Return to Mathematics IndexA FIBONACCI PRIMER

Freeman, Larry Kenwood Academy

1-312-536-8850

Objectives1) To share knowledge about an interesting but obscure facet of recreational mathematics which is accessable and interesting to junior and senior HS math students. 2) To challenge teachers and students alike to discover and perhaps prove many interesting properties of the Fibonacci sequence and its unexpected relationship with the geometry of the regular pentagon and with the theory of limits.Equipment and MaterialsWriting implements; hand calculator (scientific preferred); accurate millimeter scale. Teacher-prepared worksheets to calculate and display up to twenty elements of the classical Fibonacci Sequence ("CF") and several varieties of generalized Fibonacci- like sequences ("GS"). Teacher-prepared worksheets with several regular pentagons -- complete with all diagonals -- and with all vertices and intersections of diagonals labeled.Recommended StrategiesProvide abriefexplanation of who Leonardo of Pisa was, his dates and place in math

history -- the rabbit problem is optional. Use proper terminology which means

subscripting: We denote specific terms of GS with subscripts. The i-th term of CF is

written as F_{i}. In CF, the first two terms are always F_{1}= 1, F_{2}= 1.

The basic rule for any Fibonacci or Fibonacci-like sequence is that every term is

the sum of the two previous terms. In our symbols, F_{i+2}= F_{i}+ F_{i+1}, is an element of

the set of positive whole numbers: 1, 2, 3, ... The ith term of CF is abbreviated as

F_{i}; the first term of CF is F_{1}. The expression "F_{n+3}" means the "n + 3rd" term of the

F-sequence.

The Fibonacci sequence and its properties are generally considered part of

recreational mathematics, a category of math generally ignored by the math

"establishment."Butteachers know that play ought to precede serious study; so this

is perhaps a powerful reason to include F in our teaching! It is rich in non-trivial

arithmetic practice, calculator practice and problems, and in algebraic applications.

Fibonacci connects with Geometry -- even Pythagorean Triples -- and probability. The

ratios of successive terms of F dramatically illustrate oscillating series and the

question of convergence of infinite series (calculus).

On the worksheet with the regular pentagons, carefully draw the diagonals with a

sharp pencil. Measure lengths of sides, diagonals and segments of diagonals to the

nearest millimeter. Keep a neat record of your measurements. Check your results

against GS or CF. If they don't fit, can you devise a way to make them fit? Compute

ratios of consecutive lengths, larger to smaller. What can be discovered here?Thought Starters on FibonacciWrite the first twenty terms of CF. Here are some questions to answer as you scan this list: 1) Are odd or even terms more numerous? Whatisthe ratio of odd to even terms? Will

this continue forever? Why?

2) F_{12}= 144 and 144 = 12^{2}. There are no other perfect squares among the Fibonaccis,

but this wasn't proved until 1963. Can you find a perfect number or two among the

F's?

3) Add the first n F_{i}'s. What is special about the sum? Isita member of F? Where

is it? What should the formula be?

4) If I want to know the value of a large F_{i}such as F_{307}, is there a shortcut ormust

I list all 306 terms of F? There is a shortcut -- get your calculator out:

Let x = 0.5 + 0.5 sqrt 5 and y = 0.5 - 0.5 sqrt5

Then F_{n}= (x^{n}- y^{n})/sqrt 5 "sqrt" = "square root of"

5) What is the sum of the first noddF_{i}? Proof (not hard).....

6) Examine your answer to (1). Recompute your list of F_{i}using different values for

F_{1}and F_{2}. What do you now find for the ratio of odds to evens? Is there a

general rule working here? State it.

7) Divide F_{i+1}by F_{i}for every entry in your list of F_{i}. As i increases, what happens

to the sequence of quotients? Try this for another FS (one with different first

and second terms). Compare your result with the famed "golden ratio."

8) Can you "massage" the Pascal Triangle and get the Fibonacci Sequence out of it?

Now we can expect a relationship between Fibonacci, binomial coefficients and

probability theory!!!

+ + + + + + + + + + +

Those who would like a list of readings/references on this fascinating topic are

invited to contact me for one. Non illegitimus carborundum, y'all.

{Dedicated to Roy and Dianne Coleman by a stubborn, reluctant student.}