Growing Patterns
Growing patterns are sequences of numbers or shapes that follow a consistent rule, with each term increasing according to a specific mathematical relationship. The simplest growing patterns add the same amount each time, such as 3, 6, 9, 12, where each term increases by 3. More complex patterns might involve squares, triangles, or alternating operations that create predictable growth.
Why it matters
Growing patterns form the foundation for understanding algebraic sequences and functions in GCSE mathematics. Estate agents use linear patterns to calculate property values over time, with house prices increasing by approximately £15,000 per year in certain areas. Computer programmers rely on geometric sequences when calculating file compression rates or database growth. Triangular numbers appear in sports league tables, where a tournament with 8 teams requires 28 total matches using the pattern 1+2+3+...+7. Financial advisors use compound growth patterns to project investment returns, with a £1,000 deposit growing to £1,210 after 2 years at 10% annual interest. Pattern recognition skills directly support algebraic thinking in Key Stage 3 and GCSE Foundation topics.
How to solve growing patterns
Pattern Structures
- A pattern has a rule. Find what stays the same and what changes.
- Describe the rule in words first, then in symbols or numbers.
- Test the rule on the next term: does it predict correctly?
- Extend the pattern both forwards and backwards to check.
Example: 1, 4, 9, 16, ... The rule is square the position: 1², 2², 3², 4². Next: 5² = 25.
Worked examples
What comes next? 1, 5, 9, 13, 17, ?
Answer: 21
- Find the difference between consecutive terms → 5 - 1 = 4 — Each number increases by 4.
- Add the difference to the last term → 17 + 4 = 21 — The next number is 17 + 4 = 21.
What comes next? 1, 3, 6, 10, ?
Answer: 15
- Find the differences between consecutive terms → 2, 3, 4 — The differences are 2, 3, 4. They increase by 1 each time.
- Find the next difference and add it → 10 + 5 = 15 — The next difference is 5. So 10 + 5 = 15. These are triangular numbers.
What comes next? 2, 7, 5, 10, 8, 13, ?
Answer: 11
- Look at the pattern of changes → +5, -2, +5, -2, ... — The pattern alternates: add 5, subtract 2, add 5, subtract 2, ...
- Apply the next operation → 13 -2 = 11 — The next step is -2, so 13 -2 = 11.
Common mistakes
- Adding the last term instead of following the pattern rule, such as continuing 2, 4, 8, 16 with 32 instead of recognising the doubling pattern gives 32, not 20.
- Assuming all patterns have constant differences when some have changing differences, like treating 1, 4, 9, 16 as if it increases by 3 each time rather than recognising the square number pattern.
- Confusing position with term value in nth term calculations, writing the 5th term of 3n+1 as 51 instead of 16 because 3×5+1=16.