Exploring the Fibonacci Sequence

Middle School Teacher’s Circle

February 10, 2011

Dr. Carrie Wright

HISTORY

Leonardo of Pisa, also known as Fibonacci was born in 1170 (a

time of intense activity and growth in Europe)

Leonardo knew Arabic and studied the highly developed

mathematics of Arab scholars

Initiated the wide spread use of the Arabic numerals and the current

decimal system

In 1228 he wrote the book Liber Abacci, where he posed some “real

world” problems to students

The Fibonacci sequence was also studied independently by Indian

mathematicians

Original Problem

A man put one pair of rabbits in a certain place entirely surrounded

by a wall. How many pairs of rabbits can be produced in a year, if

the nature of these rabbits is such that every month each pair bears a

new pair, a male and a female, which from their second month on,

becomes productive?

Other Problems

In how many ways can an elf climb a set of n stairs if the elf can jump either one step or two steps at a time?

A drone bee has only one parent, a mother, whereas a female bee has both a mother and a father. Find the number of (great)n-grandmothers that a drone bee has.

How many ways can we express n as the sums of 1s and 2s? For example, the number 3 can be expressed as

3 = 1 + 1 + 1 = 1 + 2 = 2 + 1

Why does the Fibonacci sequence provide the answers to each of the above problems?

Mathematics and Nature

The numbers that appear in the Fibonacci sequence

appear in many different patterns in nature.

Let’s look for the Fibonacci numbers in pictures from

nature.

Flower Petals

Seed Heads, Pine Cones

http://www.maths.surrey.ac.uk/h

osted-

sites/R.Knott/Fibonacci/fibnat.ht

ml - plants

http://britton.disted.camosun.b

c.ca/fibslide/jbfibslide.htm

Fibonacci Rectangle1. Look at the graph paper. The first number in the Fibonacci sequence, 1, has been drawn

for you.

2. Go the square to the right of the 1. Outline that little square to represent the next number in the pattern, another 1.

3. Use the line above the two 1 squares to outline a square that is 2 squares long by 2 squares high. This represents the next number, 2.

4. Now move to the right of the 1 and 2 squares. Use the right side to draw a square that is 3 little squares high and 3 little squares long.

5. Use the bottom of both 1 squares and the bottom of the 3 square to make a square that is 5 squares by 5 squares.

6. Move to the left of the 2 square, 1 square, and 5 square. Use their left edge to make the 8 square.

7. Finally use the top of the 8 squares along with the top of the 2 and 3 square to make a 13 square

Fibonacci Spiral

All you have to do is connect one corner of each square with the opposite corner of that square with a sweeping curve.

Put your pencil in the upper right corner of the first 1 square that YOU drew. Touch it to the opposite corner and keep sweeping around.

Animals and plants that have this spiral

Flower petals

Seeds

pinecones

Cauliflower florets

Pineapples

Apples

Snail shells

Creating A Sequence (Fibonacci-esque)

Pick two numbers that are less than 30.

Add the two numbers together.

Create the sequence by adding the previous 2 numbers together

Sum up the first 10 numbers, and divide by 11.

What do you notice? Will this always happen?

What happens if you continue adding the numbers. What do you

notice about the sums? Will this always happen?

Fibonacci Sequence and Golden Ratio

ab

aa

b

6180339887.12

51

At least since the Renaissance period, many artists and architects have proportioned

their works to approximate the golden ratio – believing the proportion to be

aesthetically pleasing.