Floors, Tiles and Fibonacci Numbers
Lesson 11 of 13
Objective: Students will be able to identify and continue a recursive pattern.
I tell the students that they are about to encounter a pattern that isn’t predictable in the same way other functions are, but I also explain that we are going to approach this pattern through a puzzle.
“Today I am giving you a bunch of kitchen tiles. Your job is to tile the blueprint of my kitchen. I would like everyone to take a minute to plan how you will work as a group to complete this puzzle without talking.”
I hand out the puzzle pieces, kitchen floor blueprints, glue, scissors and color pencils as they plan.
Before we begin, I ask, “what are some basic requirements of tilling a floor?” We discuss how the entire floor needs to be covered and there can’t be either overlap or gaps. I also ask them to use all the tiles and finish with a small hint, “I would like the smaller tiles toward the middle of the floor.”
The tiles are squares with side lengths equal to the Fibonacci sequence. The idea is to help all students discover the Fibonacci sequence silently to prevent any students from shouting out, “hey this is the Fibonacci numbers!” I am excited that they have read about the numbers, but students tend to tune out an investigation if they think they know everything already.
When students are finished with the puzzle, I ask them to silently color it in (by coloring it in, they are forced to relax, slow down and think indirectly about the shapes before them).
When they are done coloring the floor pattern, I ask them to make two tables. One table should list out the areas of each square and the other table should list out the square roots of each square. They should answer the question: what patterns do you notice?
If they have time, I ask them to try different spice rack questions:
What would the next 15 tiles look like?
If I wanted to build a much bigger kitchen, how would I make sure these types of tiles will fit?
What formula describes this sequence of roots?
I love sharing the finding from this investigation. Students share how they tiled the floor without talking. They discuss their collaborative strategies. Other students share the patterns in the areas and roots of the tiles. For many students, the breakthrough for seeing the Fibonacci sequence is to write both 1 x 1 tiles in their sequence. Once they wrote 1,1 and 2 they can see that you have to add 1+1 to get 2. We have groups show their title patterns and we share how you can find the nth Fibonacci number: Fn = Fn-1 + Fn-2 and students talk about the meaning of this formula. The main take away is that you need both steps before any Fibonacci number in order to find the nth Fibonacci number. I finish by reassuring the students, “Where should you be right now with this topic? I want you to be aware that some patterns are recursive and that some patterns need you to generate each step as you move forward. What tool would really help us find large Fibonacci Numbers?” I always get blank stares here. It is interesting to me that students don’t automatically see that a computer is the perfect tool for such an investigation. We talk about it and discuss that computers aren’t just convenient for organizing data, but they are a calculating tool that allows us to look at amazingly complex sequences and to do amazing amounts of repeated calculations.