Understand the idea.
Separate fixed pieces from growth
Look for the construction rule, not just the first count. In a train of touching squares, the first square uses matchsticks. Each new square shares an existing side, so it adds only new matchsticks.
Count shared pieces once. For squares, one method is to start with and add for each of the remaining squares. This gives matchsticks.
Use a rule to jump ahead
A stage-number rule lets you jump directly to a later stage. The square train can also be counted as : three matchsticks for every square, plus one end stick. This equals .
Check that a proposed rule matches the pictured stages, but explain why the construction continues. A short list of counts alone can fit many rules; here the repeated shared-side arrangement explains the increase of each stage.
See the method in action.
A later square train
How many matchsticks are at stage 6?
Worked solution
- 1
Use the square-train construction.
The first square uses 4 sticks, and every later square adds 3 because one side is shared.
- 2
Substitute stage 6.
There are 5 additional squares after the first one.
- 3
Calculate and state the count.
Stage 6 therefore contains 19 matchsticks.
19 matchsticks.
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.
Worked solution
- 1
Count the increases from the first stage.
Starting at 5 tiles, each move to the next stage adds 2 tiles.
- 2
Substitute stage 8.
Stage 8 is 7 increases after stage 1.
- 3
Find the total.
The first 5 tiles plus 14 added tiles give 19 tiles.
19 tiles.
Try it yourself.
Check your understanding
A pattern follows . Find stage 3.
Show answer & reasoning
14 pieces.
Substitute the stage number into the given rule: . The stage therefore contains pieces.
Keep practising
6 worksheet questions · 14 marks
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
.
31 matchsticks. - A pattern follows . Find stages 1 and 4.2 marks
.
.
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
gives 4 at stage 1, so add 2.
.
; 30 pieces. - Which rule matches 3,5,7,9 at stages 1,2,3,4: or ? Explain.2 marks
At stage 2, gives 5 but 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 ↗