MATHEMATICS IN THE MODERN WORLD

(GE – 701)

MAY FLOR L. TAPOT, MST

Compiler/Editor

October 2020
Sultan Kudarat State University
May Flor L. Tapot, MST
FOREWORD

“Without Mathematics, there’s nothing you can do.

Everything around you is Mathematics.
Everything around you is numbers.”
-Shakuntala Devi
Mathematics is needed at every step of our life, and we cannot live
without it. Mathematics made our life easier and uncomplicated. That is why,
this module developed out of love, compassion, and hardwork for the student
to really understand and appreciate the simpleness of mathematics.

The content of this module begins with the nature of Mathematics that
lead to appreciation and connects the importance of Mathematics to one’s
student life.

Moreover, it includes Mathematical languages that may help better

understand the beauty and elegance of Mathematics. The book also covers
the reasoning and problem solving to develop the critical thinking skills of the
21st century learners.

It also incorporates the basic statistical concepts such measures of

central tendency, measures of relative position, measures of variation and
others so that students may develop research skills through analyzing and
interpreting data.

The book contains geometric design to develop the artistic aspects of

the students in creating designs ad promoting one’s own culture. Mathematics
of finance is also added intended for the students on how to manage their own
finances.

Each lesson in this module is design for you dear students to help you
appreciate the importance of learning Mathematics. This is also developed to
give you opportunity to express your own ideas through various exercises.

Hope you enjoy reading, and answering this module.

May Flor L. Tapot, MST

Compiler/Editor

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TABLE OF CONTENTS
Chapter 1 THE NATURE OF MATHEMATICS………………………. 1
Lesson 1: Mathematics in Our World………………….. 2
Patterns and Numbers in Nature and the Regularities in
the World……………………………………………………..
3
The Fibonacci Sequence……………………………………
6
Importance of Mathematics in Life…………………………
8
Nature of Mathematics………………………………………
13
Role of Mathematics in Other Discipline…………………..
15
Appreciating Mathematics as a Human Endeavor……….
21
Summary…………………………………………………….
25
Assessment Tasks………………………………………….
26
Lesson 2: Mathematical Language and Symbols……. 31
Language, Symbols, and Conventions of Mathematics… 32
Conversion of English expression to Mathematical
Sentences and vice-versa………………………………….
Conversion of Algebraic Expressions to English 32
Sentences……………………………………………………
The Four Basic Concepts of Mathematics………………. 35
Set…………………………………………………………… 36
Relation……………………………………………………… 36
Function…………………………………………………..…. 40
Binary Operation……………………………………………. 40
Elementary Logic…………………………………………… 42
Summary……………………………………………………. 43
Assessment Tasks…………………………………………. 46
47
Lesson 3: Problem Solving and Reasoning………….. 55
Reasoning………………………………………….………… 55
Mathematical Proofs……………………………………….. 58
Polya’s 4-Steps in Problem Solving……………………… 61
Summary..…………………………………………………... 65

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TABLE OF CONTENTS
Assessment Tasks………………………………………….. 66
Chapter 2 Mathematics as a Tool…………………………………… 72
Lesson 1: Data Management…………………………… 72
Basic Statistical Concepts…………………………………. 73
Measures of Central Tendency…………………………… 78
Measures of Relative Position……………………………. 86
Measures of Variation……………………………………… 91
Normal Distribution…………………………………………. 96
Linear Regression and Correlation ……………………….. 102
Summary..…………………………………………………... 107
Assessment Tasks………………………………………….. 111
Chapter 3 Mathematics as a Tool……………………………………. 122
Lesson 1: Geometric Designs…………………………… 123
What is Geometric Design…………………………………. 124
Mindanao Designs, Arts and Culture……………………… 136
Summary..…………………………………………………... 140
Assessment Tasks………………………………………….. 141
Lesson 2: The Mathematics of Finance……………….. 143
Simple and Compound Interest…………………………… 143
Credit Card vs Consumer Loans…………………………. 155
Stocks, Bonds, and Mutual Funds……………………….. 156
Home Ownership…………………………………………… 158
Summary..…………………………………………………... 159
Assessment Tasks………………………………………… 161
References………………………………………………….. 165
Appendices………………………………………………… 167

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Chapter 1. The Nature of Mathematics

Overview

Mathematics relies on both logic and creativity, and it is pursued both for
a variety of practical purposes and for its intrinsic interest. It reveals hidden
patterns that help us understand the world around us. For some people, and
not only professional mathematicians, the essence of mathematics lies in its
beauty and its intellectual challenge. For others, including many scientists and
engineers, the chief value of mathematics is how it applies to their own work.
Because mathematics plays such a central role in modern culture, some basic
understanding of the nature of mathematics is requisite for scientific literacy. To
achieve this, students need to perceive mathematics as part of the scientific
endeavor, comprehend the nature of mathematical thinking, and become
familiar with key mathematical ideas and skills.

This chapter focuses on identifying mathematical patterns and numbers

found in nature, it’s importance and how to appreciate mathematics as a human
endeavour. It also deals with translating mathematical expressions to English
language and vice versa; and discusses the four basic concepts of mathematics
applying elementary logic, reasoning and following Polya’s 4 – steps strategies
in problem solving.

General Objectives:

This chapter focuses on identifying, describing and performing

numerous applications of mathematics in real – life; translates mathematical
expression to English language and vice versa; how to use different types of
reasoning; writing proofs; and using Polya’s 4 – steps of strategies in solving
problems.

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Lesson 1: Mathematics in Our World

Mathematics usefulness in this modern world is necessary. Mathematics

used in different fields to calculate effectively the results of different activities,
predict the behavior of a variable when the other variables are known, identify
fully well the requirements of a particular dosage of medicine to cure a certain
illness verify whether a specific solution is applicable to general set-ups
ascertain the chronology of events in the past identify patterns of situations;
and many more.
Like any other languages, mathematics has its own symbols, syntax, and
rules characterized a precise, concise, and powerful mathematical language. It
distinguishes expressions from sentences. It discusses the conventions in the
mathematical language. It gives emphasis on four basic concepts: sets,
functions, relations, binary operations. It includes elementary logic,
connectives, quantifiers, negation and variables with formality.
Mathematics is not just about numbers. Much of it is problem solving and
reasoning-inductive and deductive. It also discusses intuition, proof, and
certainty. It utilizes Polya’s 4-steps in problem solving, varied problem-solving
strategies, mathematical problems involving patterns and recreational
problems using mathematics.

Specific objectives:
At the end of this lesson, students are expected to:
1. Identify patterns in nature and regularities in the world;
2. Articulate the importance of mathematics in your life;
3. Argue about the nature of mathematics, what it is, how it is expressed,
represented, and used; and
4. Express appreciation for mathematics as a human endeavor.

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I. Patterns and Numbers in Nature and the Regularities in the World

Patterns and counting are correlative. Counting happens when there is

pattern. When there is counting, there is logic. Consequently, pattern in nature
goes with or logical set-up. There are reasons behind a certain pattern. That’s
why, oftentimes, some people develop an understanding of patterns,
relationships, and functions and use them to represent and explain real-world
phenomena. Most people say that mathematics is the science behind patterns,
mathematics exists everywhere as patterns do in nature. Not only do patterns
take many forms within the range of school mathematics, they are also a
unifying mechanism.
In this world, a regularity (Collins, 2018), is the fact that same thing
always happens in the same circumstances. While a pattern is a discernable
regularity in the world or in a man-made design. As such, the elements of a
pattern repeat in predictable manner. Patterns in nature (Wikipedia) are visible
regularities of form found in the natural world. These patterns recur in different
contexts and can sometimes be modelled mathematically.

Some examples of Patterns in Nature

Symmetry (Wikipedia) means agreement in dimensions,

due proportion and arrangement. In everyday language, it
refers to a sense of harmonious and beautiful proportion
and balance. In mathematics, “symmetry” means that an
object is invariant to any of various transformations
including reflection, rotation or scaling.

A spiral is a curve which emanates from a point, moving

farther away as it revolves around the point. Cutaway of
nautilus shell shows the chambers arranged in an
approximately logarithmic spiral.

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A meander is one of a series of regular sinuous curves,

bends, loops, turns, or windings in the channel of a river,
stream, or other watercourse. It is produced by a stream
or river swinging from side to side as it flows across its
floodplain or shifts its channel within a valley.

A wave is a disturbance that transfers energy through

matter or space, with little or no associated mass
transport. Waves can be seen crashing on a beach, at
the snap of a rope or sound traveling through a speaker.

Foam is a substance formed by trapping pockets of gas

in a liquid or solid. A bath sponge and the head on a glass
of beer are examples of foams. Young children love to
blow bubbles or play with bubbles. They are playing with
one of the fundamental patterns in the natural world.

A tessellation of a flat surface is the tiling of plane using

one or more geometric shapes, called tiles, with no
overlaps and no gaps. In mathematics, tessellations can
be generalized to higher dimensions and a variety of
geometries.

A fracture or crack is the separation of an object or

material into two or more pieces under the action of
stress. The fracture of a solid usually occurs due to the
development of certain displacement discontinuity
surfaces within the solid.

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Stripes are made by a series of bands or strips, often

of the same width and color along the length.

A fractal is a never-ending pattern. Fractals are

infinitely complex patterns that are self-similar across
different scales. They are created by repeating a
simple process over and over in an ongoing feedback
loop.

Affine Transformations – these are the processes of

rotation, reflection and scaling. Many plant forms
utilize these processes to generate their structure.

Let us look at the things that surround us. What numbers do we see? Do they
establish a pattern? If they do, what is the pattern?

SITUATIONS NUMBERS PATTERNS

1. The number 46 – 1 46 – 1. This refers to the block number and
where your house the lot number. This indicates that the
is situated. house number after you is 46 – 3, and the
house adjacent to you is 46 – 2.
234
234. This indicates that your house is
marked the 234th and the house next to you
III-A is 235th, and so on.

III-A. This house is situated first on the third

block.

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2. The final 86 86, 75, and 99. These grades represent the
grades shown in 75 outcome of a student’s performance using
your Grade 12 99 specific criteria.
subjects.
3. The feast day Every 3rd This has no specific date since the 3rd
of the Infant Sunday of Sunday of January may fall on the 15th, 16th,
Jesus in Shrine January or even the 17th.
Hills in Davao
City

The Fibonacci Sequence

George Dvorsky (2013) highlighted that the famous Fibonacci sequence

has captivated mathematicians, artists, designers, and scientists for centuries.
Also known as the Golden Ratio, its ubiquity and astounding functionality in
nature suggests its importance as a fundamental characteristic of the universe.
Leonardo Fibonacci came up with the sequence when calculating the ideal
expansion pairs of rabbits over the course of one year. Today, its emergent
patterns and ratios (phi = 1.61803… ) can be seen from the microscale to the
macroscale, and right through to biological systems and inanimate objects.
While the Golden Ratio doesn’t account for every structure or pattern in the
universe, it’s certainly a major player. Here are some examples:

1. 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.

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2. Pine cones
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.

3. Tree branches
The Fibonacci sequence can also be seen in
the way tree branches form or split. A main trunk will
grow until it produces a branch, which creates two
growth points.

4. Shells
The unique properties of the Golden
Rectangle provide another example. This shape, a
rectangle in which the ratio of the sides a/b is equal
to the golden mean (phi), 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.

5. Spiral Galaxies and Hurricane

Not surprisingly, spiral galaxies also follow
the familiar Fibonacci pattern. The Milky Way has
several spiral arms, each of them a logarithmic
spiral of about 12 degrees. As an interesting aside,
spiral galaxies appear to defy Newtonian physics.

The Fibonacci sequence is an array of numbers that given two terms,

the next term is determined by adding the given terms. Mathematically,
fn = fn-1 + fn-2

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Though a little bit confusing, it simply means that fn can be determined if
the previous two terms, fn-1 and fn-2 are added. Ergo, we only need to have two
numbers as the first two terms in order to get the third, fourth, fifth terms.
Consider the following as examples.
Given Terms Expansion Explanation
f0 = 0, f1 = 1 fn = 0,1,1,2,3,5,8,13,... 0+1 = 1
1+1 =2
1+2=3
2+3=5
3+5=8
5+8=13, and so on …
f0 = 1, f1 = 3 fn = 1,3,4,7,11,18,29,… 1+3=4
3+4=7
4+7=11
7+11=18
11+18=29, and so on…
f0 = 2, f1 = 2 fn = 2,2,4,6,10,16,26,… 2+2=4
2+4=6
4+6=10
6+10=16
10+16=26, and so on…

II. Importance of Mathematics in Life

According to Katie Kim (2015) Math is a subject that makes students

either jump for joy or rip their hair out. However, math is inescapable as you
become adult in the real world. From calculating complicated algorithms to
counting down the days till the next Game of Thrones episode, math is versatile
and important, no matter how hard it is to admit. Before you decide to doze off
in math class, consider this list of reasons why learning math is important to
you and the world.

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1. Making Routine Budgets
How much should I spend today? When I will be able to buy a
new car? Should I save more? How will I be able to pay my EMIs?
Such thoughts usually come in our minds. The simple answer to
such type of question is maths. We prepare budgets based on
simple calculations with the help of simple mathematical
concepts. So, we can’t say, I am not going to study maths ever!
Everything which is going around us is somehow related to maths
only.

2. Construction Purpose
Mathematics is the basis of any construction work. A lot of
calculations, preparations of budgets, settings targets, estimating
the cost, etc., are all done based on maths.

3. Exercising and Training

Setting routine according to workout schedule, count the number
of repetitions while exercising, etc., just based on maths.

4. Interior Designing
Interior designers plan the interiors based on area and volume
calculations to calculate and estimate the proper layout of any
rom or building. Such concepts form an important part of maths.

5. Fashion Designing
Just like the interior designing, maths is also an essential concept
of fashion designing. From taking measurements, estimating the
quantity and quality of clothes, choosing the color theme,
estimating the cost and profit, to produce cloth according to the
needs and tastes of the customers, math is followed at every
stage.

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6. Shopping at Grocery Stores and Supermarkets
The most obvious place where you would see the application of
basic mathematical concepts is your neighborhood grocery store
and supermarket. The schemes like ‘Flat 50% off’, ‘Buy one get
one free’, etc., are seen on most of the stores. Customers visit the
stores, see such Schemes, estimate the quantity to be bought,
the weight, the price per unit, discount calculations, and finally the
total price of the product and buy it. The calculations are done
based on basic mathematical concepts. Thus, here also, maths
forms an important part of our daily routine.

7. Cooking and Baking

For cooking or baking anything, a series of steps are followed,
telling us how much of the quantity to be used for cooking, the
proportion of different ingredients, methods of cooking, the
cookware to be used, and many more. Such are based on
different mathematical concepts. Indulging children in the kitchen
while cooking anything, is a fun way to explain maths as well as
basic cooking methods.

8. Sports
Maths improves the cognitive and decision-making skills of a
person. Such skills are very important for a sport person because
by this he can take the right decisions for his team. If a person
lacks such abilities, he won’t be able to make correct estimations.

9. Management of Time
Now managing time is one of the most difficult tasks which is
faced by a lot of people. An individual wants to complete several
assignments in limited time. Not only the management, some
people are not even able to read the timings on an analog clock.
Such problems can be solved only by understanding the basic
concepts of maths. Maths not only help us to understand the
management of time but also to value it.
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10. Driving
‘Speed, Time, and distance’ all these three things ae studied in
mathematical subjects, which are the basics of driving
irrespective of any mode of transportation. Maths helps us to
answer the following questions;
▪ How much should be the speed to cover any particular
distance?
▪ How much time would be taken?
▪ Whether to turn left or right?
▪ When to stop the car?
▪ When to increase or decrease the speed?

11. Automobiles Industry

The different car manufacturing companies produce cars based
on the demands of the customers. Every company has its
category of cars ranging from microcars to luxuries SUVs. In such
companies, basic mathematical operations are being applied to
gain knowledge about the different demands of the customers.

12. Computer Applications

The fields of mathematics and computing intersect both in
computer science. The study of computer applications is next to
impossible without maths. The concepts like computation,
algorithms, and many more forms the base for different computer
applications like powerpoint, word, excel, etc. are impossible to
run without maths.

13. Planning Trip

We all are bored of our monotonous like and we wish to go for
long vacations. For this, we have to plan things accordingly. We
need to prepare the budget for the trip, the number of days, the
destinations, hotels, adjusting our other work accordingly, and
many more. Here comes the role of maths. Basic mathematical

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concepts and operations are required to be followed to plan a
successful trip.

14. Hospitals
Every hospital has to make the schedule of the timings of the
doctors available, the systematic methods of conducting any
major surgery, keeping the records of the patients, records of
success rate of surgeries, number of ambulances required,
training for the use of medicines to nurses, prescriptions, and
scheduling all tasks, etc. All these are done based on
Mathematical concepts.

15. Weather Forecasting

The weather forecasting is all done based on the probability
concept of maths. Through this, we get to know about the weather
conditions like weather it’s going to be a sunny day or rainfall will
come.

16. Base of Other Subjects

Though maths is itself a unique subject. But, you would be
surprised to know that it forms the base for every subject. The
subjects like physics, chemistry, economics, history,
accountancy, statistics, in fact, every subject is based upon
maths.

17. Music and Dance

Listening to music and dancing is one of the most common
hobbies of children. Here also, they learn maths while singing and
learning different dance steps. The coordination in any dance can
be gained by simple mathematical steps.

18. Manufacturing Industry

The parts of maths called ‘Operations Research’ is an important
concept which is being followed at every manufacturing unit. This
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concept of maths gives the manufacturer a simple idea of
performing the number of tasks under the manufacturing unit
such: quantity to be produced, methods to be followed, increase
production, and cost of production.

19. Planning of Cities

Urban planning all includes the concepts of budgeting, planning,
setting targets, and many more which all forms the part of
mathematics. No activity is possible without mathematics.

20. Problem-solving Skills

Problem-solving skills is one of the most important skills which
every individual should possess to be successful in life. Such
skills help the individual in taking correct decisions in life, let it be
professional or personal. This is all done when the person has the
correct knowledge of basic mathematical concepts.

21. Marketing
The marketing agencies make the proper plans as to how to
promote any product or service. The tasks like promoting a
product online, use of social media platforms, following different
methods of direct and indirect marketing, door to door sales,
sending e-mails, making call, providing the number of schemes
‘Buy one get one free’, Flat 50% off’, offering discounts on special
occasions, etc. are all done on the basis of simple mathematical
concepts. Thus, mathematics is present everywhere.

III. Nature of Mathematics

There are creative tensions in mathematics between beauty and utility,

abstraction and application, between a search for unity and a desire to treat
phenomena comprehensively. Keith Devlin has called mathematics a “science
of patterns”, which ties in with the ideas of beauty, abstraction and the search
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for unity. He has also said that “mathematics makes the invisible visible”,
referring to representation. Modeling and application of mathematics.

1. Patterns and Relationships

Mathematics is the science of patterns and relationship. As a
theoretical discipline; mathematics explores the possible
relationships among abstractions without concern for whether
those abstractions have counterparts in the real world. The
abstractions can be anything from strings of numbers to
geometric figures to sets of equations.

2. Mathematics, Science and Technology

Mathematics is abstract. Its function goes along well with Science
and Technology. Because of its abstractness, mathematics is
universal in a sense that other fields of human thought are not. It
finds useful applications in business industry, music, historical
scholarship, politics, sports, medicine, agriculture, engineering,
and the social and natural sciences.

3. Mathematical Inquiry
Normally, people are confronted with problems. In order to live at
peace, these problems must be solved. Using mathematics to
express ideas or to solve problems involves at least three phases:
(1) representing some aspects of things abstractly, (2)
manipulating the abstractions by rules of logic to find new
relationships between them, and (3) seeing whether the new
relationships say something useful about the original things.

4. Abstraction and Symbolic Representation

Mathematical thinking often begins with the process of
abstraction – that is noticing a similarity between two or more
objects or events. Aspects that they have in common, whether
concrete or hypothetically, can be represented by symbols such
as numbers, letters, other marks, diagrams, geometrical
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constructions, or even words. Whole numbers are abstractions
that represent the size of sets of things and events or the other of
things within a set. The circle as a concept is an abstraction
derived from human faces, flowers, wheels, or spreading ripples;
the letter A may be an abstraction for the surface area of objects
of any shape. Abstractions are made not only from concrete
objects or processes; they can also be made from other
abstractions, such as kinds of numbers (the even numbers, for
instance).

5. Manipulating Mathematical Statements

After abstractions have been made and symbolic representations
of them have been selected, those symbols can be combined and
recombined in various wats according to precisely defined rules.
Typically, strings of symbols are combined into statements that
express ideas or propositions.

6. Applications
Mathematical processes can lead to a kind of model of a thing,
from which insights can be gained about the thing itself. Any
mathematical relationships arrived at by manipulating abstract
statements may or may not convey something truthful about the
thing being modeled.

IV. Role of Mathematics in other Disciplines

Mathematics is offered in any college course. It is found in every

curriculum because its theories and applications are needed in any workplace.
That’s why students can’t stay away from attending math classes. There has to
be mathematics in the real world. This subject always bring life to any person
or professional. Every second of the day needs mathematical knowledge and
skills to perform academic activities and office routines. If ordinary people have

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to use math, then much more for students to know and master it so they will
succeed in class in the school.
A famous Jain Mathematician, 𝐴̅c𝑎̅rya Mah𝑎̅vira (19th century) write that
“What is good of saying much in vain? Whatever there is is in all three worlds,
which are possessed of moving and non-moving being all that indeed cannot
exist as apart from Mathematics.”
Here are some main disciplines in which Role of mathematics is widely
accepted:

1. Physical Sciences
In mathematical physics, some basic axioms about mass,
momentum, energy, force, temperature, heat, etc. are abstracted,
from observations and physical experiments and then the
techniques of abstraction, generalization and logical deduction
are used. It is the branch of mathematical analysis that
emphasizes tools and techniques of particular use to physicists
and engineers.

2. Fluid Dynamics
Understanding the conditions that result in avalanches, and
developing ways to predict when they might occur, uses an area
of mathematics called fluid mechanics. Many mathematicians and
physicists applied the basic laws of Newton to obtain
mathematical models for solid and fluid mechanics. This is one of
the most widely applied areas of mathematics, and is also used
in understanding volcanic eruptions, flight, ocean currents.

3. Computational Fluid Dynamics

It is a discipline wherein we use computers to solve the Navier-
Stokes equations for specified initial and boundary condition for
subsonic, transonic, hypersonic flows.

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4. Physical Oceanography
Problems of waves, tides, cyclones flow in bays and estuaries,
the effect of efflux of pollutants from nuclear and other plants in
sea water, particularly in fish population in the ocean are
important for study. From defense point of view, the problem of
under-water explosions, the flight of torpedoes in water, the
sailing of ships and submarines are also important.

5. Chemistry
Math is extremely important in physical chemistry especially
advanced topics such as quantum or statistical mechanics.
Quantum relies heavily on group theory and linear algebra and
requires knowledge of mathematical/physical topics such as
Hilbert spaces and Hamiltonian operators. Statistical mechanics
relies heavily on probability theory.

