A FIBONACCI PRIMER

Freeman, Larry                      Kenwood Academy
                                    1-312-536-8850
                           

Objectives

1) 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 Materials

Writing 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 Strategies

Provide a brief explanation 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 Fi.  In CF, the first two terms are always F1 = 1, F2 = 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,  Fi+2 = Fi + Fi+1, is an element of 
the set of positive whole numbers: 1, 2, 3, ...  The ith term of CF is abbreviated as 
Fi; the first term of CF is F1. The expression "Fn+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."  But teachers 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 Fibonacci

Write 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?  What is the ratio of odd to even terms? Will 
   this continue forever?  Why? 

2) F12 = 144 and 144 = 122.  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 Fi's.  What is special about the sum?  Is it a member of F? Where 
   is it?  What should the formula be? 

4) If I want to know the value of a large Fi such as F307, is there a shortcut or must 
   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 Fn = (xn - yn)/sqrt 5        "sqrt" = "square root of"

5) What is the sum of the first n odd  Fi?  Proof (not hard)..... 

6) Examine your answer to (1).  Recompute your list of Fi using different values for 
   F1 and F2.  What do you now find for the ratio of odds to evens? Is there a 
   general rule working here? State it. 

7) Divide Fi+1 by Fi for every entry in your list of Fi.  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.}