What You Will Learn in This Chapter

This chapter takes you on an unhurried journey through one of the oldest and most rewarding ideas in mathematics: the idea of a pattern. A pattern is simply something that repeats or grows in a way we can predict. Once you learn to notice patterns, you start to see them everywhere — in the tiles on a floor, in the petals of a flower, in the ringing of the school bell, in the numbers stacked inside Pascal's Triangle, and even in the way a computer decides whether a photograph shows a cat or a dog. Because this idea shows up in so many places, we will spend extra time exploring it slowly, from several different directions, and we will lean heavily on diagrams and charts along the way, since a pattern is often easier to feel in a picture than in a sentence.

Here is a fuller map of what lies ahead:

•        What is a pattern, and why do mathematicians and computer scientists care about them so much.

•        Counting patterns and skip-counting, and how a simple number grid reveals hidden structure.

•        Patterns hidden inside ordinary counting numbers — triangular numbers, square numbers, cube numbers, the doubling pattern, and the Virahanka numbers.

•        How square numbers can be rebuilt, layer by layer, from the odd numbers — and how the very same triangular numbers you meet early in the chapter reappear, quietly, inside Pascal's Triangle.

•        How the Virahanka numbers settle down, term after term, towards a famous number called the golden ratio.

•        Patterns made from shapes, and how a shape pattern can always be translated into a number pattern.

•        How to write the “rule” of a pattern in words, so that anyone — a classmate or a computer — could extend it without seeing every earlier term.

•        The four-step pattern-thinking process — Observe, Identify, Predict, Verify — and how this same process lies at the heart of computational thinking and Artificial Intelligence.

•        Where patterns show up in nature and in everyday technology, from sunflower spirals to traffic signals.

•        A hands-on building activity, an extension activity you can try at home, discussion questions for the classroom, and a full set of practice exercises at the end.