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MATHEMATICS IN OUR WORLD
Mathematics as the Study of Patterns )
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
MODULE 1
MODULE OVERVIEW
This module consists of two lessons: Mathematics as the Study of Patterns and Fibonacci Sequence
and Golden Ratio. Each lesson was designed as a self-teaching guide. Definitions of terms and
examples had been incorporated. Answering the problems in “your turn” will check your progress.
You may compare your answers to the solutions provided at the later part of this module for you to
be able to measure your achievement and as well as the effectiveness of the module. Individual and
group activities were prepared to apply what you had learned. Exercises were prepared as your
assignment to measure your understanding about the topics.
MODULE LEARNING OBJECTIVES
At the end of the module, you should be able to:
Identify patterns in nature and regularities in the world
Articulate the importance of mathematics in one’s life
Argue about nature of mathematics, what it is, how it is expressed, represented, and used
Express appreciation for mathematics as a human endeavor.
LEARNING CONTENTS (
Introduction
Look around you, do you notice anything that repeats or occur in a similar form?
In your life, are there any things that you tend to do over and over again? In this lesson we will
investigate patterns and regularities in nature and even in life and how mathematics come into play.
At times, consciously or unconsciously you are using mathematics in some routine transactions like
buying food, paying bills and even computing how much time do you need to come to class on time.
And you can do all of these routine effective and efficiently using your knowledge in mathematics.
You as a student taking this course, what is Mathematics for you?
Discussion
Lesson 1. Mathematics as the Study of Patterns
1.1 What is Mathematics?
Mathematics is defined as the study of numbers and arithmetic operations. Others describe
mathematics as a set of tools or a collection of skills that can be applied to questions of “how many”
or “how much”. Still, others view it as a science which involves logical reasoning, drawing
conclusions from assumed premises, and strategic reasoning based on accepted rules, laws, or
probabilities, Mathematics is also considered as an art which deals with form, size, and quantity.
In examining the development of mathematics from historical perspective, it can be seen that
much has been directed towards describing patterns of relationship that are of interest of various
individuals. Patterns arouse curiosity because they can be directly related to common human
experience. The focused of this section is, mathematics as a study of patterns.
Your turn
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
A Study of Patterns
Pattern is an arrangement which helps observers anticipate what they might see or what happens
next. Or just simply are regular, repeated, or recurring forms or designs. We see patterns around us –
layout of the floor, design of our clothes, butterflies’ wings, and even to the way we say things.
Recognizing patterns is natural to us as a rational creature because our brain is hardwired to
recognize them.
Studying patterns help you in identifying relationships and finding logical connections to
form generalizations to make predictions.
Below are examples of various patterns:
Logic Patterns. Logic patterns are usually the first to be observed. Classifying things , for
example comes before numeration. Being able to tell which things are blocks and which are not
precedes learning to count blocks.
One kind of logic pattern deals with the characteristics of various objects while another deals
with order. These patterns are seen on aptitude tests in which takers are shown a sequence of pictures
and asked to select which figure comes next among several choices.
. What comes Next?
Solution :
PA 1 The base figure rotates at an angle of
in the counterclockwise direction. Hence choice C is
the perfect match.
. What comes next?
Example 1
Example 3
Your turn
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
1 × 8 + 1 = 9 1 × 1 = 1
12 × 8 + 2 = 98 11 × 11 = 121
123 × 8 + 3 = 987 111 × 111 = 12321
1234 × 8 + 4 = 9876 1111 × 1111 = 1234321
12345 × 8 + 5 = 98765 11111 × 11111 = 123454321
123456 × 8 + 6 = 987654 111111 × 111111 = 12345654321
1234567 × 8 + 7 = 9876543 1111111 × 1111111 = 1234567654321
12345678 × 8 + 8 = 98765432 11111111 × 11111111 = 123456787654321
123456789 × 8 + 9 =? 111111111 × 111111111 =?
Have you seen the pattern? If yes, without doing calculation what do you think are the answers on
the last row?
Maybe you will agree that mathematics is the science of patterns and it’s all around us.
Recognizing number patterns is an important problem –skill. That is one reason why those who use
patterns to analyze and solve problems often find success.
