Exploring Mathematics in the Modern World
Cagayan State University
Exploring
Mathematics in the
Modern World
Avelina M. Ayuyang, PhD
Marlon T. Sinigiuan, MST
Wilfredo M. Perciano
Ariel F. Melad, DME
Marites U. Sy, PhD
Allan T. Tabuyo, MST
Joel C. Casibang, PhD
Maria Cecilia G. Uy, PhD
Manuel A. Belango, DME
2019
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Exploring Mathematics in the Modern World
Foreword
Welcome to the world of Mathematics!
Our journey to this world begins with the introduction to the nature of
Mathematics as we explore the different patterns in the environment. We will
look into the beauty of nature and appreciate the association of these concepts
to Mathematics and real-life.
Particularly, this world is divided into 12 areas namely: a.) Mathematics in
Our World, b.) Mathematical Language and Symbols, c.) Problem Solving and
Reasoning, d.) Data Management, e.) Geometric Designs, f.) Codes, g.) Linear
Programming, h.) The Mathematics of Finance, i.) Apportionment and Voting,
j.) Logic, k.) The Mathematics of Graphs, and l.) Mathematical Systems.
We then proceed with the building blocks of Mathematics – its language and
symbols. By studying these topics, we will go beyond the usual understanding
of Mathematics as merely numbers and formulas but an art that can develop
one’s critical thinking, reasoning, and problem-solving skills.
As we continue, be prepared to learn Mathematics as a tool to understand and
deal with daily activities such as budgeting, problem solving, and making
decisions. You will be facing challenges like activities, exercises, and word
problems. We hope that these can bring out your potential and improve your
Mathematical skills.
The tour ends here but your mission has just started. You are then directed to
accomplish the following:
1.) Discuss and argue about the nature of Mathematics, what it is, how it is
expressed, represented, and used;
2.) Use different types of reasoning to justify statements and arguments
made about Mathematics and mathematical concepts;
3.) Discuss the language and symbols of Mathematics;
4.) Use a variety of Statistical tools to process and manage numerical data;
5.) Analyze codes and coding schemes used for identification, privacy, and
security purposes;
6.) Use Mathematics in other areas such as finance, voting, health and
medicine, business, environment, arts and design, and recreation;
7.) Appreciate the nature and uses of Mathematics in everyday life; and
8.) Affirm honesty and integrity in the application of Mathematics to
various human endeavors.
Good luck to the rest of your journey and enjoy your learning!
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Exploring Mathematics in the Modern World
Table of Contents
Unit/ Topic
Page
Title Page
--------------------------------------------------------------- i
Foreword
--------------------------------------------------------------- ii
Table of Contents
--------------------------------------------------------------- iii
Unit 1: Mathematics in Our World ------------------------------------------- 1
Topic 1: Patterns and Numbers in Nature -------------------------------- 3
Unit 2: Mathematical Language and Symbols ------------------------------ 10
Topic 1: Nature of Mathematics as a Language -------------------------- 13
Topic 2: The Language of Variables--------------------------------------- 19
Topic 3: Language of Set ---------------------------------------------------- 22
Topic 4: The Language of Relations and Functions ---------------------- 30
Unit 3: Problem Solving and Reasoning ------------------------------------- 41
Topic 1: Inductive and Deductive Reasoning ---------------------------- 42
Topic 2: Polya’s Four Steps in Problem Solving ------------------------ 48
Topic 3: Problem Solving Strategies --------------------------------------- 52
Topic 4: Mathematical Problems Involving Patterns -------------------- 56
Unit 4: Data Management ------------------------------------------------------- 64
Topic 1: Data Gathering and Organizing Data, Representing Data Using
Graphs and Charts, and Interpreting Organized Data ----------66
Topic 2: Measures of Central Tendency ----------------------------------- 74
Topic 3:`Measures of Dispersion ------------------------------------------- 82
Topic 4: Measures of Relative Position ------------------------------------ 88
Topic 5: Probabilities and Normal Distribution --------------------------- 102
Topic 6: Linear Regression and Correlations ------------------------------ 111
Unit 5: Geometric Designs ------------------------------------------------------ 123
Topic 1: Geometric Designs ------------------------------------------------- 124
Unit 6: Codes ----------------------------------------------------------------------- 139
Topic 1: Coding --------------------------------------------------------------- 140
Topic 2: Cryptography -------------------------------------------------------- 145
Unit 7: Linear Programming --------------------------------------------------- 149
Topic 1: Linear Programming ----------------------------------------------- 150
Unit 8: The Mathematics of Finance ------------------------------------------ 159
Topic 1: Simple Interest ------------------------------------------------------ 161
Topic 2: Computing the Simple Interest using Ordinary and Exact Time
(Ordinary and Exact Interest) ------------------------------------- 166
Topic 3: Computing the Simple Interest using Actual and Approximate
Time (Interest between Dates) ------------------------------------ 169
Topic 4: Compound Interest ------------------------------------------------- 175
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Exploring Mathematics in the Modern World
Unit 9: Apportionment and Voting ------------------------------------------- 180
Topic 1: Apportionment ------------------------------------------------------ 182
Topic 2: Introduction to Voting System ------------------------------------ 191
Topic 3: Weighted Voting System ------------------------------------------ 203
Unit 10: Logic ---------------------------------------------------------------------- 210
Topic 1: Proposition and Types of Propositions -------------------------- 211
Topic 2: Truth Value and Truth Table -------------------------------------- 218
Topic 3: Arguments and Validity ------------------------------------------- 227
Unit 11: The Mathematics of Graphs ----------------------------------------- 233
Topic 1: Graph Coloring ----------------------------------------------------- 234
Unit 12: Mathematical Systems ------------------------------------------------ 240
Topic 1: Modular Arithmetic and Its Application ------------------------ 242
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Exploring Mathematics in the Modern World
Unit 1: Mathematics in Our World (5 hours)
Introduction
Did you ever wonder what the pictures above tell you about?
Quite often, people consider mathematics only as numbers and arithmetic.
Most of the time, giving more emphasis on numerical ability has brought about
a person unfavorable attitude about mathematics, and probably, you too
experience the same. One thing must be clear at this point, though, that
Mathematics is not all about numbers. Mathematics goes beyond arithmetic. It
is an art by which the universe is designed creatively. Patterns are core topics in
Mathematics. In fact, it is also known as the science of patterns- the numeric
patterns and geometric patterns. In this unit, you will appreciate that
Mathematics is not only confined to numbers but also exists in nature, on the
things that we see around us. You will learn how nature connects with numerical
patterns and sequences.
Learning Outcomes
a.
b.
c.
d.
e.
Upon the completion of this unit, you are expected to:
Identify patterns in nature and how they are related to mathematics;
Argue about the nature of mathematics;
Solve problems involving patterns and numbers;
Articulate the importance of mathematics in one’s life; and
Express appreciation for mathematics as a human endeavor.
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Exploring Mathematics in the Modern World
Activating Prior Learning
A. Directions: Study the following set of images. Can you tell what
patterns do the images exhibit?
Set A
Set B
Set C
B. Directions: The following are numerical sequences and geometric
patterns. Can you tell the next number or pattern in the following
items?
1. 2,4,6, _____
2. 2, 4, 7, 11, _____
3. iiiLL, iiL, i, _____
4.
______
5.
______
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Exploring Mathematics in the Modern World
C. Directions: Write True if the statement is correct, otherwise, write
False.
_____1.
_____2.
_____3.
_____4.
_____5.
Mathematics is exhibited only through numbers.
Mathematics can progress even without numbers.
Patterns that occur in nature are only for arts appreciation.
Nature also expresses geometric figures and designs.
Mathematics is connected with the things that we see around us.
Topic 1: Patterns and Numbers in Nature
Presentation of Content
Study each picture given below.
What does each picture above tell you about?
These are all patterns in nature. Like numbers, natural objects have also their
patterns. In some plants, spiral patterns may be found in their leaves and
flowers. The skin of some animals also exhibit fascinating designs that have
patterns.
The sunflower’s petals are carefully arranged alternately from the innermost to
the outermost petal to give a spiral pattern. The sea urchin has a pattern
where the spines are arranged in an array, where one big spine is followed by
smaller spines. The sea shell is carefully designed such that from the center,
it is exactly a reflection of the other half of the shell. This is also true to the
butterfly where one side of the wings is exactly the same with the other side.
We call these as symmetrical pattern. The shape and pattern of the left side is
the same as a mirror image of its right side.
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Exploring Mathematics in the Modern World
Other Patterns
There are a lot more patterns that we can see on the things around us such as
the following:
1. Fractal Pattern
A fractal is a never-ending pattern. It is a repeated pattern that is self-similar
across different scales. Fractal is created by repeating a process of similar pattern.
The leaf of a fern resembles a fractal pattern.
Fig. 1 is a geometric fractal
pattern where the triangles are
repeated of different scales and
they are self-similar.
Photo credit: https://www.smithsonianmag.com
Figure 1. Geometric Fractal
Figure 2. Fractal pattern in nature
Fig. 2. The fern exhibits a fractal
pattern. The leaves repeat at
different scales, and they are
self-similar, being made of little
copies of the same overall
shape.
2. Fibonacci Pattern
In arithmetic, Fibonacci pattern appears in numerical sequences such that the
sequence, is the sum of the two preceding ones, starting from 0 and 1.
For example, the sequence 1, 1, 2, 3, 5, 8, 13, … is a Fibonacci sequence
obtained by adding the two consecutive numbers starting from zero (0+1=1),
(1+1=2), (2+1=3), and so on.
The sequence starts from adding the least number to the succeeding number,
and on, giving progressing sums forming a sequence.
Fig. 3 is a Fibonacci sequence,
starting from adding 1+1 to give 2,
2+1 to give 3, 3+2 to give 5 and so
on.
Fig. 3. Fibonacci sequence
Fibonacci pattern is also exhibited amazingly in nature, such that the pattern
starts from a small loop, and becomes bigger as the spiral pattern tends to go
farther from the center of the loop.
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Exploring Mathematics in the Modern World
The formation of stars in
the galaxies and the spiral
pattern of an Aloe Vera
plant all form Fibonacci
patterns in nature.
Credit: www.scienceabc.com/eyeopeners and www.reddit.com
Fig. 4. Fibonacci patterns in nature
Natural patterns also include symmetries, trees, spirals, meanders, waves,
foams, tessellations, cracks and stripes. Early Greek philosophers
studied pattern, with Plato, Pythagoras and Empedocles attempting to explain
order in nature. ... Mathematics, physics and chemistry can
explain patterns in nature at different perspectives.
Relating Mathematics to Natural Patterns
Even clouds exhibit pattern. Clouds and cloud formations are practically used
to assess the possible occurrence of rains or storm.
Some animals and insects have also amazing patterns such as the different
patterns in the skin of a snake, feathers of a peacock, spots of a leopard, the
stripes of a tiger, the wings of a beetle, the wings of a butterfly, and many
more. How do these natural patterns then relate to mathematics?
In the book Nature’s Numbers by Ian Stewart, it was mentioned that by using
mathematics to organize and systematize our ideas out of patterns, we have
discovered a great secret: nature’s patterns are not just there to be admired,
they are vital clues to the rules that govern natural processes. The spiral
leaves of a plant for example, have drawn mathematicians to discover what is
known as the Golden Angle which measures 137.5 degrees. As they
continue to discover about the Golden Angle, they also discovered the Golden
Ratio, which later became closely attached to another numerical pattern: the
Fibonacci numbers which you will learn in the succeeding topics.
So, the original patterns in plants have been translated into numeric patterns in
mathematics through the golden angle, golden ratio and Fibonacci numbers.
Similarly, scientists have also tried to explain the connection between the
patterns in animal skin with that of mathematics. They have found out that
these designs could arrive to the formulation of equations in higher
mathematics, particularly the Differential Equation, which captures the
interaction between two chemical products that produced these designs.
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Exploring Mathematics in the Modern World
What about the volcano, do you think it has also mathematical connection?
The volcano also relates to geometric figure (symmetrical) and probability.
Based from the number of eruptions it has made in the past 100 years, it gave
the probability model predicting when and how it will erupt.
For more examples of patterns in nature that relates to mathematics, watch the
following videos:
1. God is a Mathematician
2. Mathematics in Nature
The CD for above videos will be provided to you by your teacher.
You have just learned how nature relates to mathematics. Congratulations!
Application
Activity 1. Using your cellphone, take a picture of two things that you see
around exhibiting patterns. Discuss the pattern of the pictures that you have
taken and how these relate to Mathematics using the blank sheet attached at
the end of this unit. Be able to present this in the class during our next
meeting.
Be guided by the following criteria of evaluating your output.
Indicator
Correctness
of Picture
Discussion
Good
None of the
pictures show a
pattern
Presented
incorrect
explanation/
discussion
Very Good
Only one of the
pictures shows a
pattern
Presented correct
but incomplete
explanation /
discussion
6
Outstanding
Two of the pictures
show a pattern
Presented correct
and complete
explanation /
discussion
Exploring Mathematics in the Modern World
Assessment
Directions: Supply the information being required by the following:
1. Among the images below, could you tell the pattern that each exhibit?
1
2
5
3
4
7
5
8
6
7
8
2. To address the problem on traffic in a big city, several straight roads are
being constructed. It was noted that the two roads will have at most one
junction, three roads will have at most three junctions, and so on.
a. Complete the table below:
of junctions. (5 points)
2
No. of
roads
2
No. of
junctions
3
Identify the pattern on the maximum number
4
5
6
3
b. At least how many junctions are expected to be constructed if there will
be seven roads in the city? Explain your answer. (5 points)
3. Provide the correct number in the box with a question mark. Explain how
you arrived at your answer. (5 points)
2
4
16
7
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Exploring Mathematics in the Modern World
Summary
You have just learned that mathematics is not only numerical/arithmetic in
nature. Mathematics is found in all the things that we see around us… in
plants, animals, trees and many other objects that we see in nature Geometric
designs and sequences are also found in plants. The sunflower for example
exhibit the design of spiral. The branches of some trees exhibit a fractal
pattern, and many others. Mathematics also relates to nature. It is from
nature where some scientific phenomenon came from, that brought about new
discoveries in the field of higher Mathematics.
Reflection
A. How do you articulate the importance of mathematics in your life?
B. What new ideas about Mathematics did you learn that have changed
your thoughts about it?
References
Akash Peshin: www.scienceabc.com/eyeopeners
Baltazar, E.C., Ragasa, C., and Evangelista, J., (2018) Mathematics in the
Modern World. C&N Publishing, Inc.
Earnhart, R. and Adina, E. (2018). Mathematics in the Modern World
(Outcome-Based Module). C&N Publishing, Inc. pp 1-11
New England Public Radio. Retrieved from http://www.nepr.net/post/it-takes500000-pounds-sand-throw-beach-party-north-adams#stream/0
The Science Explorer: Sunflower Spirals: Complexity Beyond the Fibonacci
Sequence. Retrieved from http://thescienceexplorer.com/nature/sunflowerspirals-complexity-beyond-fibonacci-sequence
https://www.ebay.com/p/5pcs-Spiral-Aloe-Seeds-Polyphylla-Cactus-PlantSucculents-Garden-Park-Decor-Hot/1055124369
Philstar Global. https://www.philstar.com
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Exploring Mathematics in the Modern World
Images:
www.123rf.com/photo_20751296_scallopseashell.html,ww.dolphinresearch.org.au/leadership/victorias-marineenvironment-matters/sea urchin
https://www.harrisseeds.com)
http://www.fractal.org
https://www.sciencefriday.com
www.nexusinvestments.com
https://www.smithsonianmag.com
https://www.123rf.com/stock-photo/seashell.html
https://animals.howstuffworks.com
http://phppf.blogspot.com/gallery-spiral-ginger.html
https://www.google.com/https://cdn.britannica.com/s:7
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Exploring Mathematics in the Modern World
Unit 2: Mathematical Language and
Symbols
Introduction
“The laws of nature are written in the language of mathematics.”- Galileo
Galilei
Forget everything you know about numbers. In fact, forget you even know
what a number is. This is where mathematics starts. Instead of mathematics
with numbers, we will now think about math “things” as a language.
Imagine a scenario in Math class where the instructor passes a piece of paper
to each student that contains Problems in Math written in foreign language
that they do not understand! Each student is to read it and make comments.
Is the instructor being fair?
This situation has a very strong analogy in Mathematics. People frequently
have trouble understanding mathematical ideas because they are being
presented in a foreign language – The language of Mathematics!
Like any language, Mathematics has its own symbols, syntax and rules to
follow for us to express and communicate ideas to others.
Following from the first unit of module, this second unit focuses on various
special languages as the foundation of mathematical thought, the language of
variables, sets, relations and functions. The activities and readings in this
module are quite straight-forward. However, extensive and elaborative
discussions of the concepts are expected from you.
Learning Outcomes
a.
b.
c.
d.
e.
Upon the completion of this unit, you are expected to:
Discuss the language, symbols and conventions of mathematics;
Explain the nature of mathematics as a language;
Identify conventions in the mathematical language;
Perform operations on mathematical expressions correctly; and
Acknowledge that mathematics is a useful language.
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Exploring Mathematics in the Modern World
Activating Prior Learning
A. Directions: What can you recall about your lessons in Mathematical
language or symbols and what do you want to learn? Fill up the table
below.
What do you know?
What do you want to know?
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
B. Variables: Translate each statements/phrases to mathematical symbols.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
p.
q.
r.
s.
t.
u.
v.
w.
Sum of x and 5 gives 14
2 multiplied by f is equal to 9
Difference between y and 23 is 12
Total of m and 3 is 21
F divided by 7 gives 1
20 exceeds c gives 18
11 times p is 33
The product of 4 and t is added to 1
Subtract 5 from one-half of y
Ratio of m and 7 added to 2
One-fifth of x is subtracted from 18
The ratio of y and five plus 7
The sum of 8 times d and 4
One-third of sum of z ad 4
One-fourth of the sum of n and 8 minus the product of 6 and b
Add one-fourth to 3 times c
One-sixth of y is added to 4
The sum of one-fifth of w, one-fourth of x and 7
The sum of three consecutive even integers
5 more than y is 25.
The sum of 5 less than x is 12.
The square of x is 9.
The square root of 25 is z.
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Exploring Mathematics in the Modern World
x. The sum of negative 4 and negative 5
y. The difference between 5 and negative 3
C. Variables: Translate each statements/phrases to mathematical symbols
using universal and existential quantifiers.
1. No man is an island.
2. All counting numbers are real numbers
3. Some cars have sunroofs.
4. No self-respecting person is a liar.
5. Only presidents get lifelong secret security protection.
6. Some Real numbers are integers.
7. All irrational numbers are real numbers.
8. There exist an even number that is a prime.
9. All non-repeating and non-terminating decimal numbers are
irrational numbers.
D. Variables: Which of the following is NOT true about universal
quantifiers?
1. The phase that indicates a universal quantifier is “for all”.
2. The symbol we use for universal quantifier is ∀.
3. The elements of a given set satisfy all property.
4. The symbol we use for universal quantifier is ∃.
E. Variables: Which of the following statements is true about existential
quantifiers?
1. The symbol we use for existential quantifiers is ∃.
2. A property is true for all of the elements in a set.
3. The phrase we use for existential quantifiers is 'for all.'
4. The symbol we use for existential quantifiers is ∀.
F. Relations and Functions: Which of the following sets of ordered pairs
represent functions?
M = {(0,-2), (1,4), (-3,3), (5,0)}
A = {(-4,0), (2,-3), (2,-5)}
R = {(-5,1), (2,1), (-3,1), (0,1)}
L = {(3,-4),(3,-2),(0,1),(2,-1)}
O = {(0,3),(3,0),(1,2), (2,1)}
N = {(1,3)}
G. Set: Solve using Venn Diagram.
1. In a group of 60 people, 27 like cold drinks and 42 like hot drinks
and each person likes at least one of the two drinks. How many like
both coffee and tea?
2. In a group of 100 persons, 72 people can speak English and 43 can
speak French. How many can speak English only? How many can
speak French only and how many can speak both English and
French?
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Exploring Mathematics in the Modern World
Topic 1: Nature of Mathematics as a Language
We will think of this unit as a warm up to our mental muscles before we start
to work and do our best on mathematical problems. How do we attack a
mathematical problem? The idea is the need to understand the mathematical
language clearly, precisely and unambiguously.
Mathematics is about ideas -- relationships, quantities, processes, and ways of
figuring out certain kinds of things, reasoning, and so on. It uses words.
Thus, we need to get extensive practice with mathematical language ideas, to
enhance the ability to correctly read, write, speak, and understand math. The
language of math can be learned, but requires the efforts needed to learn any
foreign language.
Learning Objectives
Upon the completion of this topic, you are expected to:
a. discuss the language, symbols and conventions of mathematics;
b. explain the nature of mathematics as a language; and
c. Perform basic unary and binary operation.
Presentation of Content
I. Characteristics of Mathematical Language
The language of mathematics makes it easy to express the kind of thoughts
that mathematicians like to express that it is precise, concise and powerful.
A. Precise means exact and accurate. It is often used in mathematical or
scientific contexts in which definite, fixed statements or measurements are
demanded. While precise and exactly are nearly synonymous, they are not
necessarily interchangeable. Exactly is preferred if you are talking about a
measurement, or a time. For instance, My alarm is set for exactly 5:30 A.M.
B. Concise use of symbols to be able to express more. It means stating
something succinctly, using as few words as possible yet still conveying the full
meaning.
C. Powerful means be able to express complex thoughts with relative ease.
Now, the following definitions will help clarify some terms.
In English, nouns are used to name things we want to talk about (like people,
places and things); whereas sentences are used to state complete thought. A
typical English sentence has at least one noun, and at least one verb. For
example, Gemma loves Mathematics.
We call mathematical analogue of NOUN as EXPRESSION. Thus an
expression is a name given to a mathematical object of interest such as number,
set, matrix and average to name a few.
