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.
Growing shape patterns have literally been built into architecture for thousands of years: the stepped pyramids (ziggurats) of ancient Mesopotamia and the Step Pyramid of Djoser in Egypt (built around 2670 BCE) are, mathematically, a growing shape pattern of stacked stone platforms, each layer smaller than the one below by a predictable amount. Repeating patterns reached extraordinary sophistication in medieval Islamic architecture, where craftsmen created "girih" tilings hundreds of years before mathematicians formally described the geometry behind them; a 2007 study in the journal Science showed that some 15th-century Islamic tilework, such as at the Darb-i Imam shrine in Iran, follows the same complex, never-repeating mathematical structure that Western mathematicians only discovered in the 1970s (known as Penrose tiling).
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.