A pattern rule is a precise description of how a pattern works — the recipe that generates every element. Rules can describe what happens between consecutive terms ("add 3 each time") or connect each term to its position ("the term is 2 times its position number"). Being able to state a clear rule is the difference between noticing a pattern and truly understanding it. Rules make patterns useful: they let you predict any term, check membership, and communicate the pattern to someone who has never seen it.
Give students patterns and ask them to write the rule in their own words. Compare student descriptions for the same pattern — are they equivalent? Introduce the distinction between "term-to-term" rules (what you do to get from one term to the next) and "position" rules (how to find any term from its position number). Use T-charts with position numbers and term values. Have students create patterns from given rules and trade with classmates.
So far you have been recognizing and extending patterns. Now you are going to focus on the most important part: stating the rule. The rule is the engine of a pattern — the precise recipe that tells someone everything they need to generate the entire sequence.
There are two kinds of rules, and both are useful. A term-to-term rule tells you how to get from one element to the next. For the pattern 4, 7, 10, 13, the term-to-term rule is "add 3 each time." This rule is easy to spot: just look at the differences between consecutive terms (7-4=3, 10-7=3, 13-10=3). A position rule tells you how to find any element from its position number. For the same pattern, the position rule is "the term equals 3 times the position plus 1" (position 1: 3x1+1=4; position 2: 3x2+1=7; position 3: 3x3+1=10).
Why have two kinds of rules? Because they have different strengths. The term-to-term rule is easy to discover — you look at the jumps between neighbors. But to find the 50th term, you would need to compute all 49 terms before it. The position rule is harder to find, but once you have it, you can jump straight to any term: the 50th term is 3(50)+1 = 151. No listing required.
A good rule passes three tests. First, it generates every element of the pattern correctly. Second, it is specific enough that someone who has never seen the pattern can recreate it. Third, it does not include unnecessary details. "Start at 4, add 3 each time" passes all three. "The numbers go up" fails the second test — it is too vague. "4, 7, 10, 13, 16, 19, 22" fails the third — it is a list, not a rule. The habit of writing clear, precise rules is the beginning of thinking like a mathematician.