§ Patterns

Growing Patterns

NO.LK20.43 min readApr 13, 2026

Growing patterns form the backbone of algebraic thinking in Key Stage 2 and 3, helping students recognise mathematical relationships before they encounter formal algebra. When Year 5 pupils spot that 3, 6, 10, 15 follows the triangular number sequence, they're developing pattern recognition skills that will serve them through GCSE mathematics.

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Why it matters

Pattern recognition underpins mathematical reasoning across all areas. In real life, growing patterns appear everywhere: a football league table where teams earn 3 points per win follows an arithmetic sequence, whilst savings accounts with compound interest follow exponential patterns. Students who master pattern analysis in primary school find algebraic expressions and sequences much easier at GCSE level. The ability to spot that 2, 6, 18, 54 multiplies by 3 each time prepares pupils for geometric sequences in Year 10. Even simple contexts like calculating bus fares (£2.50, £5.00, £7.50 for 1, 2, 3 zones) or planning school fundraising events rely on pattern recognition. These foundational skills support mathematical modelling, where students must identify relationships between variables and make predictions based on observed trends.

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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.

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Worked examples

Beginner§ 01

What comes next? 10, 12, 14, 16, 18, ?

Answer: 20

Find the difference between consecutive terms → 12 - 10 = 2 — Each number increases by 2.
Add the difference to the last term → 18 + 2 = 20 — The next number is 18 + 2 = 20.

Easy§ 02

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.

Medium§ 03

What comes next? 2, 4, 7, 11, 16, ?

Answer: 22

Calculate the differences between terms → 2, 3, 4, 5 — The differences are 2, 3, 4, 5. Each difference increases by 1.
Find the next difference and add it → 16 + 6 = 22 — The next difference is 6. So 16 + 6 = 22.

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Common mistakes

Students often assume all patterns follow simple addition rules. For the sequence 1, 4, 9, 16, they might write 25, 34, 43 (adding 9 each time) instead of recognising the square number pattern where the next term is 25.
Pupils frequently ignore changing differences in sequences. Given 2, 5, 9, 14, they might continue with 19, 24 (adding 5 repeatedly) rather than spotting that differences increase by 1 each time, making the next term 20.
Many students struggle with position-to-term relationships. When asked to find the 10th term of 5, 8, 11, 14, they count up term by term instead of using the rule '3n + 2' to calculate 3(10) + 2 = 32.

Practice on your own

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Frequently asked questions

How do I help pupils spot the difference between arithmetic and geometric patterns?

Start with the gaps between numbers. In arithmetic patterns like 4, 7, 10, 13, the difference stays constant (adding 3). In geometric patterns like 3, 6, 12, 24, look at ratios instead—each term doubles. Use concrete examples: pocket money increasing by £2 weekly versus bacteria doubling every hour.

What's the best way to teach triangular and square number sequences?

Use visual representations first. Draw dots in triangular arrangements (1, 3, 6, 10) or square grids (1, 4, 9, 16). Students can physically count the dots, then spot the algebraic relationship. Connect to the formulae n(n+1)/2 for triangular numbers and n² for squares once the visual pattern is secure.

When should I introduce algebraic notation for pattern rules?

Begin with words in Year 5-6: 'multiply the position by 4, then subtract 1'. Introduce symbols like 4n - 1 in Year 7-8 when pupils are comfortable with algebra. Always check the rule works by substituting different position values back into the original sequence.

How can I differentiate growing patterns for mixed-ability classes?

Start everyone with simple arithmetic sequences (adding the same number). Extend confident pupils to quadratic patterns or alternating sequences. Support struggling learners with number tracks and concrete materials. Use position charts where pupils can see term number against term value clearly.

What real-world contexts work well for growing pattern problems?

Sports leagues (3 points per win), savings schemes (£10 weekly deposits), mobile phone tariffs (£15 plus £5 per GB), and school fundraising targets all provide authentic contexts. Choose scenarios pupils recognise, ensuring the mathematical relationship is clear and the numbers remain manageable for the year group.

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