Finding the Rule of a Pattern (Grades 6–7)
Given a table of positions and values, the goal is a rule that turns the position (n) into the value — usually a straight-line rule like value = a·n + b.
Understanding finding the rule of a pattern
Find "a" from how much the value changes each step (the common difference), then find "b" by checking position 1. Position 1→4, 2→7, 3→10 changes by 3 each step (a = 3), and 3·1 + b = 4 gives b = 1, so value = 3n + 1.
Seeing it in action
Worked example
Find the rule: position 1→4, 2→7, 3→10.
Step (a) = 3. At n = 1: 3·1 + b = 4 → b = 1. → value = 3n + 1.
Use the step to build a straight-line rule.
A common mistake, and the fix
Using the common difference and forgetting the starting adjustment.
Position 1 → 4, 2 → 7, 3 → 10 changes by 3, but “3n” alone gives 3, 6, 9, one short every time. Use position 1 to find the constant: 3 × 1 + 1 = 4, so the rule is 3n + 1.
Try a few
1→5, 2→7, 3→9 — rule?
step 2, so 2n; at n = 1, 2 + 3 = 5.
1→2, 2→4, 3→6 — rule?
step 2, so 2n; at n = 1, 2 = 2, nothing to add.
1→0, 2→1, 3→2 — rule?
step 1, so n; at n = 1, 1 − 1 = 0.
Try it in Numeris.
A short lesson, then practice with feedback on every answer.
Put the reasoning on paper.
Try six fraction, angle, perimeter and area questions with explained answers, for grades 4–6.
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