Fibonacci Sequence and the Golden Mean

By Curtis Nichols (9-12 classroom)
Ah, so much to say... Some of it may be too advanced for your students, but may give you an idea of where the important connections lie.

> Does anybody has application of the Fibonachi numbers
> and nature.

Why not start at http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html.

> At what level and great lesson do you introduce
> Fibonachi and golden number?

I also recommend the Disney movie, "Donald in Mathmagic Land" which has a good (if fast moving) discussion of both the fibbonacci numbers and the golden mean.

For advanced elementary students who haven't shied away from working with polynomials, you can do long division of polynomials on the rational function 1/(1-x-x*x) [that is, one over 1 minus x minus x squared.] Divide it out by polynomial long division! Curious? This polynomial is called the generating function for the fibbonacci numbers because of the result you get (i.e., the infinite polynomial with fibbonacci numbers as the coefficients of x: 1 + x + 2x^2 + 3 x^3 + 5 x^4 + ... ). Curiously, the denominator has a root at (sqrt(5) - 1)/2--that is, the golden mean. This is the deep point of connection between these. This is also the source of the closed formula for the fibbonacci numbers: Fib(n) = (Phi^n – (–phi)^n)/sqrt(5) where Phi = (1 + sqrt(5))/2 and –phi = ((1 – sqrt(5))/2

Which involves the powers of the roots of the rational function mentioned above (and which formula comes from splitting that rational function into two 1st degree parts that add up to the original--you need to use the mathematical technique of partial fractions to accomplish this and that is probably beyond them.

As for introducing students (at the 9-12 level) I like to play a lot of guessing games to develop their skills with induction. This is a creative thing and most like it (I'll have to publish my follow-up books on sequences someday soon....) So you can just start writing on the board: 1, 1, 2, [can anyone guess what comes next?] 3, 5, [anyone got the pattern yet?] 8, 13, [can you explain how you get the next number? typical answer, "You add the previous two."]

I also like to give a history lesson. Fibonacci, also known as Leonardo of Piza, wrote an early (the first?) European algebra book in which he discussed these numbers in relation to rabbit reproducation. "Fibonacci" roughtly translated, comes from "Filio de bonacci" (feel free to correct my murdered Italian) which means son of the idiot. I don't think his father was well regarded (the kids love this part.)

Then you can go through the rabbit generations. Start with a pair of rabbits. So we have ONE pair. In a month they become mature and can reproduce--but they haven't yet. So we still have ONE pair. But--OOPS--next month, they give birth to a juvinile pair (so we have TWO pair) and also promptly get pregnant again. Next month, the juvinile pair grow to maturity while their parents CONTINUE to produce, so we have [everyone get this?] THREE pair--one of which is juvinile. Next month, all but the juvinile pair produce giving [still hanging in there?] FIVE pair, two of which are newly born and hence juvinile. So next month how many will we have? Right, EIGHT. And how many of those will be juvinile? Right, Three (meaning five are adults, btw--you HAVE been focusing on the adults, haven't you? It's an adult world, you know.) So what happens now.

Kids can follow this (and enjoy the humor.) It is a good time to mention cliche's like "reproduce like rabbits." You might also note that the growth is exponential--that is, the sequence will exceed any times table you might mention if you go out "far enough" in the sequence.

Teacher-to-Teacher Sharing

Edited by
Curtis J. Nichols

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Edited by Curtis J. Nichols

Edited by
Curtis J. Nichols