Operations & Algebraic Thinking

Generating Numerical Patterns

Generating two patterns from two rules and comparing them.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Picture two sequences: one starts at 0 and follows the rule 'add 3'; the other starts at 0 and follows the rule 'add 6.' Before reading on, predict — will the second sequence always be exactly double the first, only sometimes, or never?

Write out a few terms and check your prediction. The 'add 3' rule gives 0, 3, 6, 9, 12. The 'add 6' rule gives 0, 6, 12, 18, 24. Every term in the second list really is exactly double the matching term in the first list — every single time, not just occasionally.

Definition — Corresponding Terms

Terms from two different patterns that occupy the same position in their sequences — the first term of one pattern corresponds to the first term of the other, the second to the second, and so on.

Why does the doubling relationship hold at every single term, not just by coincidence for the first few? Both patterns start at the same number, 0, and one rule adds exactly twice as much as the other at every step. Since they start together and the second sequence always grows twice as fast, it stays exactly twice as large at every corresponding term.

Worked Example — Generating and Comparing Two Patterns

Generate the first five terms of the rule 'start at 0, add 4' and the rule 'start at 0, add 12.' The first pattern: 0, 4, 8, 12, 16. The second pattern: 0, 12, 24, 36, 48. Comparing corresponding terms, each term in the second pattern is exactly 3 times the matching term in the first — matching the fact that adding 12 each time is exactly 3 times adding 4 each time.

Worked Example — Patterns That Don't Start at the Same Number

Generate the first four terms of 'start at 1, add 2' and 'start at 2, add 4.' The first pattern: 1, 3, 5, 7. The second pattern: 2, 6, 10, 14. This time, each term in the second pattern is exactly 2 times the matching term in the first — even though the two patterns don't start at 0, the doubling relationship still holds because 1 doubled is 2, and adding 4 is double adding 2.

Tip

To predict the relationship between two patterns before generating every term, compare their starting numbers and their rules separately — if both relate by the same multiplier, every corresponding term will too.

Common Mistakes

Checking the relationship between two patterns using only their first term or two, and assuming it holds for the rest without verifying further.
Generate at least four or five terms of each pattern before describing a relationship between them, to make sure the pattern holds consistently and isn't a coincidence in the early terms.
Comparing terms from different positions in the two patterns, such as comparing the second term of one pattern to the third term of the other.
Always compare corresponding terms — the same position number in each sequence — when describing a relationship between two patterns.

Key Takeaways

Two number patterns generated by related rules can have a consistent relationship between their corresponding terms.
That relationship comes directly from how the rules and starting numbers relate to each other.
Checking several terms, not just the first one, confirms whether a relationship genuinely holds throughout.

Summary

Comparing patterns built from related rules reveals a consistent relationship between their terms. Later in this grade, corresponding terms like these will be graphed as points — but first, the next unit builds a deeper understanding of the place-value system these patterns and calculations rely on.

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