6. Biological Sciences
Biomathematics is a rich fertile field with open, challenging and
fascination problems in the areas of mathematical genetics,
mathematical ecology, mathematical neuron-physiology,
development of computer software for special biological and
medical problems, mathematical theory of epidemics, use of
mathematical programming and reliability theory in biosciences
and mathematical problems in biomechanics, bioengineering and
bioelectronics.

7. Social Sciences
Disciplines such as economics, sociology, psychology, and
linguistics all now may extensive use of mathematical models,
using the tools of calculus, probability, and game theory, network
theory, often mixed with a healthy dose of computing.

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8. Economics
In economic theory and econometrics, a great deal of
mathematical work is being done all over the world. In
econometrics, tools of matrices, probability and statistics are
used. A great deal of mathematical thinking goes in the task of
national economic planning, and a number of mathematical
models for planning have been developed.

9. Actuarial Science, Insurance and Finance

Actuaries use mathematics and statistics to make financial sense
of the future. For example, if an organization is embarking on a
large project, an actuary my analyze the project, assess the
financial risks involved, model the future financial outcomes and
advise the organization on the decisions to be made. Much of
their work is on pensions, ensuring funds stay solvent long into
the future, when current workers have retired. They also work in
insurance, setting premiums to match liabilities.
Mathematics is also used in many other areas of finance from
banking and trading on the stock market, to producing economic
forecasts and making policy.

10. Psychology and Archaeology

Mathematics is even necessary in many of the social sciences,
such as psychology and archaeology. Archaeologists use a
variety of mathematical and statistical techniques to present the
data from archaeological surveys and try to distinguish patterns
in their results that shed light on past human behavior. Statistical
measures are used during excavation to monitor which pits are
most successful and decide on further excavation. Finds are
analyzed using statistical and numerical methods to spot patterns
in the way the archaeological record changes over time, and
geographically within a site and across the country.
Archaeologists also use statistics to test the reliability of their
interpretations.
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11. Mathematics in Social Network
Graph theory, text analysis, multidimensional scaling and cluster
analysis, and a variety of special models are some mathematical
techniques used in analyzing data on a variety of social networks.

12. Political Science

In Mathematical Political Science, they analyze past election
results to see changes in voting patterns and the influence of
various factors on voting behavior, on switching of votes among
political parties and mathematical models for Conflict Resolution.

13. Mathematical Linguistics

The concepts of structure and transformation are as important for
linguistic as they are for mathematics. Development of machine
languages and comparison with natural and artificial language
require a high degree of mathematical ability. Information theory,
mathematical biology, mathematical psychology, etc. are all
needed in the study of Linguistics. Mathematics has had a great
influence on research in literature. In deciding whether a given
poem or essay could have been written by a particular poet or
author, we can compare all the characteristics of the given
composition with the characteristics of the poet or other works of
the author with the help of a computer.

14. Mathematics in Music

Calculations are the root of all sorts of advancement in different
disciplines. The rhythm that we find in all music notes is the result
of innumerable permutations and combinations of SAPTSWAR.
Music theorists often use mathematics to understand musical
structure and communicate new ways of hearing music. This has
led to musical applications of set theory, abstract algebra, and
number theory. Music scholars have also used mathematics to
understand musical scales, and some composers have

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incorporated the Golden ratio and Fibonacci numbers into their
work.

15. Mathematics in Management

Mathematics in management is a great challenge to imaginative
minds. It is not meant for the routine thinkers. Different
Mathematical models are being used to discuss management
problems of hospitals, public health, pollution, educational
planning and administration and similar other problems of social
decisions. In order to apply mathematics to management, one
must know the mathematical techniques and the conditions under
which these techniques are applicable.

16. Mathematics in Engineering and Technology

Mathematics has played an important role in the development of
mechanical, civil, aeronautical and chemical engineering through
its contributions to mechanics of rigid bodies, hydro-dynamics,
aero-dynamics, heat transfer, lubrication, turbulence, elasticity,
etc. It has become of great interest to electrical engineers through
its applications to information theory, cybernetics, analysis and
synthesis of networks, automatic control systems, design of
digital computers etc. The new mathematical sciences of
magneto-hydrodynamic generates and for experiments in
controlled nuclear fusion.

17. Mathematics in Computers

An important area of applications of mathematics is in the
development of formal mathematical theories related to the
development of computer science. Now most applications of
Mathematics to science and technology today are via computers.
The foundation of computer science is based only in
mathematics. It includes logic, relations, functions, basic set
theory, countability and counting arguments, proof techniques,
mathematical induction, graph theory, combinatorics, discrete
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probability, recursion, recurrence relations, and number theory,
computer-oriented numerical analysis, Operation Research
techniques, modern management techniques like Simulation,
Monte Carlo program, Evaluation Research Technique, Critical
Path Method, Development of new computer languages, study of
Artificial Intelligence, Development of automata theory etc. All
mathematical processes of use in applications are being rapidly
converted into computer package algorithms. There are computer
packages for solution of linear and non linear equations,
inversions of matrices, solution of ordinary and partial differential
equations, for linear, non linear and dynamic programming
techniques, for combinatorial problems and for graph
enumeration and even for symbolic differentiation and integration.

V. Appreciating Mathematics as a Human Endeavor

In order to appreciate mathematics much better, every person should

have the thorough understanding of the discipline as a human endeavor.
Mathematics brings impact to the life of a learner, worker, or an ordinary man
in society. The influences of mathematics affect anyone for a lifetime.
Mathematics works in the life of all professionals.
Mathematics is appreciated as human endeavor because all
professional and ordinary people apply its theories and concepts in the office,
laboratory, and marketplace. According to Mark Karadimos (2018), the
following professions use Mathematics in their scope and field of work:

Accountants
Assist businesses by working on their taxes and planning for
upcoming years. They work with tax codes. And forms, use formulas for
calculating interest, and spend a considerable amount of energy
organizing paperwork.

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Agriculturists
Determine the proper amounts of fertilizers, pesticides, and water
to produce bountiful amounts of foods. They must be familiar with
chemistry and mixture problems.

Architects
Design building for structural integrity and beauty. They must
know how to calculate loads for finding acceptable materials in design
which involve calculus.

Biologists
Study nature to act in concert with it since we are very closely tied
to nature. They use proportions to count animals as well as use
statistics/probability.

Chemists
Find ways to sue chemical to assist people in purifying water,
dealing with waste management. Researching superconductors,
analyzing crime scenes, making food products and in working with
biologists to study the human body.

Computer Programmers
Create complicated sets of instructions called programs/software
to help us use computers to solve problems. They must have strong
sense of logic and have critical thinking and problem-solving skills.

Engineers
Build products/structure/systems like automobiles, buildings,
computers, machines and planes, to name just a few examples. They
cannot escape the frequent use of a variety of calculus.

Lawyers
Argue cases using complicated lines of reason. That skill is
nurtured by high level math courses. They also spend a lot of time
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researching cases, which means learning relevant codes, laws and
ordinances. Building cases demands a strong sense of language with
specific emphasis on hypotheses and conclusions.

Managers
Maintain schedules, regulate worker performance, and analyze
productivity.

Medical Doctors
Must understand the dynamic systems of the human body. They
research illnesses, carefully administer the proper amounts of medicine,
read charts/tables, and organize their workload and manage the duties
nurses and technicians.

Meteorologists
Forecast the weather for agriculturists, pilots, vacationers, and
those who are marine-dependent. They read amps, work with computer
models, an understand the mathematical laws of physics.

Military Personnel
Carry out a variety of tasks ranging from aircraft maintenance to
following detailed procedures. Tacticians utilize a branch of mathematics
called linear programming.

Nurses
Carry out the detailed instructions doctors given them. They
adjust intravenous drip rates, take vitals, dispense medicine, and even
assist in operations.

Politicians
Help solve the social problems of our time by making complicated
decisions within the confines of the law, public opinion, and budgetary
restraints.

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Salespeople
Typically work on commission and operate under a buy low, sell
high profit model. Their job requires good interpersonal skills and the
ability to estimate basic math problems without the need of paper/pencil.

Technicians
Repair and maintain the technical gadgets we depend on like
computers, televisions, DVDs, cars, refrigerators. They always read
measuring devices, referring to manuals, and diagnosing system
problems.

Tradesmen
(Carpenters, electricians, mechanics, and plumbers) estimate job
costs and use technical math skills specific to their field. They deal with
slopes, areas, volumes, distances, and must have an excellent
foundation in math.

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Summary

✓ A regularity (Collins, 2018), is the fact that same thing always happens
in the same circumstances.
✓ A pattern is a discernable regularity in the world or in a man-made
design. As such, the elements of a pattern repeat in predictable manner.
✓ Patterns in nature (Wikipedia) are visible regularities of form found in the
natural world. These patterns recur in different contexts and can
sometimes be modelled mathematically.
✓ Some examples of Patterns in Nature are: symmetry, spiral, meander,
wave, foam, tessellation, fracture or cracks, stripes, fractal, and affine
transformation.
✓ Examples of Fibonacci sequence: seed heads, pine cones, tree
branches, shells, spiral galaxies and hurricane.
✓ The Fibonacci sequence is an array of numbers that given two terms,
the next term is determined by adding the given terms. Mathematically,
✓ fn = fn-1 + fn-2

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ASSESSMENT TASK
Exercise 1.1 – a
Mathematics in Our World
Name: __________________________ Program & Section: __________
Date: ____________________ Score: _________
A. Identification. Write the correct word on the blank provided in the right
that is being referred to in the following:
1 A series of regular sinuous curves, bends, loops,
turns, or windings in the channel of a river, stream,
or other watercourse.
2 A disturbance that transfers energy through matter
or space, with little or no associated mass
transport.
3 A substance formed by trapping pockets of gas in
a liquid or solid.
4 The tiling of a plane using one or more geometric
shapes with no overlaps and gaps.
5 A curve which emanates from a point, moving
farther away as it revolves around the point.
6 An agreement in dimensions, due proportion, an
arrangement.
7 This occurs due to the development of certain
displacement discontinuity surfaces within the
solid.
8 Is characterized by the fact that every number after
the first two is the preceding ones.
9 A series of bands or strips, often of the same width
and color along the length.
10 The infinitely complex patterns that are self-similar
across different scales.
B. Go to www.youtube.com and search for “Fibonacci Sequence” and
watch it. Write an essay of about 250 words regarding your insights into
the video. (Write your answer on the back of this paper.)
C. Make a Fibonacci sequence of ten terms out of the following given first
two terms.
a. f0 = 2, f1 = 3 d. f0 = 3, f1 = 6
b. f0 = 5, f1 = 6 e. f0 = 4, f1 = 5
c. f0 = 5, f1 = 7

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ASSESSMENT TASK
Exercise 1.1 – b
Mathematics in Our World
Name: __________________________ Program & Section: __________
Date: ____________________ Score: _________
A. From the different reasons presented why mathematics is important,
give at least 5 additional reasons with clear description of application.
Situation/Event Description
1.

2.

3.

4.

5.

B. What are the disadvantages if a person does not know and understand
mathematics?

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ASSESSMENT TASK
Exercise 1.1 – c
Mathematics in Our World
Name: __________________________ Program & Section: __________
Date: ____________________ Score: _________
Essay
1.What new ideas about mathematics did you learn?

2. What is it about mathematics that might have changed your thoughts about
it?

3. What is the most useful about mathematics for human kind?

4. What is the importance of Mathematics in your course of which you are

enrolled?

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ASSESSMENT TASK
Exercise 1.1 – d
Mathematics in Our World
Name: __________________________ Program & Section: __________
Date: ____________________ Score: _________

Cite the mathematical application that you commonly do in each of the following
place/event and state your appreciation.
Place/Event Appreciation for Mathematics
1. School

2. Market

3. Home

4. Parties (birthday, wedding,

etc.
5. Social media

6. Riding Bus/Jeep/Motorcycle

7. Gaming

8. Church

9. Watching tv

10. Fieldtrip

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References:
Daligdig, Romeo M. EdD et. al. (2019) Mathematics in the Modern World.
LORIMAR Publishing Inc., Quezon City, Metro Manila.

Alejan, Ronnie O et. al. (2018). Mathematics in the Modern World. Mutya
Publishing House Inc., Malabon City

22 Examples of Mathematics in Everyday Life (2020). Retrieved from

https://studiousguy.com/examples-of-mathematics/

Nature and Characteristics of Mathematics (2014). Retrieved from

http://drangelrathnabai.blogspot.com/2014/01/nature-characteristics-of-
mathematics.html

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Lesson 2: MATHEMATICAL LANGUAGE AND SYMBOLS

The language of mathematics is the system used

by mathematicians to communicate mathematical ideas among
themselves. This language consists of a substrate of some natural
language (for example English) using technical terms and grammatical
conventions that are peculiar to mathematical discourse, supplemented by a
highly specialized symbolic notation for mathematical formulas.

In everyday living, we may encounter and even use expressions in

English that may connote mathematical values or symbols without our knowing
it. These encounters include riding a vehicle; re-loading cellular phones in a
store; buying a particular item from a store; taking allowances from parents or
guardians; expectations of grades in a particular subject; number of friends in
Social Media who are sincere, honest, or the opposite; friends who like a photo
uploaded on Facebook; number of crushes in this School; number of hours
spent using the Internet; and many more.
The translation of these encounters into mathematical expressions and
vice-versa, however, is found to be a little complicated, especially if our
knowledge of English grammar is limited.

Specific Objectives:
At the end of this lesson, students are expected to:
1. Discuss the language, symbols and conventions of mathematics;
2. Explain the nature of mathematics as a language;
3. Perform operations on mathematical sentences;
4. Convert mathematical sentences to English expressions;
5. Convert algebraic sentences to English expressions; and
6. Appreciate that mathematics is a useful language.

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I. Language, Symbols, and Conventions of Mathematics

The language of mathematics makes it easy to express the kinds of

symbols, syntax and rules that mathematicians like to do and characterized by
the following:
i. Precise (able to make very fine distinctions)
Example:
The essential understanding of place value is that digits take on different
meanings based on their place value position. We teach students that
an 8 in the ones place has a value of 8, while an 8 in the hundreds place
has a value of 800. So how important is it, then, that we say the number
807 as eight hundred seven, rather than eight-o-seven? When we take
short cuts naming numbers, we often strip all the place value meaning
out of the number. No wonder students get confused. When numbers
are spoken correctly, we should hear the place value of every digit.
ii. Concise (able to say things briefly)
Example:
The long English sentence can be shortened using mathematical
symbols. Two x squared plus three is seven, which means 2x2 + 3 = 7.
iii. Powerful (able to express complex thought with relative ease)
Example:
The application of critical thinking and problem-solving skill requires the
comprehension, analysis and reasoning to obtain the correct solution.

II. Conversion of English expression to Mathematical Sentences and

vice-versa

We begin by describing the basic mathematical operations in the English

language. There are only 5 basic operations in mathematics excluding
exponentiation. Each of these operations has a corresponding English
translation as shown in the table below.

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Symbols/
Basic English Terms
Operations
= Equals, is equal to (most common mistake, “is equals to”),
represents, is the same as, is, are and the conjugations of
the verb “to be”, is similar to, is equivalent to, exactly,
results in.
+ Plus, sum, total, added to, added with, added by,
augmented, raised, more, more than, and, increased, put
together.
- Minus, difference, subtracted from, subtracted by, diminish,
less, less than, decreased, separated.
● or () Times, product, multiplied by, multiplied with, multiplied to,
doubled, tripled (etc.), twice, thrice (etc.)
/, ÷ Divided by, quotient, per, ratio, halved, over, over all

These symbols are used in different mathematical operations, and the

manner in which these will be read follows the English language. Consider the
following examples:
Basic Mathematical
English translations
Statements
6 + 4 = 10 • Six plus four equals ten
• Six added by four is ten.
• The sum of six and four is ten.
• Six increased by four is equal to ten.
Common errors:
• Six plus four is equals to ten. (Grammar)
• The sum of six and four equal to ten.
(Grammar)
10 – 2 = 8 • Ten minus two is eight.
• Ten diminished by two is equal to eight.
• Ten less two is eight.
• The difference of ten and two is eight.
Common Errors:

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• Ten less than two is eight. (Connotes different

meaning)
• The difference of two and ten is eight.
(Connotes different meaning)
3(4) = 12 • Three times four is twelve.
• The products of three and four is equal to
twelve.
• Thrice the number four is twelve.
Common Errors:
• Triple the four is twelve. (Grammar)
• Thrice the product of three and four is
twelve. (Redundant)
9÷3 = 3 • Nine divided by three is three.
• The quotient of nine and three is three
• The ratio of nine and three is equal to three
Common errors:
• The quotient of nine and three is three
(Connotes different meaning)
• The ratio of nine and three is to three.
(Connotes different meaning)

Let’s try this!

1. 8 – 2 = 6
2. 2(6) = 12
3. 12 +3 = 15
4. 18 ÷ 9 = 2
5. 6 – 4 = 2

Complicated expressions are not so difficult to read. Below are complicated

expressions:
Combined
Mathematical English Translations
Statements

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6+2
=1 • The sum of six and two all over eight is one.
8
• The ratio of six plus two, and eight is equal to one.
• One is the quotient when the sum of six and two is
divided by eight.
Common Errors
• Six plus two divided by eight is one. (Connotes
different meaning)
2(12 – 4) = 16 • Twice the difference of twelve and four is sixteen
• The product of twelve and two less four is sixteen.

Common Errors:
• Double the difference of twelves and four is
sixteen. (Connotes different meaning)
• Twelve minus four times two is sixteen. (Connotes
different meaning)

Let’s try this!

7−3
1. =2
2
3(4+2)
2. =4
6
5+3
3. =4
2

4. (3+7) – 5 = 5
5. 5(4 − 3) + 6 = 11

III. Conversion of Algebraic Expressions to English Sentences

By applying this concept to algebra, we use English translation of the

variable x as “the number” or “a number”. The following examples are useful.
Algebraic
English Translations
Statements
2x = 10 • Twice a number is equal to ten.
• Two times a number is ten.

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3(2x – 4) = 7 • Thrice the difference of twice a number and four is
seven.
• Three, multiplied to the difference of twice a number
less four, is seven.
2(𝑥+1)
=5 • The ratio of twice the sum of a number and one, and
3
three is equal to five.

Let’s try this!

1. 6x – 2 = 4
2. 3y = 4 – y
3. 2a – 7 = 5a + +2
2(𝑥−3)
4. 4
= 12
5𝑎− 3
5. = 2a
2

IV. The Four Basic Concepts of Mathematics

1. Set
A set is a collection of well-defined objects that contains no duplicates:
The objects in the set are called the elements of the set. To describe a set, we
use braces { }, and use capital letters to represent it.
Examples:
1. The books in the shelves in a library.
2. The bank accounts in a bank.
3. The set of natural numbers N = {1, 2, 3, …}.
4. The integer numbers Z = {…, -3, -2, -1, 0, 1, 2, 3, …}.
5. The rational numbers is the set of quotients of integers Q = {p/q : p,
q ∈ Z and q = 0}.
The three dot in enumerating the elements of the set are called ellipses
and indicate a continuing pattern. A finite set contains elements that can be
counted and terminates at certain natural number, otherwise, it is infinite set.
Example:
Set A = {1,3,5,7,11,13,17,19}

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- The set of all prime numbers less than or equal to 19. The order in
which the elements are listed is not relevant: i.e., the set
{1,3,5,7,11,13,17,19} is the same as the set {13,3,5, 11,13,17,19,1}.
There is exactly one set, the empty set, or null set, ∅ or {}, which has
no members at all. A set with only one member is called a singleton or a
singleton set. (“single of a”).

Specification of Sets
There are three main ways to specify a set:
1. List Notation / Roster Method – by listing all its members
- List names of elements of a set, separate them by commas and
enclose them in braces:
Examples:
1. {1, 12, 35}
2. {Daniela, Romina, Cassy, Marga}
3. {m, n, o, p}
4. {1, 2, …, 100}
2. Predicate Notation/Rule Method/Set-Builder Notation
- By stating a property of its elements. It has a property that members
of the set share (a condition or a predicate which holds for members
of this set).
Examples:
a) {x/x is a natural number and x< 8} means “the set of all x such
that x is a natural number and is less than 8”
b) {x/x is a letter of Korean alphabet}
c) {y/y is a student of SKSU and y is older than 20}
3. Recursive rules
- By defining a set of rules which generates or defines its members.
Examples:
a) The set E of even numbers greater than 5
b) 4 ∈ E
c) If x ∈ E, then x+2 ∈ E
d) Nothing else belongs to E

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Equal Sets
Two sets are equal if they contain exactly the same elements.
Examples:
1. {3, 8, 9} = {9, 8, 3}
2. {6, 7, 7, 7, 7,} = {6, 7}
3. {1, 3, 5, 7} ≠ {3, 5]

Equivalent Sets
Two sets are equivalent if they contain the same number of elements.
Example:
1. Which of the following sets are equivalent?
{𝜃, α, β}, {∞, ∩, ∃}, {1, 3, 5}, {a, b, c}, {€, ₸, ₢}
Solution: All of the given sets are equivalent. Note that no two of
them are equal, but they all have the same numbers of elements.

Universal Set
A set that contains all the elements considered in a particular
situation denoted by U.
Example:
The universal set
a. Suppose we list the digits only.
Then, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, since U includes all the digits.
b. Suppose we consider the whole numbers
Then U = {0, 1, 2, 3, …} since U contains all whole numbers.

Subsets
A set A is called a subset of set B if every element of A is also an
element of B. “A is a subset of B” is written as A ⊆ B.
Example:
1. A = {7, 9} is a subset of B = {6, 7, 9}
2. D = {10, 8, 6} is a subset of G = {10, 8, 6}
A proper subset is a subset that is not equal to the original set,
otherwise improper subset.

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Example:
Given {3, 5, 7} then the proper subsets are {}, {5, 7}, {3, 5}, {3, 7}.
The improper subset is {3, 5, 7}.

Cardinality of the Set

It is the number of distinct elements belonging to a finite set. It is also
called the cardinal number of the set A denoted by n(A) or card (A) and /A/.

Power Set
It is the family of all the subsets of A denoted by Power (A).
Given set A = {x, y}, the Power (A) = { ∅, {x}, {y}, {x,y} or {x/x is a subset of A}.