Geometric Patterns. Geometric pattern is a motif or design that depicts abstract shapes like lines,
polygons, and circles, and typically repeats like a wallpaper. Visual patterns are observed in nature
and in art. In art, patterns present objects in a consistent, regular manner.
Which of the figures below can be used to continue the series?
Solution:
Since it adds up two squares horizontally and vertically on each term, the correct answer is Figure 1.
Draw a figure to continue the series below.
Example 4
Your turn
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
Word Patterns. Patterns can also be found in language like morphological rules in pluralizing
nouns or conjugating verbs for tense, as well as the metrical rules of poetry. Each of these examples
supports mathematical and natural language understanding. The focus here is patterns in form and in
syntax , which lead directly to the study of language in general and digital communication in
particular.
Fill in the blank.
knife:knives life:lives wife:______
Solution :
The pattern is taking the plural form of the words involved, so wife is wives.
Fill in the blank.
meet :met lead: led feed: ____
1.2 Patterns in Nature
Patterns in nature are the regular arrangement of objects in any form found everywhere-
plants, animals, humans, earth formations, and many others. These include symmetries, spirals,
waves, arrays, cracks, stripes, etc. Some of these patterns which recur in different context can be
modelled mathematically. So, let us start looking for more patterns in nature.
Symmetric Patterns
A figure has symmetry if there is a non-trivial transformation that maps the figure onto itself
or you can draw an imaginary line across the object and the resulting parts are mirror images of each
other.
For example, a square has a vertical line symmetry. That is , the reflection about this line
maps the square onto itself.
Notice that left and right portion of the square are exactly the same. The type of symmetry, known as
line, or bilateral symmetry, which is evident in most animals, including humans. Example is the
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
spiderwort, the angle of rotation is 120 ° while the angle of rotation of the starfish is 72 °.
Consider the image of a snowflake.
It can be observed that the patterns of snowflake repeat six times. So, what is the angle of rotation
of the snowflake?
Let us try to see more patterns in nature by watching this video.
Why do honey bees love hexagons - by Zack Patterson and Andy Peterson
https://www.youtube.com/results?search_query=why+do+honey+bees+love+hexagons,
What you’ve watch is another interesting pattern in nature , the honeycomb. According to Merriam-
Webster dictionary,”a honeycomb is a mass of hexagonal wax cells built by honeybees in their nest
to contain their brood and stores honey. “, But why build hexagonal cells? Why not squares or any
other polygons?
The video had explained it well. They love it because more area will be covered using hexagon
compared to other polygons. Hexagonal formations are more optimal in making use of avail space.
These referred to as packing problem. Packing problems involve finding the optimum method of
filling up a given space such as cubic or spherical container. The bees have instinctively found that
the best solution, evident in the hexagonal construction of their hives.
Watch this
Video
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
Let us illustrate this mathematically. Suppose you have circles of radius 1 cm, each of which
will then have an area of
π
cm
2
. We are then going to fill a plane with these circles using square
packing and hexagonal packing.
Anna
For square packing, each square will have an area of 4 cm
2
. Note from the figure that for each
square, it can fit only one circle (4 quarters). The percentage of the square’s area covered by circles
will be
areaof the circles
area of the square
× 100 %=
π cm
2
4 cm
2
× 100 %=78.54 %
Anna
Now, for the hexagonal packing, we can think of each hexagonal as composed of six equilateral
triangles with side equal to 2 cm.
The area of each equilateral triangle can be computed using the formula A =
s
2
⋅ √ 3
, so
A =
( side )
2
⋅ √ 3
( 2 cm )
2
⋅ √ 3
4 cm
2
√
=√ 3 cm
2
This gives the area of the hexagon as 6 √ 3 cm
2
(multiplying the area of the equilateral triangle to 6
as the number of sides of the hexagon). Looking at figure , there are 3 circles that could fit inside one
hexagon (the whole circle in the middle, and 6 one thirds of a circle), which gives the total areas of
3 π cm
2
. The percentage of the hexagon’s are covered by circles will be
area of the circles
area of the hexagon
× 100 %=
3 π cm
2
√
3 cm
2
× 100 %=90.69 %
Comparing the two percentages, we can clearly see that using hexagons will cover a larger area than
Your turn
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
A = 30 e
0.02 t
A = 30 e
(0.02)( 0 )
Replace
t with
t =0.
