How to Teach Number Patterns and Sequences
2, 4, 6, 8, __. What comes next? Your child can probably answer: 10. But can they explain the rule? Can they find the 100th number without listing all 100?
Pattern work develops algebraic thinking — the ability to see rules in numbers and express them generally. It is the bridge from arithmetic (working with specific numbers) to algebra (working with general rules).
Start with simple patterns
Repeating patterns: A, B, A, B, A, B... "What comes next?" (A) Growing patterns: 1, 3, 5, 7... "What is the rule?" (Add 2 each time)
For young children, use physical objects before numbers: red-blue-red-blue-red-?, or 1 block, 3 blocks, 5 blocks...
Arithmetic sequences (add the same each time)
The most common pattern type:
- 3, 7, 11, 15, 19... (rule: add 4)
- 100, 90, 80, 70... (rule: subtract 10)
- 2, 5, 8, 11, 14... (rule: add 3)
Finding any term without listing all of them: For the sequence 3, 7, 11, 15... (start at 3, add 4):
- 1st term: 3
- 2nd term: 3 + 4 = 7
- 3rd term: 3 + 4 + 4 = 11
- nth term: 3 + 4(n-1)
This formula — first term + common difference × (position - 1) — is the general rule. For the 50th term: 3 + 4(49) = 3 + 196 = 199.
Key Insight: When your child can write a rule for a pattern in terms of position number (like "the nth term is 3 + 4(n-1)"), they are doing algebra — even if they do not realize it. Pattern rules are functions, and writing them is the beginning of algebraic thinking.
Geometric sequences (multiply by the same each time)
- 2, 6, 18, 54... (rule: multiply by 3)
- 1000, 100, 10, 1... (rule: divide by 10)
These grow (or shrink) much faster than arithmetic sequences. They connect to exponents.
Input-output tables
Present patterns as tables:
| Input | Output |
|---|---|
| 1 | 5 |
| 2 | 8 |
| 3 | 11 |
| 4 | 14 |
"What is the rule?" Each output is 3 more than the previous → output = 3 × input + 2.
This connects directly to linear equations and coordinate graphing.
Common mistakes
Only looking at consecutive numbers: They see 2, 5, 10, 17 and say "add 3, add 5, add 7" without seeing the bigger pattern (the differences themselves form a pattern — add 2 each time).
Confusing the rule with the terms: The rule is "add 3" but the terms are 2, 5, 8, 11. The rule describes the relationship between terms, not the terms themselves.
Not testing the rule on all given terms: They find a rule that works for the first two terms but not the third. Always verify the rule against every available term.
Number patterns bridge arithmetic and algebra. Start with "what comes next?", advance to "what is the rule?", and eventually reach "what is the formula for any term?" When your child can write a general rule for a pattern, they are thinking algebraically — even if they have never heard the word.
If you want a system that develops pattern recognition and algebraic thinking as a natural progression from arithmetic — that is what Lumastery does.