College of Information Technology, Computing and Library Science

C– GEC 4: Teaching Mathematics in the Modern World

First Semester | AY 2020-2021

Module 3 (Week 3) in C – GEC 4:

MATHEMATICS IN THE MODERN WORLD

First Semester AY 2020-2021

A.

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

Course Code /Title : C-GEC4– MATHEMATICS IN THE MODERN WORLD

B. Module Number /Title : Module 3: Fibonacci Sequence and Binet’s Formula
C. Time Frame : Three (3) hours
D. Materials : Syllabus, textbook, references, online resources

I. Overview of Module 3

This module provides a basic explanation of the Fibonacci sequence in an easy,

enthusiastic, and accessible manner.
In the module 2.1, Fibonacci number as observed in nature is introduced . It starts by
defining Fibonacci sequence and its origin and history. Followed by the discussion about
Fibonacci’ s rabbit- breeding problem or puzzle which lead to the Fibonacci Number sequence and
the simple definition of the whole never-ending series. As we dig out to the topic , many
interesting mathematical properties will be discovered on the Fibonacci sequence . Among these is
that this pattern is very visible in nature. Some of nature’s most beautiful patterns, like the spiral
arrangement of sunflower seeds, the number of petals in a flower, and the shape of a snail’s
shell—things that we looked at the surrounding or environment —all contain Fibonacci numbers.
The Golden rectangle also related to the Golden spiral, is illustrated by making adjacent squares
of Fibonacci dimensions.
In the module 2.2, Binet’s equation is also introduced for the purpose of finding any
Fibonacci number without listing down the sequence. The derivation of the explicit formula can be
browsed in the given url address or can be watched on the given youtube channel for further
understanding of the formula .

II. Desired Learning Outcomes

At the end of this module, the learner should be able to:

(1) Recognize Fibonacci number in nature and science

(2) List down the Fibonacci numbers up to nth term
(3) Construct a Golden Rectangle with Fibonacci dimensions
(4) Determine any Fibonacci number using Binet’s Formula

III. Content / Discussion

What is the Fibonacci Sequence?

The Fibonacci sequence is a series of numbers where the next value in the list is found by
adding together the preceding two values. Starting with 0 and 1, the sequence goes:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…

NOTE : Written as a rule, the mathematical expression can be defined the sequence with a recurrence
relation:
Fn = F(n-1) + F(n-2)

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

Fibonacci Sequence is named after the Italian Mathematician Leonardo of Pisa or Leonardo
Pisano , who was better known by his nickname Fibonacci . He was born in Pisa Italy in 1175
and a son of a Pisan merchant. His father was Guglielmo Bonaccion who traveled widely and
traded extensively. Fibonacci is the shortened word for the Latin term “ filius Bonacci”, which
stands for “ son of Bonaccio” .

Fibonacci introduced this sequence to European mathematics in his 1202 book Liber Abaci. It
is thought to have arisen even earlier in Indian mathematics. At a young age Fibonacci had a
passion to numbers and while growing up in North Africa, he studied knowledge of numbers
which is said to have first originated in the Hindu-Arabic arithmetic system. He wrote many
books about geometry, commercial arithmetic and irrational numbers. He also helped develop
the concept of zero.

Fibonacci's rabbits
Fibonacci first noted the sequence when pondering a mathematical problem about rabbit
breeding. Beginning with a young male and female rabbit, how many pairs of rabbits could be
born in a year? The problem assumes the following conditions:
 Begin with one male rabbit and female rabbit that have just been born.
 Rabbits reach sexual maturity after one month.
 The gestation period of a rabbit is one month.
 After reaching sexual maturity, female rabbits give birth every month.
 A female rabbit gives birth to one male rabbit and one female rabbit.
 Rabbits do not die.
This is best understood in this diagram:

After one month, the first pair is not yet at sexual maturity and can't mate. At two months, the
rabbits have mated but not yet given birth, resulting in only one pair of rabbits. After three
months, the first pair will give birth to another pair, resulting in two pairs. At the fourth month
mark, the original pair gives birth again, and the second pair mates but does not yet give birth,
leaving the total at three pair. This continues until a year has passed, in which there will be
233 pairs of rabbits.
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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

Though the rabbit question may pose completely unrealistic conditions, Fibonacci numbers do
actually appear in nature and science.

The Fibonacci spiral - Golden Rectangle

 A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares
with Fibonacci numbers for dimensions. The squares fit perfectly together because of the
nature of the sequence, where the next number is equal to the sum of the two before it. Any
two successive Fibonacci numbers have a ratio very close to the Golden Ratio, which is
roughly 1.618034. The spiral and resulting rectangle are known as the Fibonacci Rectangle or
Golden Rectangle.
 Fibonacci Rectangle - is a rectangle whose sides are consecutive Fibonacci numbers.
 Golden Rectangle- is a rectangle in which the ratio of the length to the width is the Golden
Ratio,Ø = 1.618.

