multiplying the stage by the first count even when pieces are shared.
Instead: Find the repeated change and the fixed extra part, then test the rule.
See the thinking
A later square train
How many matchsticks are at stage 6?
- Each square after the first adds 3.
- Use 3n+1 with n=6.
- 3×6+1=19.
A different growing pattern
A tile pattern has 5, 7 and 9 tiles at stages 1, 2 and 3, and keeps adding 2. Find stage 8.
- The change is 2 each stage.
- A matching rule is 2n+3; it gives 5 when n=1.
- 2×8+3=19.
Check your understanding
A pattern follows . Find stage 3.
Show answer
14 pieces.
4×3+2=14.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- For the square train, find stage 4.2 marks
- For the square train, find stage 10.2 marks
- A pattern follows . Find stages 1 and 4.2 marks
- Counts are 6,10,14,18 at stages 1,2,3,4 and keep rising by 4. Give a rule and stage 7.3 marks
- Which rule matches 3,5,7,9 at stages 1,2,3,4: or ? Explain.2 marks
- Jai says the square train uses sticks. Explain why this works at stage 1 but fails at stage 2.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- For the square train, find stage 4.2 marks
Stage 3 has 10; add 3.
13 matchsticks. - For the square train, find stage 10.2 marks
3×10+1=31.
31 matchsticks. - A pattern follows . Find stages 1 and 4.2 marks
5×1+2=7.
5×4+2=22.
7 and 22. - Counts are 6,10,14,18 at stages 1,2,3,4 and keep rising by 4. Give a rule and stage 7.3 marks
4n gives 4 at stage 1, so add 2.
4×7+2=30.
; 30 pieces. - Which rule matches 3,5,7,9 at stages 1,2,3,4: or ? Explain.2 marks
At stage 2, 2n+1 gives 5 but 3n gives 6.
. - Jai says the square train uses sticks. Explain why this works at stage 1 but fails at stage 2.3 marks
Two separate squares need 8 sticks.
Joining them shares one stick, leaving 7.
Both give 4 at stage 1. Stage 2 uses 7, not 8, because one side is shared.
Curriculum reference & lesson scope
Australian Curriculum v9.0 / South Australian Curriculum
Year 7 coordinate, pattern and authentic-graph interpretation scope. Includes prerequisite four-quadrant plotting; formal gradient/intercept techniques are not required here.
Official curriculum source ↗