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Exploring Mathematics in the Modern World
Expressions versus Sentences
MATHEMATICS
Expression
Sentence
(name given to mathematical object of
interest)
Number
Number, Set, Matrix, Ordered pair,
Average
(must state a complete
thought)
TRUE : 1+ 2 = 3
FALSE: 1 + 2 = 4
ST/SF : x =1
A Mathematical sentence is the analogue of an English sentence; it is a correct
assignment of mathematical symbols that states a complete thought. It has verbs
and connectives. Also we have to consider the notion of truth (the property of
being true or false) is of fundamental importance in the mathematical language.
Instead of writing sentences with words, we write mathematical sentences with
numbers and symbols.
Example:
a. In the mathematical sentence 6+9= 15. The equal sign is actually the
verb and indeed one of the most popular mathematical verbs.
b. The symbol “+” in 6 + 9 = 15 is a connective which is used to connect
objects of a given type.
c. Sentences can be true or false. It makes sense to ask the truth of a
sentence. Ask if it is true? Is it false? Is it sometimes true? Sometimes
false?
II. Conventions in the Mathematical Language
In mathematics, we frequently need to work with numbers, these numbers are
the most common mathematical expressions. And, numbers have lots of
different names and they are in simplified form (fewer symbols, fewer
operations, better suited to current use and preferred/ style/format)- is extremely
important in mathematics. This is the same concept as synonyms in English
(words that have the same or nearly the same) meaning.
Example: Numbers with different names and simplified form
a.
5, 2 + 3,
10÷2 ,
(6 - 2) + 1, or 1 + 1+ 1 +1+ 1
b. 3 + 1 + 5 and 9 are both names for the same number but 9 uses
fewer symbols.
c. 3 + 3+ 3 + 3 + 3 and 5×3 are both names for the same number,
but 5×3 uses fewer operation.
1
d. 3.25 units versus 3 4 unit (fraction in simplest form is necessary).
1
e. We write 2 instead of
form or simplest form.
13
. We usually write fraction in reduced
26
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Exploring Mathematics in the Modern World
III. What is the Grammar of Mathematics?
The grammar of mathematics is the structural rules governing the use of
symbols representing mathematical objects. The main reason for the
importance of mathematical grammar is that statements of mathematics are
supposed to be precise.
Mathematical sentences become highly complex if the parts that made them up
were not clear and simple which makes it difficult to understand. Some
difficulties in math language are:
(a) The word "is" could mean equality =, inequality (, ≥, , ≤) or
membership(∈, ∉ ) in a set;
(b) Different uses of a number; to express quantity (cardinal), to indicate
the order (ordinal), and as a label (nominal);
(c) Mathematical objects may be represented in many ways, such as sets
and functions; and
(d) The words "and' & "or" means different from their English uses.
Example: Express the following using mathematical symbols
a. 3 is the square root of 9
b. 15 is greater than 10
c. 103 is a prime number
Answer:
a. 3 = √9
b. 15  10
c. 103 ∈ 𝑃 where P is a prime number
IV. What are the Basic Concepts and Objects that we use in
Mathematics?
To better understand mathematical language, one must have an understanding
of at least a few of the four basic mathematical objects and concepts.
a. Objects in Mathematics are Numbers, Variables, and Operations (unary
& binary).
b. Four Basic Concepts are: sets (relationships, operations, properties),
relations (Equivalence relations), functions and binary operations.
1. Operations (Unary or Binary)
A Unary operation is an operation on a single element.
1. Unary operations
a. negative of 5
b. multiplicative inverse of 7
c. Squaring 4
d. finding the square root of 9
2. Binary Operations
A binary operation is an operation that combines two elements of a set to give
a single element.
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Exploring Mathematics in the Modern World
A binary operation on a set A is a function that takes pairs of elements of A
and produces elements of A from them.
We use the symbol * to denote arbitrary binary operation on a set A.
Example: Binary operation
a. Multiplication 3 and 4 gives 3 X 4 =12
b. Addition of 3 and 5 is 8
3+5=8
c. the difference of 7 and 2
7-2
d. Divide 21 by 3
21/3
A. Four Properties of binary operations:
1. Commutative Property: For all real numbers x and y.
𝑥∗𝑦 = 𝑦∗𝑥
 Example for addition operation: 2 + 4 = 4 + 2
 For example, multiplication on real numbers is said to be
commutative since 3 × 6 = 6 × 3.
2. Associative Property: For all real numbers x, y and z.
x* (y*z) = (x*y)* z
 For example, addition operation: 2 + (4 + 6) = (2 + 4) +
6
 Example for multiplication operation:3 × (6 × 9) = (3 × 6) ×
9
3. Existence of Identity element e for all real number x such that
𝑒 ∗ 𝑥 = 𝑥 ∗ 𝑒 = 𝑥.
 The identity element for addition is 0 such that 0 + 𝑥 = 𝑥 + 0 =
𝑥 .
 The identity element for multiplication is 1 where 1 ≠ 0 such
that
1×𝑥 =𝑥×1 =𝑥
4. Existence of Inverse element a for all real number x such that
𝑎 ∗ 𝑥 = 𝑥 ∗ 𝑎 = 𝑒.
 The additive inverse of element 𝑎 is −𝒂 such that
(−𝑎) + 𝑎 = 𝑎 + (−𝑎) = 0.
 For instance, the additive inverse element of 5 is −𝟓 such that
5 + (−5) = (−5) + 5 = 0. Adding the number and its inverse
(negative) of that number results to additive identity element 0.
The inverse of a number will always be distinct and unique.

The multiplicative inverse of element 𝑎 is
(reciprocal of 𝑎) such that
16
1
𝑎
1
𝟏
𝒂
×𝑎 = 𝑎×𝑎 = 1
where 𝒂 ≠ 0
Exploring Mathematics in the Modern World

1
1
The multiplicative inverse of 5 is 5 × (5) = (5) × 5 = 1.
Multiplying the number and its inverse (reciprocal) of that
number results to multiplicative identity element 1.
Application
A. Concept Map of Binary Operations
Let's start with some common expressions relating to the four operations
Can you think of other terms that you can add in the concept map?
Assessment
Directions: Find out how much you already know about these topics. On a
sheet of paper, write the letter of the option that best answers the question.
1. The multiplicative inverse of -1/2
A. ½
C. -2
B. 2
D. 1
2. The additive inverse of the multiplicative inverse of -3/5
A. 3/5
C. -3/5
B. -5/3
D. 5/3
3. Which property of the real numbers is used in the relation
(A + B) + C = A + (B+ C)?
A. Commutative Property
C. Closure Property
B. Associative Property
D. Transitive Property
4. What is the identity element for multiplication?
A. 0
C. -1
B. 1
D. None of them
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Exploring Mathematics in the Modern World
5. The multiplicative inverse of zero
A. 0
C. 1
B. any number
D. undefined
6. The equivalent of 6  { 3  2[5(10)] + 7}
A. 6
C. 20
B. 26
D. 28
7. Which of the following is a symmetric property of equality?
A. x = x
C. if a = b, then b = a.
B. if a = b and b = c, then a = c. D. if a = b, then a – c = b – c.
8. Which property of the real numbers is used in the relation
A + B = B + A?
A. Commutative Property
C. Closure Property
B. Associative Property
D. Transitive Property
9. Find the sum of: 89; 7004; 478; 640; 1536.
A. 8, 746
C. 8, 757
B. 9 747
D. 9, 846
10. What is the value of 3(−4) − 4(3)
A. 0
C. -24
B. 24
D. -13
11. Find the value of
A. -15
B. 9
(−12 − 3)
C. 15
D. -9
12. 38 + 12 ÷ 2 − 15 ÷ 3 + 2 = ?
A. 5 1/3
C. 37
B. 22
D. 41
13. 60 ÷ 12 + 4 × 6 − 50 ÷ 10 =?
A. 17.5
C. 0.4
B. 49
D. 24
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Exploring Mathematics in the Modern World
Topic 2: The Language of Variables
Learning Objectives
Upon the completion of this topic, you are expected to:
a. identify conventions in the mathematical language;
b. define universal and existential quantifiers; and
c. translate statements/phrases to variables or mathematical symbols using
quantifiers:
Presentation of Content
Suppose we say something like “At time t the speed of a car is S. The letters t
and S stand for real numbers and they are called variables.
More generally, a variable is any letter used to stand for a mathematical object,
whether or not one thinks of that object as changing through time. (1) it has one
or more values or (2) it is equally true for all elements in a given set.
Example: Writing sentences using variables
a. Is there a number with the property doubling it and adding 5 and
gives the same result as squaring it?
b. No matter what number, if it is greater than 2 then its square is
greater than 4.
c. Numbers with the property that the sum of their squares equals the
square of their sum?
d. Given any real number, its square is nonnegative.
Solution:
a. Let x be the number then 2𝑥 + 5 = 𝑥 2 .
b. Let 𝑦 be a number, if 𝑦 > 2 than 𝑦 2 > 4.
c. 𝑎2 + 𝑏 2 = (𝑎 + 𝑏)2.
d. For any real number z, 𝑧 2 ≥ 0.
Variables Used in a Mathematical Sentence
Two of the most important kinds of mathematical sentences are universal
statements and existential statements.
In mathematics, the order in which we write or say in words is crucial. We have
to be precise in what we want to say. If we mean that “for all x=3m -10, where
m is an integer, x is an even number then we should write the words and symbols
in the precise order.
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Exploring Mathematics in the Modern World
To help us in reading and writing mathematical statement, we must know
special words that express quantification. These are called quantifiers.
We distinguished between universal quantifiers and existential quantifiers.
Some universal quantifiers include “every”, “for all” and “any”. Some
existential quantifiers include, “for some”, “at least one” and “there exists”.
Thus, the following definition and statements illustrate the use of universal and
existential quantifiers.
Universal Statement says that a certain property is true for all elements in a
set.
Definition:
Let P be a propositional function with domain of discourse D. The statement
for all x, P(x) is said to be a Universally Quantified Statement.
The statement for all x, P(x) may be written as:
“∀𝒙, 𝑷(𝒙)". The symbol ∀ means “for all” and is called the universal
quantifier.
∀𝒙, 𝑷(𝒙)" 𝐢𝐬 𝐓𝐫𝐮𝐞 𝐢𝐟 𝐏(𝐱) 𝐢𝐬 𝐭𝐫𝐮𝐞 𝐟𝐨𝐫 𝐞𝐯𝐞𝐫𝐲 𝐱 𝐢𝐧 𝐃.
𝐈𝐭 𝐢𝐬 𝐟𝐚𝐥𝐬𝐞 𝐢𝐟 𝐏(𝐱)𝐢𝐬 𝐅𝐚𝐥𝐬𝐞 𝐟𝐨𝐫 𝐚𝐭 𝐥𝐞𝐚𝐬𝐭 𝐨𝐧𝐞 𝐱 𝐢𝐧 𝐃.
“for every 𝒙 ∈ 𝑹, |𝒙| ≥ 𝟎”
Example 7: Universal quantified statement
a. All counting numbers are greater than zero. ∀𝒄, (𝒄 ≥ 𝟎), 𝒄 ∈
Counting Numbers
b. ∀𝑥, (𝑥 2 ≥ 0), 𝑥 ∈ 𝑅
c. ∀𝑥, (𝑥 2 − 1 ≥ 0), 𝑥 ∈ 𝑍 +
d. ∀𝑥, (𝑥 2 − 1 ≥ 0), 𝑥 ∈ 𝑅
e. All birds can fly.
f. Every student in the class wear socks.
An Existential Statement says that there is at least one thing for which the
property is true.
Definition:
Let P be a propositional function with domain of discourse D. The statement
there exists x, P(x) is said to be Existentially Quantified Statement .
The statement there exists x, P(x) may be written as:
“∃𝒙, 𝑷(𝒙)".
The symbol ∃ means “there exists” and is called the existential quantifier.
There is a prime number that is even.
𝑥, 𝑃(𝑥)" is True if P(x) is true for at least one x in D. It is false if P(x)is False for
every x in D.
“There exist 𝜀 > 0. 𝑠. 𝑡. |𝐹9𝑥) − 𝐿| < 𝜀.”
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Exploring Mathematics in the Modern World
Example 8: Existentially Quantified Statement
a. ∃𝑥, (2𝑥 + 1 = 0), 𝑥 ∈ 𝑍
𝑥
b. ∃𝑥, (𝑥 2 +1 > 0) 𝑥 ∈ 𝑍
c. ∃𝑥, (𝑥 2 > 𝑥), 𝑥 ∈ 𝑍 −
d. ∃𝑥, (𝑥 > 1 → 𝑥 2 = 𝑥), 𝑥 ∈ 𝑅
e. There exists an elementary student who can vote for the national election.
Application
Direction: Write each statement using variables.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
For all real numbers x, if x is nonzero then x2 is positive.
For every real number, then its square is greater than or equal to zero.
For all real number x and y, such that x+y = y+x.
There exist an x such that x is a ballpen.
Every person who lives in Tuguegarao City lives in Cagayan.
For all x that is negative, so is its cube.
There exist a real number that is a non-positive.
There exist a counting number less than 1.
Some prime number is even.
Some professors are republican.
No triangles are rectangles.
Some guilty people re not convicted.
Some people are aggressive when they are drunk.
All songs written in a major key sound melancholy.
All teachers are intelligent.
You have done so much at this point. You are entitled to some rest before
you proceed. Why don’t you take a short break and then come back to finish the
unit module?
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Exploring Mathematics in the Modern World
Topic 3: Language of Set
We are now ready to discuss these concepts on sets and set operation in the
context of the set of numbers.
In this part of a unit, you will learn the foundational topic on Algebra from
which virtually all of mathematics can be derived. You will undertake to define
set, identify kind of sets and perform its operations
Learning Objectives
a.
b.
c.
d.
e.
Upon the completion of this topic, you are expected to:
define set, inclusive, element, object and write them using the set notations;
describe sets using the roster and rule method:
identify some kinds of sets and their properties;
perform operations on sets; and
illustrate the relationship of sets using Venn diagram.
Presentation of Content
A set is a well-defined collection of distinct objects called elements. The
elements that make up a set can be anything: numbers, people, letters of the
alphabet, other sets, and so on. It is desirable that sets be well-defined to
ensure the membership or non-membership of an object in a given set. Thus,
5 ∈ N mean 5 is an element of set N.
It is conventionally denoted by Capital letters and with braces. The braces { }
mean “the set whose elements or member are” such as:
Z = {2, 4, 6, 8, 10, 12,...} is the set of all positive even integers or
A = { 1, 3, 5, 7, …} means that A is the set of all positive odd integers or
C = {x | x = 3 * n. where n = 1, 2, 3,...} means that C is the set of all positive
multiples of 3.
NOTE: The notation {x | x...} is read as the set of all x such that x is....
Example 9:
A= the set of counting numbers.
B =the set of vowels.
C= the set of letters in the word “Ibanag”.
D= the set of 2nd Year CPAD students enrolled in GEC 103 for first
semester Academic Year 2019-2020.
E = the set letters in the English alphabet.
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Exploring Mathematics in the Modern World
Example 10: Membership or Element of a Set
a. If B is a set and x is one of the objects of B, this is denoted x ∈ B, and is
read as "x belongs to B", or "x is an element of B". If y is not a member
of B then this is written as y ∉ B, and is read as "y does not belong to B".
b. Another example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white,
red}, and F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above, 4
∈ A and 12 ∈ F; but 9 ∉ F and green ∉ B.
Notation
3 ∈𝐴
15 ∉ 𝐵
{3} ⊂ 𝐴
{15} ⊄ 𝐵
Meaning
3 is an element of set A
15 is not an element of a set B
The set consisting of 3 is a subset of set A
The set consisting of 15 is not a subset of set B
Describing Set
Sets are usually described in one of two ways:
1. Roster/Tabular Method: This first way of describing, or specifying the
members of, a set is, by listing each member of the set, separated by commas,
and enclosed by braces.
Example 11: Roster Method
Complete listing S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
C = {4, 2, 1, 3}
D = {blue, red, yellow, white}.
Partial listing S = {1, 2, 3, …, 12}
For instance, the set of the first thousand positive integers may be specified
extensionally as {1, 2, 3, ..., 1000}, where the ellipsis ("...") indicates that the
list continues in the obvious way. Ellipses may also be used where sets have
infinitely many members. Thus the set of positive even numbers can be
written as {2, 4, 6, 8, ... }.
2. Rule Method. This second way of describing and naming set is by using a
definition or semantic description. Using a set-builder notation.
For instance, S = {x | x is a counting number less than or equal to 12}. The setbuilder notation above is read as “the set of all x such that x is a counting
number less than or equal to 12.”
Example 12: Rule Method
a. A is the set whose members are the first five positive integers.
b. B is the set of colors of the Philippine flag.
c. F = {n2 − 4 | n is an integer; and 0 ≤ n ≤ 19}.
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Exploring Mathematics in the Modern World
In this notation, the vertical bar ("|") means "such that", and the description
can be interpreted as "F is the set of all numbers of the form n2 − 4, such
that n is a whole number in the range from 0 to 19 inclusive."
Now, I believe you are ready to answer the following Assessment Questions.
Application
Direction: Use the following sets. Identify each statement whether it is true or
false.
A = { 1,2 } B = { }
C = {1, 2,3,4,5} D ={-3,-2,-1,0,1,2,3} E = {0}
1. 1 ∈ 𝐴
3. {1,2} ∈ C
5. {1,2} ⊂ 𝐶
7. 𝐸 ∈ 𝐷
2. 𝐵 ⊂ 𝐴
4. E⊄ B
6. 5 ∉ 𝐷
8. {1} ⊄ 𝐴
That was a little difficult, wasn’t it? Well, the purpose of the activity was to let
you understand the importance of set.
Please continue reading the rest of the unit. In particular, kinds of sets.
Kinds of Sets
1. Finite Set. A set whose element is empty or countable. Example is Set S
= {x | x is a counting number less or equal to 12}. Some sets, however, are
infinite sets. Here’s the next definition.
2. Infinite set. A set whose elements cannot be counted. One example is the
set of counting (or natural) numbers, {1,2,3,…}. The set of whole numbers
includes all of the counting numbers, as well as the number 0 and the
set N of natural numbers.
3. Equal Sets. Two sets A and B are said to be equal if and only if they have
the same elements. We then write A=B.
4. The Universal set U is the set of all elements under discussion.
Given three sets A ={ a, b, c, d, e}, B = {a, e, i, o ,u } and C = {m, a, r, l, o ,n}.
Thus, our Universal set will be the set of letters in the English alphabet to
contain the three sets under discussion.
The cardinality n(S) of a set S is "the number of elements of S." For example,
if B = {blue, white, red}, then n(B) = 3.
5. Equivalent Set. Two sets are equivalent if and only if they have the same
number of elements. They have the same cardinality. Given two sets A
={ a, b, c, d, e}, B = {a, e, i, o ,u }, n(A) = 5 and n(B) =5 thus A and B are
equivalent sets but not precisely equal set. Note: All equal sets are
equivalent sets.
6. Empty/Void/Null Set is a unique set with no members and zero
cardinality and is denoted by the symbol ∅ (other notations are used { }).
For example, the set of all three-sided squares has zero members and thus
is the empty set.
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Exploring Mathematics in the Modern World
7. Subset-If every element of A belongs to B. we then write A ⊆ B (or A is
contained in B). Equivalently, we can write B ⊇ A, read as B is a superset
of A, B includes A, or B contains A.
If A is a subset of, but not equal to, B, then A is called a proper subset of B,
written
A B. A is strictly contained in B. The set of all men is a proper subset of the
set of all people.
Example 13: Subsets
a. {1, 3} ⊆ {1, 2, 3, 4}.
b. {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.
c. Note: In particular, every set is a subset of itself: A ⊆ A. Why?
d. The empty set or {} has no elements and is a subset of every set ∅ ⊆ A.
e. Two set are equal: A = B if and only if A ⊆ B and B ⊆ A.
Note: A set with n elements has
number of subsets!
8. Disjoint Sets. Two sets are disjoint sets if and only if they have no
elements in common. Given two sets A = {v, w, x, y, z}, B = {a, e, i,
o ,u } are disjoint sets because they have no elements in common.
9. Overlapping Sets. Two sets are overlapping sets if and only if they have
at least one element in common. Given two sets A ={ a, b, c, d, e}, B =
A, e, i, o ,u }. Sets A and B are overlapping set because they have {a, e}
in common.
10. Power sets
The power set of a set S, P(S) is the set of all subsets of S with 2𝑛 number of
subsets.
Note that the power set contains S itself and the empty set because these are
both subsets of S.
Example 14: Powerset
a. The power set of Z= {1, 2, 3} contains 23 = 8 elements is P(Z)= {{1,
2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}}.
b. Given the set A = {6, 11}, the power set of A equal to 2n(A) =22 = 4
elements, thus, P(A) = { {}, {6}, {11}, {6,11} }.
Operations on Sets
There are several fundamental operations for constructing new sets from a
given sets. Perhaps the best way to understand them is to use what are called
Venn diagrams. It is a pictorial representation of the relationship of sets. The
rectangle represents the universal set. Circle represents the given set.
1) Union. A
both:
B is the set that contains all the elements in either A or B or
25
Exploring Mathematics in the Modern World
Using Set-Builder Notation
A
Venn diagram
B = {x | x ∈ A or x ∈ B}.
Example 15: Union Set
a. If A = { 1, 2, 3} and B ={ 3, 4, 5}, then A
b. {1, 2} ∪ {1, 2} = {1, 2}.
c. {1, 2} ∪ {2, 3} = {1, 2, 3}.
d. {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
2) Intersection. A
both A and B:
A
B = { 1, 2, 3, 4, 5}
B is the set that contains all the elements common to
B = {x | x ∈A and x ∈ B}.