Operations on Sets
Union is an operation for sets A and B in which a set is formed that
consists of all the elements included in A or B both denoted by U as A U B.
Examples:
a) Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 3, 5, 7}, B = {2, 4, 6, 8} and
C = {1, 2}, find the following:
a) A U B b) A U C c) (A U B) U {8}
Solution:
a) A U B = {1, 2, 3, 4, 5, 6, 7, 8}
b) A U C = {1, 2, 3, 5, 7}
c) (A U B) U {8} = {1, 2, 3, 4, 5, 6, 7, 8}

Intersection
-is the set containing all elements common to both A and B,
denoted by ∩.
Example:
Given U = {a, b, c, d, e}, A = {c, d, e}, B = {a, c, e} and C = {a} and D =
{e}. find the following intersections of sets:
a) B ∩ C b) A ∩ C c) (A ∩ B) ∩ D
Solutions:

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a) B ∩ C = {a} c) (A ∩ B) = {c, e}, (A ∩ B) ∩ D = {e}
b) A ∩ C = ∅

Complementation
-is an operation on a set that must be performed in reference to a
universal set, denoted by A’.
Example:
Given U = {a, b, c, d, e}, A = {c, d, e}, find A’.
Solution: A’ = {a, b}

2. Relation
- A relation is a rule that pairs each element in one set, called the domain,
with one or more elements from a second set called the range. It creates a set
of ordered pairs.
Examples: 1. Given:
Regular holidays in the Philippines Month and Date
1. New Years’ Day January 1
2. Labor Day May 1
3. Independence Day June 12
4. Bonifacio Day November 30
5. Rizal Day December 30

A clearer way to express a relation is to form a set of ordered pairs;

(New Years’ Day, January 1), (Labor Day, May 1), (Independence Day,
June 12), (Bonifacio Day, November 30), (Rizal Day, December 30). This set
describes a Relation.
{ {2,3}, {4,5} is not a relation but just a set of ordered pairs.
{ {1,4}. {2,5}, {3,6} } is a relation. The domain of the relation is the set
{1,2,3} and the range is {4,5,6}

3. Function
- is a rule that pairs each element in one set, called the domain with
exactly one element from a second set, called the range. This means that for

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each first coordinate, there is exactly one second coordinate or for every first
element of x, there corresponds a unique second element y.
Remember: A one-to-one correspondence and many-to-one correspondence
are called Functions while one-to-many correspondence is not.

Examples: The function can be represented using the following:

1. Table
The perimeter of a square is four times the length of its side.
Sides (S) 1 3 5 7 9
Perimeter (P) 4 12 20 28 36

2. Ordered Pairs
{{1,4}, {3,12}, {5,20}, {7,28}, {9,36}}

3. Mapping

1 4
3 12
5 20
7 28
9 36

Therefore, this a function.

4. Graphing

Using vertical line test, that is, a set of points in the plane is the graph of
a function if and only if no vertical line intersects the graph in more than one
point. Below is not a function.

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4. Binary Operations

A binary operation on a set is a calculation involving two elements of

the set to produce another element of the set.

A new math (binary) operation, using symbol *, is defined to be

a * b = 2a + b, where a and b are real numbers.

Examples:

1. What is 5 * 3?

Solution: 5 * 3 = 2(5) + 3 = 10 + 3 = 13

2. Is a * b commutative?

Solution: Verify if a * b = b * a.

2a + b = 2b + a? Not true for all real numbers

If a = 4 and b = 2, then 2(4) + 2 = 2(2) + 4 is not true.

Therefore: The operation * is not commutative for all real numbers

3. Is a * b * c associative?

Solution: Verify if a*(b*c) = (a*b)*c

2a + (2b+c) = 2(2a+b) + c

If a = 2, b = 3, c = 4; 2●2 + (2●3 + 4) ≠ 2(2●2 + 3) + 4;

4 + 10 ≠ 2(7) + 4; 14 ≠ 18. The operation * is not associative

for real numbers.

Sometimes, a binary operation on a finite set (a set with a limited

number of elements) is displayed in a table which shows how the operation is
to be performed. A binary operation, * is defined on the set {1, 2, 3, 4}. The
table below shows the 16 possible answers using this operation.

(To read the table: read the first value from the left-hand column and the second
value from the top row. The answer is the intersection point).

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* 1 2 3 4

1 4 3 2 1

2 3 1 4 2

3 2 4 1 3

4 1 2 3 4

Examples:

1. What is 2 * 2? Answer: 1

2. Is 4 * 3 commutative? Answer: 4*3 = 3 and 3*4 = 3

3. What is the identity element for the operation *? Answer: 4 (Find the
single element that will always return the original value. The identity element is
4. You will have found the identity element when all of the values in its row and
its column are the same as the row and columns headings).

4. Is associative for these values? Answer: 4*(3*2) = (4*3)*2

4*4 = 3*2

4 =4

Answer: Yes, it is associative for values 4, 3, and 2.

V. Elementary Logic

Propositions and Connectives

A proposition (or statement) is a sentence that is either true or false

(without additional information).

The logical connectives are defined by truth tables (but have English
language counterparts).

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Logic Math English

Conjunction ^ And

Disjunction v or (inclusive)

Negation ~ Not

Conditional ⇒ If…then….

Biconditional ⇔ If and only if

A denial is a statement equivalent to the negation of a statement.

Examples:

1. The negation of P ⇒ Q is ~ (P ⇒ Q).

2. A denial of P ⇒ Q is P ^ ~ Q.

A tautology is a statement which is always true.

Examples:

1. A v (B ^ C) ⇔ (A v B) ^ (A v C) Distributive law

2. ~ (A v B) ⇔ ~ A ^ ~ B

3. P ⇔ ~ (~ P)

A contradiction is a statement which is always false.

Example: 1. (A v ~ A) ⇒ (B ^ ~B) a contradiction.

The contrapositive of the statement if P then Q is if ~Q then ~P. An

implication and its contrapositive are logically equivalent, so one can always be
used in place of the other.

A predicate (open sentence) is a sentence containing one or more

variables which becomes a proposition upon replacement of the variables.

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Examples:

1. The integer x is even.

2. y = 5.

3. Triangle ABC is isosceles.

However, a predicate is not a proposition, it does not have a truth value.

One can however use quantifiers to make propositions about predicates. For
instance, the universal/general quantifier (∀) is used to say that a given
predicate is true for all possible values of its variables. This is a proposition,
since it is either true or false. Similarly, the existential quantifier (∃) is used to
say that there is some value of the variables which makes the predicate a true
statement.

Examples:

1. Let x be a real number x2 – 1 = 0 is not a proposition, it is a

predicate.

2. “There exists an x so that x2 – 1 = 0” is a proposition (true)

3. “For all x, x2 – 1 = 0 is also a proposition (false).

4. These are written as:

(∃ x ∈ R) (x2 – 1 = 0) and (∀ x ∈ R) (x2 – 1 = 0).

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Summary

✓ The language of mathematics makes it easy to express the kinds of

symbols, syntax and rules that mathematicians like to do and characterized
by the following: Precise (able to make very fine distinctions); Concise (able
to say things briefly); and Powerful (able to express complex thought with
relative ease).
✓ The Four Basic Concepts of Mathematics: Set, Relation, Function, and
Binary operation.
✓ A Set is a collection of well-defined objects that contains no duplicates: The
objects in the set are called the elements of the set. To describe a set, we
use braces { }, and use capital letters to represent it.
✓ A relation is a rule that pairs each element in one set, called the domain,
with one or more elements from a second set called the range. It creates a
set of ordered pairs.
✓ A Function is a rule that pairs each element in one set, called the domain
with exactly one element from a second set, called the range. This means
that for each first coordinate, there is exactly one second coordinate or for
every first element of x, there corresponds a unique second element y.
✓ A binary operation on a set is a calculation involving two elements of the set
to produce another element of the set.

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ASSESSMENT TASK
Exercise 1.2 – a
Mathematical Language and Symbols

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Convert the following mathematical statements to English sentences.

1. 2 + 7 = 9
______________________________________________________________

2. 3(5) = 15
______________________________________________________________

12−2
3. =2
5

______________________________________________________________

3 (4+1)
4. =3
5

______________________________________________________________

5. 2(11 – 4) = 14
______________________________________________________________

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ASSESSMENT TASK
Exercise 1.2 – b
Mathematical Language and Symbols

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Convert the following English sentences to mathematical statement. Write your

answer on the space provided.

1. The difference of eight and three is five.

_____________________________________________________________

2. The sum of six and four, diminished by seven is equal to three.

_____________________________________________________________

3. The quotient of twenty-one and, five less two is seven.

_____________________________________________________________

4. The ratio of the sum of nine and five, and seven is two.
_____________________________________________________________

5. One thousand less 12 percent of it is eight hundred eighty.

_____________________________________________________________

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ASSESSMENT TASK
Exercise 1.2 – c
Mathematical Language and Symbols

Name: ____________________________Program & Section: __________

Date: _____________________________Score: _________

Convert the following algebraic expressions to English sentences. Write your

answer on the space provided.

1. 12x = 10 -2
_____________________________________________________________

2. 3x = 5 – 2x
_____________________________________________________________

3(𝑥+1)
3. =7
5

_____________________________________________________________

2𝑥−7
4. 3𝑥−4 = 6

_____________________________________________________________

5. 11 – 3(x – 4) = 5
_____________________________________________________________

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ASSESSMENT TASK
Exercise 1.2 – d
Mathematical Language and Symbols

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Convert the English sentences to algebraic expressions. Write your answer on

the space provided.
1. The sum of a number and four is twelve.
____________________________________
2. The difference of twice a number and eleven is fifty.
____________________________________
3. The ratio of a number and 2, diminished by five is thirteen.
____________________________________
4. The quotient of a number less five, and four is eleven.
____________________________________
5. Sixty percent of a number is fifteen.
____________________________________
6. Twice a number less four is thrice a number 5.
____________________________________
7. Four times a number minus seven is twice the difference of a number and
two.
_______________________________________
8. Four times the difference of three and a number is the same as one-
hundred twenty divided by twelve.
______________________________________
9. The number less five all over four is eleven.
______________________________________
10. One-third of a number added by two is three.

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ASSESSMENT TASK
Exercise 1.2 – e
Mathematical Language and Symbols

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Multiple Choice. Encircle the letter that corresponds to the correct answer.

1. Which is the language system that uses technical terms and grammatical
conventions peculiar to mathematical discourse and is supplemented by
a high specialized symbolic notation for mathematical formulas?
A. Mathematical Language C. Binary function
B. Set D. Singleton
2. Which is used to express a formula or to represent a constant?
A. Syntax C. Rules
B. Symbols D. Convention
3. Which of the following does NOT belong to the characteristics of the
language of mathematics?
A. Symbolic C. Concise
B. Precise D. Powerful
4. Which is a correct arrangement of mathematical symbols and is used to
represent a mathematical object of interest?
A. Mathematical expression C. Rule
B. Relation D. Function
5. Which is a collection of well-defined objects that contains no duplicates?
A. Function C. Binary
B. Relation D. Set
6. Which sets contains all the elements in a particular situation?
A. Union of sets C. Intersection of sets
B. Universal sets D. Combination of sets
7. What is a statement that is either TRUE or FALSE?
A. Proposition C. Tautology
B. Connective D. Denial

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8. Which Statement is always TRUE?
A. Proposition C. Tautology
B. Connective D. Denial
9. Which relation is described as a one-to-one correspondence and many-
to-one correspondence?
A. Tautology C. Set
B. Logic D. Function
10. Which statement is always false?
A. Tautology C. Set
B. Logic D. Contradiction

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ASSESSMENT TASK
Exercise 1.2 – e
Mathematical Language and Symbols

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Solve for the following: Write your answer in a whole sheet of paper.

1. Let A = {0, 2, 4, 6, 8}, B = {0, 1, 2, 3, 4} and C = {0, 3, 6, 9}. What are

a. A ∪ B ∪ C and b. A ∩ B ∩ C?

2. Find the union of A = {2, 3, 4} and B = {3, 4, 5}.

3. If A and B are two sets such that A ⊆ B, then what is A ∪ B?

4. Find the union, intersection and the difference (A – B) of the following

pairs of sets.

a) A = The set of all letters of the word FEAST

B = The set of all letters of the word TASTE

b. A = {x : x ∈ W, 0 < x ≤ 7}
B = {x : x ∈ W, 4 < x < 9}

c. A = {x / x ∈ N, x is a factor of 12}
B = {x / x ∈ N, x is a multiple of 2, x < 12}

d. A = {x : x ∈ I, -2 < x< 2}
B = {x : x ∈ I, -1 < x < 4}

e. A = {a, l, m, n, p}
B = {q, r, l, a, s, n}

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References:

Daligdig, Romeo M. EdD et. al. (2019) Mathematics in the Modern World.
LORIMAR Publishing Inc., Quezon City, Metro Manila.
Alejan, Ronnie O et. al. (2018). Mathematics in the Modern World. Mutya
Publishing House Inc., Malabon City

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Lesson 3: Problem Solving and Reasoning

Mathematics is not just about numbers; much of it is problem solving and

reasoning. Problem solving and reasoning are basically inseparable. The art of
reasoning is very important in mathematics. This is the skill needed in
exemplifying the critical thinking and problem-solving ability. Logic and
reasoning are very useful tools in decision making. People are also do
deductive reasoning extensively to show that certain conjectures are true as
these follow the rules of logic. A conjecture is a conclusion made from observing
data.
Inductive and Deductive reasoning are two fundamental forms of
reasoning for mathematicians. The formal theorems and proofs that we rely on
today all began with these two types of reasoning. Even today, mathematicians
are actively using these two types of reasoning to discover new mathematical
theorems and proofs. Believe it or not, you yourself might be using inductive
and deductive reasoning when you make assumptions about how the world
works.

Specific Objectives:
At the end of this lesson, students are expected to:
1. Use different types of reasoning to justify statements and arguments
made about mathematics and mathematical concepts;
2. Write clear and logical proofs;
3. Solve problems involving patterns and recreational problems involving
Polya’s 4 steps;
4. Organize one’s methods and approaches for proving and solving
problems.

I. Reasoning
Inductive Reasoning
According to www.study.com, Inductive reasoning is the logical
process in which multiple premises, all believed to be true or found true most

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of the time are combined to obtain a specific conclusion. This reasoning is used
in applications that involve prediction, forecasting, or behavior.
A conclusion that is reached by inductive reasoning may or may not be
valid. An example of inductive reasoning is when you notice that all the mice
you see around you are brown, and you make the conclusion that all mice in
the world are brown. Can you say for certain that this conclusion is correct? No,
because it is based on just a few observations. This is, however the beginning
of forming a correct conclusion or a correct proof. What this observation has
given you a starting hypothesis to test out.
We may also arrive at a conjecture whose conclusion is based on
inductive reasoning. There are instances however when even if all of the
premises are true in a statement, inductive reasoning still allows for a false
conclusion. Consider the examples below:

Examples of inductive reasoning

1. Daniela leaves for a school at 7:00 a.m. Daniela is always on time. Therefore,
Daniela assumes then that if she leaves for school at 7:00 a.m., she will always
be on time.

2. The teacher uses PowerPoint in the last three classes. Therefore, the
teacher will use PowerPoint tomorrow.

3. The chair in the living room is red. The chair in the dining room is red. The
chair in the bedroom is red. Therefore, all chairs in the house are red.

4. Cathy is a first-year college student of SKSU.

Cathy is a female.
Therefore, all first-year college students of SKSU are females.

5. Karl just moved here from General Santos City.

Karl has braces.
Therefore, all people from General Santos City have braces.

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6. All goats that we have seen have been black.
Therefore, all goats are black.

7. Fourth is an excellent lawn tennis player.

Fourth’s family has tennis court.
Therefore, the sister of Fourth named Alexa must also be an excellent lawn
tennis player.

Deductive Reasoning
Inductive reasoning typically may lead to deductive reasoning, the
process of reaching conclusions based on previously known facts. The
conclusions reached by this type of reasoning are valid and can be relied on.
For example, you know for a fact that all pennies are copper-colored. Now, if
your friend gave you a penny, what can you conclude about the penny? You
can conclude that the penny will be colored-copper. You can say this for certain
because your statement is based on facts.

Example of deductive reasoning

Examples of deductive reasoning are also called syllogism.
1. All men are mortal. (Major premise)
Senator Pacquiao is a man. (Minor premise)
Therefore, Senator Pacquiao is mortal. (Conclusion)

2. All first-year college students in the new curriculum take Mathematics in the
Modern World.
Precious is a first-year student in the new curriculum.
Therefore, Precious takes Mathematics in the Modern World.

3. All Filipinos eat rice.

Eugene is a Filipino.
Therefore, Eugene eats rice.

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II. Mathematical Proofs

A proof is a sequence of true facts (statements) placed in a logical order.

In proving, the following may be used as reasons:

• The given information (the hypothesis)

• Definition and undefined terms
• Algebraic properties
• Postulates of geometry
• Previously proven geometric conjectures (theorems)

Algebraic and Geometric Proofs

In order for us to prove properly and correctly, it is wise to remember and
understand the necessary properties to be used in writing formal proofs:

Important Properties of Algebra:

For real numbers w, x, y, and z:
Reflexive :x=x
Symmetric : If x=y, then y=x.
Transitive : If x=y and y=z, then x=z.
Substitution : If x+y = z and x = 3, then 3+y = z.
Distributive : x (y+z) = xy +yz.

Commutative Properties:
a. Addition : x+y = y+z
b. Multiplication : yz = zy

Associative Properties:
a. Addition : x + (y+z) = (x+y) +z
b. Multiplication : x(yz) = (xy)z

Addition Properties of Equality (APE)

a. If x = z, then x ± y = z ± y
b. If w = x and y = z, then w + y = x + z

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Multiplication Properties of Equality (MPE)
a. If x = z, then xy = yz or x/y = y/z
b. If w = x and y = z, then wy = xz or w/y = x/z

Example 1: Find the value of x in 2(x+1) = 6x + 4.

Proof:
Statements Reasons

1. 2( x+ 1) = 6x + 4 Given

2. 2x + 2 = 6x + 4 Distributive Property

3. 2x + 2 – 6x – 2 = 6x + 4 – 6x – 2 APE

1 1 MPE
4. -4X (- ) = 2 (- )
4 4
1 Simplification
5. x = (- )
2

Geometric Properties
The following properties may be used to justify proof of some mathematical
statements.

Reflexive Property (REF)

Statement: AB ≅ CD

Symmetric Property (SYM)

Given: AB ≅ CD
Statement: CD ≅ AB

Transitive Property (TRANS)

Given: AB ≅ CD, CD ≅ EF
Statement: AB ≅ EF

Addition property of Equality (APE)

Given 1: AB ≅ CD
Statement 1: AB ± EF = CD ± EF

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Given 2: AB = CD, EF = GH
Statement 2: AB ± EF = CD ± EF

Definition of congruent segments (DOCS)

Given 1: AB ≅ CD
Statement 1: AB = CD

Given 2: AB = CD
Statement 2: AB ≅ CD

Definition of Betweenness (DOB)

Given:

A B C

Statement: AB + BC = AC

Definition of Midpoint (DOM)

Given:

A M B
M is the midpoint of AB

Statement: AM ≅ MB

How to write proof?

There are many ways on how to write proofs. We can have top-down or
deductive reasoning or bottom-up or inductive reasoning. It can be formal or
informal procedure.

Example:

Prove the following using formal proof.

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Given: Y is the midpoint of XZ.

XY ≅ AB

X Y Z B

Prove: XY ≅ AB

Proof:

Statements Reasons

1. Y is the midpoint of XZ Given

2. XY ≅ YZ Definition of Midpoint (DOM)

3. YZ ≅ AB Given

4. XY ≅ AB Transitive

III. POLYA’s 4 – STEPS IN PROBLEM SOLVING

George Polya has had an important influence on problem solving in

mathematics education. He stated that good problem solvers tend to forget the
details and tend to focus on the structure of the problem, while poor problem
solvers focus on the opposite. He designed the following:

4 – STEPS PROCESS:

1. Understand the problem. (See)

Read and understand the problem. Identify what is the given information,
known data or values and what is the unknown and to be solved as
required by the problem.
Consider the following questions:
a. Can you restate the problem in your own words?
b. Can you determine what is known about these types of problems?

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c. Is there missing information that if known would allow you to solve
the problem?
d. Is there extraneous information that is not needed to solve the
problem?
e. What is the goal?

2. Devise a plan. (Plan)

Think of a way to solve the problem by setting up an equation, drawing
a diagram, and making a chart that will help you find the unknown and
the solution. To start devising a plan, try doing the following:
a. Make a list of the known information.
b. Make a list of information that is needed.
c. Draw a diagram.
d. Make an organized list that shows all the possibilities.
e. Make a table or a chart.
f. Work backwards.
g. Try to solve similar but simpler problem
h. Write an equation, as possible define what each variable
represents
i. Perform an experiment.
j. Guess at a solution and then check the result.

3. Carry out the plan. (Do)

Solve the equation you have to set up and observe analytical rules and
procedures until you arrive at the answer.
a. Work carefully.
b. Keep an accurate and neat record of all your attempts.
c. Realize that some of your initial plans will not work and that you
will have to devise another plan and modify your existing plan.

4. Look back. (Check)

In order to validate the obtained value, you need to verify and check if
the answer makes sense or correct based on the situation posed in the
problem. Label your final correct answer.
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a. Ensure that the solution is consistent with the facts of the problem.
b. Interpret the solution within the context of the problem.
c. Ask yourself whether there are generalizations of the solution that
you could apply to similar problems.
Example: A police station has 25 vehicles of motorcycles and cars. The
total number of wheels is 70. Find the number of motorcycles and cars
the station has.

1. Understand the problem. (See)

Given: 25 vehicles
70 wheels
Required: The number of cars and the number of motorcycles.

2. Devise a plan. (Plan)

Let x = the number of cars
y = the numbers of motorcycles
and x + y = 25 vehicles
4 wheels (x = cars) + 2 wheels (y = motorcycles) =
70 wheels
So, x + y = 25 vehicles and 4x + 2y = 70 wheels are the two
equations formed based on the problem.

3. Carry out the plan. (Do)

(1) x + y = 25
(2) 4x + 2y = 70, solving two equations with two unknown using
the process of elimination:
(1) -2 (x + y = 25) → -2x – 2y = -50
(2) 4x + 2y = 70 → 4x + 2y = 70
2x + 0 = 20
2x = 20
2 2
x = 10, since x denotes the number of cars, so, there are 10
cars. However, solving for y as the number of motorcycle is as
follows:
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Since, x + y = 25, then 10 + y = 25 – 10, finally y = 15, so there
are 15 vehicles in the police station.

4. Look back. (Check)

Therefore, there are 10 cars with 4 wheels and 15 motorcycles
with 2 wheels. The total number of wheels is 70 wheels.

Learning to solve problems is not a difficult task. It can be a huge fun and
ultimately challenging. However, it requires you to think analytically, critically,
and creatively. Practice doing and solving is the tough secret why most students
and professionals succeed in getting the problem solved and done to make the
moment of solving more enjoyable, interesting, and fulfilling.

Let’s try this!

1. Find the next number in the sequence.

a. 5, 9, 13, 17, 21, 25,…
b. 2, 6, 18, 54, 162, 486,…
2. Anne has a certain amount of money in her bank account on Friday
morning. During the day she wrote a check for 24.50, made an ATM
withdrawal of 80.00 and deposited a check for 235.00. at the end of the
day, she saw that her balance was 451.25. How much money did she
have in the bank at the beginning of the day?

3. Two cars left, at 8:00 A.M., from the same point, one travelling east at
50 mph and the other travelling south at 60 mph. at what time will they
be 300 miles apart?

