Teacher Information: Fibonacci Sequence

There are few sequences that hold more fascination than the Fibonacci sequence. Its study can be motivated in many ways, but one of the most entertaining is the question posed by the person after whom the sequence is named, Leonardo of Pisa, known as Fibonacci. He wanted to know the number of rabbit pairs we would have if we started with one male/female pair with rather peculiar reproductive habits: after reaching the age of one month, they produce another male/female pair, who also begin having bunnies at the age of one month, always in such pairs. Starting with 1 pair, at the end of the second month there will be 2 pairs, and a month later there will be 3 pairs (one from each of the earlier pairs). Then after another month, two more pairs will have matured and reproduced so that we have 2 new pairs to go with the 3, for a total of 5. The number of pairs at the end of successive months is given by the sequence, 1, 1, 2, 3, 5, 8, 13, 21,....

For students meeting the Fibonacci sequence for the first time, it is a delightful experience to discover how often the sequence appears in nature. Sunflowers and pine cones and pineapples all develop spirals as they grow. Students should have some actual models, or at least pictures, and count the number of spirals moving in one direction or the other. The numbers will be successive terms of the Fibonacci sequence. There are, of course, biological reasons for this kind of spiral packing, but most of us would not have expected such a result. The internet has a great many sources illustrating applications and appearances of the Fibonacci sequence, and unexpected side activities to explore.

This virtual manipulative does the computation to generate lots of terms of the sequence very easily. Although the original sequence begins 1,1, and then continues by adding two successive terms to get the next one, it is possible to start with any two positive numbers, and we generate a different "Fibonacci-like" sequence. The quotients of successive terms of the Fibonacci sequence rather quickly approach the "Golden Ratio," as indicated in the Instruction page. Students should be encouraged to pick their own starting values, to compute (themselves) the first eight or ten terms of their sequence, and then to calculate the ratios, F_n+1/F_n. They will then appreciate the convenience of this program to verify the remarkable fact that the ratios approach the Golden Ratio, no matter what the starting terms.