Example 16: Intersection
a. If A = { 1, 2, 3} and B ={ 3, 4, 5}, then A B = { 3 }
Note: If A ∩ B = {}, then A and B are said to be disjoint.
b. {1, 2} ∩ {1, 2} = {1, 2}.
c. {1, 2} ∩ {2, 3} = {2}.
3) Complement. A' is the set that consists of all elements in the universal set
U not contained in A:
A' = {x | x ∈ U and x
A}
Example 17: Complement
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Exploring Mathematics in the Modern World
If U= { 2, 4, 6, 8, 10, 12} and A = { 2, 4}, then A' ={ 6, 8, 10,
12}
4) Difference. A - B is the set that contains all the elements that are in A but
not in B. The relative complement of B in A:
A − B = {x | x ∈ A and not x
B}
Example 18: Difference
a. If A = { 1, 2, 3} and B ={ 3, 4, 5}, then A - B = { 1, 2 }
b. {1, 2} − {1, 2} = { }.
c. {1, 2, 3, 4} − {1, 3} = {2, 4}.
d. If U is the set of integers, E is the set of even integers, and O is the set of
odd integers, then U − E = E′ = O.
5. Symmetric Difference. A △ B is the set that contains all the elements that
are in A∪B but not in A∩B:
A △ B = {x | x ∈ A∪B and not x
A∩B }
Example 19: Symmetric Difference
a. If A = { 1, 2, 3} and B ={ 3, 4, 5}, then A △ B = { 1, 2, 4, 5 }
b. The symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set
{7, 8, 11, 12}.
Venn Diagrams
We are familiar with the use of Venn diagram to illustrate operations and
relationships on sets and a useful tool for solving certain types of problems. A
Venn diagram uses circles (or any simple closed curves) inside a rectangle to
represent relationship among groups of people or objects. Often these groups
are referred to as sets.
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Exploring Mathematics in the Modern World
For example, the following diagram shows the regions determined by A-B, BA, A∩ B, (A∪B)’.
A
A∩ B
A-B
B
B-A
(A∪B)’
Applications of Sets
Example 20: There are 25 sophomores who have seen Star Gazers, (Part I),
36 who have seen Star Gazers, (Part II), and 17 who have seen both
movies. How many sophomores saw one movie, but did not see both?
Solution: the rectangle represents all freshmen.
A
B
Circle A represents those who saw Part I.
8
Circle B represents those who saw Part I I.
17
19
The overlap represents those who saw both.
Theorem: n(𝑨 ∪ 𝑩) = 𝒏(𝑨) + 𝒏(𝑩) − 𝒏(𝑨 ∩ 𝑩)
Therefore, there are 36-17=19 sophomores who did not see Part I and 2517=8 sophomores who did not see Part II. A total of 19 +8 = 27 sophomores
saw one movie but did not see both.
How are you handling the lesson so far? We hope our discussion have been
clear. If not, we can always discuss them during tutorial sessions or re read the
unit presentation once again. In the meantime, please do the following activity.
Now, I believe you are ready to answer the following Assessment Questions.
Let us check if you have understood.
Now, I believe you are ready to answer the following Assessment Questions.
Assessment
A. Use the sets provided and complete each statement using ∈, ∉, ⊂ 𝑜𝑟 ⊄.
1
3
A = {2 , 1, 2}
1
3
E = {0,2, 1,2}
1. 1 ______A
B ={}
C = {1,2,3,4,5} D = {0}
F = { -1, 0, 1,2,3,4,5}
5. {0}________B
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1
9. {2} _______𝐹
Exploring Mathematics in the Modern World
2. A _________E
6. 𝐶 ________𝐹
3. 5 ______B
7. 2_________E
11. -1______A
4. D _________E
8. B________A
12. D_________A
10. 0 _________B
3
B. Is each set finite or infinite?
1. M={ all meter sticks in your classroom}
2. W={ All green board in your country}
3. N={0, 3, 6, 9, …, 21}
4. R={x|𝑥 is a whole number greater than 5}
5. T={ x|𝑥 is an integer less than 1}
6. Z={0, 3, 6, 9, 12, …}
C. Enumerate all the subsets of Set Q= {-1,0,1}
D. Directions: Find out how much you already know about these topics.
Given the following sets answer the following and write your answer on a
sheet of paper.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
D = {1, 3, 5, 7}
1) 𝐷′
E = {2, 4, 6, 8}
F = {1, 5, 6, 8, 9}.
6) 𝐷 ∪ 𝐸
2) 𝐷 ∩ 𝐹
7) D∆𝐹
3) (𝐸∆𝐹)′
8) (𝐷 ∩ 𝐸)′
4) 𝐹 − 𝐸
9) (𝐷 ∪ 𝐸)′ − 𝐹
5) (𝐸 − 𝐹)′
10) (𝐷 ∪ 𝐸 ∪ 𝐹)′
E. Worded Problem: Read the following information to complete the Venn
diagram and answer the questions below.
There are 128 students taking Biology (A).
There are 100 students taking Spanish (B).
A
20
B
There are 80 students taking Art (C).
10
There are 30 students taking Biology and
30
Spanish. There are
40 students taking Spanish and Art. There are
C
28 students taking Biology and Art. There are 10 students taking all three
subjects.
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Exploring Mathematics in the Modern World
a. How many students are taking Biology and Spanish, but not Art?
b. How many students are taking Biology and Art, but not Spanish?
c. How many students are taking Art, but not Biology or Spanish?
d. How many students are taking at least one of the three courses?
Topic 4: The Language of Relations and Functions
We are now ready to discuss the concepts of relations and functions.
On a digital clock, 10:12 and 12:10 represent different times. The order in
which he numbers are listed is important. This presentation deals with pair of
elements from two groups or sets and their relations between them.
Practically in every day of our lives, we pair members from two groups of
objects or numbers.
For example, we say two people are related by blood if they share a common
ancestor and that they are related by marriage. We also speak of a relationship
between student and teacher, and between people who work for the same
employer. Each hour of the day is pair with the local temperature reading by
TV station’s weatherman, and a teacher often pairs each set of score with the
number of students receiving that score to see more clearly how well the
students understood the lesson. Similarly, the objects of mathematics may be
related in various ways. Finally we shall learn about Cartesian products,
relations and special relations called functions.
Learning Objectives
Upon the completion of this topic, you are expected to:
a. determine if a given relation/mapping is a function or relation;
b. determine the domain and range of a function defined; and
c. perform operations on functions.
Presentation of Content
Definition of Relation
A relation R from a non-empty set A to a non-empty set B is a subset of the
Cartesian product set 𝑨 × 𝑩. The subset is derived by describing a relationship
between the first element and the second element of the ordered pairs in 𝑨 ×
𝑩.
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Exploring Mathematics in the Modern World
A relation is a set of ordered pairs such that the set of all first coordinates of
the ordered pairs in a Relation R is called the Domain of the relation R and the
set of all the second coordinates of the ordered pairs called images is called
the Range of R.
A relation maybe expressed as a statement, arrow diagram, table, equation, setbuilder notation and graph.
Example: Relation
1
a. The set R= {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} is a relation, expressed
in set-builder notation where the domain of
1
R ={1, 2, 3, 4, 5} and the range of R={2, 4, 6, 8, 10}.
b. The set R is expressed using arrow diagram or mapping. This mapping
1
represents the relation R= {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}.
R
1
2
1
2
4
6
3
8
4
1
c. Table form of the set R= {(1, 2), ( , 4), (3, 6), (4, 8), (5, 10)}.
2
A
B
1
2
1
4
2
3
6
4
8
5
10
1
d. The set R= {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)} is expressed using
graph. The first number is called the x-coordinate or abscissa. The
second number is called the y-coordinate or coordinate. The graph
of an ordered pair (x,y) is a point on the coordinate plane. The numbers
in an ordered pair are called the coordinates of the point they locate.
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Exploring Mathematics in the Modern World
Example: Graph (3,6) and (6,3).
Since (3,6) means x=3 and y=6,
locate by going 3 units to the
right, then 6 units up. Since (6,3)
means x-6 and y-3, locate by
going 6 units to the right, then 3
units up.
Example 23: Relation
Let A = {2, 4} and B = {2, 4, 6} and define a relation R for A to B as follows:
Given any (x, y) element𝐴 × 𝐵, (x, y) ∈R means that x + y is an integer.
a. State explicitly which ordered pairs are in 𝐴 × 𝐵 and which are in R.
b. Is (2,4) ∈ 𝑅?
c. Is (4,4) ∈ 𝑅?
d. Is (4,6) ∈ 𝑅?
e. What are the domain and co-domain of R?
Solution:
a. R={(2,2), (2,4), (2,6), (4,2), (4,4)(4,6)}
b. Yes
c. Yes
d. Yes
e. Domain of R ={2, 4} and co-domain or Range of R=(2, 4, 6) ∈ 𝑅
Types of Relations
A. One - to – One Relation
Let A={1, 2, 3} and B={2, 4, 6} defined relations S from A to B as follows. For
all (x, y) ∈ 𝐴 × 𝐵 . (x, y) ∈ 𝑆 mean that y = 2x is “twice” a relation.
S={(1, 2), (2, 4), (2, 6)} or drawn using arrow diagrams for S.
S
2
1
4
2
6
3
B. One – to – Many Relation
The first element of the relation is repeated. Simply the single element from the
first set is mapped to various elements of the second set.
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Exploring Mathematics in the Modern World
Using an arrow diagram or set notation T= {(1, 2), (1, 4), (1, 6)}
T
2
1
4
6
8
C. Many – to – One Relation
It is a reverse of one to many relation where two or more elements from the first
set are mapped to a single element of the second set. Example is the relationship
between students to a single teacher 1. Expressed in a set-builder notation
Q
Q={(𝑆1, 𝑇1), (𝑆2, 𝑇1), (𝑆3, 𝑇1)}
T1
S1
S2
S3
D. Many-to-Many Relation
It is a complicated mapping where two or more members from the first set are
mapped to two or more elements of the second set.
Example If A={2, 4, 7} and B={5, 6}, then
𝐴 × 𝐵 = {(2,5), (2,6), (4,5), (4,6), (7,5), (7,6)}
Using arrow diagram we can see multiple arrows.
W
5
2
4
6
7
E. An Equivalence Relation is a relation with the following properties:
i.
Reflexive Property: ∀𝑥 ∈ 𝑅,
𝑥~𝑥
Example 1=1, y=y
ii.
Symmetric Property : ∀𝑥, 𝑦 ∈ 𝑅, If 𝑥~𝑦 , then
For instance if y=7 then 7=y and if x=2 then 2=x.
iii.
Transitive Property: ∀𝑥, 𝑦, 𝑧 ∈ 𝑅 If 𝑥~𝑦 and y~𝑧 , then 𝑥~𝑧.
Example If x =5 and 5 =z, then x=z.
If y is divisible by z and z is divisible by w, then y is divisible by w.
33
y~𝑥.
Exploring Mathematics in the Modern World
Example 24: Equivalence Relation
Show that R = {(1,1), (1,3), (2,2), (3,1), (3,3)} is an equivalence relation from
a set A = {1, 2, 3}.
Solution:
We check the three properties of equivalence relation such as
i. Reflexive: 𝑥~𝑥 such that R = {(1,1), (2,2), (3,3)}
ii. Symmetric
: If 𝑥~𝑦 , then y~𝑥.
Such that R = {(1,1), (1,3), (2,2), (3,1), (3,3)}
If (1,3) ∈ 𝑅 then (3,1) ∈ 𝑅; If (1,1) ∈ 𝑅 then (1,1)∈ 𝑅.
iii. Transitive
: If 𝑥~𝑦 and y~𝑧 , then 𝑥~𝑧.
If (1,3) ∈ 𝑅 and (3,1) ∈ 𝑅 then (1,1) ∈ 𝑅.
Thus R is an equivalence relation because it satisfies all three conditions.∎
Some relations can also be represented by open sentence in two variables. An
open sentence in two variables has solutions that are ordered pairs. y=3x+1
represents a relation. If a replacementset is not specified for x, it is assumed to
be all real numbers.
Graph.
Functions
The concept of function provides the essential tool in applying mathematical
formulations in solving problems. For instance, the statement “the area of a
circle depends on its radius” can be denoted as A= f(r), where A represents the
area and r, the radius. This is read as “Area is a function of radius”.
Definition: A function is a relation f from a set A to a set B if every element
of set A has one and only one image in set B.
A function is a relation such that each element of the domain is paired with
exactly one element of the range. To denote this relationship, we use the
functional notation:
y = f(x)
where f indicates that a function exists between variables x and y.
The notation f : 𝐀 → 𝐁 is used to denote a function which means that f is a
function with domain A and range B or co-domain; f(x) = y means that f
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Exploring Mathematics in the Modern World
transform x (which must be an element of A) into y ( which must be an
element of B).
Note: Given an element x∈X, there is a unique element y in Y that is related to
x. The unique element y to which f relates x is denoted by f(x). And is called f
of x, of the value of f at x, or the image of x under f.
The set of values of f(x) then altogether is called the range of f or image of X
under f, symbolically
Range of f: {y∈Y |y=f(x), for some x in X}
A simple method called the vertical-line test can help you determine when a
relation is a function. If you draw a vertical line at any place on the graph and it
crosses more than one point of the graph, the relation is not a function. If a
vertical line never crosses more than one point, the relation is a function.
Example:
Vertical lines cross at only
one point. The graph does
represent a function
A vertical line crosses the graph at
more than one point. The graph
does not represent a function.
Evaluating Functions
One of the most basic activities in mathematics is to take a mathematical object
and transform into another one.
The functional notation y = f(x) allows us to denote specific values of a
function. To evaluate a function is to substitute the specified values of the
independent variable in the formula and simplify.
Example 25: Function
When f(x) = 2x – 3, (a) find f(2), (b) f(-1), (c) f(5)
Solution:
a). f(2) = 2(2) – 3 = 4 – 3 = 1 .
b). f(-1) = 2(-1) – 3 = -2 – 3 = -5
c). f(5) = 2(5) – 3 = 10 – 3 = 7.∎
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Exploring Mathematics in the Modern World
Operations of Functions
Functions with overlapping domains can be added, subtracted, multiplied and
divided. If 𝑓(𝑥) and 𝑔(𝑥) are two functions, then for all 𝑥 in the domain of
both functions the sum, difference, product and quotient are defined as follows
a. Addition of two functions
(𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥), for all 𝑥 ∈ 𝐷𝑜𝑚𝑎𝑖𝑛.
b. Subtraction of two functions
(𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥), for all 𝑥 ∈ 𝐷𝑜𝑚𝑎𝑖𝑛.
c. Multiplication by a Scalar
Then the product of 𝛼𝑓 = 𝛼𝑓(𝑥) = 𝛼𝑓(𝑥), for all 𝑥 ∈ 𝐷𝑜𝑚𝑎𝑖𝑛.
d. Multiplication of two functions
(𝑓𝑔)(𝑥) = 𝑓(𝑥)𝑔(𝑥), ∀𝑥 ∈ 𝐷𝑜𝑚𝑎𝑖𝑛.
e. Quotient of two functions
𝑓
𝑓(𝑥)
( ) (𝑥) =
, provided 𝑔(𝑥) ≠ 0, ∀𝑥 ∈ 𝐷𝑜𝑚𝑎𝑖𝑛.
𝑔
𝑔(𝑥)
Note: Domain of sum function 𝑓 + 𝑔, difference function 𝑓 − 𝑔
function 𝑓𝑔.
= {𝑥: 𝑥 ∈ 𝐷𝑓 ∩ 𝐷𝑔 }
where 𝐷𝑓 = domain of function 𝑓
𝐷𝑔 = Domain of function 𝑔
Domain of quotient funtion
𝑓
𝑔
and product
= {𝑥: 𝑥 ∈ 𝐷𝑓 ∩ 𝐷𝑔 𝑎𝑛𝑑 𝑔(𝑥) ≠ 0}.
Example 1: Let 𝑓(𝑥) = 3𝑥 + 1 and 𝑔(𝑥) = 𝑥 2 − 5
𝑓
Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑔) (𝑥).
Solution: (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)
= (3𝑥 + 1) + (𝑥 2 − 5)
= 𝑥 2 + 3𝑥 − 4.
(𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥)
= (3𝑥 + 1) − (𝑥 2 − 5)
= −𝑥 2 + 3𝑥 + 6.
(𝑓 ∙ 𝑔)(𝑥) = 𝑓(𝑥) × 𝑔(𝑥)
= (3𝑥 + 1)(𝑥 2 − 5)
= 3𝑥 3 + 𝑥 2 − 15𝑥 − 5.
and
𝑓
𝑓(𝑥)
3𝑥+1
(𝑔) (𝑥) = 𝑔(𝑥) = 𝑥 2 −5 , 𝑥 ≠ ±√5. ∎
Example 2: Let 𝑓(𝑥) = 3𝑥 + 1 and 𝑔(𝑥) = 𝑥 2 − 5
𝑓
Find specific values for: (𝑓 + 𝑔)(1), (𝑓 − 𝑔)(0), (𝑓 ∙ 𝑔)(−1), and (𝑔) (2).
Solution:(𝑓 + 𝑔)(1) = 𝑓(1) + 𝑔(1)
= (4) + (−4)
= (1)2 + 3(1) − 4
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Exploring Mathematics in the Modern World
= 0.
(𝑓 − 𝑔)(0) = 𝑓(0) − 𝑔(0)
= −(0)2 + 3(0) + 6
= 6.
(𝑓 ∙ 𝑔)(−1) = 𝑓(−1) × 𝑔(−1)
= 3(−1)3 + (−1)2 − 15(−1) − 5
= −3 + 1 + 15 − 5
= 8.
𝑓
𝑓(2)
3(2)+1
(𝑔) (2) = 𝑔(2) = (2)2 −5
and
=
7
−1
= −7. ∎
Example 3: If 𝑓 and 𝑔 are real functions defined by 𝑓(𝑥) = 𝑥 + 7 𝑎𝑛𝑑
𝑔(𝑥) = 3𝑥 2 + 2, find each
a.
b.
c.
d.
e.
f.
𝑓(1) + 𝑔(3)
2 ∙ 𝑓(1)
𝑓(2) + 𝑔(0)
𝑓(−1) − 𝑔(−2)
𝑓(3) ∙ 𝑔(5)
𝑓(−2)
𝑔(−2)
𝑓(3)
g. 𝑓(2) + 𝑔(1)
Solution:
a. 𝑓(1) = 1 + 7 = 8 and 𝑔(3) = 3(3)2 + 2 = 3(9) + 2 = 29
Thus, 𝑓(1) + 𝑔(3) = 8 + 29 = 37.
b. 2 ∙ 𝑓(1) = 2 ∙ (1 + 7)
= 2 ∙ (8)
= 16.
c. 𝑓(2) = 2 + 7 = 9 and 𝑔(0) = 3(0)2 + 2 = 2
Thus, 𝑓(2) + 𝑔(0) = 9 + 2
= 11.
d. 𝑓(−1) = −1 + 7 = 6 and 𝑔(−2) = 3(−2)2 + 2 = 3(4) +
2 = 14
Thus, 𝑓(−1) − 𝑔(−2) = 6 + 14
= 20.
e. 𝑓(3) ∙ 𝑔(5) = (3 + 7)[3(5)2 + 2]
= (10)[75 + 2]
= (10)(77)
= 770.
𝑓(−2)
−2+7
f. 𝑔(−2) = 3(−2)2 +2
5
= 3(4)+2
5
= 14.
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Exploring Mathematics in the Modern World
𝑓(3)
3+7
g. 𝑓(2) + 𝑔(1) = (2 + 7) + 3(1)2 +2
=9+
=9+2
= 11∎.
10
5
In this part of lesson we have tried to show the definitions and examples of
relations and function, in briefly discussing how to differentiate a function from
simply a relation. We also discuss the operations on functions. We are now
ready to solve mathematical problems anytime in the succeeding units.
Assessment
Assessment 7: Relations and Functions
1. Find x and y if
a. (4𝑥 + 3, 𝑦) = (3𝑥 + 5, −2)
b. (𝑥 − 𝑦, 𝑥 + 𝑦) = (6, 10)
2. If 𝐴 = {3, 5, 7, 9} and 𝐵 = {4, 6, 25,27,54, 100} , 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵,
find the set of ordered pairs such that "𝑎" is a factor of "𝑏" and 𝑎 <
𝑏.
3. Find the domain and range of the relation R given by
𝑅 = {(𝑥, 𝑦): 𝑦 = 𝑥 + 3𝑥; 𝑤ℎ𝑒𝑟𝑒 𝑦 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 20}
4. Is the following relation a function? Justify your answer
a. 𝑅1 = {(2,3), (3,0), (2,7), (−4,5)}
b. One - to – One Relation S={(1, 2), (2, 4), (3, 6)}
c. One – to – Many Relation T= {(1, 2), (1, 4), (1, 6)}
d. ℎ = {(4,6), (9,3), (5,9), (8,3)}
e. Many – to – One Relation Q={(𝑆1, 𝑇1), (𝑆2, 𝑇1), (𝑆3, 𝑇1)}
f. Many-to-Many Relation 𝐴 × 𝐵 = {(2,5), (2,6), (4,5), (4,6),
(7,5), (7,6)}
g. Each person is assigned a birth date.
h. Each course in a degree program is assigned a tuition fee.
i. Each faculty in a college is assigned to only one particular
parking slot
5. Find the domain for which the function are equal.
𝑓(𝑥) = 2𝑥 2 − 1 𝑎𝑛𝑑 𝑔(𝑥) = 1 − 3𝑥.
6. Let 𝑓(𝑥) = 𝑥 − 1 and 𝑔(𝑥) = 𝑥 2 .
𝑓
Find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), (𝑓 ∙ 𝑔)(𝑥), and (𝑔) (𝑥).
