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Fibonacci-Sequence

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					TMTA 2006 Fall Conference University of Tennessee – Martin “Implementing Tablet PC’s in the Standards-based Mathematics Classroom”

Thomas E. Hodges R. Lee Collins

Fibonacci Sequence
An Investigation Original Investigation by Fibonacci in 1202 – Fibonacci’s Rabbits Ideal Circumstances: A newly born pair of rabbits, one male and one female, is placed in a setting such that: 1. Rabbits become fertile at the end of the second month of life. 2. Each month every pair of fertile rabbits produces another pair (one male and one female). 3. Rabbits do not die. How many pairs of rabbits are there at the end of one year?

Do you recognize the pattern? Is there a way for you to generalize the next number in the sequence? Do so in words, then in symbolic form.

TMTA 2006 Fall Conference University of Tennessee – Martin “Implementing Tablet PC’s in the Standards-based Mathematics Classroom”

Thomas E. Hodges R. Lee Collins

To construct a Golden Rectangle:  Construct a square labeled ABCD.  Construct the midpoint of the segment AB and name it point F.  Construct a circle with center F passing through point C.  Extend segment AB and label the points of intersection with the circle G and H.  Construct a line perpendicular to line AB passing through the point H.  Construct a line perpendicular to segment BC and passing through point C.  Construct the intersection of these last two lines and name it point I. Construct segments BH, HI, and CI.  Hide the circle; lines GH, HI, and DI; and points G and F. The resulting figure ADHI is a golden rectangle.
Directions from http://teacherlink.org/content/math/activities/skpv4-goldenrec/guide.html . Retrieved July ll, 2005.

Import the image to this document and add the Fibonacci numbers to the image. This new image is an approximation of a tiling with Fibonacci numbers. Describe how the picture of the golden rectangle relates to the tiling of the Fibonacci numbers:

TMTA 2006 Fall Conference University of Tennessee – Martin “Implementing Tablet PC’s in the Standards-based Mathematics Classroom”

Thomas E. Hodges R. Lee Collins

Let Fn be the nth term of the Fibonacci sequence. Find several values of Start with

Fn . Fn 1

F2 . As you begin to generate several ratios of each term over the F1

 F  previous  n  , what do you notice about the ratios?  Fn 1 

To the right is an image of a Nautilus sea shell. It shows the spiral curve of the shell. Right-click on the picture and copy it to Sketchpad. Draw a line from the center out in any direction and place a point at two places where the shell crosses it so that the shell spiral has gone round just once between them. Measure the distance between the points on the segment.

Investigate the spirals on the sea shell and describe any relationship you see with the Fibonacci numbers:

TMTA 2006 Fall Conference University of Tennessee – Martin “Implementing Tablet PC’s in the Standards-based Mathematics Classroom”

Thomas E. Hodges R. Lee Collins

Extensions 1. There are many occurrences of Fibonacci numbers in nature. Find as many different occurrences as you can and discuss them.

2. Do the Fibonacci numbers occur in Pascal’s Triangle? Investigate the triangle and justify your conclusion. 3. Some have researched Tribonacci and Tetranacci numbers (the sum of the three and four previous terms, respectively). Write several numbers from each sequence and approximate the ratio for each. Use the Internet to investigate the ratios’ relationship to certain polynomial functions.

Some Interesting Websites: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/ http://www.mathacademy.com/pr/prime/articles/fibonac/


				
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