Looks like this. It also has some interesting connections in the natural world

This
is a nautilus shell. The shapes of spiral galaxies, such as Messier 74,

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

hurricanes, such as Hurricane Irene, follow the Sunflower seeds grow in Fibonacci spirals.
Fibonacci sequence.

Rose petals follow and grow in Fibonacci spirals.

Uncanny Examples of the Fibonacci sequence found in Nature and Science
1. FLOWER PETALS:
The number of petals in a flower consistently follows the Fibonacci sequence. .Famous
examples include the lily and iris, which have three petals, buttercups, wild rose, larkspur,
columbine (aquilegia), pinks (shown below) which have five petals, 8 petals: delphiniums, 13
petals: ragwort, corn marigold, cineraria, some daisies, 21 petals : aster, black-eyed susan,
chicory, 34 petals: plantain, pyrethrum and so on.

Spiderwort (Tradescantia ohiensis) - Buttercups

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

Clematis : 8 petals Common Ragwort : 13 petals

2. LEAF ARRANGEMENTS

Also, many plants show the Fibonacci numbers in the arrangements of the leaves around their
stems. If we look down on a plant, the leaves are often arranged so that leaves above do not
hide leaves below. This means that each gets a good share of the sunlight and catches the most
rain to channel down to the roots as it runs down the leaf to the stem.

The sunflower here when viewed from the top shows the same pattern. It is the same plant
whose side view is above. Starting at the leaf marked "X", we find the next lower leaf turning
clockwise. Numbering the leaves produces the patterns shown here on the right.

The leaves here are numbered in turn, each exactly 0.618 of a clockwise turn (222.5°) from the
previous one.

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

3. SEED HEADS
The head of a flower is also subject to Fibonaccian processes. Typically, seeds are produced at
the center, and then migrate towards the outside to fill all the space. Sunflowers provide a
great example of these spiraling patterns. When you count the clockwise and
counterclockwise spirals in the sunflower seed head, it is interesting to know that the
numbers 34 and 55 come — which are successive numbers in Fibonacci sequence.
In some cases, the seed heads are so tightly packed that total number can get quite high — as
many as 144 or more. And when counting these spirals, the total tends to match a Fibonacci
number. Interestingly, a highly irrational number is required to optimize filling (namely one
that will not be well represented by a fraction). Phi fits the bill rather nicely.

Sunflower seeds grow in Fibonacci spirals.

4. PINECONES:
The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps
the spirals take tend to match Fibonacci numbers. Similarly, the seed pods on a pinecone are
arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiraling upwards
in opposing directions. The number of steps will almost always match a pair of consecutive
Fibonacci numbers. For example, a 3-5 cone is a cone which meets at the back after three steps
along the left spiral, and five steps along the right.

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

5. FRUITS AND VEGETABLES

Likewise, similar spiraling patterns can be found on pineapples and cauliflower. Pineapples
also have spirals formed by their hexagonal nubs. The nubs on many pineapples form eight
spirals that rotate diagonally upward to the left ,13 that rotate diagonally upward to the right,
and 21 parallel rows of scales spiraling slooply again these are consecutive Fibonacci numbers
(Aufmann, 2015). The same is also observed in the clockwise and counterclockwise spirals of
a pine cone.

Here is a picture of an ordinary cauliflower. Note how it is almost a pentagon in outline.

Looking carefully, you can see a center point, where the florets are smallest. Look again, and
you will see the florets are organized in spirals around this center in both directions.

6. TREE BRANCHES
The way tree branches form or split is an example of the Fibonacci sequence. A main trunk
will grow until it produces a branch, which creates two growth points. Then, one of the new stems
branches into two, while the other one lies dormant. This pattern of branching is repeated for each
of the new stems. A good example is the sneezewort. Root systems and even algae exhibit this
pattern.

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

7. SHELLS
Many shells, including snail shells and nautilus shells, are perfect examples of the Golden
spiral. It can result in a nesting process that can be repeated into infinity — and which takes
on the form of a spiral. It's called the logarithmic spiral, and it abounds in nature. Snail shells
and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can
also be seen in the horns of certain goats, and the shape of certain spider's webs.
Spiral galaxies: Not surprisingly, spiral galaxies also follow the familiar Fibonacci pattern. The
Milky Way has a number of spiral arms, each of which has a logarithmic spiral of roughly 12
degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can
be drawn over any spiral galaxy.
Hurricanes: Much like shells, hurricanes often display the Golden spiral.

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

8. REPRODUCTIVE DYNAMICS
If you follow the family tree of honeybees, it follows the fibonacci sequence perfectly. Males
have one parent (a female), whereas females have two (a female and male). Thus, when it
comes to the family tree, males have 2, 3, 5, and 8 grandparents, great-grandparents, gr-gr-
grandparents, and gr-gr-gr-grandparents respectively. Following the same pattern, females
have 2, 3, 5, 8, 13, and so on. And as noted, bee physiology also follows along the Golden Curve
rather nicely. It would look like this:

Lesson 2.2- Binet’s Formula

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

Considering finding the 18th and 20th term of the Fibonacci sequence.

