Download Fibonacci - The Cathedral Grammar School

Specific learning outcome
Explain Fibonacci numbers
and their origin.
Activities/Learning Experiences
Watch this introduction video:
http://www.sciencekids.co.nz/videos/math/numbersofnature.html
This math video is about the numbers of nature, featuring Fibonacci numbers,
golden spirals, interesting shapes, and mathematical formulas that appear in
natural objects such as flowers and shells. They come together in an inspirational
video that shows just how beautiful mathematics can be.
The Fibonacci sequence is the sequence of numbers that starts off with 1 and 1, and then after that
every new number is found by adding the two previous numbers. So if we start with 1, and have 1
next, then the third number is 1 + 1 = 2, the fourth number is 1 + 2 = 3, the fifth number is 2 + 3 = 5,
and so on. The first 20 Fibonacci numbers are listed below
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987, 1597, 2584, 4181, 6765
In this unit we will use F(n) to represent the n-th Fibonacci number. This means that F(1) = 1, F(2) = 1,
F(3) = 2; F(4) = 3 and F(5) = 8.
Using the F notation, the relation that ‘every new number is found by adding the two previous
numbers’ is expressed by the recurrence relation
F(n + 2) = F(n + 1) + F(n).
This sequence is named after Fibonacci because he used it in his book Liber Abaci (1202) to show
how much easier it was to compute with the ‘new’ Arabic number system compared to the Roman
system that was then commonly used in the Western World. (For more on this see John Stillwell’s
article in Newsletter No. 26).
The problem that he posed went something like this. Suppose that rabbits mate when they are one
month old and when a female gets to be two months old she produces a pair of baby rabbits, one
male and one female. If we start off with a new-born male and female rabbit, how many pairs of rabbits
will there be in after 12 months? How many will there be after n months?
Assessment
Specific learning outcome
Activities/Learning Experiences
Understand the concept of
Fibonacci numbers and how
they are generated
Draw up the numbers of pairs of rabbits in each generation.
What patterns can you see in these numbers?
Gather information about
Fibonacci and places where
the Fibonacci numbers
appear in nature.
What does the name mean?
What other names did he have?
When did he live?
What was the Liber Abaci?
How was it printed?
Why was it written?
What relevant pictures are there?
What else did you find out about Fibonacci or his family?
We call these numbers the Fibonacci numbers and define F(n) to be
the n-th Fibonacci number.
what is: F(3), F(5), F(6), etc., are.
Find the values of n in the equations F(n) = 8; F(n) = 144; F(n) =
377; etc.
Nature: Where in nature can you find Fibonacci numbers appearing?
Are there any other occurrences like the rabbits or the bees?
What occurrences are related to flowers?
What relevant pictures can you find?
Where do spirals come into this?
Are any special names given to these spirals?
Can you explain what is going on in these situations?
What else have you discovered about this topic?
Assessment
Completed Copy Master 1/2
Specific learning outcome
Activities/Learning Experiences
Assessment
Explore the Golden Ratio and
the Golden Angle through
natural and human made
structures.
Read the short illustrated novel Fibonacci’s Cows by Ray Galvin
To be able to identify
examples of natural and human
made structures that represent
the Fibonacci sequence.
Put together a slideshow (?) of Fibonacci evidence at your school or within
the community with slides explaining the mathematics of the plants and
buildings.
Photographs/descriptions
Reflect
Complete an ongoing PMI reflecting on your learning,
successes, difficulties/brick walls and discoveries.
PMI
Collect photographic examples of the Fibonacci numbers in nature. One
place that is a rich source of these is the plant world with the number of
petals on different flowers and the branching of stems or leaf organisation.
There are many examples on the internet through Google Images that can
inspire you and guide what they will be looking for in their environment.
Following the reading, explore and discuss the Golden Number, Golden
Ratio and the Golden Angle. Explore the ratio within the school structures
(the hall, the classroom, etc) and the angle within natural examples
(sunflowers, pinecones, shells, etc). Record examples with photos.