A = 30 e
0
e
0
A = 30 ( 1 )
A = 30
Therefore, the city population in 1995 was 30,000.
b. We need to find A for the year 2017. To find t , we subtract 2017 and 1995 to get t = 22 , which we
then plug in to our exponential growth model.
A = 30 e
0.02 t
A = 30 e
0.02 ( 22 )
Replace
t
with
t =22.
A = 30 e
0.0.
A = 30 (1.55271)
e
0.0.
A =46.
Therefore, the city population would be about 46,581 in 2017
The exponential growth model A = 50 e
0.07 t
describes the population of a city
in the Philippines in thousands, t years after 1997.
a. What is the population after 20 years?
b. What is the population in 2037?
Mathematics as the Study of Patterns
Pattern is an arrangement which helps observers anticipate what they might see or what happens
next. Or just simply are regular, repeated, or recurring forms or designs. Example of various patterns
are : logic patterns, number patterns, geometric patterns, word pattern. Patterns in nature are the
regular arrangement of objects in any form found everywhere-plants, animals, humans, earth
formations, and many others
Exponential Growth Model Population can be modeled by the exponential growth formula
A = P e
rt
LEARNING ACTIVITY 1
- Select a suitable figure from the four alternatives that would complete the figure matrix. Encircle
the letter corresponding to the missing pattern.
a. b.
LEARNING POINTS
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
c. d.
e. f.
- Calculate
2 + 4 + 6 + … + 2 n for
n =1,2 , ...
- Calculate 1 + 3 + 5 + … +( 2 n − 1 ) for n =1,2 , .. , 6.
- Calculate 1 + 3 + 7 + … +( 2
n
for n =1,2 , .. , 6.
- What is the missing number in each of these sequences?
a) _______, 17 , 15, 13, …
b) 8, 11, ______, 17 , …
c) 5, ______, 27, 38, ….
d) 84, _____, 76, 72,…
e) 98, 109, ______, 131, …
- Determine the pattern and find out the numbers which will complete the sequence.
a. 58, 68, 57, 67, 56 , __________
b. 3, 4, 6, 10, 18, ___________
c. 10, 54, 98, 1312, 1716 _________
- Draw Fig.5 following the given pattern.
Fig. 5
Fig. 1 Fig. 2 Fig. 3 Fig. 4
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
He first observed the pattern while investigating how fast rabbits could breed under ideal
circumstances. The problem goes like this.
“At the beginning of a month, you are given a pair of newborn rabbits. After a month the rabbits
have produced no offspring; however, every month thereafter, the pair of rabbits produces another pair of
rabbits. The offspring reproduce in exactly the same manner. If none of the rabbits dies, how many pairs of
rabbits will there be at the start of each succeeding month?”
The solution of this problem is a sequence of numbers that we now call the Fibonacci
sequence. The following figure shows the numbers of pairs of rabbits on the first day of each of the
first six months. The larger rabbits represent mature rabbits that produce another pair of rabbits each
month. The numbers in the blue region—1, 1, 2, 3, 5, 8—are the first six terms of the Fibonacci
sequence.
By definition, the first two numbers in the Fibonacci sequence are 1 and 1, and each subsequent
number is the sum of the previous two.
The position of each number in the sequence is indicated by a subscript , so that
F
1
= 1 , F
2
= 1 , F
3
and so forth , with
F
n
denoting the n th Fibonacci number.
The Fibonacci sequence exhibits the following property.
¿ F
3
F
2
F
1
1 + 2 = 3 ∨ F
4
F
3
F
2
2 + 3 = 5 ∨ F
5
= F
4
+ F
3
3 + 5 = 8 ∨ F
6
= F
5
+ F
4
The general rule is given by
F
1
= F
2
, F
n
= F
n − 1
+ F
n − 2
for n ≥ 3
Example 6
Your turn
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
To find the n
th
Fibonacci number without using recursion formula, , the following is evaluated using a
calculator.
F
n
√
n
√
n
√
This form is known as the Binet form of the Fibonacci number.
Determine the 10
th
th
and 30
th
term in a Fibonacci sequence.
Solution
F
10
= F
9
+ F
8
Using recursive formula
F
n
= F
n − 1
+ F
n − 2
For finding
F
25
and
F
30
we will now use Binet’s formula since it would take a while using the
recursive formula.