7. Let 𝑓(𝑥) = 𝑥 2 + 1 and 𝑔(𝑥) = 𝑥.
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Exploring Mathematics in the Modern World
Find specific value: (𝑓 + 𝑔)(1), (𝑓 − 𝑔)(0), (𝑓 ∙ 𝑔)(−1) , and
𝑓
( ) (2).
𝑔
8. If 𝑓 and 𝑔 are functions defined by 𝑓(𝑥) = 𝑥 2 + 7 𝑎𝑛𝑑 𝑔(𝑥) =
3𝑥 + 2, find each
h. 𝑓(2) + 𝑔(−3)
i. 𝑓(−1) + 𝑔(−4)
1
j. 𝑓 (2) − 𝑔(13)
k. 𝑓(2) ∙ 𝑔(3)
𝑓(2)
l. 𝑔(−2)
𝑓(3)
m. 𝑓(5) + 𝑔(1)
This video shows the importance/ advantage of introducing the language of
mathematics to a child at an early stage of development.
Video Watching Math isn't hard_ it's a language _ Randy Palisoc _
TEDxManhattanBeach.mp4
Using mathematical language can be a barrier to student’s learning because of
particular conventions in expressing mathematical ideas. For many students,
learning to use language to express mathematical ideas will be similar to
learning to speak a foreign language. Math is a human language just like
Filipino, English, Spanish or Chinese because it allows people to communicate
with each other. Even in the ancient of time people need the language of math
to conduct trade, to build monuments and to measure the length of farming.
This idea of math as a language isn’t exactly new. Students have to learn
specific vocabulary, but also means of expression and phrasing that are
specifically mathematical and which make it possible to explain mathematical
ideas. To express their mathematical ideas clearly enables students to know
that they understand and use mathematical ideas. Randy Palisoc. Math isn't
hard_ it's a language _ Randy Palisoc _ TEDxManhattanBeach.mp4
Summary
You just have learned how mathematics is connected with language.
Congratulations! You did a lot in this unit. Recall that we began by defining
important terms in mathematics. We summarized the four basic concepts and
languages of mathematics namely, variables, set, relations and functions that
eventually helped you in checking mathematical sentences. Those concepts
were applied to the language of mathematics. In particular you analyzed
English statements and transformed it to mathematical statement using
symbols, syntax and rules. With all these you are now quite ready to move on
to the next unit.
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Exploring Mathematics in the Modern World
References
Aufmann, R. et. al. Mathematical Excursion Chapter 4.
Jamison, R.E. (2000). Learning the Language of Mathematics. Language and
Learning Across Disciplines, 4(1), 45-54
Randy Paliso. Math isn't hard_ it's a language _ Randy Palisoc _
TEDxManhattanBeach.mp4
The language of Mathematics (from One Mathematical cat, Please! by Carol
Burns Fisher)
https://study.com/academy/practice/quiz-worksheet-quantifiers-in-mathlogic.html
https://www.varsitytutors.com/hotmath/hotmath_help/topics/operations-onfunctions
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Exploring Mathematics in the Modern World
UNIT 3: Problem Solving and Reasoning
(6 hours)
Introduction
Everyday in our life, whether we’re students, a parent, an ordinary person, a
business person, or the president of the country, we always face so many
problems that need solving. For example, you want to buy your favorite pair
of shoes but you don’t have enough money, or you want to travel from one
place to another by taking the shortest distance, if possible. Whether the
problem is big or small, we all set our objectives for ourselves, face hardships,
and make every effort to overcome them. But you might not know is there’s
an easy way to arrive over and over again at effective and satisfying solutions.
There is a common and essential way to answering problems, and that is
through problem solving.
But why do we really need to learn problem solving? In this unit, problem
solving, its benefits and the different strategies that can be used to solve
problems will be discussed.
Problem solving 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 bunch 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.
Learning Outcomes
a.
b.
c.
d.
Upon the completion of this unit, you are expected to:
Use different types of reasoning to justify statements and arguments
made about mathematics and mathematical concepts;
Write clear and logical proofs;
Solve problems involving patterns and recreational problems following
Polya’s four steps; and
Organize one’s methods and approaches for proving and solving
problems.
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Exploring Mathematics in the Modern World
Activating Prior Learning
Directions: Use inductive reasoning to predict the most probable next number
in each of the following lists.
a. 10, 15, 20, 25, 30, ?
b. 3, 5, 8, 12, 17, ?
c. 21, 16, 12, 9, 7, ?
d. 5, 20, 35, 50, 65, ?
e. 1, 2.5, 4, 5.5, 7, ?
Topic 1: Inductive and Deductive Reasoning
Learning Objectives
Upon the completion of this topic, you are expected to:
1. define inductive and deductive reasoning;
2. use different types of reasoning; and
3. write clear and logical proofs.
Presentation of Content
Inductive Reasoning
The type of reasoning that forms a conclusion based on the examination of
specific examples is called inductive reasoning. The conclusion formed by
using inductive reasoning is often called a conjecture, since it may or may not
be correct. When you examine a list of numbers and predict the next number
in the list according to some pattern you have observed, you are using
inductive reasoning. What kind of thinking is used when solving problems?
Example 1.
Use inductive reasoning to predict the most probable next number in each of
the following lists.
a. 3, 6, 9, 12, 15, ?
b. 1, 3, 6, 10, 15, ?
Solution:
a. Each successive number is 3 larger than the preceding number. Thus
we predict that the most probable next number in the list is 3 larger
than 15, which is 18.
b. The first two numbers differ by 2. The second and the third numbers
differ by 3. It appears that the difference between any two numbers is
always 1 more than the preceding difference. Since 10 and 15 differ by
5, we predict that the next number in the list will be 6 larger than 15,
which is 21.
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Exploring Mathematics in the Modern World
Solution:
Suppose we pick 5 as our original number. Then the procedure would produce
the following results:
Original number 5
Multiply by 8:
8(5) = 40
Add 6:
40 + 6 = 46
Divide by 2:
46 ÷ 2 = 23
Subtract 3:
23 − 3 = 20
We started with 5 and followed the procedure to produce 20. Starting with 6 as
our original number produces a final result of 24. Starting with 10 produces a
final result of 40. Starting with 100 produces a final result of 400. In each of
these cases the resulting number is four times the original number. We
conjecture that following the given procedure will produce a resulting number
that is four times the original number.
Example 2:
Consider the following procedure:
1. Pick a number.
2. Multiply the number by 9
3. Add 15 to the product.
4. Divide the sum by 3 and subtract 5.
Complete the above procedure for several different numbers. Use inductive
reasoning to make a conjecture about the relationship between the size of the
resulting number and the size of the original number.
Solution:
If the original number is 2, then
original number.
If the original number is 7, then
original number.
2(9)+15
3
7(9)+15
3
−5=6
which is three times the
– 5 = 21 which is three times the
12(9)+15
If the original number is 12 then
− 5 = 36 which is three times the
3
original number.
It appears, by inductive reasoning, that the procedure produces a number that
is three times the original number.
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Exploring Mathematics in the Modern World
Scientists Use Inductive Reasoning
Scientists often use inductive reasoning. For instance, Galileo Galilei (1564–
1642) used inductive reasoning to discover that the time required for a
pendulum to complete one swing, called the period of the pendulum, depends
on the length of the pendulum. Galileo did not have a clock, so he measured
the periods of pendulums in “heartbeats.” The following table shows some
results obtained for pendulums of various lengths. For the sake of
convenience, a length of 10 inches has been designated as 1 unit.
Length of Pendulum (in units)
1
4
9
Period of Pendulum (in heartbeats)
1
2
3
The conclusion formed by using inductive reasoning is often called a
conjecture, since it may or may not be correct.
Examples:
1. A baby cries, then cries, then cries to get a milk. We conclude that if a
baby cries, he/she gets a milk.
2. Here is a sequence of numbers: 3, 6,
9, 12, ____. What is the 5th
number? We can easily conclude that the next number is 15.
3. You are asked to find the 6th and 7th term in the sequence: 1, 3, 6, 10,
15, ______, _____. The first two numbers differ by 2. The 2nd and 3rd
numbers differ by 3. The next difference is 4, then 5. So, the next
difference will be 6 and thus, the 6th term is 15+ 6 = 21 while the 7th
is21 + 7 = 28.
Take note!
1. Inductive reasoning is not used just to predict the next number in a list.
2. We use inductive reasoning to make a conjecture about an arithmetic
procedure.
Counterexample
A statement is a true statement if and only if it is true in all cases. If you can
find one case for which a statement is not true, called a counterexample, then
the statement is a false statement. In Example 4 we verify that each statement
is a false statement by finding a counterexample for each.
Example:
Verify that each of the following statements is a false statement by finding a
counterexample. For all numbers x:
a. |𝑥| > 0
b. 𝑥 2 > 𝑥
c. √𝑥 2 = 𝑥
Solutions:
A statement may have many counterexamples, but we need only find one
counterexample to verify that the statement is false.
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Exploring Mathematics in the Modern World
a. Let x=0. Then because |0| is not greater than 0, we have found a
counterexample. Thus “for all numbers x, |x| ˃ 0” is a false statement.
b. For x = 1 we have 1² =1. Since 1 is not greater than 1, we have found a
counterexample. Thus “for all numbers x, x² ˃ x ” is a false statement.
c. Consider x = ˗3. Then √(−3)2 = √9 = 3. Since 3 is not equal to ˗3, we
have found a counterexample. Thus “for all numbers x, √x² = x” is a false
statement
Deductive Reasoning
Another type of reasoning is called deductive reasoning. Deductive reasoning
is distinguished from inductive reasoning in that it is the process of reaching a
conclusion by applying general principles and procedure.
Example:
Use deductive reasoning to show that the following procedure produces a
number that is four times the original number.
1. Pick a number.
2. Multiply the number by 8
3. Add 6 to the product
4. Divide the sum by 2
5. Subtract 3.
Solution:
Let n represent the original number.
Multiply the number by 8:
Add 6 to the product:
Divide the sum by 2:
Subtract 3:
8𝑛
8𝑛 + 6
(8𝑛 + 6)/2 = 4𝑛 + 3
4𝑛 + 3 – 3 = 4𝑛
We started with n and ended with 4n. The procedure given in this example
produces a number that is four times the original number. Complete the above
procedure for several different numbers. Use inductive reasoning to make a
conjecture about the relationship between the size of the resulting number and
the size of the original number.
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Exploring Mathematics in the Modern World
Application
Activity 1
Directions: Complete the procedure for several different numbers. Use
inductive reasoning to make a conjecture about the relationship between the
size of the resulting number and the size of the original number.
Consider the following procedure.
1. Pick a number.
2. Multiply the number by 8,
3. Add 6 to the product.
4. Divide the sum by 2 and subtract 3.
Activity 2
Directions: Verify that each of the following statements is a false statement
by finding a counterexample for each item. For all numbers x:
𝑥
a. 𝑥 = 1
b.
𝑥+3
3
=𝑥+1
c. √𝑥² + 16
= x+4
Activity 3
Directions: Determine whether each of the following arguments is an example
of inductive reasoning or deductive reasoning.
a. During the past 10 years, a tree has produced plums every other year.
Last year the tree did not produce plums, so this year the tree will
produce plums.
b. All home improvements costs more than the estimate. The contractor
estimated that my home improvement will cost Php 53,000.00. Thus
my home improvement will cost more than Php 53,000.00.
c. I know I will win a jackpot on this slot machine in the next 10 tries,
because it has not paid out any money during the last 45 tries.
Assessment
Test 1
Directions: Use inductive reasoning to predict the most probable next number
in the following lists.
a. 5, 10, 15, 20, 25, ?
b. 2, 5, 10, 17, 26, ?
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Exploring Mathematics in the Modern World
Test 2
Directions: Complete the following procedure for several different numbers.
Use inductive reasoning to make a conjecture about the relationship between
the size of the resulting number and the size of the original number.
A. Consider the following procedure.
1. Pick a number.
2. Multiply the number by 8
3. Add 6 to the product.
4. Divide the sum by 2 and subtract 3.
B. Consider the following procedure
1. List 1 as the first odd number
2. Add the next odd number to 1.
3. Add the next odd number to the sum.
4. Repeat adding the next odd number to the previous sum.
5. Construct a table to summarize the result. Use inductive reasoning
to make a conjecture about the sum obtained.
Test 3
Directions: Use deductive reasoning to show that the following procedure
produces a number that is three times the original number.
Procedure: Pick a number. Multiply the number by 6, add 10 to the product,
divide the sum by 2, and subtract 5. Hint: Let n represent the original number.
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Exploring Mathematics in the Modern World
Topic 2.
Polya’s Four Steps in Problem Solving
Learning Objectives
Upon the completion of this topic, you are expected to:
1. define the terms and concept in problem solving; and
2. identify the four steps of Polya in problem solving;
Presentation of Content
Problem Solving
What is a problem? Generally, it is a situation you want to change!
A problem is a situation that conforms the learner, that requires resolution, and
for which the path of the answer is not immediately known. There is an
obstacle that prevents one from setting a clear path to the answer.
What is a Problem Solving?
Problem Solving has been defined as higher-order cognitive process that
requires the modulation and control of more routine or fundamental skills"
(Goldstein & Levin, 1987).
Mathematical Reasoning
It refers to the ability of a person to analyze problem situations and construct
logical arguments to justify his process or hypothesis, to create both
conceptual foundations and connections, in order for him to be able to process
available information
Pólya’s How to Solve It
George Pólya was a great champion in the field of teaching effective problem
solving skills. He was born in Hungary in 1887, received his Ph.D. at the
University of Budapest, and was a professor at Stanford University (among
other universities). He wrote many mathematical papers along with three
books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1
Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC
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Exploring Mathematics in the Modern World
In 1945, Pólya published the short book How to Solve It, which gave a fourstep method for solving mathematical problems:
First, you have to understand the problem.
Second, after understanding, then make a plan.
Third, carry out the plan.
Fourth, look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1
and 2 are particularly mysterious! How do you “make a plan?” That is where
you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the
truth is that they are more art than science. This is where math becomes a
creative endeavor (and where it becomes so much fun). We will articulate
some useful problem solving strategies, but no such list will ever be complete.
This is really just a start to help you on your way. The best way to become a
skilled problem solver is to learn the background material well, and then to
solve a lot of problems!
Four Steps in Problem Solving
How do we perform Problem Solving? To be able to solve problem
systematically, we follow the four basic steps enunciated by George Polya in
1945 through all of these steps where known already and used well before
then. The ancient Greek mathematicians like the Euclid and Pythagoras
certainly knew how it was done.
1. Understand the Problem
First you have to understand the problem. Study the essential
mathematical concepts by considering the terminology and notation
used in the problem. Rephrase the problem in your own words, if
needed. Then write down specific example of the condition given in
the problem. Ask yourself these questions:
a. What kind of a problem is it?
b. What is the known?
c. What information is given?
d. What do the terms mean?
e. Is this enough information or is more information needed?
f. What is or are the conditions in the problem? Is possible to satisfy
the condition/s? Is/Are the condition sufficient to determine the
unknown?
2. Devise a Plan
First find the link between the data and the unknown. You must start
somewhere, so try something. But if an immediate connection cannot
be found, then it would be necessary to consider more problems. You
should obtain eventually a plan of the solution. Think of ways on how
you are going to attack the problem, that is, try using strategies that
could help you solve the problem.
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Exploring Mathematics in the Modern World
Here are some of the possible strategies that can be used:
1. Identify a Sub-goal
2. Making a Table
3. Make an organized List (Tree Diagram, Venn Diagram)
4. Making an illustration/Drawing
5. Eliminating Possibilities
6. Writing an Equation/ Using a Variable
7. Solving a simpler version of the problem
8. Trial and Error / Guest and Check
9. Work Backwards
10. Look for a Pattern/s
3. Carry out the Plan
As soon as you have an idea for the solution of the problem, write it
down instantly then carry out your plan of the solution. Just make sure
that each step in the solution is logically correct. However, if the plan
does not seem to be working well, then start over again then try
another strategy. Sometimes, the first approach will not work. But do
not worry because if the strategy does not work, it does not mean you
did wrong. It could be that there is more appropriate strategy that you
can use for the particular problem. Remember, the secret here is to
keep trying until something works.
4. Look Back
Once you have a potential solution, check to see if it works. Ask the
following to yourself:
1. Did you answer the question?
2. Is your result reasonable?
Then, double check your solution to make sure that all of the
conditions related to a problem are satisfied. Make sure that any
computation involved in finding your solution is correct. If you find
that your solution does not work or satisfy the problem, there may only
be a simple mistake. Try to fix or modify your existing solution before
disregarding it. Remember what you tried- It is likely that at least part
of it will end up useful.
Another way of checking your solution is to make of another concepts
or formulas or given strategies to solve the problem.
If the answer that you will get using that new concept, formula or
strategy is the same as you first attempt, then it means that your answer
is right. Remember, there are different way of solving a problem.
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Exploring Mathematics in the Modern World
Application
Directions: George Polya introduced the four steps of problem solving. In
your own words, discuss each step to solve a problem.
Understand
the Problem
• Your Explanation
• Your Explanation
Devise a
Plan
Carry Out
the Plan
Look Back
• Your Explanation
• Your Explanation
Assessment
Directions: Solve the following problems using Polya’s four-step problem
solving strategy. Label your work so that each of Polya’s four step is
identified.
1. During one semester, Rica Mae was given P25 for each math test that
she passed and was fined P50 for each math test that she failed. By the
end of the semester, Rica Mae passed seven times as many times tests
as she failed and she had a total of P375. How many tests did she fail?
2. A rancher decides to enclose a rectangular region and 2240 feet of the
new fence on the other three sides. The wants the length of the
rectangular region to be five times as long as its width. What will be
the dimensions of the rectangular region?
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Topic 3: Problem Solving Strategies
Learning Objectives
Upon the completion of this topic, you are expected to:
a. Identify problem solving strategy that is appropriate to solve a given
problem; and
b. Employ the strategy to solve word-problems.
Presentation of Content
Problem-Solving Strategies
Ancient mathematicians such as Euclid and Pappus were interested in solving
mathematical problems, but they were also interested in heuristics, the study
of the methods and rules of discovery and invention. In the seventeenth
century, the mathematician and philosopher René Descartes (1596–1650)
contributed to the field of heuristics. He tried to develop a universal problemsolving method. Although he did not achieve this goal, he did publish some of
his ideas in Rules for the Direction of the Mind and his better-known work
Discourse de la Methode.
Another mathematician and philosopher, Gottfried Wilhelm Leibnitz (1646–
1716), planned to write a book on heuristics titled Art of Invention. Of the
problem solving process, Leibnitz wrote, “Nothing is more important than to
see the sources of invention which are, in my opinion, more interesting than
the inventions themselves.” One of the foremost recent mathematicians to
make a study of problem solving was George Polya (1887–1985). He was
born in Hungary and moved to the United States in 1940. The basic problemsolving strategy that Polya advocated consisted of the four steps presented in
the previous topic.
Problem Solving Strategy (Guess and Test)
Make a guess and test to see if it satisfies the demands of the problem. If it
doesn't, alter the guess appropriately and check again. Keep doing this until
you find a solution.
Example:
Mr. Jones has a total of 25 chicken and cows on his farm. How many of each
does he have if all together there are 76 feet?
Solution:
Step 1: Understanding the problem
We are given in the problem that there are 25 chickens and cows. All together
there are 76 feet. Chickens have 2 feet and cows have 4 feet.
We are trying to determine how many cows and how many chickens Mr. Jones
has on his farm.
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Exploring Mathematics in the Modern World
Step 2: Devise a plan
We are going to use Guess and Test along with making a table.
Make a table and look for a pattern:
Procedure: Make a table reflecting the data in the problem. If done in an
orderly way, such a table will often reveal patterns and relationships that
suggest how the problem can be solved.
Step 3: Carry out the plan:
Chickens
Cows
Number of
chicken feet
Number of
cow feet
Total number
of feet
20
5
40
20
60
21
4
42
16
58
Notice we are going in the wrong direction! The total number of feet is
decreasing!
19
6
38
24
62
Better! The total number of feet are increasing!
15
10
30
12
13
24
40
70
52
76
Step 4: Looking back:
Check: 12 + 13 = 25 heads
24 + 52 = 76 feet.
We have found the solution to this problem. I could use this strategy when
there are a limited number of possible answers and when two items are the
same but they have one characteristic that is different.
Problem Solving Strategy (Using an Organized List)
It is done by applying Polya’s Strategy (make an organized list).
Example:
A baseball team won two out of their four games. In how many different
orders could they have two win and two losses in four games?
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Exploring Mathematics in the Modern World
Solution:
Step 1: Understand the Problem
There are many different orders. The team may have won two straight games
and lost the last two (WWLL). They may also lost the first two games and
won the last two (LLWW). Of course there are other possibilities, such as
WLWL.
Step 2: Devise a Plan
We will make an organized list of all possible orders. An organized list is a list
that is produce using a system that ensures that each of the different order will
be listed once and only once.
Step 3: Carry out the Plan
Each entry in our list must contain two Ws and two Ls. We will use a strategy
that makes sure each order is considered, with no duplications. One such
strategy is to always write a W unless doing so will produce too many Ws or a
duplicate of one previous orders. If it is not possible to write a W, then and
only then do we write is L. This strategy produces the six different orders
shown below.
1. WWLL(start with two wins)
2. WLWL(Start with one win)
3. WLLW
4. LWWL(Start with one loss)
5. LWLW
6. LLWW(Start with two losses)
Step 4: Review the Solution
We have made an organized list. The list has no duplicates and the list
consider all possibilities, so we are confident that there are six different orders
in which a baseball team can win exactly win two out of four games.
Application
Directions: Solve each problem using Polya’s four-step problem solving
strategy.
1. True- False Test In how many ways can you answer a 15-question test
if you answer each question with either a “true,” a “false,” or an two
“always false”?