4. An algebra test consists of ten multiple choice questions. Ten points are
given for each correct answer and three points are deducted for each
incorrect answer. If Joshua did all questions and scored 48, how many
incorrect answers did he have?

5. a. Find the next term of the sequence 7/2, 19/2 31/2, 43/2, 55/2…
b. Find the next term for 1, 5, 12, 22, 35,..

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Summary

✓ Inductive reasoning is the logical process in which multiple premises, all

believed to be true or found true most of the time are combined to obtain
a specific conclusion. This reasoning is used in applications that involve
prediction, forecasting, or behavior.
✓ Deductive reasoning, the process of reaching conclusions based on
previously known facts. The conclusions reached by this type of
reasoning are valid and can be relied on.
✓ A proof is a sequence of true facts (statements) placed in a logical order.
✓ Polya’s 4 – steps in problem solving: Understand the problem. (See):
Devise a plan. (Plan); Carry out the plan. (Do); and Look back. (Check)

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ASSESSMENT TASK
Exercise 1.3 – a
Reasoning
Name: __________________________ Program & Section: __________
Date: ___________________________ Score: _________

Answer the problem comprehensively:

Can you definitely tell the truthfulness to the examples of inductive reasoning?
Write your arguments below:
1.

2.

3.

4.

5.

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ASSESSMENT TASK
Exercise 1.3 – b
Reasoning
Name: __________________________ Program & Section: __________
Date: ___________________________ Score: _________

Give five (5) syllogisms that manifest a real-life scenario. Explain the
truthfulness of your major premise. Write your answers on the space
provided.
1.

2.

3.

4.

5.

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ASSESSMENT TASK
Exercise 1.3 – c
Reasoning
Name: __________________________ Program & Section: __________
Date: ___________________________ Score: _________
Write the proof of the following on the space provided:

1. If 5x – 8 = 12, then x = 4.
Statement Reasons

2. If -2(3x + 5) = -17 + x, then x = 1.

Statement Reasons

3. Write the geometric proof of the following:

Given: IE bisects < MIK
< EIK ≅ < XYZ
Prove: <MIE ≅ < XYZ
Statement Reasons

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ASSESSMENT TASK
Exercise 1.3 – d
Reasoning
Name: __________________________ Program & Section: __________
Date: ___________________________ Score: _________
Find the next term in the following sequences by using a difference table:
1. 2, 7, 24, 59, 118, 207,…

2. 5, 14, 27, 44, 65,…

3. 1, 14, 51, 124, 245, 426,…

4. 10, 10, 12, 16, 22, 30,…

5. 1, 7, 17, 31, 49, 71,…

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Assessment Task
Exercise 1.3 – e
Reasoning
Name: __________________________ Program & Section: __________
Date: ___________________________ Score: _________
Solve the following problems using the 4 steps of George Polya.

1. Mrs. Dizon withdrew ¼ of her savings in July and later deposited a total
of 1,500.00 on four separate days. If her bank statement showed a
balance of 3,500.00 after four deposits, what was the balance
immediately before her withdrawal?

2. Manny rode his bicycle 6 km east, 4 km west, and then 5 km east. How
far is he from his starting point?

3. T, U, W, X, Y, and Z are points on a circle. Each of these points is

connected to each other by a line segment. How many line segments
are there?

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References:

Daligdig, Romeo M. EdD et. al. (2019) Mathematics in the Modern World.
LORIMAR Publishing Inc., Quezon City, Metro Manila.pp.44 - 46
Deductive Reasoning vs. Inductive Reasoning (2017, July, 5). Retrieved from
https://www.livescience.com/21569-deduction-vs-induction.html
Inductive vs. deductive Reasoning (2019, November, 11). Retrieved from
https://www.scribbr.com/methodology/inductive-deductive-reasoning/
Immediate Algebra Tutorial 8: Introduction to Problem Solving (2011, July, 1).
Retrieved from
https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg
_tut8_probsol.htm

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Chapter 2: Mathematics as a Tool (Part I)

Overview

The field of statistics is the science of learning from data. Statistical

knowledge helps you use the proper methods to collect the data, employ
analyses, and effectively present the results. Statistics is a crucial process
behind how we make discoveries in science, make decisions based on data,
and make predictions. Statistics allows you to understand a subject much more
deeply.

In performing all these processes involved, the application of statistical

tools and techniques is necessary. Statistical tools derived from mathematics
are useful in processing and managing numerical data in order to describe a
phenomenon and predict values.

General Objectives:

This chapter emphasize the use of different statistical tools to process

and manage numerical data.

Lesson 1: Data Management

Data come in different forms and from different sources. You read them
in a daily newspaper, hear them over the radio, see them on television, and find
them on the internet. We have great quantities of data related to sports,
business, education, traffic, law enforcement, and hundreds of some other
human activities. These vast data are made available to assist us in our
decision-making. If these data are not properly managed and analyzed,
everything will be meaningless and void or lead us to false and unreliable
information.

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Specific objectives:

At the end of this lesson, the students are expected to:

1. Organize and present data in forms that are both meaningful and
useful to decision makers;
2. Use a variety of statistical tools to process and manage numerical
data;
3. Use the methods of linear regression and correlation to predict the
value of a variable given certain conditions; and
4. Advocate the use of statistical data in making important decisions.

I. Basic Statistical Concepts

The study of statistics can be organized in different ways. One way is to

subdivided statistics into two branches: descriptive statistics and inferential
statistics. To understand the difference between the two, definitions of
population and sample are helpful.

A population generally consists of the totality of the observations,

individuals, or objects in which the investigator is interested. One should not
start collecting data without carefully defining the population to be considered
in the study. It should be in agreement with the objective, and its statistical
elements should be properly identified. A sample is a portion of a population.
This is a small but representative cross section of the population. It is used to
give inferences on the population from which it was extracted.

If a researcher is using data gathered on a group to describe or reach

conclusions about that same group, the statistics are called descriptive
statistics. For example, if an instructor produces statistics to summarize a class’
examination performance and uses those statistics to reach conclusions about
that class only, the statistics are descriptive. The instructor can use these
statistics to discuss class average or talk about the range of class scores.

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If a researcher gathers data from a sample and uses the statistics
generated to reach conclusions about the population from which the sample
was drawn, it is called inferential statistics. For example, a soft drink company
asked 500 students from a certain university about the number of bottles of soft
drink they consumed in a week and infer from the data the number of bottles of
soft drink consumed by all 50,000 students in the campus.

Statistician and researchers are interested in particular variables of a

sample or population. A variable is a characteristic of interest about an object
under investigation that can take on different possible outcomes, such as age,
color, height, weight, and religious preference.

There are basically two kinds of variables:

1. Qualitative variables are variables that can be placed into distinct categories,
according to some characteristic or attribute. For example, if subjects are
classified according to sex (male or female), then the variable gender is
qualitative. Some other examples of qualitative variables are religious
preference and geographic location.

2. Quantitative variables are numerical and can be ordered or ranked. For

example, the variable age is numerical, and people can be ranked according to
their ages. Some other examples of quantitative variables are height, weight,
and body temperature.

Quantitative variables can be further classified into two groups: discrete and
continuous.

Discrete variables can be assigned values such as 0, 1, 2, an 3 and are said to

be countable. The data are obtained by means of counting. Example of discrete
variables are the number of children in a family and the number of calls received
by a telephone operator each day for a month. Continuous variables, on the
other hand, can assume an infinite number of values in an interval between any
two specific values. The data for this variable are obtained by means of direct
or indirect measuring. Temperature, for example, is a continuous variable, since
the variable can assume an infinite number of values between any two given
temperatures.

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The bulk of data gathered every day should not be analyzed the same way
statistically because the values are represented differently. For this reason,
data are categorized according to four levels of data measurement:

1. Nominal level – is the lowest level of data measurement. The numbers

representing nominal data are used only for identification of classification.
These numbers may serve as labels and have no meaning attached to their
magnitude. Examples are ID number of students, numbers on the uniform
jerseys of basketball players, and plate numbers of vehicles.

2. Ordinal level – is higher that the nominal level. The numbers are used not
only to classify items but also to reflect some rank or order of the individuals or
objects. It indicates that objects in one category are not only different from those
in the other categories of the variable, but they may also be ranked as either
higher or lower. Bigger or smaller, better or worse than those in the other
categories. Examples are ranks given to the winners in a singing contest, hotel
classifications, and military ranks.

3. Interval level – is second to the highest of data measurement. The

measurements have all the properties of ordinal data; in addition, the distance
between consecutive numbers have meaning. The zero point value on this level
is arbitrary; that is, zero is just another point on the scale relative to a certain
concept and does not mean the absence of the phenomenon. Examples are
temperature reading in Celsius scale, scores in intelligence tests, and
scholastic grades of students.

4. Ratio level – is the highest level of data management. It has the same
properties as interval level but the zero point value of this level is absolute; that
is, the zero value represents the absence of the characteristic being
considered. Examples are height, weight, time, and volume.

Data that are collected must be organized and presented effectively for
analysis and interpretation. They can be presented in different forms as follows:

1. Textual presentation – presents data in a paragraph form which combines

text and figures. Examples are data in business, finance, economics, or

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industries which are used to make emphasis or to make comparisons,
contrasts, syntheses, generalizations, or findings.

2. Tabular presentation – presents data in tables. Tabulation is a process of

summarizing classified and arranging them in table. It gives a more precise,
systematic, and orderly presentation of data in rows and columns. It makes
comparison of figures easy and comprehensible. The table below displays the
Philippine population by region based on the 2010 and 2015 censuses.

Table 1. Population Enumerated in 2010 & 2015

REGION 2010 2015

NATIONAL CAPITAL REGION 92,337,852 100,981,437
CORDILLERA ADMINISTRATIVE 11,855,975 12,877,253
REGION
REGION I – ILOCOS 1,616,867 1,722,006
REGION II – CAGAYAN VALLEY 3,229,163 3,451,410
REGION III – CENTRAL LUZON 10,137,737 11,218,177
REGION IV – A – CALABARZON 12,609,803 14,414,774
MIMAROPA REGION 1 2,744,671 2,963,360
REGION V – BICOL 5,420,411 5,796,989
REGION VI – WESTERN VISAYAS 4,194,579 4,477,247
REGION VII – CENTRAL VISAYAS 5,513,514 6,041,903
NEGROS ISLAND REGION 2 4,194,525 4,414,131
REGION VIII – EASTERN VISAYAS 4,101,322 4,440,150
REGION IX – ZAMBOANGA PENINSULA 3,407,353 3,629,783
REGION X – NORTHERN MINDANAO 4,297,323 4,689,302
REGION XI – DAVAO 4,468,563 4,893,318
REGION XII – SOCCSKSARGEN 4,109,571 4,545,276
REGION XIII – CARAGA 2,429,224 2,596,709
ARMM 3,256,140 3,781,387
Source: Philippine Statistics Authority

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3. Graphical presentation – is an effective method of presenting statistical
results and can present clear pictures of the data. There are several kinds of
graphs, and some of these are as follows:

Bar Graph consists of bar either vertically or

horizontally and usually constructed for comparative
purposes. The lengths of the bars represent the
frequencies or magnitudes of the quantities being
compared.

Line Graph shows the relationship between two or

more sets of quantities. It may show the relationship
between two variables, and it is best used to establish
trends.

Pie Chart is used to represent quantities that make

up a whole. It is a circular diagram cut into subdivisions.
The size of each section indicates the proportion of each
component part of the whole. The pie chart can be
constructed using percent or the actual figures. The slices
of the pie must be drawn in proportion to the different
values of the items.

Data analysis techniques enabling to meaningfully describe data with

numerical or in graphic form. This technique includes the following:

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II. Measure of Central Tendency

Central tendency determines a numerical value in the central region of

a distribution of scores. Central tendency refers to the center of a distribution of
observations. There are three measures of central tendency: the mean, the
median, and the mode. These are used when general or over-all performance
of the class is compared to other classes.
1. Mean
̅ is also called the arithmetic mean or average. It can be
The mean, 𝒙
affected by extreme scores. It is stable, and varies less from sample to sample.
It is used if the most reliable measure is desired and when there are a few with
very high values and a few with very low values. The mean is the balance point
of a score distribution.
A. Ungrouped Data
a. Arithmetic mean (denoted by 𝑥̅ ) or simply mean is the sum
of all values in a data set divided by the number of values that are summed. It
is written mathematically as
∑𝑥
𝑥̅ = 𝑛

Where x = individual value

n = total number of values

The mean is a more stable or reliable measure of central tendency in

which its value is dependent upon every item in the data set. It is preferred for
interval or ratio measurements and is used in the computation of some other
advanced statistical measures.

Example. The following are the scores in a quiz by ten students in Algebra. Find
the mean score of the data set.

5 12 20 16 15 23 10 18 7 11

Solution: From the given data set, n = 10

Solve for the mean

5+12+20+16+15+23+10+18+7+11 137
𝑥̅ = = = 13.7
10 10

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Sometimes each value in the data set is associated with a certain weight
or degree of importance. In such cases, the weighted mean is computed.

The weighted mean (denoted by 𝑥̅ w) of a set of values can be computed

by multiplying each value with its corresponding weight and taking the sum of
the products and then divided by the total number of weights written as

∑𝑛
𝑖=1 𝑤𝑖 𝑥𝑖
𝑥̅ w = ∑𝑛
𝑖=1 𝑤𝑖

Where 𝑥𝑖 = individual value

𝑤𝑖 = weight of each value

Example: The final grades of a student in six courses were taken and are shown
below. Compute the student’s weighted mean grade.

COURSE NO. OF UNITS FINAL GRADE

GE 701 3 2.5
GE 702 6 2.0
GE703 3 1.5
GE 706 3 1.5
GE 709 5 2.5
PE 101 2 1.0

Solution: Solve for the weighted grade of each course.

NO. OF UNITS FINAL GRADE

COURSE 𝒘𝒊 𝒙𝒊
(𝒘𝒊 ) (𝒙𝒊 )
GE 701 3 2.5 7.50
GE 702 6 2.0 12.00
GE703 3 1.5 4.50
GE 706 3 1.5 4.50
GE 709 5 2.5 12.50
PE 101 2 1.5 3.00

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∑ 𝑤𝑖 = 22 ∑(𝑤𝑖 𝑥𝑖 ) =
44.00
Thus, the weighted mean is

∑𝑛
𝑖=1 𝑤𝑖 𝑥𝑖 44
𝑥̅ w = ∑𝑛
= 22 = 2.00
𝑖=1 𝑤𝑖

b. Grouped Data
There are two ways on how to solve for the value of mean given the
grouped data or frequency distribution.

a. by midpoint method

∑𝒌𝒊=𝟏 𝒇𝒊 𝒙𝒊
̅=
𝒙 𝒏

Where:
𝑥̅ = sample mean
𝑓𝑖 = frequency of the ith class
𝑥𝑖 = midpoint of the ith class
n = sample size

Weights of the Frequency Midpoint 𝒇𝒊 𝒙 𝒊

Cubs (𝑓𝑖 ) (𝑥𝑖 )
201 – 210 3 205.5 616.5
191 – 200 8 195.5 1564.0
181 – 190 12 185.5 2226.0
171 – 180 11 175.5 1930.5
161 – 170 9 165.5 1489.5
151 - 160 2 155.5 311.0
Total 45 ∑ 𝒇𝒊 𝒙𝒊 = 8137.5
Note: n = 45
∑𝟔𝒊=𝟏 𝒇𝒊 𝒙𝒊 = 3(205.5)+8(195.5)+…+2(311.0) = 8137.5
Therefore:
∑𝒌𝒊=𝟏 𝒇𝒊 𝒙𝒊
̅=
𝒙 𝒏
8137.5
= 45

= 180.83

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b) by coded deviation method

∑𝒌𝒊=𝟏 𝒇𝒊 𝒅𝒊
̅ = ̅̅̅
𝒙 𝒙𝟎 + i [ ]
𝒏

Where:
𝑥̅ = sample mean
𝑥0
̅̅̅ = assumed mean or the midpoint where the zero code is
assigned
𝑓𝑖 = frequency of the ith class
𝑑𝑖 = code assigned to the ith class
𝑛 = sample size

Rule of Thumb: Assign a zero code to the class with the highest frequency.
Negative codes with one as the deviation are assigned to the classes from a
class with zero code going down and positive codes with one as the deviation
to classes going up. This is with the assumption that the distribution starts from
the highest class to the lowest class.

Sample solution of finding the mean score of the grouped data below
using the coded deviation method:
Weights Frequency Midpoint 𝒅𝒊 𝒇𝒊 𝒅 𝒊
of the Cubs (𝒇𝒊 𝑖 ) (𝒙𝒊 )
201 – 210 3 205.5 2 6
191 – 200 8 195.5 1 8
181 – 190 12 185.5 0 0
171 – 180 11 175.5 -1 -11
161 – 170 9 165.5 -2 -18
151 - 160 2 155.5 -3 -6
Total 45 -21

Note: n = 45
i = 10
̅̅̅
𝒙𝟎 = 185.5

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∑𝒌𝒊=𝟏 𝒇𝒊 𝒅𝒊 = (6)+(8)+(0)+…+(-6) = -21

Therefore:
∑𝒌𝒊=𝟏 𝒇𝒊 𝒅𝒊
̅ = ̅̅̅
𝒙 𝒙𝟎 + i [ ]
𝒏
−21
= 185.5 + 10 ( 45 ) = 180.83

2. Median
The median (𝒙̃ ), is the value in the distribution that divides an arranged
(ascending/descending) set into two equal parts. It is the midpoint or
middlemost of a distribution of scores. Fifty percent of the scores falls above it
and 50% falls below it. It is also known as the 50th percentile. It is not affected
by extreme scores. This is used when the distribution of scores is skewed. The
median separates the distribution into two equal parts.

a. Ungrouped data
The median is obtained by inspecting the middlemost value of the
arranged distribution either in ascending or descending order. It can also be
(𝑁+1)
solved using the formula position after being arranged.
2𝑡ℎ

Example. Find the median of the following set of measurements.

25 41 56 34 28 67 49 37 52

Solution: Arrange the data in ascending order

25 28 34 37 41 49 52 56 67

Locate the middlemost value. The middlemost value is the median.

𝑥̃ = 41

Example. Find the median of the given data set.

4.5 2.8 5.6 9.2 3.5 6.7 3.9 8.4

Solution. Arrange the data in ascending order.

2.8 3.5 3.9 4.5 5.6 6.7 8.4 9.2

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Locate the middlemost value.

In this case, there are two middle values in the distribution. Obtain the
mean of the middle values and the mean is the median of the distribution.

4.5+5.6
𝑥̃ = = 5.05
2

b. Grouped data
In computing the median of the grouped data, determine the median
class which contains the (N/2)th score under CF of the cumulative frequency
distribution. To solve for the median, we use the formula:
𝑵
−𝑪𝑭𝒃
𝟐
̃ = XLB + i [
𝒙 ]
𝒇𝒎

𝑵
Where: median class = a class where lies
𝟐

XLB = the lower boundary or true lower limit of the median class
N = total frequency
CFb = cumulative frequency before the median class
fm = frequency of the median class
i = size of the class interval
Example:
Solve for the median for the following data.
Scores Frequency Midpoint Class boundary Cumulative frequency
(fi) (xi)

35 – 40 4 37.5 34.50 – 40.5 30

29 – 34 5 31.5 28.50 – 34.50 26
23 – 28 8 25.5 22.50 – 28.50 21 median class

17 – 22 6 19.5 16.50 = 22.50 13

11 – 16 4 13.5 10.50 – 16.50 7
5 – 10 3 7.5 4.50 – 10.50 3
Total 30

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Solution for the median of the distribution:
𝑵
−𝑪𝑭𝒃
𝟐
̃ = XLB + i [
𝒙 ]
𝒇𝒎

30
−13
2
= 22.50 + 6 [ ] = 24.00
8

3. Mode
̂) is the value with the largest frequency. It is the
The mode (𝒙
value that occurs most frequently in the distribution. This is used when
the quickest estimate of typical performance is wanted. A distribution can
be unimodal with one mode value, bimodal with two mode values or
trimodal with three mode values. In other words, it can have more than
one mode.
a. Ungrouped data

Example. Find the mode of the following data sets.

a. 12 15 13 12 14 17 16 12 13 19

b. 3.4 2.2 3.5 3.4 2.2 2.6 2.1 3.9 2.2 3.4

c. 105 200 159 110 225 170 115 250 285 190

Solution:

a. In the first data set, 12 has the highest frequency in the distribution;
therefore, the mode is

̂ = 12
𝒙

b. In the second data set, two values have the highest frequency;
therefore, there are two modes and the distribution is called bimodal. The
modes are

̂ = 3.4
𝒙 and ̂ = 2.2
𝒙

c. In the third data set, there is no value that occurs most often;
therefore, there is NO mode in the distribution.

̂ = Does not exist

𝒙

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b. Grouped data

To find the mode of the grouped data, determine first the modal class.
The modal class is the class with the highest frequency.

𝒅𝒇𝟏
̂ = XLB + i [
𝒙 ]
𝒅𝒇
𝟏 + 𝒅𝒇𝟐

Where: XLB = lower boundary of the modal class

df1 = difference between the frequency of the modal
class and the frequency below it
df2 = difference between the frequency of the modal
class and the frequency above it
i = size of the class interval

Example: Find the mode of the following data:

Scores Frequency Midpoint Class boundary Cumulative
(fi) (xi) frequency

35 – 40 4 37.5 34.50 – 40.5 30

29 – 34 5 31.5 28.50 – 34.50 26
23 – 28 8 25.5 22.50 – 28.50 21 modal class

17 – 22 6 19.5 16.50 = 22.50 13

11 – 16 4 13.5 10.50 – 16.50 7
5 – 10 3 7.5 4.50 – 10.50 3
Total 30

Solution for the mode of distribution:

𝒅𝒇𝟏
̂ = XLB + i [
𝒙 ]
𝒅𝒇
𝟏 + 𝒅𝒇𝟐

(𝟖−𝟔)
̂ = 22.50 + 6 [( )
𝒙 ]
𝟖−𝟔 +(𝟖−𝟓)

= 24.90 or 25

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III. Measure of Relative Position
As median divides the set of scores into two equal parts, there are other
measures that divide the distribution into one hundred, four, or ten equal parts.
These are the other measures of position: the percentiles, the quartiles,
and the deciles.

A. The Percentiles
a. Ungrouped data
To approximate the percentile rank of value x in the distribution, then

(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑥)+0.5

Percentile = ●100
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠

Example. A 30-point quiz was given to 10 students and the scores are shown
below. What is the percentile rank of 24?

23 25 19 21 28 15 20 24 22 27

Solution. Arrange the data in ascending order.

15 19 20 21 22 23 24 25 27 28

There are 6 values below 24.

Determine the percentile using the formula.