F
25
√
25
√
25
√ 5
F
30
√
30
√
30
√
Find and evaluate the following.
a. If
F
22
and
F
24
, what is
F
23
b. Find
F
50
Fibonacci numbers appears everywhere – from the leaf and flower arrangement in plants, to
the animal skin , to the scales of pineapples, and many others. The Fibonacci numbers can be found
in the growth of living things and in human beings. Let us look at the few examples.
Take a look at sunflower. In particular , pay attention to the arrangement of seeds in its head. Do you
notice they form spirals? In certain species, there are 21 spirals in clockwise direction and 34 spirals
in the counterclockwise direction.
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
Notice how the squares fit neatly in the snail shell which implies that Fibonacci numbers are very
much present in any spirals.
The Golden Ratio [ Phi = φ ]
The value approached by dividing two consecutive Fibonacci numbers, that is
Bigger F
n
Smaller F
n
is
called the Golden ratio. It is symbolized by the Greek letter Phi “ φ and is approximately equal to
1.6180339887. Below is the geometric explanation of the Golden Ratio.
Let x the longer part and y be the shorter part
x + y
We divide a line into two parts so that the longer part ÷ the smaller part = the whole length ÷ longer
part will have the Golden ratio.
x
y
x + y
x
= φ ≈ 1.6180339887 ….
Let us investigate the ratio of two adjacent Fibonacci numbers as
n becomes large
n
F
n
F
n − 1
n
F
n
F
n − 1
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
It is interesting to note that the ratio of two adjacent Fibonacci numbers approaches
the golden ratio ; that is
F
n
F
n − 1
As seen in the preceding discussion, Fibonacci numbers appears in many places. The
golden ratio does too. It shows up in art, architecture, music and nature. For example , the ancient
Greeks thought that rectangles whose side form a golden ratio were pleasing to look.
Many buildings and artworks follow golden ratio such as Parthenon in Greece, but it is not really
known if they are designed that way.
The Parthenon
Leonardo da Vinci has incorporated geometry in many of his paintings, with the golden ratio just
being one of his mathematical tools. Experts agree that he probably thought that the golden ratio
made his paintings more attractive. Below are
just some
of his
artwork.
The Mona
Lisa
The
Vitruvian
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
try. Do we have any takers?
Oh
great.
One more
metrical
syllable-counting
challenge. Haiku and tanka rules
are all I want. Picture me counting on my fingers.
Group Activity
Are you Golden?
This group activity will further your appreciation of Fibonacci sequence and the Golden ratio,
enhance your prediction and measurement skills, and enjoy mathematics. Each group should
compose of five members.
Direction: List down several body measurements which you suspect have golden or nearly golden.
Determine how many people in your group are golden or nearly golden.(See Activity Sheet on the
next page)
Materials. Activity sheets with direction, meter stick/tape measure, and calculator.
Activity Sheet
LEARNING ACTIVITY 3
Study Guide in Mathematics in the Modern World
GE 7 Mathematics in the Modern World Module 1 : Mathematics in our World
Take note of the data you gathered here. Express each ratio R up to 3 –decimal places.
Name A/B C/D E/F G/H I/J K/L M/N O/P Are you
Golden?
Chief
Learner:
Fraction
Decima
l
V. Chief
Learner:
Fraction
Decima
l
Recorder: Fraction
Decima
l
Analyzer: Fraction
Decima
l
Prompter: Fraction
Decima
l
On each team member , encircle the ratios 1.500 ≤ R ≤ 1.800_._ `(Greater than of equal to 1.5 but less
than or equal to 1..
Interpretation:
Golden : if s/he has at least five highlighted ratios within 1.500 ≤ R ≤ 1.
Nearly Golden : if s/he has at most three or four highlighted ratios within 1.500 ≤ R ≤ 1.
Far from Golden : if s/he has at most two highlighted ratios within
1.500 ≤ R ≤ 1.
Findings (in paragraph form): Who among your group members is golden or nearly golden?
Which body parts are the most golden?
Conclusion:
Reflection:
LEARNING ACTIVITY 4
Use the concepts of Fibonacci sequence and Golden Ratio which we learned in this module to help
