Shape Patterns - Repeating and Growing Arrangements
Shape patterns use geometric figures arranged according to a rule. As the pattern grows, you can count sides, shapes, or dots to build a number sequence – then apply the same nth term skills you learned with number patterns.
Types of Shape Pattern
| Type | Description | Example |
|---|---|---|
| Repeating pattern | A block of shapes that cycles over and over | Circle, Square, Triangle, Circle, Square, Triangle … |
| Growing pattern | Each stage adds more shapes according to a rule | Staircase of squares: 1, 3, 6, 10 … squares |
| Rotating pattern | A shape turns by a fixed angle at each step | Arrow pointing N, E, S, W, N … (rotating 90°) |
| Dot / grid pattern | Dots arranged to form larger versions of the same shape | Triangular or square dot arrays |
Connecting Shape Patterns to Number Sequences
Every growing shape pattern hides a number sequence. Count a measurable feature (number of shapes, perimeter in units, number of dots) at each stage and you get a sequence you can analyse with nth term rules.
Worked Examples
Sequence: 2, 5, 8, … Common difference = 3. First term = 2.
nth term = 2 + (n−1) × 3 = 3n − 1.
Stage 10: 3(10) − 1 = 29 squares.
Sequence: 3, 5, 7, … Common difference = 2. First term = 3.
nth term = 3 + (n−1) × 2 = 2n + 1.
Check n=1: 3 ✓. 10 triangles: 2(10)+1 = 21 sticks.
The pattern repeats every 3 shapes. 29 ÷ 3 = 9 remainder 2. The 2nd shape in the block is Blue.
Square Dot Patterns
| Stage (n) | Dots across | Total dots | Formula |
|---|---|---|---|
| 1 | 1 | 1 | 1² |
| 2 | 2 | 4 | 2² |
| 3 | 3 | 9 | 3² |
| n | n | n² | n² |
Key Takeaways
- Shape patterns are either repeating or growing – identify which type first.
- For repeating patterns, use remainder (mod) arithmetic to find any term.
- For growing patterns, count a measurable feature at each stage to get a number sequence, then find the nth term.
- Always check your formula by substituting known stage numbers.
Practice Questions
- A pattern of squares: Stage 1 has 1 square, Stage 2 has 4, Stage 3 has 9. Write the nth term and find Stage 8.
- Hexagons are joined edge to edge. 1 hexagon has 6 sides visible; 2 joined have 10; 3 have 14. Write the nth term for visible sides.
- A repeating pattern: Star, Circle, Star, Square, Star, Circle, Star, Square, … What is the 50th shape?
- Matchstick squares: 1 square = 4 sticks, 2 squares in a row = 7, 3 = 10. Write the nth term and find how many sticks for 20 squares in a row.
- A growing pattern has nth term = n² + 1. Find the first five terms and describe what is happening to the differences between terms.