6+0.5
Percentile = . ●100
10

Percentile = 65 percentile

This means that a student with a score 24 did better than 65% of the
class

b. Grouped data

The general formula for the percentile is:

𝒏𝑵
− 𝑪𝑭𝒃
𝟏𝟎𝟎
Pn = XLB + i[ ]
𝒇𝑷𝒏

Where: n = the rank in decimals

Pn = nth percentile
XLB = lower class boundary of the nth percentile class

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CFb = cumulative frequency of the class before the nth
percentile
𝑓𝑃𝑛 = frequency of the nth percentile class
i = class size interval
N = the total frequency
𝑛𝑁
nth percentile class = a class where cases lie.
100

Example:
Scores of 30 Students in a 45 – item Quiz.
10 7 5 15 11 18
16 17 20 22 18 21
24 28 25 23 27 28
26 29 34 33 32 29
24 40 35 36 35 16
For the grouped data given below, solve for the 50th percentile (P50).

Scores Frequency Midpoint Class boundary Cumulative frequency

(fi) (xi)
35 – 40 4 37.5 34.50 – 40.5 30
29 – 34 5 31.5 28.50 – 34.50 26
23 – 28 8 25.5 22.50 – 28.50 21 50th percentile class

17 – 22 6 19.5 16.50 = 22.50 13

11 – 16 4 13.5 10.50 – 16.50 7
5 – 10 3 7.5 4.50 – 10.50 3
Total 30

Solution:
𝒏𝑵 50(30)
= = 15
𝟏𝟎𝟎 100

With the result, look at the cumulative frequency and see where these
15 cases belong. So, we see that 15 is incorporated in CF = 21. Therefore, 23
– 28 is the 50th percentile class.
Pn = P50
n = 50
XLB = 22.50

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𝑓𝑃𝑛 = 8
CFb = 13
i=6
N = 30
Solve for P50:
𝒏𝑵
− 𝑪𝑭𝒃
𝟏𝟎𝟎
Pn = XLB + i[ ]
𝒇𝑷𝒏

(𝟓𝟎)(𝟑𝟎)
− 𝟏𝟑
𝟏𝟎𝟎
P50 = 22.50 + 6[ ]
𝟖

= 24.00
The result indicates that 50% or around 15 of those who took the exam
got the scores of 24 and below. This result is also equal to the median since
the 50th percentile is equal to the median, second quartile and the 5th decile.

B. Quartiles
The quartiles are points that divide a distribution into four equal
parts. Consider that Q1 = P25, Q2 = P50, Q3 = P75, Q4 = P100. The lower quartile
is Q1 and the upper is Q3.
The general formula for the nth quartile is:

𝒏𝑵
− 𝑪𝑭𝒃
𝟒
Qn = XLB + i [ ]
𝒇𝑸𝒏

Where:
Qn = nth quartile
XLB = lower class boundary of the of the nth quartile class
CFb = cumulative frequency of the class before the nth quartile
class
𝒇𝑸𝒏 = frequency of the nth class
i = class size interval
N = the total frequency
𝑛𝑁
nth quartile class = a class where 100 case

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Example: For the grouped data given below, solve for the third quartile (Q 3).
Scores Frequency Midpoint Class boundary Cumulative
(fi) (xi) frequency
35 -40 4 37.5 34.50 – 40 50 30
29 – 34 5 31.5 28.50 – 34.50 26 3rd quartile class

23 – 28 8 25.5 22.50 – 28.50 21

17 – 22 6 19.5 16.50 – 22.50 13
11 – 16 4 13.5 10.50 – 16.50 7
5 – 10 3 7.5 4.50 – 10.50 3
Total 30

Solution:
XLB = 28.50
𝑛𝑁 3(30)
= = 22.50, it is incorporated in CF = 26
4 4

𝐶𝐹𝑏 = 21
𝑓𝑄𝑛 =5
i =6

𝒏𝑵
− 𝑪𝑭𝒃
𝟒
Qn = XLB + i [ ]
𝒇𝑸𝒏

𝟑(𝟑𝟎)
− 𝟐𝟏
𝟒
Q3 = 28.50 + 6 [ ]
𝟓

= 30.40 or 30
The result implies that ¾ or around 23 of those who took the exam got
the scores of 30 and below.

C. Deciles
The deciles are points that divide a distribution into ten equal parts.
Each part is called a decile. So, D1 = P10, D2 = P20, …, D10 = P100.
For the nth decile, the formula is:

𝒏𝑵
− 𝑪𝑭𝒃
𝟏𝟎
Dn = XLB + i [ 𝒇𝑫𝒏
]

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Where: Dn = nth decile
XLB = lower class boundary of the of the nth decile class
CFb = cumulative frequency of the class before the nth quartile
class
𝒇𝑫𝒏 = frequency of the nth class
i = class size interval
N = the total frequency
𝑛𝑁
nth quartile class = a class where 100 case

Example: For the grouped data given below, solve for the 4th decile (D4)
Scores Frequency Midpoint (xi) Class boundary Cumulative frequency
(fi) (CF)
35 -40 4 37.5 34.50 – 40 50 30
29 – 34 5 31.5 28.50 – 34.50 26
23 – 28 8 25.5 22.50 – 28.50 21
17 – 22 6 19.5 16.50 – 22.50 13 4th decile class

11 – 16 4 13.5 10.50 – 16.50 7

5 – 10 3 7.5 4.50 – 10.50 3
Total 30

Solution:
XLB = 16.50
CFb =7
𝒇𝑫𝟒 =6
i =6
𝑛𝑁 4(30)
=
10 10

𝒏𝑵
− 𝑪𝑭𝒃
D4 = XLB + i [ 𝟏𝟎 ]
𝒇𝑫𝒏

𝟒(𝟑𝟎)
−𝟕
𝟏𝟎
D4 = 16.50 + 6 [ ]
𝟔

= 21.50 or 22

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The result indicates that 4/10 or around 12 of those who took the
exam got the scores of 22 and below.

IV. Measures of Variation

The degree of variation measures the degree of the spread of the
values. The measures of spread are commonly called measures of
dispersion or measures of variation. There are six measures of variation:
the range, the quartile deviation, the interquartile deviation, the mean
deviation, the variance, and the standard deviation.

A. Range
The range is the difference between the highest scores (h.s.) and
the lowest scores (l.s.). It gives us the quickest estimate. It shows the
two extreme scores of a set of data. For grouped data, the range can be
calculated by subtracting the lower boundary (l.b.) of the lowest class
interval from the upper boundary (u.b.) of the highest class interval.
Examples:
1. Find the range of the following data:
a. 10, 12, 12, 14 R = 14 – 10 = 4
b. 45, 50, 50, 55 R = 55 – 45 = 10
2. Find the range of the frequency distribution below
Class frequency f
75 – 79 6
70 – 74 7
65 – 69 2
60 – 64 8
55 – 59 12
50 – 54 7
45 – 49 10
40 – 44 8
N = 60

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Solution:
Range = u.b. – l.b.
= 79.50 – 39.50
= 40.0

B. The Interquartile Range

The interquartile range is a more reliable measure of variability. It
is the difference between the 75th percentile or Q3 and the 25th percentile
or Q1 hence, the 50 percent of the distribution will be falling within the
interquartile range, 25 percent will fall below Q 1 and 25 percent will fall
above Q3.
I.R. = Q3 – Q1

Example: Solve for the interquartile range, given Q1 = 45.25 and Q3 =

52.3.
Solution: IR = 52.3 – 45.25 = 7.05

C. The Quartile Deviation

If we want to get half of the distance or interquartile range, then we
simply divide the difference between Q3 and Q1 by two. This value is
called quartile deviation.
𝑸𝟑 − 𝑸𝟏
Q.D. = 𝟐

Example: Solve for the quartile deviation, given Q1 = 45.25 and Q3

= 52.3.
52.3 − 45.25
Solution: Q.D. = = 3.525
2

D. The Mean Deviation

The mean deviation is a measure of variation that makes use of all
the scores in a distribution. This is more reliable than the range and
quartile deviation.
a. Ungrouped data
To solve the mean deviation for ungrouped data, we use the
formula:

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∑/𝑿 −𝒙
̅/
MD =
𝑵

where: X = the score in the distribution

𝑥̅ = the mean
N = is the number of observations
Example: Find the mean deviation of the following distribution: 4, 8,
12.
Solution:
24
Calculate the mean. 𝑥̅ = =8
3

X 𝑥̅
4 4
8 0
12 4
∑/𝑿 − 𝒙
̅/ = 8

∑/𝑿 −𝒙
̅/ 𝟖
MD = = 𝟑 = 2.67
𝑵

b. Grouped data
For group frequency distribution, the formula is,
∑ 𝒇/𝑿 −𝒙
̅/ ∑/𝑿𝒊 −𝒙
̅/
MD = or MD =
𝑵 𝑵

Example: Find the mean deviation of the following:

X Frequency (f)
30 – 34 4
25 – 29 5
20 – 24 6
15 – 19 2
10 – 14 3
N = 20

Solution:
∑ 𝒇𝑿𝒊
̅=
Calculate the mean deviation by using the formula, 𝒙 , midpoint
𝑵

method, add columns for 𝑿𝒊 and 𝒇𝑿𝒊 .

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X Frequency (f) 𝑿𝒊 𝒇𝑿𝒊
30 – 34 4 32 128
25 – 29 5 27 135
20 – 24 6 22 132
15 – 19 2 17 34
10 – 14 3 12 36
N = 20 ∑ 𝒇𝑿𝒊 = 465
∑ 𝒇𝑿𝒊 𝟒𝟔𝟓
̅=
𝒙 = = 23.25
𝑵 𝟐𝟎

̅/ and f/X - 𝒙
Add the columns /𝑿𝒊 - 𝒙 ̅/

X frequency (f) 𝑿𝒊 /𝑿𝒊 - 𝑥̅ / 𝒇/𝑿𝒊 - 𝑥̅ /

30 – 34 4 32 8.75 35
25 – 29 5 27 3.75 18.75
20 – 24 6 22 1.25 7.50
15 – 19 2 17 6.25 12.50
10 – 14 3 12 11.25 33.75
N = 20 ∑ 𝒇/𝑿𝒊 − 𝒙
̅ / = 107.50

∑ 𝒇/𝑿𝒊 −𝒙
̅/ 𝟏𝟎𝟕.𝟓𝟎
MD = = = 5.375
𝑵 𝟐𝟎

E. The Variance and the Standard Deviation

The variance is the average of the squared deviation of the values about
the arithmetic mean. The differences of the values from the mean will produce
negative differences if the values are below the mean. To avoid this, variance
is developed as an alternative mechanism for overcoming the zero-sum
property of deviations from the mean.

The population variance is denoted by σ2 and can be obtained using

∑(𝒙− 𝒖)𝟐
the formula σ2 = 𝑵

where x = individual value

𝑢 = population mean

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N = population size

The standard deviation is the square root of the variance. It is popular,

and it is the most reliable measure of variability expressed in the same units as
the raw data, unlike the variance, which is expressed in square units. The
population standard deviation is denoted by σ can be computed as follows

∑(𝒙− 𝒖)𝟐
σ = √𝛔𝟐 = √ 𝑵

Sample variance is denoted by s2 and the sample standard deviation by

s. These are used as estimates of population variance and population standard
deviation. Using n – 1 in the denominator of a sample variance or sample
standard deviation, rather than n, results in a better estimate of the population
values. The sample variance can be obtained using the formula

∑𝒕𝒊=𝟏 𝒇𝒊(𝒙𝒊 − 𝒙
̅ )𝟐
s2 = 𝒏−𝟏

where s2 = sample variance

𝑓𝑖 = frequency of the 𝑖 th class
𝑥𝑖 = midpoint of the 𝑖 th class
𝑥̅ = sample mean
n = sample size

Sample solution of variance using the formula:

Scores Frequency Midpoint
[𝒙 𝒊 − 𝒙
̅] [𝒙𝒊 − 𝒙
̅]2 𝑓𝑖 [𝒙𝒊 − 𝒙
̅ ]2
(𝒇𝒊) (𝒙𝒊 )
35 -40 4 37.5 14.0 196 784
29 – 34 5 31.5 8.0 64 320
23 – 28 8 25.5 2.0 4 32
17 – 22 6 19.5 -4.0 16 96
11 – 16 4 13.5 -10.0 100 400
5 – 10 3 7.5 -16.0 256 768
Total 30 ̅]𝟐 = 2400
∑ 𝒇𝒊 [𝒙𝒊 − 𝒙

∑𝒕𝒊=𝟏 𝒇𝒊(𝒙𝒊 − 𝒙
̅ )𝟐 𝟐𝟒𝟎𝟎
s2 = = 𝟑𝟎−𝟏 = 82.7568
𝒏−𝟏

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While the sample standard deviation is
∑𝒕𝒊=𝟏 𝒇𝒊(𝒙𝒊 − 𝒙
̅ )𝟐
s = √𝒔𝟐 = √ = √𝟖𝟐. 𝟕𝟓𝟔𝟖 = 9.0972
𝒏−𝟏

Coefficient of Variance
𝒔
CV = [𝒙̅ (𝟏𝟎𝟎)] %
𝟗.𝟎𝟗𝟕𝟐
=[ (𝟏𝟎𝟎)] %
𝟐𝟑.𝟓

= 38.71%

V. Normal Distribution

The normal distribution is an extremely important concept because it

occurs so often in the data we collect from the natural world. If you measure the
height, the weight, or the age of students in class, probably you will find that
there are some students with very low measurements while others have very
high, and the majority of them are centered on a particular value. These show
a typical pattern that seems to be part of many real-life phenomena. In statistics,
this pattern is formally called the normal distribution.
Every normal distribution has a bell-shaped curve that is symmetric
about a vertical line though the mean of the distribution. Because of the exact
symmetry of a normal curve, the center of a normal distribution is located at the
highest point of the distribution and therefore, the mean, median, and mode are
all equal. In an idealized normal distribution of a continuous random variable,
the distribution continues infinitely in both directions. The area under the curve
is being associated with probabilities or proportions of the distribution with the
total area equal to 1.0 or 100%.

Probability and the Normal Curve

The normal distribution is a continuous probability distribution. This has several

implications for probability.

▪ The total area under the normal curve is equal to 1.

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▪ The probability that a normal random variable X equals any particular
value is 0.
▪ The probability that X is greater than a equals the area under the normal
curve bounded by a and plus infinity (as indicated by the non-
shaded area in the figure below).
▪ The probability that X is less than a equals the area under the normal
curve bounded by a and minus infinity (as indicated by the shaded area
in the figure below).

Using the empirical rule of a normal distribution, approximately

• 68% of the data lie within 1 standard deviation of the mean.
• 95% of the data lie within 2 standard deviations of the mean.
• 99.7% of the data lie within 3 standard deviations of the mean.

Example: 95% of students at school are between 1.1m and 1.7m tall.

Assuming this data is normally distributed can you calculate the mean and
standard deviation?

The mean is halfway between 1.1m and 1.7m:

Mean = (1.1m + 1.7m) / 2 = 1.4m

95% is 2 standard deviations either side of the mean (a total of 4 standard

deviations) so:

1 standard deviation = (1.7m-1.1m) / 4

= 0.6m / 4

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= 0.15m

It is good to know the standard deviation, because we can say that any value
is:

• likely to be within 1 standard deviation (68 out of 100 should be)

• very likely to be within 2 standard deviations (95 out of 100 should be)
• almost certainly within 3 standard deviations (997 out of 1000 should
be)

Standard Scores

The number of standard deviations from the mean is also called the
standard Score", "sigma" or "z-score".

Example 1. In that same school one of your friends is 1.85m tall.

You can see on the bell curve that 1.85m is 3 standard deviations from the
mean of 1.4, so:

Solution: Your friend's height has a "z-score" of 3.0

It is also possible to calculate how many standard deviations 1.85 is from the
mean

How far is 1.85 from the mean?

It is 1.85 - 1.4 = 0.45m from the mean

How many standard deviations is that? The standard deviation is 0.15m, so:
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0.45m / 0.15m = 3 standard deviations

To convert a value to a Standard Score ("z-score"):

• first subtract the mean,

• then divide by the Standard Deviation

And doing that is called "Standardizing":

We can take any Normal Distribution and convert it to The Standard Normal
Distribution. (See Appendix B)

Example 2: A survey of daily travel time had these results (in minutes):

26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34

The Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes.
Convert the values to z-scores ("standard scores").

To convert 26:

first subtract the mean: 26 − 38.8 = −12.8,

then divide by the Standard Deviation: −12.8/11.4 = −1.12

So 26 is −1.12 Standard Deviations from the Mean

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Here are the first three conversions

Original Value Calculation Standard Score

(z-score)

26 (26-38.8) / 11.4 = −1.12

33 (33-38.8) / 11.4 = −0.51

65 (65-38.8) / 11.4 = +2.30

... ... ...

And here they are graphically:

The z-score formula that we have been using is:

𝒙− 𝝁
z=
𝝈

• z is the "z-score" (Standard Score)

• x is the value to be standardized
• μ ('mu") is the mean
• σ ("sigma") is the standard deviation

And this is how to use it:

Example 3: Travel time (continued)

Here are the first three conversions using the "z-score formula":

𝒙− 𝝁
z= 𝝈

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• μ = 38.8
• σ = 11.4

𝒙− 𝝁 z
X 𝝈
(z-score)

26 26 − 38.811.4 = −1.12
33 33 − 38.811.4 = −0.51
65 65 − 38.811.4 = +2.30
... ... ...

The exact calculations we did before, just following the formula.

Standard Normal Distribution

If the data set is a normal distribution, it follows that the corresponding

distribution of z-scores is also a normal distribution which is known as standard
normal distribution. The mean of the transformed z-scores is equal to 0 and the
standard deviation is 1.

Example: A 60-item test was conducted among First year BSIT students. The
following scores were obtained: 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17

Most students didn't even get 30 out of 60, and most will fail.

The test must have been really hard, so the teacher decides to Standardize all
the scores and only fail people more than 1 standard deviation below the
mean.

The Mean is 23, and the Standard Deviation is 6.6, and these are the
Standard Scores:

-0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91

Now only 2 students will fail (the ones lower than −1 standard deviation)

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VI. Linear Regression and Correlation

Correlation (Diane Keirnan, 2014) refers to the statistical association

between two variables. A correlation exists between two variables when one of
them is related to the other in some way. A scatterplot is the best place to start.
A scatterplot (or scatter diagram) is a graph of the paired (x, y) sample data
with a horizontal x-axis and a vertical y-axis. Each individual (x, y) pair is plotted
as a single point.
A scatterplot can identify several different types of relationships between
two variables.
A relationship has no correlation when the points on a scatterplot do
not show any direction or pattern.
A relationship is non-linear when the points on a scatterplot follow a
pattern but not a straight line.
A relationship is linear when the points on a scatterplot follow a
somewhat straight line pattern. This is the relationship that we will examine.
Linear relationships can be either positive or negative. Positive
relationships have points that incline upwards to the right. As x values increase,
y values increase. As x values decrease, y values decrease. For example,
when studying plants, height typically increases as diameter increases.
Correlation coefficients are computed and the most widely used
measure of correlation is the Pearson Product Moment Correlation Coefficient
or simply Pearson r.
𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)
r=
√[𝑛 ∑ 𝑥 2 −(∑ 𝑥)2 ]−[𝑛 ∑ 𝑦 2 −(∑ 𝑦)2 ]

where x = the observed data for the independent variable

y = the observed data for the dependent variable
n = the sample size
∑ 𝑥 = the summation of x values
∑ 𝑦 = the summation of y values
∑ 𝑥 2 = the summation of the square of each of x values
∑ 𝑦 2 = the summation of the square of each of y values
∑ 𝑥𝑦 = the summation of the product of the x and y values

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Range of correlation coefficients Degree of correlation
±1 Perfect linear relationship
±0.81 – ±0.99 Very strong linear relationship
± 0.61 - ± 0.80 Strong linear relationship
±0.41 - ±0.60 Moderate linear relationship
±0.21 - ±0.40 Weak Linear relationship
±0.01 - ±0.20 Very weak linear relationship
0 No linear relationship

Example:
A study was conducted to investigate the relationship existing between
the grade in Statistics and the grade in Computer subject. A random sample of
10 computer students in a certain college were taken and the data are as
follows:

Student A B C D E F G H I J
Statistics 75 83 80 77 89 78 92 86 93 84
Computer 78 87 78 76 92 81 89 89 91 84

Is there a relationship between the performance of the students in Statistics

and Computer subjects?

Solution:
Student Statistics Computer xy x2 y2
(x) (y)
A 75 78 5850 5625 6084
B 83 87 7221 6889 7569

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C 80 78 6240 6400 6084
D 77 76 5852 5929 5776
E 89 92 8188 7921 8464
F 78 81 6318 6084 6561
G 92 89 8188 8464 7921
H 86 89 7654 7396 7921
I 93 91 8463 8649 8281
J 84 84 7056 7056 7056

∑𝒙𝒚 = ∑ 𝒙𝟐 = ∑ 𝒚𝟐 =
n = 10 ∑ 𝒙 = 837 ∑ 𝒚 = 845 71717
71030 70413

𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)

r=
√[𝑛 ∑ 𝑥 2 −(∑ 𝑥)2 ]−[𝑛 ∑ 𝑦 2 −(∑ 𝑦)2 ]

10(71030)−(837)(845)
=
√[10 70413)−(837)2][10(71717)−(845)2 ]
(

3035
=
√(3561)(3145)

= 0.906906226
= 0.91
Therefore: There exists a very positive relationship between the
performance of the students in Statistics and Computer.

A simple linear regression model is a mathematical equation that

allows us to predict a response for a given predictor value. This is used in the
process of prediction. Prediction is calculating scores of the criterion variables
(𝑦̂) on the basis of the knowledge of the predictor (x). One example is the
prediction of job performance of an applicant using information available during
the time of his application.
Linear regression can be computed using the equation,
̂ = a + bx
𝒚 which is called the least square line or the simple regression line
Where a = the y-intercept
b = the slope
x = the predictor variable, and
̂ = estimate of the mean value of the response variable for any
𝒚

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value of the predictor variable.
The y-intercept is the predicted value for the response (y) when x = 0. The slope
describes the change in y for each one unit change in x.

The value of a and b can be obtained by using the following:

𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)
b= 𝑛 ∑ 𝑥 2 −(∑ 𝑥)2

a = Mny - bMnx
where Mny = the mean of the y values
Mnx = the mean of the x values

Example:
Problem: Compute and interpret the correlation coefficient for the
following grades of ten students selected at random.

Student A B C D E F G H I J
Statistics 75 83 80 77 89 78 92 86 93 84
Computer 78 87 78 76 92 81 89 89 91 84
Is there a significant degree of relationship between Statistics and computer
grades of the student?

