Level 4: Rules with position numbers

Patterns and sequences · level 4 of 8 · rough age 11-12 (a guide for parents; anyone can start anywhere)

In level 3 you found far-away terms by counting jumps. In this level you write a pattern's rule with a letter, \(n\) for the position, so that one short line like \(t_n = 3n + 2\) holds the whole infinite sequence. You learn to find that rule from a list of numbers, from a graph and from a growing picture, to compare two patterns and find where they meet, and you finish with a picture proof that works for every \(n\) at once.

Lesson	You will be able to
Two kinds of rule	Write a sequence with a "next term" (recursive) rule and a "position" (explicit) rule, and switch between them
Finding the position rule	Find the rule for any sequence with a constant difference, using the difference and the zero term
Graphing a pattern	Plot position against term, and explain why a constant difference gives points on a straight line
Growing pattern pictures	See the rule inside a picture pattern (L-shapes, crosses, square borders) in more than one way
Comparing two patterns	Decide which of two plans gives more, and find exactly when they are equal
Sums of odd numbers	Prove that the first \(n\) odd numbers add to \(n^2\), with a picture and with pairing

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