8 – Sequences and Functions

• Winter, spring, summer, fall, winter, spring…What comes next? Experience tells you it’s summer. The seasons form a pattern, and elements of a pattern repeat in a predictable manner.
• Patterns occur in both nature and in man-made designs.
• We can use all of our senses to observe patterns.
• Which senses would you use to identify these patterns?
• Start looking and you’ll find patterns all around you. All patterns have a repeating mathematical structure that can be modeled by formulas.
• Using these formulas allows you to extend the pattern infinitely, or identify an element at any point in the pattern. You could easily figure out the ten thousandth term without actually extending the pattern that far.
• That’s pretty powerful!
• Can you guess the next number in this pattern? 1, 1, 2, 3, 5, 8, 13, 21, 34,…
• This set of numbers is called the Fibonacci sequence. It may be hard to believe, but this strange pattern often shows up in real life.
• You can see it in the perfect spirals on a snail shell…
• …the pattern of petals on a rose…
• …the number of seeds in the spirals of a sunflower…
• …or the scales on a pinecone.
• In this unit, you’ll learn about arithmetic and geometric sequences of numbers and take a closer look at the Fibonacci sequence.