Fibonacci Numbers
Martha Torstenson, Math Teaching Fellow

In math class, during the week of October 15th, each class spent a half hour exploring the Fibonacci numbers. In small groups, students began by investigating how many possible ways there are to tile a 1 by n board using dominos (2x1 tiles) and monominos (1x1 tiles). This was a fairly straightforward task for the boards of length 1, 2, 3 and 4, but counting the possible tilings for boards of lengths 7 or 8 proved a little tedious. There had to be a better way! 

Math students learning about Fibonacci numbers

We then ventured into the world of recursion to figure out the patterns in these numbers that might make counting possible tilings for longer boards simpler. We recognized that the total possible tilings of a board can be broken up into those tilings that end with a domino and those that end with a monomino. Further, we recognized that the number of tilings of a board of length n that end with a domino is equal to the number of tilings of a board of length n-2 and that the number of tilings that end with a monomino is equal to the number of tilings of a board of length n-1. We had found our pattern! The sum of the two previous terms gives us the term in the sequence we are looking for! This looked awfully familiar to students who had seen the Fibonacci numbers before since it is exactly the recursive formula that defines the Fibonacci sequence! We finished up the class with a brief discussion of the places the Fibonacci numbers are found in nature, looking at spirals characterized by the Golden Ratio. Math is all around us!

Math students working with monimos

Photos contributed by Caitlin Lemley, Math Teacher.