Freeman, Larry Kenwood Academy

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

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

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.}

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