Solution:
Student Statistics Computer xy x2 y2
(x) (y)
A 75 78 5850 5625 6084
B 83 87 7221 6889 7569
C 80 78 6240 6400 6084
D 77 76 5852 5929 5776
E 89 92 8188 7921 8464
F 78 81 6318 6084 6561
G 92 89 8188 8464 7921
H 86 89 7654 7396 7921
I 93 91 8463 8649 8281

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J 84 84 7056 7056 7056

∑𝒙𝒚 = ∑ 𝒙𝟐 = ∑ 𝒚𝟐 =
n = 10 ∑ 𝒙 = 837 ∑ 𝒚 = 845 71717
71030 70413

n = 10 ∑ 𝑥 = 837 ∑ 𝑦 = 845 ∑ 𝑥 2 = 70413 ∑ 𝑥 𝑦 = 71030

10(71030)−(837)(845) 3035
b= = 3561 = 0.85
10(70413)−(837)2
837
Mnx = = 83.7
10
845
Mny = = 84.5
10

a = 84.5 – (0.850)(83.7) = 13.36

The regression equation is 𝑦̂ = 13.36 + (0.85)x.

Based on the qualitative interpretation of r, the result indicates that there

is a very strong linear relationship between the Computer and Statistics
grades of the students. That is, the higher is the student’s Computer grade, the
higher is his/her Statistics grade.

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Summary

✓ The study of statistics can be organized in different ways. One way is to

subdivided statistics into two branches: descriptive statistics and
inferential statistics.
✓ A population generally consists of the totality of the observations,
individuals, or objects in which the investigator is interested.
✓ A sample is a portion of a population. This is a small but representative
cross section of the population. It is used to give inferences on the
population from which it was extracted.
✓ There are basically two kinds of variables: Qualitative variables are
variables that can be placed into distinct categories, according to some
characteristic or attribute. Quantitative variables are numerical and can
be ordered or ranked.
✓ Quantitative variables can be further classified into two groups: discrete
and continuous. Discrete variables can be assigned values such as 0, 1,
2, an 3 and are said to be countable. The data are obtained by means
of counting. Continuous variables, on the other hand, can assume an
infinite number of values in an interval between any two specific values.
The data for this variable are obtained by means of direct or indirect
measuring.
✓ The four levels of measurement are: nominal, ordinal, interval, and ratio.
✓ Data are best presented in different forms such as: Textual presentation,
tabular presentation and graphical presentation (bar graph, line graph,
pie graph).

✓ Descriptive Statistics is a technique enabling to meaningfully describe

data with numerical or in graphic form. This technique includes the
following: Measure of Central Tendency; Measures of Relative Position;
and Measures of Variation.
✓ Measure of Central Tendency determines a numerical value in the
central region of a distribution of scores. Central tendency refers to the
center of a distribution of observations. There are three measures of
central tendency: the mean, the median, and the mode.

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❖ The mean, 𝑥̅ is also called the arithmetic mean or average. It can
be affected by extreme scores. It is stable, and varies less from
sample to sample.
∑𝑥
o The mean 𝑥̅ formula: 𝑥̅ =
𝑛
∑𝑛
𝑖=1 𝑤𝑖 𝑥𝑖
o The weighted mean formula 𝑥̅ w = ∑𝑛
𝑖=1 𝑤𝑖

❖ There are two ways on how to solve for the value of mean given
the grouped data or frequency distribution:
∑𝑘
𝑖=1 𝑓𝑖 𝑥𝑖
o by midpoint method: 𝑥̅ =
𝑛
∑𝑘
𝑖=1 𝑓𝑖 𝑑𝑖
o by coded deviation method: 𝑥̅ = ̅̅̅
𝑥0 + i [ ]
𝑛

❖ The median (𝑥̃ ), is the value in the distribution that divides an

arranged (ascending/descending) set into two equal parts. It
is the midpoint or middlemost of a distribution of scores.
o The median for ungrouped data is obtained by
inspecting the middlemost value of the arranged
distribution either in ascending or descending order.
(𝑁+1)
It can also be solved using the formula position
2𝑡ℎ

after being arranged.

o The formula to solve for the median given the
𝑁
−𝐶𝐹𝑏
grouped data: 𝑥̃ = XLB + i [ 2 𝑓 ]
𝑚

❖ The mode (𝑥̂) is the value with the largest frequency. It is the
value that occurs most frequently in the distribution.
❖ To find the mode of the grouped data, determine first the
modal class. The modal class is the class with the highest
frequency.
𝒅𝒇𝟏
o 𝒙
̂ = XLB + i [ ]
𝒅𝒇𝟏 + 𝒅𝒇𝟐

✓ Measure of Relative Position, as median divides the set of scores into

two equal parts, there are other measures that divide the distribution into
one hundred, four, or ten equal parts.
o For ungrouped data:

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(𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑥)+0.5
Percentile = ●100
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠

o For grouped data:

𝒏𝑵
− 𝑪𝑭𝒃
➢ Percentile is: Pn = XLB + i[𝟏𝟎𝟎𝒇 ]
𝑷𝒏

𝒏𝑵
− 𝑪𝑭𝒃
➢ Quartile: Qn = XLB + i [ 𝟒 𝒇 ]
𝑸𝒏

𝒏𝑵
− 𝑪𝑭𝒃
➢ Decile: Dn = XLB + i [ 𝟏𝟎
]
𝒇𝑫𝒏

✓ The degree of variation measures the degree of the spread of the values.
The measures of spread are commonly called measures of dispersion or
measures of variation.

o Range: R= u.b. – l.b.

o Interquartile Range: I.R. = Q3 – Q1
𝑄3 − 𝑄1
o Quartile Deviation Q.D. = 2
∑ 𝑓/𝑋 −𝑥̅ /
o Mean Deviation: MD = 𝑁

❖ The variance is the average of the squared deviation of the

values about the arithmetic mean.
∑(𝑥− 𝑢)2
o The population variance formula: σ2 =
𝑁

❖ The standard deviation is the square root of the variance.

o The population standard deviation formula:
∑(𝑥− 𝑢)2
σ = √σ2 = √ 𝑁

𝑠
❖ Coefficient of Variance: CV = [𝑥̅ (100)] %

✓ The normal distribution has a bell-shaped curve that is symmetric about

a vertical line though the mean of the distribution.
✓ Correlation (Diane Keirnan, 2014) refers to the statistical association
between two variables. A correlation exists between two variables when
one of them is related to the other in some way.
❖ Pearson Product Moment Correlation Coefficient or simply
𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)
o Pearson r formula: r=
√[𝑛 ∑ 𝑥 2 −(∑ 𝑥)2 ]−[𝑛 ∑ 𝑦 2 −(∑ 𝑦)2 ]

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❖ A simple linear regression model is a mathematical equation that
allows us to predict a response for a given predictor value.
o Linear regression formula: 𝑦̂ = a + bx

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ASSESSMENT TASK
Exercise 3.1 – a
Data Management

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

1. The size of pants sold during one business day in a department store
are 32, 38, 34, 42, 36, 34, 40, 44, 32, 34. Find the average size of the
pants sold.

2. Given the frequency distribution for the weights of 50 pieces of luggage.

Compute the mean

WEIGHT (kg) NUMBER OF PIECES (f)

7–9 2
10 – 12 8
13 – 15 14
16 – 18 19
19 – 21 7
N 50

3. The ages of 10 administrators in a certain college are given as follows:

Compute the median.

40 38 45 51 53 59 45 56 45 44

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4. Compute the median given the following data:
SCORES IN (f)
STATISTICS
75 – 79 6
70 – 74 7
65 – 69 2
60 – 64 8
55 – 59 12
50 – 54 7
45 – 49 10
40 – 44 8
N 60

5. Find the mode in the following data:

1 5 6 9 11 15 17
2 5 7 9 12 15 17
3 5 7 9 12 15 18
4 6 8 12 10 16 18
4 6 9 12 11 16 18

6. Solve for the mode, given the frequency distribution:

SCORES IN ALGEBRA (f)

75 – 79 6
70 – 74 7
65 – 69 2
60 – 64 8
55 – 59 12
50 – 54 7
45 – 49 10
40 – 44 8
N 60

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ASSESSMENT TASK
Exercise 3.1 – b
Data Management

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Given the frequency distribution below, calculate the following: P25, Q3, D4

Statistics Test Results

Scores Frequency Cumulative
Frequency
60 – 62 2 40
57 – 59 2 38
54 – 56 4 36
51 – 53 5 32
48 – 50 11 27
45 – 47 8 16
42 – 44 4 8
39 -41 2 4
36 – 38 1 2
33 – 35 1 1
N = 40

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ASSESSMENT TASK
Exercise 3.1 – c
Data Management

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________
1. Find the range:
1 5 6 9 11 15 17
2 5 7 9 12 15 17
3 5 7 9 12 15 18
4 6 8 12 10 16 18
4 6 9 12 11 16 18

2. Solve for the range.

Class interval f
25 – 29 5
20 – 24 6
15 – 19 7
10 – 14 8
5–9 4
N = 30

3. Solve for the IR, given Q1 = 69.81 and Q3 = 87.9

4. Compute the IR using the following data,

Class interval frequency (f)
25 – 29 5
20 – 24 6
15 – 19 7
10 – 14 8
5–9 4
N = 30

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5. Solve for QD.
Class interval frequency (f)
25 – 29 5
20 – 24 6
15 – 19 7
10 – 14 8
5–9 4
N = 30

6. Find the mean deviation: 32, 35, 26,15

7. Given:
Class interval frequency (f)
25 – 29 5
20 – 24 6
15 – 19 7
10 – 14 8
5–9 4
N = 30

8. Solve for the variance, standard deviation and coefficient of variance.

Class interval frequency (f)
25 – 29 5
20 – 24 6
15 – 19 7
10 – 14 8
5–9 4
N = 30

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ASSESSMENT TASK
Exercise 3.1 – d
Data Management

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Answer the following:

1. For a certain type of computers, the length of time between charges of

the battery is normally distributed with a mean of 50 hours and a
standard deviation of 15 hours. John owns one of these computers and
wants to know the probability that the length of time will be between 50
and 70 hours.

2. Entry to a certain University is determined by a national test. The scores

on this test are normally distributed with a mean of 500 and a standard
deviation of 100. Tom wants to be admitted to this university and he
knows that he must score better than at least 70% of the students who
took the test. Tom takes the test and scores 585. Will he be admitted to
this university?

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ASSESSMENT TASK
Exercise 3.1 – e
Data Management

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

A. Multiple Choice. Encircle the letter of the correct answer.

1. Given below are the daily high temperatures for one April, week in Albay.
What is the mode of these temperatures? 37, 38, 45, 46, 38, 40, 41
A. 37 B. 46 C. 38.5 D. 38

2. The mean of a set of numbers is 148. The sum of the numbers is 3,552. How
many numbers are in the set?
A. 24 B. 30 C. 20 D. 28

3. A car dealer is recording a table indicating the number of cars of each color
sold during the last 6 months. Which measure of central tendency is used to
describe the bestselling color of the car sold?
A. Mean B. Median C. Mode D. Range

4. Joshua’s academic target is to get his test average in the top half of his
Statistics class. Which measure can he use to determine if he has achieved his
goal?
A. Mean B. Median C. Mode D. Range

5. The time Jepoy worked last week was recorded in the table below. Which
measure of central tendency would change if Jepoy worked two days less than
on Wednesday?
Monday Tuesday Wednesday Thursday Friday

8 hours 6 hours 7 hours 4 hours 6 hours

A. Mean B. Median C. Mode D. Range

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6. The median score is also the _______________.
A. 75th percentile C. upper quartile
B. 5th decile D. first quartile

7. Mary obtained a score of 45 which is equivalent to a 60th percentile rank in

Statistics test. Which of the following is NOT true?
A. She scored above 60% of her classmates
B. Forty percent of the class got scores of 45 and above
C. If the passing grade is the lower quartile, she passed the test.
D. Her score is below the 5th decile.
8. Juvy’s score in a 75-item was the median score. What is her percentile rank?
A. 75 B. 35 C. 37 D. 50

9. If you have the variance, how do you get the standard deviation?
A. Square it. C. Take the reciprocal
B. Take the square root D. Subtract the mean

10. The variance and mean for a given set of data is 25 and 64 respectively.
Calculate the standard deviation.
A. 625 B. 5 C. 2.56 D. 8

11. When the correlation coefficient, r, is close to one ______________.

A. there is no relationship between the two variables
B. there is a strong linear relationship between the two variables
C. it is impossible to tell if there is a relationship between the two
variables
D. the slope of the regression line will be close to one.

12. Let x be a normal random variable with a mean of 50 and a standard

deviation of 3. A z-score was calculated for x, and the z-score is -1.2. What is
the value of x?
A. 53.6 B. 0.1151 C. 46.4 D. 0.8849

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13. Mr. Z has just given a chemistry exam and wanted to calculate the ‘range’
of performance of his students. What must he do?
A. Identify the central score.
B. Find the score that occurs most often.
C. Identify how far from average each score is.
D. Subtract the lowest score from the highest score.

14. To determine the mean of a set of data, the first step is to _________.
A. calculate the median
B. add the sum of the values in the set
C. Determine the standard deviation of the set of values
D. Calculate the mode first and then the sum

15. When the correlation coefficient, r, is close to one ___________.

A. there is no relationship between the two variables
B. there is a strong linear relationship between the two variables
C. it is impossible to tell if there is a relationship between the two
variables
D. the slope of the regression line will be close to one

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ASSESSMENT TASK

Exercise 3.1 – f
Data Management

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

B. Solve the given problem.

1. Using the data below which are the number of kilos of newspapers
contributed by one section in the Newspaper Fund Drive.
25 64 15 42 64

30 30 12 45 63

45 25 70 48 82

16 17 42 33 75

10 12 48 35 67

13 18 16 47 80

12 34 15 50 83

52 48 35 15 57

a. Construct a frequency distribution

b. Calculate for:
1. Mean 6. D8
2. Median 7. IR
3. Mode 8. QD
4. Q1 9. P90
5. P75 10. Standard Deviation

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References:
Daligdig, Romeo M. EdD et. al. (2019) Mathematics in the Modern World.
LORIMAR Publishing Inc., Quezon City, Metro Manila. Pp

Alejan, Ronnie O et. al. (2018). Mathematics in the Modern World. Mutya
Publishing House Inc., Malabon City pp. 50 – 56

Tagaro, Cosuelo PhD (2014). Advanced Statistics 22nd edition

Normal Distribution (2019). Retrieved from

https://www.mathsisfun.com/data/standard-normal-distribution.html
Understanding Probability Distributions (2018). Retrieved from
https://www.statisticsbyjim.com/basics/probability-distributions/.

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Chapter 3. Mathematics As A Tool (Part II)

Overview

Mathematical ideas contribute to art like patterns, symmetry, tiling,

geometry, Islamic art, T’boli weaving, higher order geometry and topology;
fractals, Fibonacci numbers and golden ratio. Mathematical operations occur
everyday in arts and even in science.

In our culture, mathematics has really influenced a lot. The weaving of

localized materials employs finite designs and repeating patterns in art forms
that are based on symmetry and colored symmetrical structures. The different
algebraic structures can be used as a framework to distinguish the artwork from
a particular cultural community focusing on mathematics inherent in local
designs that can promote better appreciation for Philippine heritage and culture.

On the other hand, according to Vijay Joshi (2019) “Money is not

everything in life but money is the most need of everyone’s life.” In this chapter,
the mathematics of Finance will describe the value of money which change over
a period of time. It will discuss the simple interest; compound interest; credit
cards versus consumer loans; stock; bonds; and mutual funds from home
ownership.

General Objectives:

This chapter aim to discuss the application and appreciation of geometric

designs; compute simple and compound interest; and differentiate credit cards
from consumer loans, stocks, bonds and mutual funds from home ownership.

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Lesson 1. Geometric Designs
Geometry (from the Ancient Greek: geo- "earth", -
metron "measurement") is a branch of mathematics concerned with questions
of shape, size, relative position of figures, and the properties of space. A
mathematician who works in the field of geometry is called a geometer.

Geometry arose independently in a number of early cultures as a

practical way for dealing with lengths, areas, and volumes. Geometry began to
see elements of formal mathematical science emerging in Greek
mathematics as early as the 6th century BC. By the 3rd century BC, geometry
was put into an axiomatic form by Euclid, whose treatment, The Elements, set
a standard for many centuries to follow. Geometry arose independently in India,
with texts providing rules for geometric constructions appearing as early as the
3rd century BC. Islamic scientists preserved Greek ideas and expanded on
them during the Middle Ages. By the early 17th century, geometry had been put
on a solid analytic footing by mathematicians such as René
Descartes and Pierre de Fermat. Since then, and into modern times, geometry
has expanded into non-Euclidean geometry and manifolds, describing spaces
that lie beyond the normal range of human experience.

While geometry has evolved significantly throughout the years, there are
some general concepts that are fundamental to geometry. These include the
concepts of point, line, plane, distance, angle, surface, and curve, as well as
the more advanced notions of topology and manifold.

Geometry has applications to many fields,

including art, architecture, physics, as well as to other branches of
mathematics.

Specific Objectives:

At the end of this lesson, the students are expected to:

1. Apply geometric concepts in describing and creating designs; and

2. Contribute to the enrichment of the Filipino culture and the arts using the
concept in geometry.

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I. What is Geometric Design?

Geometrical design (GD) (Wikipedia) is a branch of computational

geometry. It deals with the construction and representation of free-form curves,
surfaces, or volumes and is closely related to geometric modeling. Core
problems are curve and surface modelling and representation. GD studies
especially the construction and manipulation of curves and surfaces given by a
set of points using polynomial, rational, piecewise polynomial, or piecewise
rational methods. The most important instruments here are parametric
curves and parametric surfaces, such as Bézier curves, spline curves and
surfaces. An important non-parametric approach is the level-set method.

Application areas include shipbuilding, aircraft, and automotive

industries, as well as architectural design. The modern ubiquity and power of
computers means that even perfume bottles and shampoo dispensers are
designed using techniques unheard of by shipbuilders of 1960s.

Geometric models can be built for objects of any dimension in

any geometric space. Both 2D and 3D geometric models are extensively used
in computer graphics. 2D models are important in
computer typography and technical drawing. 3D models are central
to computer-aided design and manufacturing, and many applied technical
fields such as geology and medical image processing.

Geometric models are usually distinguished from procedural and object-

oriented models, which define the shape implicitly by an algorithm. They are
also contrasted with digital images and volumetric models; and with
mathematical models such as the zero set of an arbitrary polynomial. However,
the distinction is often blurred: for instance, geometric shapes can be
represented by objects; a digital image can be interpreted as a collection of
colored squares; and geometric shapes such as circles are defined by implicit
mathematical equations. Also, the modeling of fractal objects often requires a
combination of geometric and procedural techniques.

Geometric problems originating in architecture can lead to interesting

research and results in geometry processing, computer-aided geometric
design, and discrete differential geometry.

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In architecture, geometric design is associated with the pioneering
explorations of Chuck Hoberman into transformational geometry as a design
idiom, and applications of this design idiom within the domain of architectural
geometry.

Uses of Geometric Patterns

Geometric patterns are widely considered as one of the most visually

appealing styles when it comes to design.

They give a sense of stability, futurism, minimalism while adding

boldness to any creative.

Geometric patterns include many different shapes that can help you
enforce the message you want to send.

Although you might expect simplicity when it comes to geometric

designs, you’ll see in the examples that you can create complex visuals as well.

The Most Common Types of Geometric Shapes

At the core of any geometric designs stand five geometric shapes and
each of them symbolizes something:

➢ The square/rectangle. This is the most commonly used shape design

and gives a sense of balance and tradition.
➢ The circle. It’s a reminder of harmony, love, and perfection and is also
known as a protective symbol.
➢ The triangle. This shape suggests stability, energy, and power, and it’s
also associated with motion and direction.
➢ The hexagon. It communicates unity and balance and can be linked to
cooperation.

You can only use one of these shapes in your designs, but you can also
experiment, get creative, and combine several of them to create neat geometric
patterns.

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Shapes can also be used to add some balance to your design. As you’re
about to see, incorporating different geometrical shapes in your design can be
quite easy.

You can use patterns to highlight specific elements in your design as

they have the power to really draw people’s attention.

Regardless if you’re thinking of geometric circle patterns or square ones,

it’s essential to know which one would best accompany the message that you
can want to send out.

Here are a few tips that you can get inspired by:

1. Use shapes to create an image

Using different shapes to create an image can be a great way of
making your design stand out. It will look fresh, as well as elegant.
The clear lines, as well as the color selection, can definitely make
a difference, so pay extra attention when choosing these elements

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2. Create an appealing background

Geometric design patterns can very well be the centerpiece of

any design. At the same time, they can successfully be used as a
background.

In the image below, for example, a combination of several shapes

was used to create a geometric background.

The colors that were chosen manage to give off a vibe of stability
and vibrancy at the same time.

3. Use real-life elements

When creating a poster, for example, using real-life elements that
you encounter every day geometric patterns can lead to a fantastic
design.
However, you should keep in mind that the elements used should
be carefully chosen. You can opt for simple ones and combine them with
either geometric patterns or simple shapes.

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4. Make a collage

A collage made out of shapes is also a valid option. Create

mesmerizing design by combining different shapes and colors.

The benefit of using geometric shape patterns is that they fir

almost in any design, as long as you can choose the right colors and the
right theme.

Combining shapes, flowers, and vibrant colors can result in a

fascinating design that’s sure to draw attention.

However, you wouldn’t want to just throw in some patterns

together and hope for the best.

You should make sure, above anything else, that the colors work
perfectly together. So, if you go for pastel colors, it’s best to stick to that
color palette and not add any other colors that might look like they don’t
belong there.

5. Create depth

A geometric shape pattern, whatever shapes it consists of, can

look great in any design.

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The following example is a simple as it can be, using only a
geometric square pattern. Still, with the use of the different shades blue,
it looks futuristic and quite appealing.

6. Make it abstract
As you can see in the image below, all shapes are different and
kind of funky looking, but somehow connected to each other.
They might all have different colors and decorations, with some
having a geometric circle design or a circle design, but they need to have
a connecting piece.
In this case, it’s the little lines and dots that fill each shape, as well
as the pastel color pallete.

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7. Get creative with lines
Lines can also be a great way of drawing attention to your design.
They can be used by themselves or in combination with circles or
rhombuses.
Lines can help you add to touch of sleekness and elegance, and
most of all, are highly noticeable.
If you want your design to look stunning, without being
unnecessarily complex, go for some straight lines like the one below.
Choosing to play with colors and lines will result in some very eye-
catching patterns.

8. Combine patterns with photos

It goes without saying that designs that combine real-life images with
geometric patterns will definitely make an impression, primarily if we’re
referring to nature photos.

However, feel free to use any kind of photos that you think work
best for your campaign, the main goal here being to combine these two
elements as seamlessly as possible.

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Black and White Geometric Patterns

Black and white have always been a staple of elegance. Combining

these two colors with geometric patterns can only result in a very sophisticated
and appealing design.

What’s great about them is that black and white geometric shapes are
always bound to look complex and refined.

Just choose the shapes you want to create a geometric black and white
pattern and leave the rest to the imagination.

You can play with them however you want, there’s no wrong way to go
about it.

Here are some ideas that you can use:

a. Use geometric swirls

Since black and white designs are bound to stand out only due to
the shapes used and not the colors, make sure that you create
something unusual such as the pattern below.
The overlapped use of circle adds motion to the design, as well
as the balance.
Used by itself or incorporated in a more sophisticated design, a
circle pattern can definitely look compelling.
It can also play the small part of a large picture, and represent a
pattern within a more prominent, more complex pattern.