2. Number of Skyboxes The skyboxes at a large sports arena are equally
spaced around a circle. The 11th skybox is directly opposite the 35th
skybox. How many skyboxes are in the sports arena?
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Exploring Mathematics in the Modern World
Assessment
Directions: Solve the following problems using Polya’s four-step problem
solving strategy and discuss how the strategy works.
1. Magic squares are squares grids which are to be arrangement of
numbers in them. These numbers are special because every row,
column and diagonal adds up to the same number. Arrange the
numbers from 1 to 9 in a 3x3 magic square so that the sum of every
row, column and diagonal adds up to the same number.
2. A person’s age was a square number last year, and next year will be a
cubic number. How old is the person?
3.
A housewife goes to the market once every two days and her neighbor
goes to the same market every after 5 days. One Sunday, the two
housewives met at the market. When will they meet again at the
market?
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Exploring Mathematics in the Modern World
Topic 4: Mathematical Problems Involving Patterns
Learning Objectives
Upon the completion of this topic, you are expected to:
a. Identify the pattern in a given sequence of numbers;
b. Solve problems involving patterns and recreational mathematical
problems.
Presentation of Content
Sequence
A sequence is a pattern involving an ordered arrangement of numbers.
We first need to find a pattern. Ask yourself as you search for a pattern – are
the numbers growing steadily larger? Steadily smaller? How is each number
related?
Example 1: 1, 4, 7, 10, 13…
Find the next 2 numbers. The pattern is each number is increasing by 3. The
next two numbers would be 16 and 19.
Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each
successive number is increase by the next odd number. 1 + 3 = 4.
4+5=9
9 + 7 = 16
So the next number would be
16 +9 = 25
25 + 11 = 36
Terms of a Sequence
An ordered list of numbers such as 5, 14, 27, 44, 65, . . . is called a
sequence. The numbers in a sequence that are separated by commas are the
terms of the sequence. In the above sequence, 5 is the first term, 14 is the
second term, 27 is the third term, 44 is the fourth term, and 65 is the fifth term.
The three dots “…” indicate that the sequence continues beyond 65, which is
the last written term. It is customary to use the subscript notation to designate
the nth term of a sequence.
That is,
𝑎1 represents the first term of a sequence.
𝑎2 represents the second term of a sequence.
𝑎3 represents the third term of a sequence.
.
.
𝑎𝑛 represents the nth term of a sequence.
In the sequence 2, 6, 12, 20, 30, … 𝑛2 + 𝑛, …
𝑎1 = 2,
𝑎2 = 6,
𝑎3 = 12, 𝑎4 = 20, 𝑎5 = 30, and 𝑎𝑛 = 𝑛2 + 𝑛,
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When we examine a sequence, it is natural to ask:
 What is the next term?
 What formula or rule can be used to generate the terms?
To answer these questions we often determine the differences between
successive terms of the sequence. The following shows the common
difference of the sequence 2, 5, 8, 11, 14, …
Sequence:
2
5
8
11
14
…
First difference
3
3
3
…
3
(1)
Each of the numbers in row (1) is the difference between the two closest
numbers just above it (upper right number minus upper left number). The
differences in row (1) are called the first differences of the sequence. In this
case the first differences are all the same. Thus, if we use the pattern above to
predict the next number in the sequence, we predict that is the next term of the
sequence. This prediction might be wrong; however, the pattern shown by the
first differences seems to indicate that each successive term is 3 larger than the
preceding term.
The following shows the differences for the sequence 5, 14, 27, 44, 65, …
Sequence:
5
14
27
44
65
…
First differences:
9
Second differences:
13
4
17
4
21
4
…
(1)
…
(2)
In this example the first differences are not all the same. In such a situation it
is often helpful to compute the successive differences of the first differences.
These are shown in row (2). These differences of the first differences are
called the second differences. The differences of the second differences are
called the third differences.
To predict the next term of a sequence, we often look for a pattern in a row of
differences. For instance, the second differences are all the same constant,
namely 4. If the pattern continues, then a 4 would also be the next second
difference, and we can extend the table to the right as shown.
Sequence:
5
14
27
44
65 …
First differences:
Second differences:
9
13
4
17
4
21
4
…
(1)
…
(2)
Now we work upward. That is, we add 4 to the first difference 21 to produce
the next first difference, 25.We then add this difference to the fifth term, 65, to
predict that 90 is the next term in the sequence. This process can be repeated
to predict additional terms of the sequence.
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Sequence:
First differences:
5
14
9
Second differences:
27
13
4
44
17
4
65
21
4
…
90
25
4
…
(1)
…
(2)
Example 1
Determine the next term in the sequence.
2, 7, 24, 59, 118, 207, …
Solution:
Determine the differences as shown below.
Sequence:
2
7
24
59
First differences:
Second differences:
5
17
12
Third difference
35
18
6
118
59
24
6
207
89
30
6
332
125
36
6
…
… (1)
… (2)
… (3)
The third differences, are all the same constant, 6. Extending this row so that it
includes an additional 6 enables us to predict that the next second difference
will be 36. Adding 36 to the first difference 89 gives us the next first
difference, 125.Adding 125 to the sixth term 207 yields 332.Using the method
of extending the differences, we predict that 332 is the next term in the
sequence.
nth Term Formula for a Sequence
In Example 1 we used a difference table to predict the next term of a sequence.
In some cases we can use patterns to predict a formula, called an nth term
formula that generates the terms of a sequence. As an example, consider the
formula 𝑎𝑛 = 3𝑛2 + 𝑛. This formula defines a sequence and provides a
method for finding any term of the sequence. For instance, if we replace n
with 1, 2, 3, 4, 5, and 6, then the formula 𝑎𝑛 = 3𝑛2 + 𝑛 generates the
sequence 4, 14, 30, 52, 80, 114. To find the 40th term, replace each n with 40.
𝑎𝑛 = 3(40)2 + 40 = 4840
In Example 2 we make use of patterns to determine an nth term formula for a
sequence given by geometric figures.
Example 2
Assume the pattern shown by the square tiles in the following figures
continues.
a. What is the nth term formula for the number of tiles in the nth figure of the
sequence?
b. How many tiles are in the eighth figure of the sequence?
c. Which figure will consist of exactly 320 tiles?
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Exploring Mathematics in the Modern World
Solution
a. Examine the figures for patterns. Note that the second figure has two tiles
on each of the horizontal sections and one tile between the horizontal sections.
The third figure has three tiles on each horizontal section and two tiles
between the horizontal sections. The fourth figure has four tiles on each
horizontal section and three tiles between the horizontal sections.
Thus the number of tiles in the nth figure is given by two groups of n plus a
group of n less one. That is,
an = 2n + ( n – 1 )
an = 3n – 1
b. The number of tiles in the eighth figure of the sequence is 3 (8) – 1 =23.
c. To determine which figure in the sequence will have 320 tiles, we solve the
equation
3n – 1 = 320
3n – 1 = 320
3n = 321
Add 1 to each side
n = 107
Divide each side by 3
The 107th figure is composed of 320 tiles.
Sequences on the Internet
If you find it difficult to determine how the terms of a sequence are being
generated, you might be able to find a solution on the Internet. One resource is
Sloane’s On-Line Encyclopedia of Integer Sequences_ at:
http://www.research.att.com/~njas/sequences/
The Fibonacci Sequence
At the beginning of a month, you are given a pair of newborn rabbits. After a
month the rabbits have produced no offspring; however, every month
thereafter, the pair of rabbits produces another pair of rabbits. The offspring
reproduce in exactly the same manner. If none of the rabbits dies, how many
pairs of rabbits will there be at the start of each succeeding month?
The solution of this problem is a sequence of numbers that we now call the
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Fibonacci sequence. The following figure shows the numbers of pairs of
rabbits for the first 5 months. The larger rabbits represent mature rabbits that
produce another pair of rabbits each month. The numbers in the blue region—
1, 1, 2, 3, 5, 8—are the first six terms of the Fibonacci sequence.
Fibonacci discovered that the number of pairs of rabbits for any month after
the first two months can be determined by adding the numbers of pairs of
rabbits in each of the two previous months.
A recursive definition for a sequence is one in which each successive term of
the sequence is defined by using some of the preceding terms. If we use the
mathematical notation to represent the nth Fibonacci number, then the
numbers in the Fibonacci sequence are given by the following recursive
definition.
𝐹1 = 1, 𝐹2 = 1, 𝑎𝑛𝑑 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 𝑓𝑜𝑟 𝑛 ≥ 3
Application
Activity 1
Directions: Determine the 5th, 7th, and 12th terms of the sequences given by
the formulas below:
A. 3𝑎(𝑎 − 2)
B. (𝑛 − 1)(𝑛 + 1)
C.
D.
E.
F.
𝑥 2 −𝑥−6
𝑥+2
2𝑣 2
31−𝑣
1−1𝑚
4
𝑚3 +1
𝑚2 +2𝑚
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Activity 2
Directions: Use the definition of Fibonacci numbers to find the seventh to
twentieth Fibonacci numbers.
Assessment
A. Directions: Find the nth term of the sequences below:
1. 10, 7, 4, 1, −2, … find the next 2 numbers.
2. 1, 2, 4, 8, … find the next two numbers.
B. Directions: Find the nth term of the sequences below:
1. 3, − 5, − 7, 9, 11, …
2. 0, 2, 6, 12, 20, …
3. 99, 199, 299, 399, 499, …
4. 1/2, 2/3, 3/4, 4/5, …
5. −1/2, 2/5, −3/8, 4/11, …
6. If the first three Fibonacci numbers are given as 𝑥1 = 1, 𝑥2 = 1,
and 𝑥3 = 2, then what is the least value of n for which 𝑥𝑛 > 500?
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Summary
 Inductive reasoning is the process of reaching a general conclusion by
examining specific examples.
 A conclusion based on inductive reasoning is called a conjecture.
 A conjecture may or may not be correct.
 Deductive reasoning is the process of reaching a conclusion by
applying general assumptions, procedure, or principles.
 If you can find one case in which a statement is not true, it is called a
counterexample.
 A sequence is an ordered list of numbers. Each number in a sequence
is called a term of the sequence.
 A formula that can be used to generate all the terms of a sequence is
called an nth-term formula.
 Polya’s four-step problem-solving strategy includes: Understand the
problem, Devise a plan, Carry out the plan, and Review the solution.
 Bar graphs, circle graphs, and broken-line graphs are often used to
display data in a visual format.
“Mathematics is not a deductive science – that’s a cliché. When you try to
prove a theorem, you don’t just list the hypotheses, and the start to reason.
What you do is trial and error, experimentation, guess work.”
- Paul R. Halmos (19162006)
References
Mathematical Excursions Second Edition by Aufmann Lockwood Nation
Clegg
https://webstockreview.net/explore/document-clipart-labreport/#gal_post_993_clipart-pen-reflection-paper-12.png
https://pngimage.net/objectif-png-4/
http://onlineresize.club/pictures-club.html
https://www.iconfinder.com/icons/1296370/book_note_icon
https://www.pinterest.ph/pin/215469163399087874/?lp=true
https://iconscout.com/icon/book-folder-pen-pencil-notebook-education-logoffice-1
http://www.mathstories.com/strategies.htm
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Exploring Mathematics in the Modern World
http://www.mathinaction.org/problem-solving-strategies.html
https://garyhall.org.uk/maths-problem-solving-strategies.html
http://www.mathstories.com/strategies.htm
https://www.youtube.com/watch?v=5FFWTsMEeJw
http://www.mathstories.com/strategies.htm
http://www.mathinaction.org/problem-solving-strategies.html
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Exploring Mathematics in the Modern World
Unit 4: Data Management (10 hours)
Introduction
Statistics is very important especially in academic endeavors like research
writing. Data management is one of those processes involved to come up with
an accurate findings and conclusion.
This unit will broaden your understanding of Mathematics as it relates to
managing data. You are expected to apply methods for organizing and
analyzing large amounts of information and carry out a culminating
investigation that integrates statistical concepts and skills.
More so, this unit covers important statistical tools in data management. It
presents data gathering and organizing data, representing data using graphs
and charts, interpreting organized data, measures of central tendency,
measures of dispersion and relative position, the normal distribution curve,
and linear correlation.
In this unit, you are expected gain a practical, legal and ethical understanding
of how to access, query and manage data collections, using real-world
datasets, standard software packages and data visualization techniques. They’ll
learn how to organize and analyze data collections to answer questions about
the world, as well as developing an appreciation of user needs surrounding
data systems.
Do your best in accomplishing the different tasks provided in this unit and
answer the questions honestly by considering your previous experiences and
prior knowledge.
Enjoy your learning!
Learning Outcomes
a.
b.
c.
d.
e.
f.
Upon the completion of this unit, you are expected to:
Use variety of statistical tools to process and manage numerical data;
Calculate the measures of central tendency and measures of dispersion for a
set of discrete data;
Identify the location of data in a given set of observations;
Determine the relationship that exists between two quantitative variables;
Use the methods of linear regression and correlations to predict the value of
a variable given certain conditions; and
Advocate the use of statistical data in making important decisions.
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Exploring Mathematics in the Modern World
Activating Prior Learning
Directions: Find out how much you already know about these topics. On a
sheet of paper, write the letter of the option that best answers the question.
1. For the set of data consisting of 8, 8, 9, 10, and 10, which of these is
TRUE?
A. Mean = Mode
C. Mean = Median
B. Median = Mode
D. Mean < Median
2. Nine people contributed 100, 200, 100, 300, 300, 200, 200, 150, 100,
and 100 pesos for a door prize. What is the median contribution?
A. 100
C. 175
B. 150
D. 200
3. Which of the following indicates how many standard deviations a data
point is from the mean?
A. Z-Score
C. Quantiles
B. Box Whisker’s Plot
D. Skewness
4. Which of these is equivalent to the median of a distribution?
A. First Quartile
C. Fifth Decile
B. Tenth Percentile
D. Second Quartile
5. Which of the following Statistical tests allows us to determine the
strength of association of two quantitative variables?
A. T-test
C. Chi-square
B. Linear Correlation
D. Regression Analysis
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Exploring Mathematics in the Modern World
Topic 1: Data Gathering, Organization, Presentation and
Interpretation
Learning Objectives
Upon the completion of this topic, you are expected to:
a. summarize and present data using the different methods of data
presentation;
b. construct graphs and tables to present given data; and
c. interpret the data presented.
Presentation of Content
I. Data Gathering
Research is only valuable if you can share the data effectively. In this topic,
you will learn how to organize data and construct various charts and graphs to
represent the same.
What is a Data?
Data is a collection of information from facts, statistics, numbers,
characteristics, observations, and measurements that represent an idea. There
are two forms of data.
1. Quantitative data deals with the quantity (for example, the number of
whales at Sea World).
2. Qualitative data is another form of data that deals with the description
of things. It can be observed but not measured (such as the color of
your eyes).
What are the Levels of Measuring Data?
When grouped, data can be formed into a single variable. Variables in
quantitative analysis are usually classified by their level of measurement, as
indicated below.
1. Nominal data are categorical variables and has lowest level of
measurement. Category means that the values are not numerical.
Examples are civil status, ID number, religion, sex, etc.
When you are asked about your civil status, you will not answer 1,2,3
etc. But rather your answer would either be single, married, widow or
widower. These data (single, married, widow, widower) are called
categorical data.
Sex is either be male or female, but not 4 or 5. the category is either
female or male.
2. Ordinal variables are categorical variables with order. (e.g. level of
satisfaction, quality of life indices)
3. Interval are quantitative variables but has no true zero point. (e.g.
temperature in degree Celsius, Intelligence Quotient)
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Exploring Mathematics in the Modern World
4. Ratio is the highest level of measurement and has true zero point.
(e.g. weight of child, number of vaccinations)
A. Methods of Gathering Data
There are different methods that you can use to collect data and they are the
following:
1. Direct method is data collection through the use of interviews. The
enumerator talks to the subject personally. He gets the data through a
series of questions asked from the subject of the interview.
2. Indirect Method is data collection through the use of questionnaires.
These questionnaires may be sent through the postal or electronic mail.
3. Observation is done through observation with the use of our senses.
For example, the MMDA gives report every week on the number of
accidents happening at EDSA. To do this, an MMDA personnel will
just count the number of accidents through their CCTV.
4. Experimentation is usually done through experiment in laboratories
and classrooms.
5. Registration is acquiring data from private and government agencies
such as from the National Statistics Office, the Bangko Sentral ng
Pilipinas, Department of Finance, etc.
II. Organization of Data
After data has been collected, it can be consolidated and summarized in tables.
When the variable of interest is qualitative, the statistical table is a list of the
categories being considered, along with a measure of how often each value
occurred.
The data can be summarized through the following ways:
A. The frequency or number of measurements in each category
B. The relative frequency, or proportion, of measurements in each
category
C. The percentage of measurement in each category
III. Presentation of Data
Once the measurements are summarized in a statistical table, you can either
use graphs or charts to display the distribution of the data.
A. Ways of Presenting Data
These are the different ways of presenting data.
1. Textual Form– Data and information are presented in paragraph and
narrative form.
2. Tabular Form– Quantitative data are summarized in rows and columns.
3. Graphical Form– Data are presented in charts, graphs or pictures.
Textual Form
Have you seen data presented in textual form? Below is an example.
Study revealed that Mathematics teachers always used chalkboard (4.62) and
textbooks (4.37); and they sometimes used geometric figures (3.29), graphs
(3.16), graphing board (3.12), pictures (3.02), flash cards (3.01), and
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Exploring Mathematics in the Modern World
whiteboard (3.00). The respondents seldom used geometry board (2.19),
advance organizers (2.12), and realia (2.12). The overall weighted mean of
2.93 indicates that the Mathematics teachers sometimes used the given
traditional instructional materials in teaching mathematical concepts.
Tabular Form
We can present data using stem and leaf plot or frequency distribution table.
Stem and Leaf Plot
In the stem and leaf plot, data are displayed using the actual numerical values
of each data point.
Steps in constructing a stem and leaf plot:
A. Divide each measurement into two parts: stem and leaf.
B. List the stem in column, with the vertical line to the right.
C. For each measurement, record the leaf portion in the same row as its
corresponding stem.
D. Order the leaves from the lowest to highest in each stem.
Example:
Daily sales of ream of bond papers of MARS Paper Company for the forty days:
34
40
31
33
20
25
51
62
45
30
38
45
61
42
30
28
35
31
28
42
39
40
52
43
36
46
48
51
52
47
42
39
40
31
29
33
47
36
45
21
Below shows the presentation of data using stem and leaf plot.
2 0588
2
0588
3 41308051969136
3
00111334566899
4 05522036872075
4
00022235556778
5 1212
5
1122
6 21
6
12
Frequency Distribution Table
The frequency distribution is an arrangement of numerical data according to
size or magnitude, with corresponding frequencies and class mark.
How can we present data using frequency distribution?
Constructing the Frequency Distribution Table
Refer to the guidelines below in constructing the table.
1. Construct the stem and leaf plot of the set of data.
2. Determine the range of the data (the difference between the highest
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Exploring Mathematics in the Modern World
and lowest figure).
3. Divide the range by the number of classes to determine the class
interval. To determine the number of classes, we can use the formula:
𝑘 = 1 + 3.3 𝑙𝑜𝑔 𝑛
Where:
4.
5.
6.
7.
8.
𝑘 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑙𝑎𝑠𝑠𝑒𝑠
𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
The result is rounded off to the nearest whole number.
Start the first class with the lowest observation or a multiple of the
class interval. This is the lower limit of the first class. The highest
observation is the upper limit of the last class.
Determine the other lower limits by adding the class interval until we
reach the computed number of classes (k).
Write the upper limits by subtracting 1 from the lower limit of the
upper class.
Count the number of values that fall under each class.
Example:
Construct a frequency distribution from the sales volume of 50 medical sales
representatives.
723
735
720
765
779
788
745
757
819
767
767
755
781
800
812
796
753
728
740
753
770
793
786
775
760
801
793
786
794
781
738
744
757
769
752
735
746
769
777
766
750
771
730
745
783
779
805
788
768
760
Solution:
1. Construct a stem and leaf plot.
72
038
73
0558
74
04556
75
0233577
76
005677899
77
015799
78
1136688
79
3346
80
015
81
29
2. Compute the range (R).
𝑅 = 819 − 720 = 99
3. Find the number of classes (k).
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Exploring Mathematics in the Modern World
𝑘 = 1 + 3.3 𝑙𝑜𝑔 𝑛
𝑘 = 1 + 3.3 𝑙𝑜𝑔 (50)
𝑘 = 6.6 𝑜𝑟 7
4. Compute the class interval (𝑖).
𝐶𝑙𝑎𝑠𝑠 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 99/7 = 14.14 𝑜𝑟 14
After computing the required values, we can now construct the frequency
distribution table.
Amount of Sales
(Classes)
720 – 733
734 - 747
748 - 761
762 - 775
776 - 789
790 - 803
804 - 819
Boundaries
Number of
Sales
(Frequency)
Relative
Frequency
(Percentage)
719.5 – 733.5
733.5 – 747.5
747.5 – 761.5
761.5 – 775.5
775.5 – 789.5
789.5 – 803.5
803.5 – 819.5
4
8
9
10
10
6
3
8%
16%
18%
20%
20%
12%
6%
Note: The lower boundaries for the classes is 0.5 unit below the smallest
observation of the class. The upper boundary for the class is 0.5 unit above the
largest observation of the class. The data can be summarized in the table by
recording the number (frequency) and the percentage (relative frequency) of
observations in each category or class.
Graphical Form
We can present data using charts and graphs. For instance, pie chart displays
how the total quantity is distributed among the categories while the bar chart
uses the height of the bar to display the amount in a particular category.
