Geometric (growing) patterns
Theory
A growing pattern adds the same number of parts at each step, so the rule is that constant times the shape number, plus a fixed starting piece.
A growing pattern is made of shapes that get bigger by the same amount each step.
The rule links the shape number to the count of parts.
For the squares below, each shape adds \(3\) matchsticks with \(1\) fixed upright, so the rule is \(3 \times \text{shape} + 1\).
From the pattern to the rule
| Shape | Matchsticks |
|---|---|
| 1 | \(4\) |
| 2 | \(7\) |
| 3 | \(10\) |
| \(n\) | \(3n + 1\) |
How to find and use the rule
- Count how many parts are added each step.
- Find the fixed piece that never changes.
- Write the rule and use it for any shape.
Use \(3 \times \text{shape} + 1\).
| \(3 \times 4 + 1\) | \(=\) | \(13\) |
Use the rule for shape \(10\).
| \(3 \times 10 + 1\) | \(=\) | \(31\) |
Undo the rule: \(25 - 1 = 24\), then \(24 \div 3 = 8\).
| \(24 \div 3\) | \(=\) | \(8\) |
Each shape adds \(4\), so \(4 \times \text{shape} + 1\).
| \(4 \times 5 + 1\) | \(=\) | \(21\) |
Common pitfalls
Frequently asked questions
How do I find the rule of a growing pattern?
Count how many parts are added each step; that is the number you multiply by. Then add the fixed piece that never changes.
Why is there a “plus 1”?
It is the starting piece that is there before any repeats are added, such as the single upright matchstick at the left.
Can I jump to a far shape?
Yes. Use the rule directly. For \(3 \times \text{shape} + 1\), shape \(10\) needs \(31\) matchsticks.
How do I find which shape has a given count?
Undo the rule. For \(25\) matchsticks, \(25 - 1 = 24\), then \(24 \div 3 = 8\).