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b. Mix and match different shapes

Without the use of color, your design must stand out by relying
only on the shapes used.
So you need to get creative.
Play with several shapes in a new way and create a simple
abstract design, but at the same time attractive.
This poster is the perfect example of how sophisticated a design
can look using just black and white geometric patterns.

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c. Keep it simple
Whatever your idea, don’t let yourself think that using as many
colors as possible is the only way to go.
Great design relies on the creative idea behind it, and as it turns
out, some ideas look better illustrated using monochrome colors.
Also, remember that less is more, and even the simplest design
can bring amazing results.
Great ideas take time and creativity, so make sure you put in as
much thought as possible in the entire process.
Use geometric shapes in whole new creative ways, add them
sparingly to a design, or base the entire background on them.
Either way, you’re sure to come up with something great that’s
not only original but also new and refreshing.

Playing with Colorful Geometric Patterns

Check out these ideas on how you can successfully use patterns and colors in
your designs:

1. Choose unusual shapes

A colorful geometric pattern might be precisely what your design
needs to be popping.
Take Portuguese azulejos as an example.
They’re probably the most recognizable tile pattern in the world
because of their unique geometrical and colorful designs.

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They’re fun, good-looking, and definitely draw anyone’s attention.
Most of all, colorful geometric designs require creativity, and you
can start from there.
Choose your colors, choose your shapes, and start creating.
However, don’t limit yourself to those five commonly used shapes, use
others from real life, such as the shape pf plants or flowers.

2. Make it bold
Colorful geometric shapes do a great job of creating a very
appealing and playful design, even in combination with shapes that are
not so commonly used.
In fact, the combination of several patterns with a pop of color will result
in a much more lasting impression.

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3. Use complementary colors
Geometric color patterns such as the one below, use a variety of
shapes and colors that complement each other beautifully.
It’s essential to pay extra attention to this since the color choice
can impact the overall aspect of your design.
When using the right colors and the most suitable shapes, your
design will look very well put together, with elements that give it a certain
sense of fluidity and structure.
The design below looks like it’s organized chaos, both
aesthetically pleasing and well balanced.

4. Combine different images

Think about the main feeling or emotion that you want to display
through your design and what the highlight should be.
Do you want your design to be contemporary, to reflect stability,
or suggest motion?
Or, on the contrary, do you want a more traditional design that
focuses on simplicity and clarity?

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II. Mindanao Designs, Arts, and Culture

The T’boli Tribe of South Cotabato

More than 40 different ethnic groups can be found in the Philippines.

There are 18 tribal groups on the island of Mindanao. The most well-known is
the T'boli Tribe, that lives in the province of South Cotabato, around lake Sebu.
Since the arrival of settlers originating from the other islands of the Philippines,
they gradually moved to the mountain slopes to live in scattered settlements in
the Highlands.

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This Tribal group is still living in a traditional way, comparable with how
their ancestors lived centuries ago. The T'boli distinguish themselves from other
Tribal Groups by their colorful clothes, bracelets and earrings, this tribe is
famous for their complicated beadwork, wonderful woven fabrics and beautiful
brass ornaments.

A T'boli legend tells that the T'boli are descendants of the survivors of a
great flood. A man named Dwata warned the people of an impending great
flood. But the tribe refused to listen, except for two couples, La Bebe and La
Lomi, and Tamfeles and La Kagef. Dwata told them to take shelter in a bamboo
so huge they could fit inside and, in this way, survive the flood. The story tells
that the first couple are the ancestors of the T'boli and other highland ethnic
groups. The second couple descended the other Filipino indigenous groups.

The T'boli culture is richly connected with and inspired by nature, their
dances are a mimick from the action of animals such as monkeys and birds.
The T'boli have a rich musical culture with a variety of musical instruments, but
the T'boli music and songs are not meant for entertainment only. The Tribal
songs are a living contact with their ancestors and a source of ancient wisdom.
The T'boli believe that everything has a spirit which must be respected for good
fortune. Bad spirits can cause illness and misfortune.

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The T'nalak, the T'boli sacred cloth, made from abaca is the best known
T'boli craft and is one of the tribe’s traditional textile, this cloth is exchanged
during marriages and used as a cover during births. The T'boli women are
named dreamweavers, another legend tells us that the T'nalak weaving was
taught by a goddess named Fu Dalu in a dream and that women learn this
ethnic and sacred ritual, based on tribal designs and cloth patterns through their
dreams. These unique patterns are made with centuries-old practices and
passed down from generation to generation. This typical T'boli textile is history
held in the hands of their makers and the rich cultural heritage can be seen
through their creations, it shows the tribe's collective imagination and cultural
meanings.

The weaving is a very tedious job and requires much patience, a lot of
creativity and a good memory to remember the particular designs. Men are not
allowed to touch the chosen abaca fiber and materials used in the weaving
process and the weaver should not mate with her husband in the time the cloth
is woven, for it may break the fiber and destroy the design. At present the
T'nalak products have become the signature product of the province of South
Cotabato.

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When visiting the friendly T'boli tribe in South Cotabato you will be in the
midst of a distinctive and very well-preserved culture that is keeping their
characteristic, ancient traditions alive. A visit that will be an unforgettable and
inspiring experience!

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Summary:

✓ Geometrical design (GD) (Wikipedia) is a branch of computational geometry. It

deals with the construction and representation of free-form curves, surfaces, or
volumes and is closely related to geometric modeling.
✓ The most common types of geometric shapes are the: squares/
rectangles, circle, triangle and hexagon.
✓ The different tips using different geometric designs are: use shapes to
create an image; create an appealing background; use real – life
elements; make a collage; create a depth; make it abstract; get creative
with lines; and combine patterns in photos.
✓ Black and white geometric patterns are: use of geometric swirls; mix and
match different shapes; and keep it simple.
✓ Some uses of patterns and colors in a design: choose unusual shapes;
make it bold; use complementary colors; and combine different images.
✓ T’boli tribes lives in the province of South Cotabato around Lake Sebu.
They can be distinguished from other Tribal groups by their colorful
clothes, bracelets, and earrings.
✓ The T’nalak, the T’boli sacred cloth, made from abaca which considers
their craft and one of the tribes’ traditional textile.

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ASSESSMENT TASK

Exercise 3.1 – a
Geometric Designs

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

1. Aside from the given geometric patterns, give atleast 5 other patterns
and apply each one in coming up with your own design. Be ready to
create it on a ¼ size of illustration board and submit for classroom
exhibit.

2. In what other ways can we utilize geometric and other mathematical

patterns in enriching Philippine culture and arts. Cite 5 situations to prove
this.

3. How do we apply math transformations in enhancing the Philippine

culture and arts?

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References:

Ronald de Jong, Oct 23, 2008 | Destinations: Philippines / Mindanao

Scott, William Henry. Barangay: Sixteenth-Century Philippine Culture and

Society. Ateneo de Manila University Press.
Eugenio, Damiana. Philippine Folk Literature Vol. VIII: The Epics. University of
the Philippines Press.
Geometry Wikipedia (2020). Retrieved from
https://en.wikipedia.org/wiki/Geometry

40 Brilliant Geometric Patterns (And How To Use them in Your Designs)

(2020). Retrieved from https://blog.bannersnack.com/author/ana-darstaru/

The T’boli Tribe of South Cotabato (2018, October, 23). Retrieved from
thingsasian.com

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Lesson 2: The Mathematics of Finance

Your future is dependent on what you are doing now and how you are
preparing for it. A better future is associated with money because we believe
that money affects how we lead our life. If you have money you can do many
things you want in life. But, if you have money and do not know how to manage
it, you will end up of nothing. That is why personal finance is one of the most
important aspects in your life, and being able to manage your money is one f
the most important accomplishments you can achieve.

You are responsible for your personal finance. The concepts you will
learn in this lesson will be a great help to your personal financial planning.

Specific Objectives:

At the end of this lesson, the students are expected to:

1. Distinguish simple interest from compound interest;

2. Solve problem on simple and compound interest; and
3. Differentiate credit cards from consumer loans, stock, bonds and mutual
funds from home ownership.

History of Interest Rates

This cost of borrowing money is considered commonplace today.

However, the wide acceptability of interest became common only during the
Renaissance.

Interest is an ancient practice; however, social norms from ancient

Middle Eastern civilizations, to Medieval times regarded charging interest on
loans as a kind of sin. This was due, in part because loans were made to people
in need, and there was no product other than money being made in the act of
loaning assets with interest.

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The moral dubiousness of charging interest on loans fell away during the
Renaissance. People began borrowing money to grow businesses in an
attempt to improve their own station. Growing markets and relative economic
mobility made loans more common, and made charging interest more
acceptable. It was during this time that money began to be considered
a commodity, and the opportunity cost of lending it was seen as worth charging
for.

Political philosophers in the 1700s and 1800s elucidated the economic

theory behind charging interest rates for lent money, authors included Adam
Smith, Frédéric Bastiat and Carl Menger. Some of those titles included
the Theory of Fructification by Anne-Robert-Jacques Turgo, and Interest and
Prices by Knut Wicksell.

Iran, Sudan and Pakistan removed interest from their banking and
financial systems, making it so lenders partner in profit and loss sharing instead
of charging interest on the money they lend. This trend in Islamic banking—
refusing to take interest on loans—became more common toward the end of
the 20th century, regardless of profit margins.

Today, interest rates can be applied to various financial products

including mortgages, credit cards, car loans, and personal loans. In 2017, the
Fed increased rates three times, due to low unemployment and growth in
the GDP. Because of these numbers, interest rates are expected to continue to
increase in 2018.

I. Simple and Compound Interest

Interest may be defined as the charge for using the borrowed money. It is
an expense for the person who borrows money and income for the person who
lends money. Interest is charged on principal amount at a certain rate for a
certain period. For example, 10% per year, 4% per quarter or 2% per month
etc. Principal amount means the amount of money that is originally borrowed

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from an individual or a financial institution. It does not include interest. In
practice, the interest is charged using one of two methods. These are:

1. simple interest method; and

2. compound interest method

These two methods are briefly explained below:

Simple Interest Method

Under this method, the interest is charged only on the amount originally
lent (principal amount) to the borrower. Interest is not charged on any
accumulated interest under this method. Simple interest is usually charged on
short-term borrowings.

Simple interest can be easily computed using the following formula:

I = Prt

Where;

• I = Simple interest
• P = Principal amount
• i = rate of interest
• n = time/number of periods

Example 1: A person deposits 5,000.00 in a bank account which pays 6%

simple interest per year. Find the value of his deposit after 4 years.

Solution : Formula for simple interest is

I = Prt

Here, P = 5000, t = 4, r = 6%

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Let us plug these values in the above formula

I = 5000 ⋅ 6/100 ⋅ 4

I = 1200

The formula to find the accumulated value is

= Principal + Interest

= 5000 + 1200

= 6200

Hence, the value of his deposit after 4 years is 6,200.00.

Example 2: Glen received 2,250.00 loan from bank. After six months, he paid
back 2,295.00 and closed the loan. Find the rate of interest.

Solution: Interest = Amount - Principal

I = 2295.00 – 2250.00

I = 45.00

Formula for simple interest is I = Prt

Given: Time period is 6 months.

In simple interest formula, we use time period in years. But, the time period
given in the question is in months.

So, let us change the given time period in years.

6
6 months = year
12

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1
6 months = 2
year

1
So, the time period is 2 year.

Formula for simple interest is I = Prt

Here, I = 45, P = 2250, t = 1/2

Let us plug these values in the above formula

1
45 = 2250 • r • 2

45 = 1125 • r

Divide both sides by 1125.

45
= r
1125

0.04 = r

To convert the decimal 0.04 into percentage, multiply it by 100.

0.04 • 100 % = r

4% = r

Hence, the rate of interest is 4%.

Example 3: A man invests 16,500.00 in two kinds treasury notes, which yield
7.5% and 6% annually. After two years year, he earns 2,442.00 in interest. How
much does he invest at the 6 % rate?

Solution: Let "x" be the amount invested at 6% rate.

Then, the amount invested in 7.5% account is

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= 16500 - x

Given: After two years, total interest earned in both the accounts is
2,442.00.

So, we have Interest at 6% rate + Interest at 7.5% rate = 2442

6 7.5
x • 100 • 2 + (16500 - x) • 100 • 2 = 2442

x • 0.06 • 2 + (16500 - x) • 0.075 • 2 = 2442

0.12x + (16500 - x) • 0.15 = 2442

0.12x + 2475 - 0.15x = 2442

2475 - 0.03x = 2442

2475 - 2442 = 0.03x

33 = 0.03x

Divide both sides by 0.03

33
0.03
= x

3300
= x
3

1100 = x

Hence, the amount invested at 6% rate is 1,100.00.

Example 4: A person invested 25,200.00 in two accounts, which pay 5% and

10% interest annually. The amount invested at 10% rate is 110% of the amount
invested at 5% rate. After three years year, he earns 2,442.00 in interest. How
much did he invest at the 5% rate?
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Solution: Let "x" be the amount invested at 5% rate.

Then, the amount invested in 10% account is

= 110% of x

= 1.10 ⋅ x

= 1.1x

Given: After three years, total interest earned in both the accounts is 5,760.00.

So, we have

Interest at 5% rate + Interest at 10% rate = 5760

5 10
x • 100 • 3 + 1.1x • 100 • 3 = 5760

x • 0.05 • 3 + 1.1x • 0.1 • 3 = 5760

0.15x + 0.33x = 5760

0.48x = 5760

Divide both sides by 0.48

5760
x = 0.48

576000
x = 48

x = 12000

Hence, the amount invested at 5% rate is 12,000.00

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Example 5: In simple interest, a sum of money doubles itself in 10 years. Find
the number of years it will take to triple itself.

Solution:

Let P be the sum of money invested.

Given: Sum of money doubles itself in 10 years

Then, P will become 2P in 10 years.

Now, we can calculate interest for ten years as given below

From the above calculation, P is the interest for the first 10 years.

In simple interest, interest earned will be same for every year.

So, interest earned in the next 10 years also will be P.

It has been explained below.

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Hence, it will take 20 years for the principal to become triple itself.

Compound Interest Method:

Compounding of interest is very common. Under this method, the

interest is charged on principal plus any accumulated interest. The amount of
interest for a period is added to the amount of principal to compute the interest
for next period. In other words, the interest is reinvested to earn more interest.
The interest may be compounded monthly, quarterly, semiannually or annually.
Consider the following example to understand the whole procedure of
compounding.

Example 1: Suppose, you have deposited 100.00 with a bank for five years at
a rate of 5% per year compounded annually. The interest for the first year will
be computed on 100.00 and you will have 105.00 (100.00 principal + 5.00
interest) at the end of first year. The interest for the second year will be
computed on 105.00 and at the end of second year you will have 110.25 (105
principal + 5.25 interest). The interest for the third year will be computed on
110.25 and at the end of third year you will have 115.76 (110.25 principal +
5.51 interest). The following table shows the computation for 5-year period of
investment.

Principal Rate of
Year Interest Compound Amount
Amount Interest
1 100.00 5% 100.00 × 0.05 = 100.00 + 5.00 = 105.00
5.00
2 105.00 5% 105.00 × 0.05 = 105.00 + 5.25 = 110.25
5.25
3 110.25 5% 110.25 × 0.05 = 110.25 + 5.51 = 115.76
5.51
4 115.76 5% 115.76 ×0.05 = 115.76 + 5.79 = 121.55
5.79

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5 121.55 5% 121.55 × 0.05 = 121.55 + 56.08 =
6.08 127.63

Under compound interest system, when interest is added to the principal

amount, the resulting figure is known as compound amount. In the above
table, the compound amount at the end of each year have been computed in
the last column. Notice that the compound amount at the end of a year becomes
the principal amount to compute the interest for the next year.

Compound amount and compound interest formula:

The above procedure of computing compound amount and compound

interest is lengthy and time consuming. Fortunately, the formulas are available
to compute compound amount and compound interest for any number of
periods.

(i) Compound amount formula:

A = P(1 + i)n

Where;

• A = Compound amount
• P = Principal amount
• i = rate of interest
• n = number of periods

(ii) Compound interest formula

Compound interest = Compound amount – Principal amount

Example 2: The City Bank has issued a loan of 10,000.00 to a sole proprietor
for a period of 5-years. The interest rate for this loan is 5% and the interest is
compounded annually. Compute
1. compound amount
2. compound interest

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1. Computation of Compound Amount:

A = P(1 + i)n

= 10,000 × (1 + 5%)5

= 10,000 × (1 + .05)5

= 10,000 × (1.05)5

= 10,000 × 1.276

= 12,760.00

2. Computation of Compound Interest:

Once the compound amount has been computed, the amount of interest
earned over the investment period can be computed by subtracting principal
amount from the compound amount. In this example, the principle amount is
10,000 and the compound amount computed above is 12,760.00. The amount
of compound interest for the fiver year period can be computed as follows:

Compound interest = Compound amount – Principle amount

= 12,760.00 – 10,000.00

= 2,760.00

Use of future value of $1 table to compute compound amount: (see

Appendix A)

The shortest and easiest method to compute compound amount is to

use the future value of $1 table (See Appendix 1). This table contains the value
of (1 + i)n for a given value of i and n. After locating the value of (1 + i)n in the
table, the principal amount is simply multiplied with the value to find the
compound amount. The principal amount is then subtracted from compound

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amount to get the amount of compound interest for the given interest rate and
time period.

= 10,000.00 × (1 + 5%)5

= 10,000.00 × 1.276*

= 12,760.00

Compound interest: 12,760.00 – 10,000.00 = 2,760.00

*Value of (1 + 5%)5 from future value of $1 table: 5 periods; 5% interest rate.

The future value tables are widely used in accounting and finance to save time
and avoid unnecessary computations.

Compound interest is greater than simple interest:

Compound interest is greater than simple interest. The reason is very

simple. Under simple interest system, the interest is computed only on principal
amount whereas under compound interest system, the interest is computed on
principle as well as on accumulated interest. Consider the following example
for the explanation of this point:

Example3: A woman has deposited 6,000.00 in a saving account. Bank pays

interest at a rate of 9% per year.

Required: Compute the amount of interest that will be earned over 12-year
period:

1. if the interest is simple?

2. if the interest is compounded annually?
Solution:

(1) Simple interest:

= 6,000 × 0.09 × 12

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= 6,480.00

(2) Compound interest:

= 6,000.00 × (1 + 9%)12

= 6,000.00 × 2.813*

= 16,878.00

Compound interest = 16,878.00 – 6,000.00 = 10,878.00

Notice that compound interest is more than simple interest by 4,398 (10,878 –
6,480).

*Value of (1 + 9%)12 from future value of $1 table: 12 periods; 9% interest rate.

II. Credit card vs. Consumer loans

A credit card and a consumer loan are two different ways of borrowing
money and they provide different benefits. Which of the two is best suited for
you depends on your need and purchasing pattern.

A consumer loan is a good alternative to a credit card if you want

predictability with your monthly expenses. A consumer loan provides a set plan
for your monthly down payments which gives many a sense of security. You
can arrive back from a vacation paid with a consumer loans and not expect any
surprises. You will simply start paying back a pre decided amount each month.

A credit card provides a lot more flexibility in that you can decided how
much you want to borrow and how much you want to pay back every month.
As long as you stay within your credit limit you have the freedom to decided
how much you want to borrow and how much to pay back each month.
However, flexibility can tempt some to spend beyond their means. A credit card

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is therefore more demanding since you need to be in charge of your own
spending and what you owe.

Summed up a consumer loan is the best choice for large purchases that
you plan to pay down over a longer period of time, while a credit card is best
for smaller purchases that can be paid back relatively quick.

About consumer loans:

• Good for someone who wants to borrow a one-time amount that you pay
back following a set down payment plan. A consumer loan provides
structure and predictability in your finances and you know exactly how
much you need to pay back each month.
• Lower interest than credit card debt.
• A consumer loan enables you to refinance smaller and more expensive
loans.

About credit cards:

1. Good for someone who prefers flexibility as it gives you the opportunity
to borrow as you go as long as you stay within your credit limit provided.
2. Offers a no interest period between 30 and 52 days
3. You decided how much you want to pay within a certain time frame
4. The card may give you additional benefits such as bonuses and
discounts in stores, restaurants, online and other partner benefits
5. Most often a credit card includes travel and cancellation insurance

III. Stocks, Bonds and Mutual Funds

Stocks, or shares, are units of equity (or ownership stake) in a company.

The value of a company is the total value of all outstanding stock of the
company. The price of a share is simply the value of the company also called
market capitalization, or market cap divided by the number of shares
outstanding.

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Bonds are simply loans made to an organization. They are a form of debt
and appear as liabilities in the organization's balance sheet. While stocks are
usually offered only in for-profit corporations, any organization can issue bonds.
Indeed, the governments of United States and Japan are among the largest
issuers of bonds. Bonds are also traded on exchanges but often have a lower
volume of transactions than stocks.

More specifically, here are the key differences between stocks and bonds:

• Priority of repayment. In the event of the liquidation of a business, the

holders of its stock have the last claim on any residual cash, whereas the
holders of its bonds have a considerably higher priority, depending on the
terms of the bonds. This means that stocks are a riskier investment than
bonds.

• Periodic payments. A company has the option to reward its shareholders

with dividends, whereas it is usually obligated to make periodic interest
payments to its bond holders for very specific amounts. Some bond
agreements allow their issuers to delay or cancel interest payments, but this
is not a common feature. A delayed payment or cancellation feature reduces
the amount that investors will be willing to pay for a bond.

• Voting rights. The holders of stock can vote on certain company issues,
such as the election of directors. Bond holders have no voting rights.

There are also variations on the stock and bond concept that share
features of both. In particular, some bonds have conversion features that
allow bondholders to convert their bonds into company stock at certain
predetermined ratios of stocks to bonds. This option is useful when the price
of a company's stock rises, allowing bondholders to achieve an immediate
capital gain. Converting to stock also gives a former bond holder the right
to vote on certain company issues.

Both stocks and bonds may be traded on a public exchange. This is

a common occurrence for larger publicly-held companies, and much more
rare for smaller entities that do not want to go through the inordinate
expense of going public.

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A mutual fund is an open-end professionally managed investment fund
that pools money from many investors to purchase securities. These investors
may be retail or institutional in nature. The term is typically used in the United
States, Canada, and India, while similar structures across the globe include the
SICAV in Europe ('investment company with variable capital') and open-ended
investment company (OEIC) in the UK.

Mutual funds have advantages and disadvantages compared to direct

investing in individual securities. The advantages of mutual funds include
economies of scale, diversification, liquidity, and professional management.
However, these come with mutual fund fees and expenses.

Primary structures of mutual funds are open-end funds, unit investment

trusts, closed-end funds and exchange-traded funds (ETFs).

Mutual funds are often classified by their principal investments as money

market funds, bond or fixed income funds, stock or equity funds, hybrid funds,
or other. Funds may also be categorized as index funds, which are passively
managed funds that match the performance of an index, or actively managed
funds. Hedge funds are not mutual funds as hedge funds cannot be sold to the
general public.