Example:
Four thousand new students were admitted at a university in Metro Manila for
the school year, 2011-2012. The students were enrolled in the following
programs:
Program
Number of students
Accounting
320
Actuarial Science
440
Banking and Finance
720
Entrepreneurial Management
1,080
Economics
800
Marketing
400
Tourism
240
4,000
Total
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Exploring Mathematics in the Modern World
How do we present these data using pie chart and bar graph?
Below are the calculations for the construction of the pie chart.
Program
Frequency Relative Percent
Accounting
Actuarial Science
Banking and Finance
Entrepreneurial Management
Economics
Marketing
Tourism
Total
320
440
720
1,080
800
400
240
4,000
.08
.11
.18
.27
.20
.10
.06
1.00
8%
11%
18%
27%
20%
10%
6%
100%
Angle
28.8
39.6
64.8
97.2
72.0
36.0
21.6
3600
From the given calculations, this is how to present the data using pie chart.
Program Preference of the New Students
Acc
As
Bf
Em
Eco
M
This can also be represented by a solid diagram:
This is how to present the data using graph.
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T
Exploring Mathematics in the Modern World
Application
Activity 1
Below is a summary of color preference of 400 randomly selected car buyers
in Cebu:
BLACK
RED
BLUE
GRAY
WHITE
320
180
195
155
250
A. Construct a percentage of relative distribution.
B. Construct a pie chart to describe the data.
C. Construct a bar chart to describe the data.
Activity 2
Three hundred eighty students are grouped into four categories: W, X, Y, and
Z. the number of female and male students who fall in each category is shown
in the table:
Category
Female
Male
Total
W
24
20
44
X
36
45
81
Y
74
60
134
Z
66
55
121
Total
200
180
380
1. Construct a pie chart and a bar graph to describe the data on female students.
2. Construct a pie chart band a bar chart to describe the data on male students.
Activity 3
The Table below shows the number of items bought daily at the MAI Computer
Shop. Construct both the pie chart and the bar chart to describe the data.
Items
Number of pieces
LED monitor
5
Desktop computer
7
Laptop computer
6
Printer
4
Fax machine
3
Total
25
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Exploring Mathematics in the Modern World
Assessment
Directions: Construct the graph or table that is required by each item.
1. The table below shows the monthly sales of Adora Generics, Inc.
Month
Number of Machines
January
60
February
65
March
58
April
44
May
58
June
63
July
61
August
42
September
37
November
46
December
88
Construct a pie chart to describe the data.
2. The table below shows the length of time a piano player practices
seven days before a concert.
Day
Number of minutes
1
90
2
88
3
66
4
94
5
56
Construct a bar graph to describe the data.
3. Consider the following data:
31
39
70
60
27
24
75
67
41
55
61
41
50
72
25
Present the data by construct a graph or table.
73
40
56
27
66
59
50
38
42
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Exploring Mathematics in the Modern World
Topic 2: Measures of Central Tendency
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
define and differentiate the measure of central tendency: mean, median and
mode;
give the advantage of mean, median and mode; and
calculate mean, median and mode for a grouped and ungrouped data;
identify the most appropriate measure of central tendency in a certain
distribution.
Presentation of Content
I. Mean
Are you familiar with the averages? One of them is the mean.
The mean is the most popular and well known measure of central tendency. It
can be used with both discrete and continuous data, although its use is most
often with continuous data. The mean is equal to the sum of all the values in the
data set divided by the number of values in the data set.
So, if we have n values in a data set and they have values x1, x2, ..., xn, the
sample mean, usually denoted by 𝑥̅ (pronounced x bar), is:
(𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛 )
𝑥̅ =
𝑛
This formula is usually written in a slightly different manner using the Greek
capital letter , pronounced "sigma", which means "sum of".
∑𝑥
𝑥̅ =
𝑛
You may have noticed that the above formula refers to the sample mean.
Why have we called it a sample mean?
In statistics, samples and populations have very different meanings and these
differences are very important, even if, in the case of the mean, they are
calculated in the same way.
To acknowledge that we are calculating the population mean and not the
sample mean, we use the Greek lower case letter "mu", denoted as µ:
∑𝑥
𝜇 =
𝑁
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Exploring Mathematics in the Modern World
Characteristics of the Mean
These are some of the characteristics of the mean.
1. The mean is essentially a model of your data set.
2. It includes every value in your data set as part of the calculation.
3. Mean is the only measure of central tendency where the sum of the
deviations of each value from the mean is always zero.
4. The mean is a reliable or a more stable measurement to use when
sample data are being used to make inferences about populations.
5. The mean is sensitive or is greatly affected by the values, high or low
and this makes in appropriate average to use.
6. The mean is the most commonly used, easily understood, easily
calculated, and generally recognized average.
7. It is best measure to use when the distribution is symmetrical.
8. It is useful measure for inferential statistics.
9. It is used to obtain an average value of a series of value after each item
is weighted. This is referred to as weighted mean.
Mean Computation for Ungrouped Data
For ungrouped data, the mean is computed by simply adding all the values and
dividing the sum by the total number of items. For the sample mean, the formula
is:
∑𝑛𝑖=1 𝑥𝑖
𝑥̅ =
𝑛
Where:
𝑥 ̅ = 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛
𝑥 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑖𝑡𝑒𝑚
𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑡𝑒𝑚𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒
𝛴 = 𝑡ℎ𝑒 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓
In simpler form, the formula for the sample mean may be presented as:
∑𝑥
𝑥̅ =
𝑛
And for the population mean, it is:
∑𝑥
𝜇 =
𝑁
Where:
µ = 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑚𝑒𝑎𝑛 𝑜𝑓 𝑎 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑁 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑥 𝑖𝑡𝑒𝑚𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
Example:
Let us consider the scores of Michael in his statistics class. The scores have been
arrayed in descending order.
76
76
62
51
45
75
27
12
6
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Exploring Mathematics in the Modern World
Solution:
Since in the case of Michael’s scores, Σx = 357, Michael’s mean score is
∑𝑥
357
𝑥̅ =
=
= 39.67
𝑛
9
Example:
The grade in Geometry of 10 students are 87, 84, 85, 85, 86, 90, 79, 82, 78,
and 76. What is the average grade of the 10 students?
Solution:
∑𝑥
832
=
= 83.2
𝑛
10
Hence, the average grade of the 10 students is 83.2.
𝑥̅ =
Example:
The weight of four bags of wheat (in kg) are 103, 105, 102, and 104. Find the
mean weight.
Solution:
𝑥̅ =
∑𝑥
414
=
= 103.5 𝑘𝑔
𝑛
4
II. Median
The median (𝑥̃) of a set of data is a measure of central tendency that occupies
the middle position in an array of values.
It is the number that divides the bottom 50% of the data from the top 50%, that
is, half the data items fall below the median and half above that value. In an odd
number of items the median is simply the middle value. If n is even, the median
is the average of the two middle data values in its ordered list.
The middle value or term in a set of data arranged according to size/ magnitude
(either increasing or decreasing) is called the median.
Uses of Median
The median is used whenever an average of position is desired. It is used when
open– ended intervals are involved. Since the median divides a distribution in
half, it is also frequently used as an average in testing general abilities, like in
intelligence test.
Characteristics of Median
The median is another widely used average, easy to understand, and easy to
compute. It cannot be found unless the items are arranged in an ascending or
descending order. It is the point that divides the frequency distribution into two
halves. The median is not affected by the extremely high or low values, so it is
better choice when a distribution is in p badly skewed. It may be determined in
open– ended distribution.
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Exploring Mathematics in the Modern World
Median Computation for Ungrouped Data
The median is computed as follows:
1. Arrange the items in an array.
2. Identify the middle value.
Example 1:
The library logbook shows that 58, 60, 54, 35, and 97 books, respectively,
were borrowed from Monday to Friday last week. Find the median.
Solution:
Arrange the (58, 60, 54, 35, and 97) data in increasing order.
35, 54, 58, 60, 97
We can see from the arranged numbers that the middle value is 58. Thus, the
median is 58.
Example 2:
The amount of money a balut vendor earned on five randomly selected days
are:
₱ 86, ₱ 109, ₱ 141, ₱ 74, ₱ 123
Solution:
Making an array, we have:
₱ 74, ₱ 86, ₱ 109, ₱ 123, ₱ 141
Since there are 5 (odd) items,
𝑥̃ = ₱ 109
Example 3:
Andrea’s scores in 10 quizzes during the first quarter are 8, 7, 6, 10, 9, 5, 9, 6,
10, and 7. Find the median.
Solution:
Arrange the scores in increasing order. 5, 6, 6, 7, 7, 8, 9, 9, 10, 10
Since the number of measures is even, then the median is the average of the
two middle scores.
𝑥̃ =
7+8
2
= 7.5
Hence, the median of the set of scores is 7.5.
III. Mode
The mode (𝑥̂), by definition, is the most commonly occurring value in a series.
A series may have more than one or none at all.
For the grouped data, the class with the greatest frequency is called the modal
class.
A distribution with only one mode is said to be unimodal. In case wherein there
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Exploring Mathematics in the Modern World
are two class limits with the highest frequency, the distribution is referred to as
bimodal. Further, the distribution is multimodal when there are three or more
modes.
Uses of Mode
It is used when a quick estimate of the average is needed. It helps us spot a trend.
Being the most frequently occurring value, if you are a shoe producer or a
clothing manufacturer and you want to know the size that will fit the greatest
number of people, you would seek the modal size.
Obviously, the shoe producer or clothing manufacturer will produce more shoes
or dresses in the most commonly purchased size than in other sizes.
The mode therefore provides information to businessman and producers that
would help them in business and decision making.
The mode is the measure or value which occurs most frequently in a set of data.
It is the value with the greatest frequency.
Characteristics of Mode
It is the simplest central tendency. It is not affected by extreme values in a
distribution but unreliable measure of central tendency.
It is not affected by extreme values in a distribution. It is not necessary to
arrange the item before the mode is known.
The mode may not exist in some set of data or there maybe more than one mode
in other data set.
Mode Computation for Ungrouped Data
For ungrouped data, the most frequent occurring score is the mode.
To find the mode for a set of data:
1. Select the measure that appear most often in the set;
2. If two or more measures appear the same number of times, then each of
these values is a mode; and
3. If every measure appears the same number of times, then the set of data
has no mode.
Example 1
Find the mode of the following values.
3, 4, 7, 7, 7, 8, 11, 11, 14, 18, 19
Answer
𝑥̂ = 7
Example 2
Determine the mode of the following set of data.
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6, 6, 6, 9, 9, 9, 9, 12, 12, 12, 12, 12, 12, 15, 15, 15, 15, 15, 21, 21, 35, 35
Answer
𝑥̂ = 12 and 15
Application
Activity 1
Directions: Supply the information reqiured by each item.
1. The enrolment in a school in last five years was 605, 710, 745, 835 and
910. What was the average enrolment per year?
2. A random sample of six cashiers in a department store shows the
following balances at the end of the day: ₱ 16,640.39; ₱ 26,915.59; ₱ 6,
827.08; ₱ 101,791.17, ₱ 61,811.75, and ₱ 20,244.12. Compute the mean
balance.
3. A group of students obtained the following scores in a math quiz: 8, 7,
9, 10, 8, 6, 5, 4, 3. Arranging these scores in increasing order, find the
median.
4. If the score 5 of another student is included in the list: 3, 4, 5, 5, 6, 7, 8,
8, 9, 10. What is the middle score?
5. Find the mode in the given sets of scores. (10, 12, 10, 9, 13, 11, 10)
Assessment
Directions: Read and answer the following questions carefully. Write the letter
that corresponds to your answer on the space provided.
_____ 1. Four friends went shopping for school clothes. Kim bought 5 shirts,
Jill bought 4 shirts, Leslie bought 6 shirts, and Crystal bought only 1
shirt. Which of the following choices represents the mean of the shirts
purchased?
A. 2
C. 4
B. 3
D. 6
_____ 2. What is the median of the following numbers? {1, 2, 2, 8, 9, 14}
A. 2
C. 13
B. 5
D. 6
_____ 3. Which set of data has a mean of 15, a median of 14, and a mode of 14?
A. 3, 14, 19, 25, 14
C. 25, 15, 14, 3, 7
B. 14, 22, 15, 15, 9
D. 14, 22, 14, 15, 4
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_____ 4. The following are scores in a Math test. 80, 90, 90, 85, 60, 70, 75, 85,
90, 60 and 80. What is the mode of these scores?
A. 80
C. 90
B. 70
D. 60
_____ 5. Which number is NOT the mean, median or mode of the data set 4, 3,
15, 11, 3, 8, 7, 5?
A. 5
C. 7
B. 3
D. 6
_____ 6. The mode score on the Grade 8 Math test was 94. Which of the given
interpretations must be correct?
A. More students got 94 than any other score.
B. A score of 91 was slightly below average.
C. 99 is the highest score in the class.
D. No one scored below 50.
_____ 7. The quiz papers of 7 students were arranged according to their score.
What will be the score of the student in the middle if the scores are 7,
6, 4, 8, 2, 5, and 11?
A. 6
C. 11
B. 12
D. 4
_____ 8. Which number occurs most frequently in the following set of numbers?
12, 17, 16, 14, 13, 16, 11, 14
A. 13
C. 14
B. 11
D. 14 and 16
_____ 9. If the mean of 6 numbers is 41, what is the sum of the numbers?
A. 250
C. 134
B. 246
D. 456
_____ 10. The mean score of 5 students is 23, what is the total score of the 5
students?
A. 125
C. 115
B. 120
D. 110
_____ 11. Which of the following data set has a mean of 15, a median of 14,
and a mode of 11?
A. 11, 11, 13, 15, 20
C. 11, 11, 14, 19, 20
B. 5, 11, 14, 14, 31
D. 6, 10, 14, 15, 15
_____ 12. Cory received the following grades this year: 75, 87, 90, 88, and 79.
If she wishes to earn an 85 average, what must be her score on her
final test?
A. 91
C. 85
B. 87
D. 88
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_____ 13. What is the mode of these values? 15, 21, 26, 25, 21, 23, 28, 21
A. 23
C. 25
B. 21
D. No mode
_____ 14. Find the mode of 4, 8, 15, 21, and 23.
A. 8
C. 21
B. 15
D. No mode
_____ 15. Cassandra sold the following number of candles over the last 6 days:
25, 48, 25, 33, 57, 50. What was the mean number of candles sold
each day?
A. 39.67
C. 43.36
B. 36.87
D. 45.33
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Topic 3: Measures of Dispersion
Learning Objectives
Upon the completion of this topic, you are expected to:
a. define range, standard deviation, and variance;
b. calculate range, standard deviation, and variance for ungrouped data; and
c. describe the given set of data using the computed measures of dispersion.
Presentation of Content
I. Range
The range is the simplest measure of variability. It is the difference between
the largest value and the smallest value. The formula for the range is:
𝑅 =𝐻−𝐿
Where:
𝑅 = 𝑅𝑎𝑛𝑔𝑒
𝐻 = 𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒
𝐿 = 𝐿𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒
Test scores of 10, 8, 9, 7, 5, and 3, will give us a range of 7 from 10 – 3.
Characteristics of Range
It is easy to compute and understand. It emphasizes the extreme values.
However, it is the most unstable measure because its values easily change or
fluctuates with the change in the extreme values.
Uses of Range
The range is used to report the movement of stock process over a period of time
and the weather reports typically state the high and low temperature readings
for a 24– hour period.
Example 1:
The following are the daily wages of 8 factory workers of two garment
factories. Factory A and factory B. Find the range of salaries in peso (Php).
Factory A: 400, 450, 520, 380, 482, 495, 575, 450.
Factory B: 450, 400, 450, 480, 450, 450, 400, 672
Solution:
Finding the range of wages: Range = Highest wage – Lowest wage
𝑅𝑎𝑛𝑔𝑒 𝐴 = 575 − 380 = 195
𝑅𝑎𝑛𝑔𝑒 𝐵 = 672 − 350 = 322
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Comparing the two wages, you will note that wages of workers of factory B
have a higher range than wages of workers of factory A. These ranges tell us
that the wages of workers of factory B are more scattered than the wages of
workers of factory A.
The range tells us that it is not a stable measure of variability because its value
can fluctuate greatly even with a change in just a single value, either the
highest or lowest.
Example 2:
Find the range in the sets A, B, and C.
Set A : 81, 83, 87, 90, 94
Set B : 84, 86, 87, 88, 90
Set C
: 85, 86, 87, 88, 89
Solution:
𝑆𝑒𝑡 𝐴: 𝑅𝑎𝑛𝑔𝑒 = 𝐻𝑉 − 𝐿𝑉 = 94 − 81 = 13
𝑆𝑒𝑡 𝐵: 𝑅𝑎𝑛𝑔𝑒 = 𝐻𝑉 − 𝐿𝑉 = 90 − 84 = 6
𝑆𝑒𝑡 𝐶: 𝑅𝑎𝑛𝑔𝑒 = 𝐻𝑉 − 𝐿𝑉 = 89 − 85 = 4
Based on the computed range for sets A, B, C, it can be concluded that A has
greater variability as compared top B and C.
III. Variance
The variance of a set of data is denoted by the symbol s2. It determines how
spread out the data is. To find the variance (s2), we use the formula:
∑(𝑥 − 𝑥̅ )2
𝑠 =
𝑛−1
2
Where:
n
x
̅
𝒙
= the total number of data
= is the raw score
= the mean of the data
Variance Computation for Ungrouped Data
Calculate the variance follow these steps:
1. Work out the mean (the simple average of the numbers)
2. For each number, subtract the mean and square the result (the squared
difference).
3. Work out the average of those squared differences.
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Example:
You and your friends have just measured the heights of your dogs (in
millimeters). The heights (at the shoulders) are: 600mm, 470mm, 170mm,
430mm, and 300mm. Find out the value of the variance.
Solution:
Step 1. Work out the mean (the simple average of the numbers)
Mean =
600+470+170+430+300
5
=
1,970
5
= 394
Step 2. For each number, subtract the mean and square the result
(the squared difference) and work out the average of those squared
differences.
2062 + 762 + (−224)2 + 362 + (−94)2
5−1
42,436
+
5,776
+
1,296 + 8,836
𝑠2 =
4
108,520
𝑠2 =
4
𝑠 2 = 27,130
So the value of the variance is 27,130.
𝑠2 =
III. Standard Deviation
While the range is about how much your data covers, standard deviation has
to do more with how much difference there is between the scores. It is defined
as a number representing how far from the mean each score is.
Simply, the standard deviation is the square root of the variance.
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Characteristics of Standard Deviation
Standard deviation is a number used to tell how measurements for a group are
spread out from the mean or expected value.
A low standard deviation means that most of the numbers are very close to the
average. A high standard deviation means that the numbers are spread out.
Standard Deviation Computation for Ungrouped Data
To find the standard deviation, follow the steps below.
1. Calculate the mean.
2. Calculate the deviations, which are the scores minus the average.
3. Square the deviations.
4. Sum up the squared deviations.
5. Divide the sum of the squared deviations by the number of scores in
your data set minus 1.
6. Take the square root of the result.
Formula:
𝑠= √
Where:
𝑠 =
𝑥 =
𝑥̅ =
𝑛 =
∑(𝑥 − 𝑥̅ )2
𝑛−1
𝑡ℎ𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑡ℎ𝑒 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑠𝑐𝑜𝑟𝑒
𝑡ℎ𝑒 𝑚𝑒𝑎𝑛
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠
Example 1
Sam has 20 rose bushes, but only counted the flowers on 6 of them! The
"population" is all 20 rose bushes and the "sample" is the 6 bushes that Sam
counted the flowers of. Let us say Sam's flower counts are: 9, 2, 5, 4, 12, and
7, find the value of the standard deviation.
Solution
Step 1. Work out the mean.
Using sampled values 9, 2, 5, 4, 12, 7
The mean is (9 + 2 + 5 + 4 + 12 + 7) / 6 = 39/6 = 6.5
So, 𝑥̅ = 6.5
Step 2. Then for each number, subtract the mean and square the result.
(9 − 6.5)2 = (2.5)2 𝑜𝑟 6.25
(2 − 6.5)2 = (−4.5)2 𝑜𝑟 20.25
(5 − 6.5)2 = (−1.5)2 𝑜𝑟 2.25
(4 − 6.5)2 = (−2.5)2 𝑜𝑟 6.25
(12 − 6.5)22 = (5.5)2 𝑜𝑟 30.25
(7 − 6.5)2 = (0.5)2 𝑜𝑟 0.25
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Exploring Mathematics in the Modern World
Step 3. Then work out the mean of those squared differences.
𝑆𝑢𝑚 = 6.25 + 20.25 + 2.25 + 6.25 + 30.25 + 0.25 = 𝟔𝟓. 𝟓
65.5
65.5
𝑜𝑟 13.1
6−1
5
This value is called the sample variance.
=
Step 5. Take the square root of that.
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = √13.1 = 3.62
Application
Directions: Find the range, variance, and standard deviation of the following
quantitative frequency distributions.
The following data represent the difference in scores between the winning and
losing teams in a sample of 15 college football bowl games from 20018-2019.
12
24
12
25
12
15
15
9
8
10
16
15
12
11
10
11
12
9
8
9
13
15
16
17
18
Assessment
Directions: Read and answer the following questions carefully. Write the letter
that corresponds to your answer on the space provided. After answering the
problems, check whether your answers are correct on the given key to
corrections.
______1. Ten friends scored the following marks in their end-of-year math
exam: 23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%.
What was the standard deviation of their marks?
A. 15.1%
C. 15.5%
B. 16.9%
D. 18.6%
______2. What is the standard deviation of the first 10 natural numbers (1 to
10)?
