1. Fibonacci Numbers Based on the following PowerPoint's
The Fibonacci Numbers and The Golden Section
M Golenberke 6/2012

2. Who Was Fibonacci? ~ Born in Pisa, Italy in 1175 AD
~ Full name was Leonardo Pisano
~ Grew up with a North African education under the Moors
~ Traveled extensively around the Mediterranean coast
~ Met with many merchants and learned their systems of arithmetic
~ Realized the advantages of the Hindu-Arabic system

3. The Fibonacci Numbers ~ Were introduced in The Book of Calculating
~ Series begins with 0 and 1
~ Next number is found by adding the last two numbers together
~ Number obtained is the next number in the series
~ Pattern is repeated over and over
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
Fn + 2 = Fn Fn

4. The Fibonacci Numbers }+ 1 1 2
The number pattern that you have been using is known as the Fibonacci sequence. }+
1 1 2

5. The Fibonacci Numbers }+ 1 1 2 3
The number pattern that you have been using is known as the Fibonacci sequence. }+ 3

6. The Fibonacci Numbers }+ 1 1 2 3 5 8 13 21 34 55
The number pattern that you have been using is known as the Fibonacci sequence. }+ 5 8 13 21 34 55
These numbers can be seen in many natural situations

7. Fibonacci’s Rabbits Problem:
Suppose a newly-born pair of rabbits (one male, one female) are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

8. Fibonacci’s Rabbits Continued
~ End of the first month = 1 pair
~ End of the second month = 2 pair
~ End of the third month = 3 pair
~ End of the fourth month = 5 pair
~ 5 pairs of rabbits produced in one year
1, 1, 2, 3, 5, 8, 13, 21, 34, …

9. The Fibonacci Numbers in Nature
~ Fibonacci spiral found in both snail and sea shells

10. The Fibonacci Numbers in Nature Continued
~ Sneezewort (Achillea ptarmica) shows the Fibonacci numbers

11. Fibonacci’s sequence… in nature
One plant in particular shows the Fibonacci numbers in the number of "growing points" that it has.
Suppose that when a plant puts out a new shoot, that shoot has to grow two months before it is strong enough to support branching.
If it branches every month after that at the growing point, we get the picture shown here.
A plant that grows very much like this is the "sneezewort“.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

12. The Fibonacci Numbers in Nature Continued
~ Plants show the Fibonacci numbers in the arrangements of their leaves
~ Three clockwise rotations, passing five leaves
~ Two counter-clockwise rotations

13. The Fibonacci Numbers in Nature Continued
~ Pinecones clearly show the Fibonacci spiral

14. Fibonacci’s sequence… in nature
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

15. The Fibonacci Numbers in Nature Continued
Lilies and irises = 3 petals
Buttercups and wild roses = 5 petals
Corn marigolds = 13 petals
Black-eyed Susan’s = 21 petals

16. The Fibonacci Numbers in Nature Continued
~ The Fibonacci numbers are found in the arrangement of seeds on flower heads
~ 55 spirals spiraling outwards and 34 spirals spiraling inwards

17. Fibonacci’s sequence… in nature
On many plants, the number of petals is a Fibonacci number:
Buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
13 petals: ragwort, corn marigold, cineraria, some daisies petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the asteraceae family.
Some species are very precise about the number of petals they have - eg buttercups, but others have petals that are very near those above, with the average being a Fibonacci number.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

18. The Fibonacci Numbers in Nature Continued
~ Fibonacci spiral can be found in cauliflower

19. Fibonacci’s sequence… in nature
Take a look at a cauliflower. Count the number of florets in the spirals. The number in one direction and in the other will be Fibonacci numbers.
If you look closer at a single floret, it is a mini cauliflower with its own little florets all arranged in spirals around a centre. If you can, count the spirals in both directions. How many are there?
Now look at the stem. Where the florets are rather like a pinecone or pineapple. The florets were arranged in spirals up the stem. Counting them again shows the Fibonacci numbers. Try the same thing for broccoli.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584…

20. The Fibonacci Numbers in Nature Continued
~ The Fibonacci numbers can be found in pineapples and bananas
~ Bananas have 3 or 5 flat sides
~ Pineapple scales have Fibonacci spirals in sets of 8, 13, 21

21. The Fibonacci Numbers in Nature Continued
~ The Fibonacci numbers can be found in the human hand and fingers
~ Person has 2 hands, which contain 5 fingers
~ Each finger has 3 parts separated by 2 knuckles

22. Fibonacci in Nature
The lengths of bones in a hand are Fibonacci numbers.

23. Class Activity

24. Patterns in Fibonacci 1+1 = 2 1+2 3 2+3 5 3+5 8 5+8 13 8+13 21 13+21
Complete the ‘Fibonacci Numbers’ sheet using column addition
1+1 = 2
1+2 3
2+3 5
3+5 8
5+8 13
8+13 21
13+21 34
Colour all the even numbers in blue and all the even numbers in red.
What do you notice about the patterns in the colouring?

25. Patterns in Fibonacci On another completed Fibonacci sheet,
use your knowledge of divisibility rules to
a) colour in blue the multiples of 5
b) colour in red the multiples of 3
c) underline the multiples of 6
What patterns can you see?
What if you extend the sequence?
Are there any patterns in the multiples of 10?

26. Square Spiral 29 little boxes down 1 little square
Side 1
1 little square
15 little boxes across

27. 29 little boxes down 15 little boxes across 1 more little square Box
Side 1 2
1 more little square
15 little boxes across

28. 29 little boxes down 2 x 2 square 15 little boxes across Box Side 1 23
2 x 2 square
15 little boxes across

29. Box
Side 1 2 3 4
3 x 3 square

30. Now complete the pattern to fill your page as much as possible.
Box
Side 1 2 3 4
3 x 3 square
Now complete the pattern to fill your page as much as possible.

31. Box
Side 1 2 3 4

32. Box
Side 1 2 3 4

33. Box
Side 1 2 3 4

34. Box
Side 1 2 3 4

35. Draw your own “Nautilus” shell by drawing quarter circles in each square and joining to form the spiral

36. Fibonacci in Nature
Spirals, like the one you drew earlier are common in nature.