Patterns

Builds on

Required prerequisites, all the way down: 2

You will learn to

• Copy, extend and create repeating patterns such as red, blue, red, blue, … and identify the unit that repeats.
• Extend simple growing patterns such as 2, 4, 6, 8, … and describe the rule in words.
• Predict an item further along a repeating pattern without drawing every step.

Definition

• A repeating pattern is made by repeating the same unit again and again.
• A growing pattern changes by the same rule at each step, such as 'add 2'.

Theorem

• In a repeating pattern whose unit has k items, the item k places further on is always the same.
• The first few items do not always decide a pattern: 1, 2, 4 could continue with 8 (doubling) or with 7 (add 1, then 2, then 3), so the rule must be stated.

Common misconceptions

• Thinking a pattern must continue the way you first guessed.
• Confusing the repeating unit with a single item of the pattern.

Optional discoveries

• Clock arithmetic: remainders and modular arithmetic Which colour is the 100th bead in a red–blue–green pattern? The answer depends only on the remainder when 100 is divided by 3.
• Expressions, unknowns and rules Growing patterns lead to rules such as 'the n-th term is 2 × n', the first algebraic expressions.

Further reading

• Keith Devlin, Mathematics: The Science of Patterns (1994)
• Dorothy K. Washburn and Donald W. Crowe, Symmetries of Culture (1988)