A. 2.45
C. 3.16
B. 2.87
D. 8.25
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______3. Nine friends each guessed the number of marbles in a jar. When the
answer was revealed they found they had guessed well (and one was
the winner!) Here is how close they each got: -9, -7, -4, -1, 0, 2, 7, 9,
12. What is the range?
A. 3.0
C. 6.2
B. 5.5
D. 6.8
______4. What is the variance for the numbers: 75, 83, 96, 100, 121 and 125?
A. 216.9
C. 332.66
B. 127.1
D. 18.2
______5. What is the standard deviation for the numbers: 4,5,5,4,4,2,2,6
A. 1.32
C. 16.2
B. 5.45
D. 2.8
_____ 6. Given the following set of data, what is the variance? {2, 6, 8, 3, 7, 9,
1, 4}
A. 40
C. 2.47
B. 5
D. 7.5
______7. An instructor gave students a 20-item quiz on a course topic. The
distribution of scores on the quiz was: 8, 8, 10, 10, 11, 12, 12, 13, 13,
13, 15, 15, 16, 17, 19, 19, 20, 20, 20. What is the range for these
scores?
A. 20
C. 16
B. 12
D. 18
______8. Data were collected on the number of minutes spent cooking a
meal. The data are as follows: 8, 10, 15, 25, 30, 40, 12, 20, and
19. What is the range of this data?
A. 24
C. 32
B. 22
D. 48
_____ 9. The mean of a distribution is 14 and the standard deviation is 5. What
is the value of the coefficient of variation?
A. 60.4%
C. 35.7%
B. 48.3%
D. 27.8%
_____ 10. Price of gasoline for three days are as 98, 96, 97, 100 then value of
standard deviation with assumed mean method is
A. 15
C. 1
B. 10
D. 11
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Topic 4: Measures of Relative Position
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
determine the corresponding z-score of a given set of raw data;
identify the location of a given data in terms of corresponding z-score and
quantiles;
interpret the location of raw data in the Box Whisker’s plot; and
solve word problems involving the concepts of the measures of relative
position.
Presentation of Content
I. Z-score
What do you know about z-score? Do you know how z-score is determined?
A z-score indicates how many standard deviations a data point is from the
mean. A given raw data can be converted in terms of z-score using the
formula:
𝒛=
̅
𝒙− 𝒙
𝒔
Where:
𝑥 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑤 𝑑𝑎𝑡𝑎
𝑥̅ = 𝑚𝑒𝑎𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑑𝑎𝑡𝑎 𝑏𝑒𝑙𝑜𝑛𝑔𝑠
𝑠 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑑𝑎𝑡𝑎
𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑑𝑎𝑡𝑎 𝑏𝑒𝑙𝑜𝑛𝑔𝑠
𝑧 = 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑧 − 𝑠𝑐𝑜𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑤 𝑑𝑎𝑡𝑎
Hence, we can convert a given raw data if you know the mean and standard
deviation of the set where the raw data belongs.
How will you utilize the formula in converting a given raw score into its
corresponding z score?
Interpreting Z-scores
After determining the corresponding z-score of a given raw score, we need to
interpret it to be able to identify its location.
Here is how to interpret z-scores.
1. A z-score less than 0 represents a data below the mean.
2. A z-score greater than 0 represents a data above the mean.
3. A z-score equal to 0 represents a data equal to the mean. Thus, it is
found at the middle of the distribution.
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Exploring Mathematics in the Modern World
Example:
a. A z-score equal to 1 represents a data that is 1 standard deviation
above the mean; a z-score equal to 2, 2 standard deviations above the
mean; etc.
b. A z-score equal to -1 represents a data that is 1 standard deviation
below the mean; a z-score equal to -2, 2 standard deviations below the
mean; etc.
II. Quantiles
Do you know anything about quantiles? Aside from z scores, we can use
quantiles as measure of location. It is an extension of median concept where
items in the distributions are divided into equal parts.
Types of Quantiles
There are three types of quantiles namely: quartiles, deciles, and percentiles.
A. Quartiles divide the distribution into four equal parts. The values that
divide the parts are called first, second, and third quartiles. These are
denoted by Q1, Q2, and Q3 respectively. Below shows a representation
of a set of observations divided into quartiles.
Q1
Q2
Q3
B. Deciles divide the distribution into 10 equal parts. The values divide
the parts are called first, second, third, fourth, fifth, sixth, seventh,
eight, and ninth deciles. These are denoted by D1, D2, D3, D4, D5, D6,
D7, D8, and D9 respectively.
D1
D2
D3
D4
D5
D6
D7
D8
D9
C. Percentiles divide the set of observations into 100 divisions. These are
the points or values separating the scores into 100 parts. A percentile
indicates the value below which a given percentage of observations in
a group of observations fall.
P10
P20
P30
P40
P50
89
P60
P70
P80
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Exploring Mathematics in the Modern World
Note: Quantiles are used in reporting scores from norm-referenced tests. For
example, if a score is at the 60th percentile, where 60 is the percentile rank, it
is equal to the value below which 60% of the observations may be found.
Procedure in Determining Quantile Measures
To determine the value of the quantile of interest, the following guidelines can
help you.
1. Arrange the given observations from lowest to highest.
2. Determine the ordinal rank (n) or location by applying the formulas
below:
For Quartiles
𝑛=
𝑄
(𝑁 + 1)
4
Where:
n = the ordinal rank
Q = the nth quartile
N = the number of
observations
For Deciles
𝑛=
𝐷
(𝑁 + 1)
10
Where:
n = the ordinal rank
D = the nth decile
N = the number of
Observations
For Percentiles
𝑛=
𝑃
(𝑁 + 1)
100
Where:
n = the ordinal rank
P = the nth percentile
N = the number of
Observations
3. Locate the score corresponding to the obtained ordinal rank (n) or
location in the distribution.
4. If the obtained location is not a whole number, interpolate.
5. Interpolate by subtracting the values of the upper and lower scores.
6. Multiply the difference by the decimal part of the obtained location.
7. Add the product to the lower score.
III. Box Whisker’s Plot
The Box Whisker’s Plot is a type of graph used to display patterns of
quantitative data. It is a graphical method of displaying variation in a set of
data.
In most cases, a histogram provides a sufficient display; however, a Box
Whisker’s Plot can provide additional detail while allowing multiple sets of
data to be displayed in the same graph. Some types are called Box Whisker’s
Plots with outliers.
It makes use of the median, first quartile, and third quartile. Since Q1 is the
value of the data wherein 25% of the scores are below it and Q3 is above 75%
of the scores when they are arranged in ascending order.
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Below is an example of a Box Whisker’s Plot.
How comparable is the Box Whisker’s Plot to the other measures of relative
location?
Note: The percentage of data between Q1 and Q3 is about 50%. Thus, only
about 25% of the data are found on both ends of the distribution.
Skewness of the Distribution
We can determine the skewness of the distribution depending on the location
of the box on the line.
If the box is situated on the upper portion of the line, then the distribution is
skewed to the left and if it is situated on the lower portion, them the
distribution is skewed to the right. The line inside the box represents the
location of the median of the distribution.
The Box Whisker’s Plots below show distributions with different skewness.
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From the previous Box Whisker’s Plot, we can say that the distribution of
Group A is skewed to the right, the distribution of Group B is symmetric with
the mean, and the distribution of Group C is skewed to the left.
How can we use the Box Whisker’s Plot in determining the location of
observation in the distribution?
Procedure
A Box Whisker’s Plot is developed from five statistics.
1. Minimum value – the smallest value in the data set
2. First quartile – the value below which the lower 25% of the data are
contained
3. Median value – the middle number in a range of numbers
4. Third quartile – the value above which the upper 25% of the data are
contained
5. Maximum value – the largest value in the data set
For example, given the following 16 data points, the five required statistics are
displayed.
Number
Raw Data
Statistics
1
50
Minimum (50)
2
51
3
52
4
54
1st Quartile (54.5)
5
55
6
55
7
56
8
58
Median (58)
9
58
10
59
11
60
12
62
3rd Quartile (62.5)
13
63
14
63
15
64
16
65
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Maximum (65)
Exploring Mathematics in the Modern World
Note: Note that for a data set with an even number of values, the median is
calculated as the average of the two middle values.
From the observations given in the previous page, the values of the five
Statistics are:
1. Minimum value = 50
2. First quartile
= 54.5
3. Median value
= 58
4. Third quartile
= 62.5
5. Maximum value = 65
Here are their representation in Box Whisker’s Plot format.
A boxplot splits the data set into quartiles. The body of the boxplot consists of
a "box" (hence, the name), which goes from the first quartile (Q1) to the third
quartile (Q3).
Within the box, a horizontal line is drawn at the Q2, the median of the data
set. Two vertical lines, called whiskers, extend from the front and back of the
box. The front whisker goes from Q1 to the smallest non-outlier in the data set,
and the back whisker goes from Q3 to the largest non-outlier.
If the data set includes one or more outliers, how will they be plotted on the
Box Whisker’s Plot?
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Exploring Mathematics in the Modern World
Application
Activity 1
Directions: Based on what we have learned about z-score, convert the
following raw scores to their corresponding z-scores. Use the formula
presented in the discussion and identify its location.
Raw
Score
Mean
Standard
Deviation
1. 24
20
2
2. 16
16
1
3. 18
24
8
Z-score
Location
Solution:
After accomplishing the previous activity, compare your answers to the
following solutions.
1. x = 24
𝑥̅ = 20
𝑠 =2
𝑥 − 𝑥̅
𝑧=
𝑠
24 − 20
𝑧=
2
4
𝑧=
2
𝑧= 2
Interpretation: The corresponding z-score of the raw score is 2. It represents
that the data can be found 2 standard deviations above the mean.
2. x = 16
𝑥̅ = 16
𝑠 =1
𝑥 − 𝑥̅
𝑧=
𝑠
16 − 16
𝑧=
1
0
𝑧=
1
𝑧= 0
Interpretation: The corresponding z-score of the raw score is 0. It represents
that the data is equal to the value of the mean.
3. x = 18
𝑥̅ = 24
𝑥 − 𝑥̅
𝑧=
𝑠
18 − 24
𝑧=
8
𝑠 =8
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Exploring Mathematics in the Modern World
−6
8
𝑧 = −0.75
Interpretation: The corresponding z-score of the raw score is -0.75. It
represents that the data can be found 0.75 standard deviation below the mean.
𝑧=
Good job! Did you get the same answers? If not, what part do you need to
improve?
Can you determine the raw score given the standard deviation and the z-score?
In what way?
Activity 2
Let us have another activity. This time, you can seek the help of your friends
to answer the following problems.
Directions: Given the mean and standard deviation of the distribution, convert
the following raw scores to their corresponding z scores and interpret their
location relative to the distribution. Good luck!
1. mean = 120
standard deviation = 10 raw score = 100
2. mean = 50
standard deviation = 5
raw score = 55
3. mean = 35
standard deviation = 4
raw score = 40
You just have learned to measure relative position of data through z-score.
Congratulations!
Activity 3
Let us try to follow the procedure in determining a quantile value. Do your
best in answering the following:
1. Find the 30th percentile of the set {12, 15, 17, 20, 25, 27, 29, 30, 30,
34, 36, 36, 37, 38, 39, 40, 41, 42, 43}
2. Determine the 2nd quartile from the set {30, 34, 36, 36, 37, 38, 39, 40,
41, 42, 12, 15, 17, 20, 25, 27, 29, 30}
3. Determine the 2nd decile from the set {20, 25, 27, 29, 30, 30, 34, 36,
36, 12, 15, 17, 37, 38, 39, 40, 41, 42}
Have you determined the values of the quantiles? Good job!
Solutions:
You may compare yours to the following solutions.
1. Note: The observations are already arranged from lowest to highest.
The given are:
P = 30
N = 19
n = unknown
Value of the 30th percentile = unknown
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Exploring Mathematics in the Modern World
𝑛=
𝑃
(𝑁 + 1)
100
𝑛=
30
(19 + 1)
100
𝑛= 6
The 30th percentile is the 6th observation from the set of data which is 27.
2. We arrange the observations from lowest to highest as: {12, 15, 17, 20,
25, 27, 29, 30, 30, 34, 36, 36, 37, 38, 39, 40, 41, 42}.
The given are:
Q=2
N = 18
n = unknown
Value of the 2nd quartile = unknown
𝑄
𝑛 = (𝑁 + 1)
4
2
𝑛 = (18 + 1)
4
𝑛 = 9.5
The whole number part of the ordinal part (n) is 9 and the 9th observation is
30. The value next to 30 is 34 and their difference is 4. The product of their
difference and the decimal part of the ordinal rank (n) which is 0.5 is 2. Thus,
the value of the 2nd quartile is 32.
3. We arrange the observations from lowest to highest as: {12, 15, 17, 20,
25, 27, 29, 30, 30, 34, 36, 36, 37, 38, 39, 40, 41, 42}
The given are:
D=2
N = 18
n = unknown
Value of the 2nd decile = unknown
𝐷
𝑛=
(𝑁 + 1)
10
2
𝑛=
(18 + 1)
10
𝑛 = 3.8
The whole number part of the ordinal part (n) is 3 and the 3rd observation is
17. The value next to 17 is 20 and their difference is 3. The product of their
difference and the decimal part of the ordinal rank (n) which is 0.8 is 2.4.
Thus, the value of the 2nd decile is 19.4.
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Exploring Mathematics in the Modern World
Activity 4
Let us have another activity to test your understanding and mastery of the
topic.
This time, call for a friend to help you answer the items. Good luck!
1. The following are the scores of 19 students of the College of
Agriculture: 40, 32, 32, 30, 45, 44, 43, 35, 39, 23, 25, 36, 37, 28, 33,
27, 30, 29, and 20. Calculate Q1, D3, and P40.
2. Determine the 3rd quartile, 2nd decile, and 10th percentile of the
number of siblings of the 11 students of the College of Teacher
Education.
2
1
3
7
2
6
4
5
3
4
2
You just have learned to determine the quantiles of observations to identify their
location relative to the set of data. Congratulations!
Activity 5
Identifying the Location of Observation
Given the Box Whisker’s Plot below:
A. Identify the location of the following scores relative to the five
Statistics presented.
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B. Determine the skewness of the distribution.
1. 48
2. 40
3. 35
4. 30
5. 20
Do your best in determining the location of the scores before proceeding so
you can compare your answers to the solution. Good luck!
If you are done answering the activity, you can compare now your answers to
the solutions.
Solution:
A. Identifying the Location of Data
1. 48 is found above the third quartile and below the maximum score
2. 40 is the median of the distribution
3. 35 is located above the first quartile and below the median
4. 30 is positioned just below the first quartile
5. 20 is the lowest score in the distribution
B. Skewness of the Distribution
The box of the Box Whisker’s Plot is situated on the upper portion of the
line. Thus, we can say that the distribution is skewed to the left.
Activity 6
With the concepts that you have learned, interpret the following Box
Whisker’s Plot. This time you can ask the help of your friends in this activity.
Good luck!
1. Determine the skewness of the three groups.
2. Identify the location of the score 20 in the three distributions.
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Exploring Mathematics in the Modern World
Assessment
Now, let us test your understanding on the measures of relative position.
Goodluck!
Test I.
Directions: Supply the information being required by each item.
1. Given the mean of the distribution as 30 with a standard deviation of 5,
determine the
corresponding z-score of the following raw scores.
a. 15
b. 30
c. 35
2. Interpret the location of the following z-scores.
a. 0.5
b. 2.2
c. 1.8
Test II.
Directions: Determine whether the following are correct or not. Write True if
the statement is true and False if it is false.
_____1. The 3rd quartile corresponds to the 30th percentile.
_____2. The 25th percentile is the observation below which 75% of the
observations may be found.
_____3. The Box Whisker’s Plot uses five Statistics namely: Q1, maximum,
mean, Q3, and minimum.
_____4. If the box of the Box Whisker’s Plot is situated on the upper part of
the line, then the distribution is skewed to the right.
_____5. Outliers can lie inside the box of the Box Whisker’s Plot.
Test III.
Directions: Below is the list of daily allowances (in peso) of 29 first year
students in Cagayan State University. Determine the value of:
1. 10th percentile
2. 3rd decile
3. 1st quartile
50
55
55
60
60
65
65
70
70
75
75
80
80
80
90
95
95
100
100
100
110
110
120
130
140
99
150
170
180
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Exploring Mathematics in the Modern World
The rubric below will be used to evaluate your answers.
Criteria
Exceeds
Expectation
(3 points)
Understanding The given and the
unknown were
identified and
properly labelled.
Meets
Expectation
(2 points)
Approaches
Expectation
(1 point)
The given were
identified.
Some of the given
were not identified.
Solution
The problem was
solved efficiently
and systematically
with the use of
appropriate
solution.
The problem was
solved with the use
of appropriate
solution.
The problem was
solved inefficiently
with the use of
inappropriate
solution.
Answer
The problem was
answered
accurately.
The requirements
of the problem
were provided.
The problem was
not answered.
Test IV.
A. Directions: Given the Box Whisker’s Plot in the next page:
1. Identify the location of these observations; and
a.) 30
b.) 35
c.) 40
2. Determine the skewness of the distribution.
The rubric below will be used to evaluate your answers.
Criteria
Identifying
the Location
of
Observation
Exceeds
Expectation
(3 points)
Provided a detailed
and accurate
information on the
location of the
observation relative
to the five
Statistics.
Meets Expectation
(2 points)
Provided
information on the
location of the
observation.
Determining Provided a detailed Determined the
skewness of the
the Skewness and accurate
information on the distribution.
skewness of the
distribution.
100
Approaches
Expectation
(1 point)
The information
was insufficient to
identify the
location of the
observation.
The information
provided is
insufficient to
determine the
skewness of the
distribution.
Exploring Mathematics in the Modern World
B. Directions: Provide the information required by each item. Show all
pertinent solutions. The rubric below will be used to evaluate your answers.
Criteria
Exceeds
Expectation
(3 points)
Understanding The given and the
unknown were
identified and
properly labelled.
Meets
Expectation
(2 points)
Approaches
Expectation
(1 point)
The given were
identified.
Some of the given
were not identified.
The problem was
solved inefficiently
with the use of
inappropriate
solution.
Solution
The problem was
solved efficiently
and systematically
with the use of
appropriate
solution.
The problem was
solved with the
use of appropriate
solution.
Answer
The problem was
answered
accurately.
The requirements The problem was
of the problem
not answered.
were provided.
Problem: Albert’s teacher revealed that the mean score of their previous exam
is 60 with a standard deviation of 10. Instead of their raw scores, she gave the
z-scores instead.
Albert’s got a z-score of –0.5. If the passing score is 50, did he pass the exam?
Why or why not?
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Topic 5: Probabilities and Normal Distribution
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
identify the properties of the normal distribution curve;
determine the areas under the normal distribution curve given a portion of
the z table;
determine the probability of cases in the normal distribution curve; and
solve word problem involving the concepts of normal distribution.
Presentation of Content
I. Normal Distribution
A random variable x whose distribution has the shape of normal curve is
called a normal random variable. Its equation is as follow:
𝑓(𝑥) =
1
2 /2𝑒 2
𝑒 −(𝑥−𝜇)
𝜎√2𝜋
Note: The random variable x is said to be normally distributed with mean and
standard deviation if its probability distribution is the above equation.
The normal curve is represented by a bell-shaped curve and its probability
distribution is termed as the normal distribution. The values in the curve are
clustered around the average value and fewer values are found at increasing
distances from the average.
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Properties of a Normal Distribution
The following are the properties of a normal distribution. Do not forget about
them.
1. The mean, median, and mode are equal and are located at the center of
the distribution.
2. The distribution is symmetrical about the mean.
3. The total area under the normal curve is 1 or 100%.
4. The tails extend infinitely but will never touch the horizontal line.
5. The location of the distribution is determined by the mean and the
standard deviation determines the dispersion of the distribution.
6. For a normal curve, the area within:
a. one standard deviation from the mean is about 68%;
b. two standard deviation from the mean is about 95%; and
c. three standard deviations from the mean is about 99%.
-2𝝈
1𝝈
𝝁
-1𝝈
2𝝈
Note: The shape of the normal distribution depends only on two parameters:
the population mean and the population standard deviation.
What variables are normally distributed?
You can observe how the mean and standard deviation of different
distributions affect the size and location of the curve.
The first figure in the next page shows normal distributions with the same
mean but different standard deviations while the second figure presents
distributions with different means but the same standard deviation.
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Figure I: Two distributions with equal mean but different standard deviation
Figure 2: Two distributions with different means but the same standard
deviations
How can the mean and standard deviation of the distribution affect its shape?
Standard Scores
It is the position of raw score values in terms of the standard deviation relative
to the mean of the distribution.
Given the raw scores, we can convert them to their corresponding standard
scores or z scores. This means that the empirical distribution will be
standardized to the theoretical normal curve.
We can use the formula:
𝑧=
𝑥 − 𝑥̅
𝑠
Where:
𝑧 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑠𝑐𝑜𝑟𝑒
𝑥̅ = 𝑚𝑒𝑎𝑛
𝑠 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑥 = 𝑟𝑎𝑤 𝑠𝑐𝑜𝑟𝑒
Note: This is the same with our previous topic on z scores.
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A standard normal curve is a normal distribution with a mean of 0 and a
standard deviation of 1 and all its raw scores are expressed in terms of
standard scores below or above the mean.
When the standard score is positive, it means that the raw score is above or
higher than the mean; if negative, it means that the raw score is below or lower
than the mean of the distribution.
Areas under the Normal Distribution Curve
To determine the areas under the normal curve, we shall convert the raw score
into its corresponding standard or z-score. Again, we will be using the formula:
𝑧=
Where:
𝑧=
𝑥̅ =
𝑠=
𝑥=
𝑥 − 𝑥̅
𝑠
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑠𝑐𝑜𝑟𝑒
𝑚𝑒𝑎𝑛
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑟𝑎𝑤 𝑠𝑐𝑜𝑟𝑒
After determining the corresponding value of the raw score, we need a z table
to determine the area between the given two values. Here is a portion of the
table.
z (±)
0.0
0.1
0.2
0.3
0.4
0.5
0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.01
0.0040
0.0438
0.0832
0.1217
0.1519
0.1950
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
How will we use the z table to determine the area under the normal curve?