Listing down: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,_____, ______, ______
Fn = F(n-1) + F(n-2)
F18 = F17 + F16
F18 = 987 + 1597
F18 = 2 584

Q: Since the solution above is quite tedious and tenacious, is there a more convenient way to
compute or a formula to find the nth term in the Fibonacci sequence without the need of listing
down all the numbers starting from 0 to 1 and repeatedly turn the crank in the recursive
formula?

A: YES. With the use of Binet's formula .

Binet's formula is a mathematical expression used to find the th term of the Fibonacci
sequence. It was named after Jacques Philippe Marie Binet,
a French mathematician, physicist and astronomer born in Rennes because he derived and
published the in 1843 though it was already known in the 18th century, by Leonhard Euler,
Daniel Bernoulli, and Abraham de Moivre.

If Fn is the th Fibonacci number, then the Binet’s formula in finding a term in Fibonacci
sequence specially the n is large is :

Since this formula is quite complicated and complex and even with a good calculator
you might have difficulty in finding an exact value when n is large, but there is a simplified
version of the formula that makes the computation a bit easier. In this simplified version,
the second half of the numerator which is a very small number enough to be neglected)
will be basically disregarded.

Where : the “≈” means close enough that you can round to the nearest integer.

NOTE : For the derivation of the proof of the general formula for the Fibonacci numbers,
access the following url; https://www.youtube.com/watch?v=ZiGW6PbB8wA
https://artofproblemsolving.com/wiki/index.php/Binet%27s_Formula

Illustrations: Using Binet's formula, Find F18 , F20, and F 32

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

Activity# 1
Now that you had gone through The Fibonacci Sequence you are now ready to do the
following. Submit the Word output following this format.
Name _____________________________________________ Date____________________
Course& Yr.& Section:______________________________ SCORE: ________________
Activity #1 : The Fibonacci Sequence
1. How are Fibonacci numbers expressed in nature?

2. Take 10 photographs that follow the Fibonacci number you can find in surroundings. .
Explain how each follows the Fibonacci number. Table them below.
EXPLANATION
PICTURES

3. A fib is a poem where the number of syllabus per line follow the Fibonacci sequence. Invented
by Greg Pincus, it caught on so well that the New York Times actually wrote an article about it,
which naturally spawned many more fibs. Here are the examples. Write two fibs of your own.

I Oh
think great.
these might One more
be kind of metrical
fun- let’s give them a syllable- counting
try. Do we have any takers? challenge. Haiku and tanka rules
- A. E. Stallings are all I want. Picture me counting on my fingers.

- Maryann Corbett

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

4. Lucas numbers are obtained just like fibonacci number- each term is obtained by adding two
immediate previous terms- except that the first two terms are 2 and 1. Thus the Lucas sequence
starts as 2, 1, 3,4,…

a. Give the first eighteen terms of the Lucas sequence

b. Find the ratios of each term to the preceeding term
c. Do the number approach any number?

5. Make your own “ Fibonacci like number sequence”. Name your Sequence accordingly.

A. Give the first fifteen terms of your sequence

B. Find the ratios of each term to the preceding term.
C. Do the number approach any number?

References

Aufmann, R. et.al. (2018). Mathematics in the Modern World. Rex Book Store Inc. Sampaloc
Manila Philippines
Aufmann, R. et.al. (2014). Mathematical Excursions. 3rd Edition(International Edition) Brooks
Cole.
Baltazar, et.al. (2018). Mathematics in the Modern World. C & E Publishing Inc. 839, EDSA South
Triangle, Quezon City
Camarista G, & Gabinete S. (2016). The Art of Problem Solving. Lorimar Publishing Inc. Cubao
Quezon City
Carpio, J. & Peralta B. (2018). Mathematics in the Modern World. Books Atbp. Publishing Corp.
Barangka Drive, Mandaluyong City
MathIsFun (2016). Nature, the Golden Ratio anfd Fibonacci too…
https://www.mathisfun.com/numbers/Fibonacci-sequence.html
Meisner, G. (2012). The Golden Number. https://www.goldennumber.net/nature

https://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2020.

https://www.phimatrix.com/.
https://bit.ly/goldenratiobook
https://www.youtube.com/watch?v=nt2OlMAJj6o
https://www.youtube.com/watch?v=dREpRHgkjsg
https://io9.gizmodo.com/15-uncanny-examples-of-the-golden-ratio-in-nature-5985588
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#section4
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

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 College of Information Technology, Computing and Library Science
C– GEC 4: Teaching Mathematics in the Modern World
First Semester | AY 2020-2021

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html

Congratulations for having completed this C-GEC4 Module 3!

Meet you in the next Module.

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