IV. Home Ownership

Owner-occupancy or home-ownership is a form of housing

tenure where a person, called the owner-occupier, owner-occupant,
or home owner, owns the home in which they live. This home can be house,
like a single-family house, an apartment, condominium, or a housing
cooperative. In addition to providing housing, owner-occupancy also functions
as a real estate investment. Owner-occupancy or home-ownership is a form
of housing tenure where a person, called the owner-occupier, owner-
occupant, or home owner, owns the home in which they live. This home can
be house, like a single-family house, an apartment, condominium, or a housing
cooperative. In addition to providing housing, owner-occupancy also functions
as a real estate investment.

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Summary:

✓ Simple Interest Method, the interest is charged only on the amount

originally lent (principal amount) to the borrower. Interest is not charged
on any accumulated interest under this method. Simple interest is
usually charged on short-term borrowings.
✓ Simple interest can be easily computed using the following formula: I =
Prt

✓ Compound Interest Method, the interest is charged on principal plus

any accumulated interest. The amount of interest for a period is added
to the amount of principal to compute the interest for next period. In other
words, the interest is reinvested to earn more interest. The interest may
be compounded monthly, quarterly, semiannually or annually. Consider
the following example to understand the whole procedure of
compounding.
✓ Compound amount formula: A = P(1 + i)n
✓ Interest may be defined as the charge for using the borrowed money. It
is an expense for the person who borrows money and income for the
person who lends money.
✓ A consumer loan is a good alternative to a credit card if you want
predictability with your monthly expenses. A consumer loan provides a
set plan for your monthly down payments which gives many a sense of
security.
✓ A credit card provides a lot more flexibility in that you can decided how
much you want to borrow and how much you want to pay back every
month. A credit card is more demanding since you need to be in charge
of your own spending and what you owe.
✓ Stocks, or shares, are units of equity (or ownership stake) in a
company. The value of a company is the total value of all outstanding
stock of the company.
✓ A mutual fund is an open-end professionally managed investment fund
that pools money from many investors to purchase securities. These
investors may be retail or institutional in nature.

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✓ Owner-occupancy or home-ownership is a form of housing
tenure where a person, called the owner-occupier, owner-occupant,
or home owner, owns the home in which they live.

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ASSESSMENT TASK

Exercise 3.2 – a
The Mathematics of Finance

Name: ______________________________ Program & Section:

__________
Date: _______________________________ Score: _________

Answer the following problems:

1. A loan of 10,000 has been issued for 6-years. Compute the amount to
be repaid by borrower to the lender if simple interest is charged @ 5%
per year.
2. Suppose 7,000 is divided into two bank accounts. One account pays
10% simple interest per year and the other pays 5%. After three years
there is a total of 1451.25 in interest between the two accounts. How
much was invested into each account (rounded to the nearest cent)?
3. John wants to have an interest income of 3,000 a year. How much must
he invest for one year at 8%?
4. Jane owes the bank some money at 4% per year. After half a year, she
paid 450.00 as interest. How much money does she owe the bank?
5. A student borrowed some money from his father at 2% simple interest
to buy a car. He paid his father 3,600.00 in interest after 3 years, how
much did he borrow?
6. A credit union loaned out 500,000, part at an annual rate of 6% and the
rest at an annual rate of 12%. The collected combined interest was
36,000 that year. How much did the credit union loan out at each rate?

7. A fireman invests 40,000 in a retirement account for 2 years. The interest

rate is 6%. The interest is compounded monthly. What will his final
balance be?

8. Calculate the amount of this investment after 5 years with interest

compounded yearly.

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Principal = 3,000.00
Rate = 4%

9. A deposit of 21,500.00 earns 6% interest compounded quarterly. How

much money is in the bank after for 6 years?

10. The compound interest and simple interest on a certain sum for 2 years
is 12,300.00 and 12,000.00 respectively. The rate of interest is same for
both compound interest and simple interest and it is compounded
annually. What is the principal?

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ASSESSMENT TASK
Exercise 3.2 – b
The Mathematics of Finance

Name: __________________________ Program & Section: __________

Date: ___________________________ Score: _________

Research Project:

1. Make a survey among your teachers and classmates regarding

the use of credit cards. How many of them have stopped and have
continued their subscription? List the reasons why they stopped
and yet others are still using credit cards today. What about the
charging of the interest, how many percent? Is it simple or
compound interest? Is it practical or reasonable to have the credit
card?
2. Given the banks and lending institutions in the country, visit the
bank and inquire to gather the data on the base lending rate of
the housing loan program for each of the following and present it
on bar graph bank (x-axis) and interest rate (y-axis). Describe the
trend and report your analysis and interpretation.
a) BDO
b) Landbank
c) Eastwest bank
d) Penbank
e) BPI
f) Metrobank
g) Security Bank
h) Citysavings Bank
i) Sta. Catalina Lending Institution
j) Makilala Cooperative
k) Others

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References:

Interest (2018, February, 8). Retrieved from

https://www.investopedia.com/terms/i/interest.asp

Simple and Compound Interest (2018). Retrieved from

https://www.accountingformanagement.org/simple-and-compound-interest/

Simple Interest Problems with Solutions (2012). Retrieved from

https://www.onlinemath4all.com/simple-interest-problems-with-solutions.html

Credit vs. consumer loan (2020). Retrieved from www.enterrcard.com

The difference between stocks & bonds (2020, March, 4). Retrieved form
https://www.accountingtools.com/articles/what-is-the-difference-between-
stocks-and-bonds.html

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Bibliography

1. Daligdig, Romeo M. EdD et. al. (2019) Mathematics in the Modern

World. LORIMAR Publishing Inc., Quezon City, Metro Manila.

2. Alejan, Ronnie O et. al. (2018). Mathematics in the Modern World. Mutya
Publishing House Inc., Malabon City

3. Tagaro, Cosuelo PhD (2014). Advanced Statistics 22nd edition

4. Ronald de Jong, Oct 23, 2008 | Destinations: Philippines / Mindanao

5. Scott, William Henry. Barangay: Sixteenth-Century Philippine Culture

and Society. Ateneo de Manila University Press.

6. Eugenio, Damiana. Philippine Folk Literature Vol. VIII: The Epics.

University of the Philippines Press.

e – References:

7. 22 Examples of Mathematics in Everyday Life (2020). Retrieved from

https://studiousguy.com/examples-of-mathematics/

8. Nature and Characteristics of Mathematics (2014). Retrieved from

http://drangelrathnabai.blogspot.com/2014/01/nature-characteristics-of-
mathematics.html

9. Deductive Reasoning vs. Inductive Reasoning (2017, July, 5).

Retrieved from https://www.livescience.com/21569-deduction-vs-
induction.html
10. Inductive vs. deductive Reasoning (2019, November, 11). Retrieved
from https://www.scribbr.com/methodology/inductive-deductive-
reasoning/
11. Immediate Algebra Tutorial 8: Introduction to Problem Solving (2011,
July, 1). Retrieved from
https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/i
nt_alg_tut8_probsol.htm

12. Normal Distribution (2019). Retrieved from

https://www.mathsisfun.com/data/standard-normal-distribution.html

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13. Understanding Probability Distributions (2018). Retrieved from
https://www.statisticsbyjim.com/basics/probability-distributions/.
14. Geometry Wikipedia (2020). Retrieved from
https://en.wikipedia.org/wiki/Geometry
15. 40 Brilliant Geometric Patterns (And How To Use them in Your Designs)
(2020). Retrieved from https://blog.bannersnack.com/author/ana-
darstaru/
16. The T’boli Tribe of South Cotabato (2018, October, 23). Retrieved from
thingsasian.com
17. Interest (2018, February, 8). Retrieved from
https://www.investopedia.com/terms/i/interest.asp
18. Simple and Compound Interest (2018). Retrieved from
https://www.accountingformanagement.org/simple-and-compound-
interest/
19. Simple Interest Problems with Solutions (2012). Retrieved from
https://www.onlinemath4all.com/simple-interest-problems-with-
solutions.html
20. Credit vs. consumer loan (2020). Retrieved from www.enterrcard.com
21. The difference between stocks & bonds (2020, March, 4). Retrieved form
https://www.accountingtools.com/articles/what-is-the-difference-
between-stocks-and-bonds.html

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Appendix A: z-table

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Appendix B: Future Value of $1

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Appendix C

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Appendix D

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Appendix E

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Republic of the Philippines

SULTAN KUDARAT STATE UNIVERSITY
Isulan, Sultan Kudarat
College of Industrial Technology
S.Y. 2020 - 2021

GE701
MATHEMATICS IN THE MODERN WORLD
Syllabus

1st Semester
A.Y 2020 – 2021

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Republic of the Philippines

SULTAN KUDARAT STATE UNIVERSITY
Isulan, Sultan Kudarat
College of Industrial Technology
S.Y. 2020-2021

UNIVERSITY VISION
A trailblazer in arts, science and technology in the region. UNIVERSITY OBJECTIVES
a. Enhance competency development, commitment,
UNIVERSITY MISSION professionalism, unity and true spirit of service for public
The University shall primarily provide advanced instruction accountability, transparency and delivery of quality
and professional training in science and technology, agriculture, services;
fisheries, education and other relevant fields of study. b. Provide relevant programs and professional trainings that
It shall also undertake research and extension services, and will respond to the development needs of the region;
provide progressive leadership in its areas of specialization. c. Strengthen local and international collaborations and
partnerships for borderless programs;
UNIVERSITY GOAL d. Develop a research culture among faculty and students;
To produce graduates with excellence and dignity in arts, e. Develop and promote environmentally-sound and
science and technology. market-driven knowledge and technologies at par with
international standards;
f. Promote research-based information and technologies
for sustainable development;
g. Enhance resource generation and mobilization to sustain
financial viability of the university.

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Program objectives and its relationship to University Objectives:

PROGRAM OBJECTIVES (PO) OBJECTIVES
A graduate of Bachelor of Science in Industrial Technology can: a b c d e f g
a) assume professional, technical, managerial and leadership roles in industrial organizations with the
desired competence in the fields of practiced such as Automotive, Architectural Drafting, Civil,
Electrical, electronics, food and its allied discipline.
b) innovate explicit and modern technologies in the advancement of economy, society, technology and
environmental sustainability.
c) generate research-based information and technologies at par from international standards; and
d) promote and transfer knowledge and technologies for effective and efficient school-industry
partnership.

1. Course Code : GE701 5. Course Description:

2. Course Title : Mathematics in the Modern This course deals with nature of mathematics, appreciation of
World its practical, intellectual and aesthetic dimensions, and
3. Pre-requisite : None application of mathematical tools in daily life.
4. Credit : 3 units
The course begins with an introduction to the nature of
mathematics as an exploration of patterns (in nature and the
environment) and as an application of inductive and deductive
reasoning. By exploring these topics, students are encouraged
to go beyond the typical understanding of mathematics as
merely a set of formulas but as a source of aesthetics in
patterns of nature, for example, and a rich language in itself
(and of science) governed by logic and reasoning.

The course then proceeds to survey ways in which

mathematics provides a tool for understanding and dealing with
various aspects of present-day living, such as managing

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personal finances, making social choices, appreciating

geometric designs, understanding codes used in data
transmission and security, and dividing limited resources fairly.
These aspects will provide opportunities for actually doing
mathematics in a broad range of exercises that bring out the
various dimensions of mathematics as a way of knowing, and
test the students’ understanding and capacity. (CMO No. 20,
series of 2013)

6. Course Learning Outcomes and Relationships to program Educational Objectives

Course Learning Outcome Program Objectives

At the end of the semester, the students can: a b c d
a) Discuss and argue about the nature of mathematics, what it is, how it is expressed, represented, and
used.
b) Discuss the language and symbols of mathematics.
c) Use different types of reasoning to justify statements and arguments made about mathematics and
mathematical concepts.
d) Apply strategies for effective problem solving
e) Use a variety of statistical tools to process and manage numerical data, and be able to formulate
significant decision.
f) Analyze codes and coding schemes used for identification, privacy, and security purposes;
g) Use mathematics in other areas such as finance, voting, health and medicine, business, graphs,
environment, arts and design, and recreation.
h) Appreciate the nature and uses of mathematics in everyday life.
i) Affirm honesty and integrity in the application of mathematics to various human endeavor.

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7. Course Content

Desired Student Outcome- Evidence Course Program Values

Topics, Learning Objectives Based of Outcom Objectiv Integrat
Course Objectives
Time Allotment Assessment Outcomes es es ion
(OBA)
Activities

Topic: VMGO, Classroom Policies, Course Overview, Course Requirements, Grading System (1.5 hour)

Explain VMGO of Student can be aware Class Individual Value of

the SKSU, of the SKSU VMGO, Discussion Recitation a,c,h,i a,b,c,d Respons
classroom policies, classroom policies, ibility
VMGO of SKSU
scope of the course, scope of the course, Student’s Feed
course requirements course requirements backing
and grading system. and grading system.

Chapter 1: The Nature of Mathematics (22 hours)

Lesson 1. Mathematics in our World (4 hours)

Patterns and 1.1 Identify the The students can Video-watching Group a, h, i a, b, c, d Value of
Numbers in mathematical identify nature that Heads Output Self-
Nature and patterns and exhibits different Together: Small- Presentatio confiden
the numbers found patterns and Group Sharing n of ce,
Regularities in nature and regularities in the Selected Open-
in the World the World such world. Pattern mindedn
ess and

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as symmetry in Insightfu
The Fibonacci snowflake and Rubric lness
Sequence tessellation in
honeycomb;
tiger’s stripes
and hyena’s
spots; Fibonacci
Importance of sequence in the
Mathematics sunflower,
in Life snail’s shell,
flower petals;
Nature of Exponential
Mathematics Growth Model in
world’s
population, the
weather,
fractals in ferns
,etc.

Value of
Role of Describe how The students can (Individual Individual Creativit
Mathematics Mathematics helps articulate the Collage) Output y,
in Other organize patterns importance of Collage of daily Presentatio Hardwor
Discipline and regularities and mathematics in one’s life, new n k,
predict behaviour of life. discoveries, Patienc
nature and technological Rubric e
phenomena and discoveries,
control its phenomenon
occurrences

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Appreciating Perform numerous The students can (Group Activity) Group

Mathematics applications of perform numerous Concept Output Value of
as a Human Mathematics in the applications of Mapping Presentatio Respons
Endeavor world making it Mathematics and Group 1: daily n iveness,
indispensable. express appreciation Life Collabor
for mathematics as a Group 2: New Rubric ation
human endeavour. Discoveries
Group3:
Technological
Breakthroughs
Group 4:
Natural
Phenomenon

Lesson 2. Mathematical Language and Symbols (10 hours)

Language, Explain the The students can Heads Group b, h, i a, b, c, d Value of

Symbols, and characteristics of discuss the language, Together: Output Activen
Conversion of mathematical symbols, and Individual or Rubric ess and
Mathematics language (precise, convention of small group Teamw
concise, powerful), mathematics and exercises ork
formality and explain and including games
convention appreciate the nature
of mathematics as a
language.
Conversion of Perform the proper Heads Group Value of
English translation and Together- Report Particip

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expression to writing of The students can Divergent ation,

Mathematical mathematical perform operations on Thinking: Rubric Teamw
Sentences and expressions and mathematical Concept Map ork and
vice versa sentences expressions correctly Unity
and acknowledge that
Conversion of it is a useful language
Algebraic
Expressions to
English
Sentences

The Four Basic Explain the four The students can Group Report Group Value of
Concepts of basic concepts: explain the four basic Assignment: Output Account
Mathematics: sets, functions, concepts such as Group 1: Sets ability
Set relations, and binary sets, functions, Group2: Rubric and
Relation operations relations and binary Relations Teamw
Function operations. Group 3: Quiz ork
Binary Operation Functions
Group 4: Binary
Operations

Elementary Logic Apply elementary The students can Group Reporting Value of
logic: correctly apply Self-
connectives, connectives, Board work reliance
quantifiers, quantifiers, negation
negation and and variables in
variables making valid
arguments.

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Lesson 3. Problem Solving and Reasoning (8 hours)

Reasoning Use the two types of The students can use Group Activity Group c, d, h, i a, b, c, d Value of
reasoning- different types of (Brainstorming Presentatio Collabor
inductive and reasoning to justify and Argument- n of ation
deductive to statements and Construction) Constructed
justify arguments made Arguments
statements and about mathematics
arguments and mathematical Rubric
concepts.

Mathematical Writing basic kinds The students can Sticky Notes Students’ Value of
Proofs of mathematical write clear and logical Graph: Organized Logical
statements and proofs. Organizing Proof Thinking
construction of Statements in
their logical Forming the
proofs. Proof of
Mathematical
Statement

Polya’s 4-Steps in Solve different The students can Cooperative Presentatio Value of
Problem mathematical and solve problems Learning (Group n of Group Cooper
Solving recreational involving patterns and Effort in Solving Output ation
problems and recreational problems Problems) and
following Polya’s following Polya's four Rubric Interdep
four steps of steps. endenc
e

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problem solving
strategies The students can
organize one's
methods and
approaches for
proving and solving
problems.

Chapter 2: Mathematics as a Tool (27.5 hours)

Lesson 1. Data Management (12 hours)

Basic 2.1 Perform the The students can use Practical Survey e, h, i a, b, c, d Value of
Statistical basic concepts variety of statistical Activity: Data Result Persiste
Concepts in Descriptive tools to process and Gathering and MS Excel nce and
Statistics, and manage numerical Data Output Effectiv
Measures of discuss Normal data. Description/Inter e
Central Distribution, pretation with Commu
Tendency Hypothesis Computer nication
Testing, Application
Measures of Regression and
Relative Correlation,
Position Least Squares
Lines and Chi-
Measures of square
Variation

Normal
Distribution

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Linear 2.2 Plan or conduct The students can plan• Group Group Value of
Regression an experiment or or conduct their own Action Output Accurac
and study (optional) experiment or study Research y and
Correlation and make important Rubric Explorat
decisions with the use ion
of statistical data.

Chapter 3. Mathematics as a Tool (15.5 hours)

Lesson 1. Geometric Design (7.5 hours)

What is 1.1 Recognize and The students can Brainstorming g, h, i a, b, c, d Value of

Geometric analyze apply geometric Awaren
Design? geometric concepts in ess
shapes describing and
creating designs

Mindanao 1.2 Identify different The students can Create Output may Value of
Designs, Arts transformations identify different geometric be in a form Creativit
and Culture , patterns and transformations of designs using of stitching, y and
diagrams, geometric figures and transformations drawing or Hardwor
designs, arts contribute to the any form of k
and culture enrichment of the artwork
Filipino culture and
Rubric

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arts using the

concepts in Geometry

Lesson 2: The Mathematics of Finance (8hours)

Simple and 2.1 Compute The students can Pair-Share Pair Output g, h, i a, b, c, d Value of
Compound simple and apply the different Activity Cooper
Interest compound concepts of (Problem ation
interest, credit mathematics of Solving)
Credit card vs. cards and finance in making
Consumer Loans consumer wise decisions related
loans, stocks, to personal finance.
Stocks, Bonds, bonds and
and Mutual mutual funds
Funds and home
ownership The students can
Home Ownership support the use of
Mathematics in
financial aspects and
endeavors in life.

TOTAL: 54 hours
Lectures: 51 hours
Examination (Midterm and Final): 3 hours

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8. Course Evaluation

Course Requirements:
• Attendance
• Major Exams (Midterm and Final)
• Recorded Problem Sets, Quizzes and
all other outputs
• Research Paper (Optional)

Grading System:

MIDTERM FINAL TERM

1. Quizzes - 30 % 1. Quizzes - 30 %
2. Class Participation/Seatworks/ - 15 % 2. Class Participation/Seatworks/ - 15 %
Assignments Assignments
3. Midterm Exam - 50 % 3. Final Exam - 50 %
4. Attendance - 5% 4. Attendance - 5%
Total - 100% Total - 100%

(Midterm Grade + Final Term Grade)/2= Final Grade

Schedule of Examination
Midterm - ____________________
Final Term - ____________________

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9. Bibliography
1. Daligdig, Romeo M. EdD et. al. (2019) Mathematics in the Modern World. LORIMAR Publishing Inc., Quezon City, Metro
Manila.
2. Alejan, Ronnie O et. al. (2018). Mathematics in the Modern World. Mutya Publishing House Inc., Malabon City
3. Tagaro, Cosuelo PhD (2014). Advanced Statistics 22nd edition
4. Ronald de Jong, Oct 23, 2008 | Destinations: Philippines / Mindanao
5. Scott, William Henry. Barangay: Sixteenth-Century Philippine Culture and Society. Ateneo de Manila University Press.
6. Eugenio, Damiana. Philippine Folk Literature Vol. VIII: The Epics. University of the Philippines Press.

e – References:
7. 22 Examples of Mathematics in Everyday Life (2020). Retrieved from https://studiousguy.com/examples-of-mathematics/
8. Nature and Characteristics of Mathematics (2014). Retrieved from http://drangelrathnabai.blogspot.com/2014/01/nature-
characteristics-of-mathematics.html
9. Deductive Reasoning vs. Inductive Reasoning (2017, July, 5). Retrieved from https://www.livescience.com/21569-deduction-
vs-induction.html
10. Inductive vs. deductive Reasoning (2019, November, 11). Retrieved from https://www.scribbr.com/methodology/inductive-
deductive-reasoning/
11. Immediate Algebra Tutorial 8: Introduction to Problem Solving (2011, July, 1). Retrieved from
https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut8_probsol.htm
12. Normal Distribution (2019). Retrieved from https://www.mathsisfun.com/data/standard-normal-distribution.html
13. Understanding Probability Distributions (2018). Retrieved from https://www.statisticsbyjim.com/basics/probability-
distributions/.
14. Geometry Wikipedia (2020). Retrieved from https://en.wikipedia.org/wiki/Geometry

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15. 40 Brilliant Geometric Patterns (And How To Use them in Your Designs) (2020). Retrieved from
https://blog.bannersnack.com/author/ana-darstaru/
16. The T’boli Tribe of South Cotabato (2018, October, 23). Retrieved from thingsasian.com
17. Interest (2018, February, 8). Retrieved from https://www.investopedia.com/terms/i/interest.asp
18. Simple and Compound Interest (2018). Retrieved from https://www.accountingformanagement.org/simple-and-compound-
interest/
19. Simple Interest Problems with Solutions (2012). Retrieved from https://www.onlinemath4all.com/simple-interest-problems-
with-solutions.html
20. Credit vs. consumer loan (2020). Retrieved from www.enterrcard.com
21. The difference between stocks & bonds (2020, March, 4). Retrieved form https://www.accountingtools.com/articles/what-is-
the-difference-between-stocks-and-bonds.html

Prepared by:

SGD
MAY FLOR L. TAPOT, MST
Subject Teacher

Reviewed by: Approved by:

SGD SGD
ARNEL Y. CELESTE, MIT RANDY T. BERINA, MAT
BSIT Program Head Dean

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