Mathematics 7–10 · Year 7
Matchstick growing patterns: from a table of values to a linear rule
Number and algebra
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The idea
A growing row of squares adds the same number of sticks each time, so the count follows a rule of the form M = 3n + 1 that can be read from the structure, tabulated and plotted as points on a straight line that does not pass through the origin.
What you need
- 100 headless wooden craft sticks per pair (no match heads in the classroom)
- Recording table with columns for the number of shapes n and the number of sticks M
- 1 cm grid paper for the graph
How to do it
- Build 1, 2, 3 and 4 squares in a row, each new square sharing a side with the last, and record the stick counts.
- Describe in words how each new square is added, then write the rule as an expression in n; build seven squares (the NRICH Seven Squares case) to test it.
- Plot the table as points (n, M), describe how the points lie, and extend the pattern to predict the sticks needed for 20 squares.
- Use the rule backwards to find how many squares exactly 100 sticks make, then build as many as the sticks allow to check.
- Repeat for triangles in a row and hexagons in a row, and compare the three rules and graphs.
- Test the claim that doubling the number of squares doubles the number of sticks by comparing n = 5 with n = 10.
What you should see
Squares in a row use 4, 7, 10 and 13 sticks for n = 1 to 4, so M = 3n + 1: each new square adds 3 sticks and the extra 1 is the first upright. Seven squares use 22 sticks, 20 squares use 61, and 100 sticks make exactly 33 squares (3 × 33 + 1 = 100). Triangles in a row follow M = 2n + 1 (3, 5, 7, 9) and hexagons M = 5n + 1 (6, 11, 16, 21). The points lie on straight lines with gradients 3, 2 and 5 that all meet the vertical axis at 1, not at 0. Doubling from 5 to 10 squares takes the count from 16 to 31, not 32. The learner knows it worked when the rule predicts a count before building and the built pattern matches it.
What changes
- What you change
- number of shapes n
- What you measure
- number of sticks M
- What you keep the same
- each new shape shares one side with the previous shape
- one stick per side
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Each square needs 4 sticks, so the rule is 4n.
- A pattern that grows steadily must pass through the origin on a graph.
- Doubling the number of shapes doubles the number of sticks.
Safety card
Hazards
- wooden sticks can splinter or be used to poke
Controls
- headless craft sticks only
- sticks counted back into their box at the end
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 4 Linear relationships; page read 2026-09-22MA4-LIN-C-01
- Australian Curriculum v9AC9M7A05AC9M7A02
Sources
The pages the author read to write this activity.