Fibonacci Numbers
This activity will re-introduce students to sequences, and allow them to discover the Fibonacci Sequence based on a question posed by a mathematician.
Resources: handout 1 , handout 2
Part 1
Discovering Patterns: This part of the activity was used as a warm-up to get the students used to patterns.
Rabbit Population: Leonardo of Pisa posed a model for rabbit breeding. Starting with one pair of immature rabbits, how many pairs of rabbits would there be after x months? A pair of rabbits takes one month to mature, and one pair of mature rabbits will give birth to one pair of immature rabbits every month. The sequence of growth after each new month is exactly the Fibonacci sequence (Fibonacci was a pseudonym of Leonardo of Pisa). The idea behind the activity is to discover the Fibonacci sequence through the hands on modeling of rabbit population growth.
We used plastic chips that were white on one side and red on the other. We gave each group of students enough chips to determine the number of rabbit pairs in the first six months. After that, they had to find that pattern in order to fill in the remaining months. We made sure that each group had found the correct pattern, and then discussed the results as a class. At this point we told them that this was called the Fibonacci Sequence.
Part 2
Warm-Up: As with the previous activity, we had the students find these sequences as a warm-up.
Convergence: After introducing the concept of convergence, we went over the first few with them and they did the rest on their own. We used a number line to give a visual representation that would help them understand the idea of convergence.
Important to Remember: This space we left so that they could take notes. We discussed the fact that sequences can converge to a number even though this number is not a term in the sequence. For example, the sequence in problem 2 converges to 0, but there is no natural number n where 1/n = 0.
The Golden Ratio: The students fill in the terms of the Fibonacci Sequence using the rule they discovered in the last part of the activity. Next to each term of the sequence, they then fill in the ratio of that term over the previous term. To make the convergence of that sequence easier to see, we had the students calculate each ratio as a decimal.
We gave them the geometric meaning of phi (the golden ratio) and they used that formula to set up a quadratic equation. Then, they used the quadratic formula to solve for phi. Notice that this is the number our sequence of ratios is converging to.