Application
Activity 1
Directions: Answer the following problem.
Problem: Suppose that in a given test, the mean is 45 and the standard
deviation is 5. If Mario obtained a score of 50, what is his standard score?
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Exploring Mathematics in the Modern World
Solution:
Given:
𝑥̅ = 45
x = 50
s=5
z = unknown
𝑥 − 𝑥̅
𝑧=
𝑠
50 − 45
𝑧=
5
5
𝑧=
5
𝑧= 1
Interpretation: A standard score of 1 means that the score 50 is one standard
deviation above 45. This further indicates that the given observation is greater
than the mean of the distribution.
Did you get the answer correctly? Good job!
Activity 2
Directions: Given the mean and standard deviation of the distribution as 20
and 4 respectively, convert the following raw scores into their corresponding
standard scores and interpret their location relative to the mean.
1.
2.
3.
4.
5.
18
20
16
24
28
Activity 3
Using the z table, let us determine the areas of the following:
1. Between 0.1 and 0
2. Between 0.03 and 0
3. Between 0.3 and 0
4. Between 0.45 and 0
5. Between 0.32 and 0
Note: Remember that the standard score for the mean is 0.
Answers to Activity 3
Have you tried to answer the activity? Here are the answers.
1. Between 0.1 and 0
= 0.0398
2. Between 0.03 and 0
= 0.0120
3. Between 0.3 and 0
= 0.1179
4. Between 0.45 and 0
= 0.1736
5. Between 0.32 and 0
= 0.1255
Did you get all the items? You’re doing great!
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Guidelines in Determining the Areas under the Normal Distribution Curve
Here are the guidelines to remember when determining areas under the normal
distribution curve. Read them carefully.
1. To determine the area to the right of a positive z score, subtract the area
between the z-score and the mean from 0.5 (The area of the half of the
curve is 0.5).
2. To determine the area to the left of a positive z score, add the area
between the z-score and the mean to 0.5.
3. To determine the area to the right of a negative z score, add the area
between the z-score and the mean to 0.05.
4. To determine the area to the left of a negative z score, subtract the area
between the z-score and the mean from 0.05.
5. To determine the area between two positive z scores, subtract the areas
formed by the two z scores and the mean.
6. To determine the area between two negative z scores, subtract the areas
formed by the two z scores and the mean.
7. To determine the area between a positive and a negative z score, add
the areas formed by the two z scores and the mean.
Can you follow the guidelines? Are there items that you are not sure with?
Activity 4
To understand the guidelines, let us determine the areas of the following. Try to
answer them before comparing your answers to answers provided in the next
page.
1. To the right of 0.1
2. To the left of –0.3
3. Between –0.2 and –0.4
How many of the guidelines did you apply? Congratulations! You can compare
now your answers to see how much you have understood.
Solutions to Activity 4
The following are the solutions to the previous activity.
1. Area to the right of 0.1 = unknown
Area between 0.1 and 0 = 0.0398
Subtract it from 0.5
= 0.4602
2. Area to the left of –0.3 = unknown
Area between 0.3 and 0 = 0.1179
Subtract it from 0.5
= 0.3821
3. Area between –0.2 and –0.4
Area between 0.4 and 0
Area between 0.2 and 0
Subtract 0.1554 and 0.0793
= unknown
= 0.1554
= 0.0793
= 0.0761
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Exploring Mathematics in the Modern World
Note: The areas between 0.2 and 0.4 and –0.2 and –0.4 are equal since they
are symmetrical about the mean.
II. Probability Distribution
Do you know that we utilize the concepts of the areas under the normal
distribution curve in determining the proportion of cases in the curve?
Note: The probability of the occurrence of a case is its area under the curve!
Example:
If x is a normal random variable with a mean of 90 and standard deviation of 4,
find the probability that x is:
1. Greater than 92
2. Less than 89
3. Between 89 and 92
Solution:
Given: mean = 90
standard deviation = 4
1. Given: mean = 90 standard deviation = 4
P(x > 92) = unknown
Convert the raw score 92 to z-score:
𝟗𝟐 − 𝟗𝟎
𝟐
𝒛=
= = 𝟎. 𝟓
𝟒
𝟒
Determine the area to the right of 0.5: P(x > 0.5) = 0.3085
Subtract 0.3085 from 0.5.
The probability that x is greater than 92 is 19.15%.
2. Given: mean = 90 standard deviation = 4
P(x < 89) = unknown
Convert the raw score 89 to z-score:
𝟖𝟗 − 𝟗𝟎
𝟏
𝒛=
= − = −𝟎. 𝟐𝟓
𝟒
𝟒
Determine the area to the left of —0.25: P(x > —0.25) = 0.4013
Subtract 0.4013 from 0.5.
The probability that x is less than 89 is 9.87%.
3. Given:
mean = 90 standard deviation = 4
P(89 < x < 92) = unknown
Convert the raw score 89 to z-score:
𝟖𝟗 − 𝟗𝟎
𝟏
𝒛=
= − = −𝟎. 𝟐𝟓
𝟒
𝟒
Convert the raw score 92 to z-score:
𝟗𝟐 − 𝟗𝟎
𝟐
𝒛=
= = 𝟎. 𝟓
𝟒
𝟒
Determine the area between –0.25 and 0.5: P (–0.25 < x < 0.5) = 0.7098
The probability that x is between 89 and 92 is 70.98%.
Congratulations! You finished learning the topics. Did you enjoy it?
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What are other applications of the areas under the normal distribution curve in
the real life setting?
Assessment
Test I.
Directions: Supply the information being required by each item.
_____1. It has the same value and location as the median and mode of the
normal distribution.
_____2. It is the total area under the normal distribution curve.
_____3. It is the shape of a normal distribution curve.
_____4. The shape of the normal curve depends on these parameters.
_____5. It is the equivalent standard score of the mean of the distribution.
Test II.
Directions: Given a portion of the z table, determine the areas of the
following z scores.
z (±)
0.0
0.1
0.2
0.3
0.4
0.5
0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.01
0.0040
0.0438
0.0832
0.1217
0.1519
0.1950
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
1. Between 0.11 and 0
2. To the right of 0.12
3. Between 0.12 and 0.34
Test III.
Directions: Provide the information required by each item. Show all pertinent
solutions. The rubric below will be used to evaluate your answers.
Criteria
Exceeds
Expectation
(3 points)
Understanding The given and the
unknown were
identified and
properly labelled.
Solution
The problem was
solved efficiently
and systematically
with the use of
appropriate
Meets
Expectation
(2 points)
Approaches
Expectation
(1 point)
The given were
identified.
Some of the given
were not identified.
The problem was
solved with the
use of appropriate
solution.
The problem was
solved inefficiently
with the use of
inappropriate
solution.
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Exploring Mathematics in the Modern World
solution.
Answer
The problem was
answered
accurately.
The requirements The problem was
of the problem
not answered.
were provided.
Problem 1: Juan and Jimmy took a test in Geometry. Their teacher revealed
their scores but in terms of standard scores. Juan has z score of 1 while Jimmy
has 1.5. If the mean score of the class is 60 with a standard deviation of 6, who
got a higher raw score and how much higher?
Problem 2: In the Prelim Examination in the College of Teacher Education,
the mean score of the 30 students of Mr. Aguinaldo is 20 with a
standard deviation of 4. Assuming normality:
1. What is the percentage of cases fall between the mean and 22?
2. What is the probability that a score lie above 21?
3. What is the probability that a score lie between 18 and 21?
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Topic 6: Linear Regression and Correlations
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
recall concepts on linear correlation and least square line;
describe the set of data using the computed correlation coefficient;
identify what relationship that exists between two variables; and
estimate a value of the dependent variable based on the derived regression
equation.
Presentation of Content
I. Linear Correlation
The coefficient measures the strength and direction of linear coefficient
between two variables (Larson and Farber, 2000; Pagala, 2011). We will use
the formula below to determine the value of linear coefficient.
𝑛 ∑ 𝑥𝑦 − ∑ 𝑥 ∑ 𝑦
𝑟=
√[𝑛 ∑ 𝑥2 − (∑ 𝑥)2 ][𝑛 ∑ 𝑦2 − (∑ 𝑦)2 ]
Where:
𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟𝑒𝑑 𝑝𝑎𝑖𝑟𝑠
𝑥 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑦 = 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
How will you use the formula to determine the relationship of two variables?
From the formula, we will follow the following procedure:
1. Multiply x and y values and compute for the sum of the products.
∑ 𝒙𝒚
2. Multiply the sum of the products by the number of ordered pairs.
𝒏 ∑ 𝒙𝒚
3. Determine the sum of x values. ∑ 𝒙
4. Determine the sum of y values. ∑ 𝒚
5. Multiply the totalled values of x and totalled values of y. ∑ 𝒙 ∑ 𝒚
6. Square the values of x and take the sum. ∑ 𝒙𝟐
7. Multiply the sum of the squares of the values of x by the number of
ordered pairs. 𝒏 ∑ 𝒙𝟐
8. Square the values of y and take the sum. ∑ 𝒚𝟐
9. Multiply the sum of the squares of the values of y by the number of
ordered pairs. 𝒏 ∑ 𝒚𝟐
10. Square the total value of x. (∑ 𝒙)𝟐
11. Square the total value of y. (∑ 𝒚)𝟐
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12. Substitute the values in the formula to determine the value of the
coefficient.
Note: We can only employ correlation when data are in interval or ratio scale.
II. Simple Regression Analysis
We start with the concept of simple regression analysis.
When only one independent variable is used, the analysis is referred to as simple
regression analysis.
The formal statements of the simple linear regression model is:
𝑦 = α + βx + ε
Where:
𝑦 = 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑎 = 𝑡ℎ𝑒 𝑦— 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
𝛽 = 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑙𝑖𝑛𝑒
𝑥 = 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝜀 = 𝑡ℎ𝑒 𝑟𝑎𝑛𝑑𝑜𝑚 𝑒𝑟𝑟𝑜𝑟 𝑡𝑒𝑟𝑚
How can we apply the formula to predict values of the dependent variable?
Method of Least Square
Since α and β are generally not known in a regression problem, they must be
estimated from a sample data taken on the dependent variable y for a number of
values of the independent variable x.
Note: The standard approach to estimating α and β is using the least squares
(minimizing the sum of the squared errors for your data points.)
Sample estimates of α and β are denoted by α and β, respectively, and the
resulting regression line is called sample least squares regression equation.
y = α + βx + ε
The sum of the squared deviation between the line and the scatter of points
should be minimized. Statisticians have found that the formulas for α and β are
shown below:
𝛽=
∑(𝑥 − 𝑥̅ )(𝑥 − 𝑦̅)
∑(𝑥 − 𝑥̅ )
𝑎 = 𝑦̅ − 𝛽𝑥̅
Note: Here, 𝑥̅ and 𝑦̅ denote the sample means of x and y.
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Alternative Formulas
The alternative formulas for α and β are as follow.
𝑛 ∑ 𝑥𝑦 − (∑ 𝑥)(∑ 𝑦)
𝑛 ∑ 𝑥 2 − (∑ 𝑥)2
𝛽=
𝑎=
∑𝑦 − 𝛽∑𝑥
𝑛
Application
Activity 1
Now, let us apply what we have learned. Here is an activity where we can
utilize the formula given. Remember to follow the guidelines in determining
the linear coefficient. Try to solve the problem independently before
comparing your answers to the answers provided.
Problem: The list of height and weight of 10 basketball players is given
below. Determine the value of the linear coefficient.
The list of height and weight of 10 basketball players.
X
(Height in
Inches)
67
70
71
70
66
69
72
78
64
65
Y
(Weight in
Kilograms)
71
70
69
68
66
65
71
70
64
65
Have you tried answering the problem? Great! Now, we can compare your
answers.
Solution:
We determine the values of the variables.
Height (X)
Weight (Y)
XY
X2
Y2
67
71
4,757
4,489
5,041
70
70
4,900
4,900
4,900
71
69
4,899
5,041
4,761
70
68
4,760
4,900
4,624
66
66
4,356
4,356
4,356
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69
65
4,485
4,761
4,225
72
71
5,112
5,184
5,041
78
70
5,460
6,084
4,900
64
64
4,096
4,096
4,096
65
65
4,225
4,225
4,225
The values of the variables are:
∑ 𝒙𝒚 = 47,050
∑ 𝒚 = 697
𝒏 ∑ 𝒙𝟐 = 480,360
(∑ 𝒙)𝟐 = 478,864
𝒏 ∑ 𝒙𝒚 = 470,500
∑ 𝒙 ∑ 𝒚 = 469,868
∑ 𝒚𝟐 = 46,169
(∑ 𝒚)𝟐 = 46,104
∑ 𝒙 = 692
∑ 𝒙𝟐 = 48,036
𝒏 ∑ 𝒚𝟐 = 461,690
We are now ready to substitute them in the formula.
𝑛 ∑ 𝑥𝑦 − ∑ 𝑥 ∑ 𝑦
𝑟=
√[𝑛 ∑ 𝑥 2 − (∑ 𝑥)2 ][𝑛 ∑ 𝑦2 − (∑ 𝑦)2 ]
𝑟=
𝑟=
𝑟=
𝑟=
𝑟=
(470,500) − (692)(697)
√[480,360 − (692)2 ][461,690 − (697)2 ]
470,500 − 469,868
√[480,360 − 478,864][461,690 − 461,041]
632
√(1,496)(649)
632
√970,909
632
985.35
𝑟 = 0.64
The value of the linear coefficient is 0.64.
What could be the meaning of the value we computed?
Interpreting the Correlation Coefficient
After determining the correlation coefficient, we need to interpret the value.
The quantitative interpretation of the degree of linear relationship existing is
shown below.
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Values
Interpretation
±1.00
Perfect positive/ negative correlation
±0.91 to ±0.99
Very high positive/ negative correlation
±0.71 to ±0.90
High positive/ negative correlation
±0.51 to ±0.70
Moderately positive/ negative correlation
±0.31 to ±0.50
Low positive/ negative correlation
±0.01 to ±0.30
Slight positive/ negative correlation
0
No correlation
From the previous activity, the correlation coefficient is 0.64 which can be
interpreted as a moderately positive correlation. There is a substantial degree
of correlation between the height and weight of the ten basketball players.
Awesome! Keep up the good work!
Activity 2
Let us put your understanding into practice. Below are the test results of 10
students in their Mathematics and English examinations. With a partner,
determine the linear correlation coefficient and interpret its value.
X
(Score in
Mathematics)
34
23
45
44
37
46
23
41
40
35
Y
(Score in
English)
35
21
43
42
32
45
23
47
43
37
Activity 3
Using the given formulas, try to determine the values of the variables to come
up with the least squares regression equation.
Problem:
The Cagayan State University officials wished to determine if the CSU—
College Admission scores is a good indicator of the General Weighted
Average (GWA) of the 16 scholars selected at random from the first year
class. Their GPA and CSU-CAT scores are shown in the next page.
What will the estimated GWA of a student with the CAT score of 83?
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Student
CAT Raw Score (x)
GWA (y)
1
80
85
2
82
87
3
90
90
4
87
88
5
80
84
6
85
89
7
95
97
8
97
98
9
98
98
10
90
92
11
82
85
12
81
83
13
85
87
14
86
88
15
88
88
16
92
95
How can one predict and estimate GWA from CAT scores?
Solution
Now, we need to obtain the equation for the line that best fits the sample
data.
Student
CAT Raw
Score (x)
GWA (y)
xy
x2
y2
1
80
85
6,800
6,400
7,225
2
82
87
7,134
6,724
7,569
3
90
90
8,100
8,100
8,100
4
87
88
7,656
7,569
7,744
5
80
84
6,720
6,400
7,056
6
85
89
7,565
7,225
7,921
7
95
97
9,215
9,025
9,409
8
97
98
9,506
9,409
9,604
9
98
98
9,604
9,604
9,604
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10
90
92
8,280
8,100
8,464
11
82
85
6,970
6,724
7,225
12
81
83
6,723
6,561
6,889
13
85
87
7,395
7,225
7,569
14
86
88
7,568
7,396
7,744
15
88
88
7,744
7,744
7,744
16
92
95
8,740
8,464
9,025
Total
1,398
1,434
125,720
122,670
128,892
Solution:
Using the formulas:
1,434
𝑦̅ =
= 89.625
16
1,398
𝑥̅ =
= 87.375
16
𝛽=
16(125,720) − (1,398)(1,434)
= 0.8163
16(122,670) − (1,398)2
𝑎 = 89.625 − (0.8163)(87.375) = 18.3008
The fitted equation describing the relationship between GWA and CAT scores
is: GWA = 18.3008 + 0.8163x
To predict the future GWA of a student with a CAT score of 83:
GWA = 18.3008 + 0.8163(83) = 86
Congratulations! You just learned to predict the future General Weighted
Average of the student.
Activity 4
With a partner, determine the equation that would fit the following set of
observations.
Age
(x)
10
12
11
26
28
21
22
18
16
15
Score
(y)
32
30
34
39
38
32
29
28
25
20
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Exploring Mathematics in the Modern World
Assessment
We will now test your understanding on linear regression and correlations.
Good luck!
Test I.
Directions: Identify the term being described by each item.
_____1. It is employed to determine the existence of relationship between
variables in interval or ratio scale.
_____2. It is the type of relationship that exists when the value of the
correlation coefficient is zero.
_____3. It is the value of the coefficient with perfect negative correlation.
_____4. It is the symbol used in the formula for correlation coefficient that
represents the independent variable.
_____5. It is the symbol used in the formula for correlation coefficient that
represents the dependent variable.
Test II.
Directions: Write TRUE if the statement is correct and FALSE if the statement
is wrong on the space provided before each question.
_____1. Beta is the y-intercept in regression analysis.
_____2. In the regression analysis, it is the dependent variable that we want to
predict.
_____3. The slope of the regression line is denoted by alpha.
_____4. The ultimate goal of regression analysis is to predict or estimate the
value of one variable corresponding to a given value of another
variable.
_____5. The sample regression equation may be used to predict or estimate
outside the range of values of the independent variable represented in
the sample.
Test III.
Directions: Provide the information required by the problem in the next page.
The rubric below will be used to evaluate your answers.
Criteria
Exceeds
Expectation
(3 points)
Understanding The given and the
unknown were
identified and
properly labelled.
Solution
The problem was
solved efficiently
and systematically
with the use of
appropriate
Meets
Expectation
(2 points)
Approaches
Expectation
(1 point)
The given were
identified.
Some of the given
were not identified.
The problem was
solved with the
use of appropriate
solution.
The problem was
solved inefficiently
with the use of
inappropriate
solution.
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Exploring Mathematics in the Modern World
solution.
The problem was
answered
accurately.
Answer
The requirements The problem was
of the problem
not answered.
were provided.
Problem 1: The raw scores obtained by 10 students in a quiz are given below.
What is the relationship that exist in their performance in Biology and
Chemistry?
X (Biology)
12
11
19
20
15
17
18
12
14
15
Y
(Chemistry)
16
17
13
19
15
16
19
10
15
13
Problem 2: The Dean of the College of Education wants to determine if GPA
could be used to estimate the performance of the students in the Board Licensure
Examination for Professional Teachers. The scores are shown below.
GPA
(x)
89
92
91
86
88
91
92
88
86
85
BLEPT
Score
(y)
80
82
80
85
82
87
88
82
83
81
Test IV.
Directions: Come up with a research proposal which focuses on solving
environmental issues with the use of regression analysis. Present this before
the class. The rubric below will be used to evaluate your outputs.
Criteria
Content
Exceeds
Expectation
(3 points)
The output is
profoundly written
and accurately
prepared.
Meets
Expectation
(2 points)
Approaches
Expectation
(1 point)
The content is free The output has
from errors.
grammatical lapses
and misinformation.
Significance There is a need and
urgency for the
issue to be studied.
It is not only a
national issue but a
global as well.
The environmental
issue presented is
very crucial to
study.
The environmental
issue presented is
not so important to
be studied.
Presentation The study was
presented
excellently.
The presentation
was satisfactorily
presented before
the class.
The study was
poorly presented.
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Summary
 Averages such as the mean, median, and mode summarize a given set of
data into a single value.
 The extent to which the median and mean are good representatives of the
values in the original dataset depends upon the variability or dispersion in
the original data.
 Datasets are said to have high dispersion when they contain values
considerably higher and lower than the mean value.
 Dispersion within a dataset can be measured or described in several ways
including the range, standard deviation, and variance.
 The location of data in a given distribution can be determined using the zscore, quantiles, and Box Whisker’s plot.
 The normal curve is represented by a bell-shaped curve and its probability
distribution is termed as the normal distribution.
 The proportion of cases in a distribution can be determined through the areas
under the normal curve.
 Linear correlation tests the direction and strength of relationship of two
quantitative variables.
 Linear regression analysis allows us to predict or estimate the value of a
given variable that corresponds to another variable.
Reflection
A. How much have you learned in this unit? Are there things that you
didn’t understand?
o I cannot understand the topic on _____________________.
o Now, I understand what the topics are all about.
I think that these topics are:
o Easy
o Moderate
o Difficult
B. Directions: Write your thoughts on the things that you have learned
and what you still need to improve by completing the following.
I have learned that …
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
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I still need to improve myself on ...
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
I can understand the topic better if …
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
_______________________________________________________________
References
Asaad, A. (2008). Statistics Made Simple for Researchers. Rex Book Store,
Inc.
ASQ (Box and Whisker’s Plot). Retrieved from http://asq.org/learn-aboutquality/data-collection-analysis-tools/overview/box-whisker-plot.html
Bourne, M. (n.d.). Interactive Mathematics (Normal Probability Distributions).
Retrieved from https://www.intmath.com/
Everitt, B. (1999). Chance Rules: An Informal Guide to Probability, Risk, and
Statistics. Copernicus.
Galfian (2016). 50 Popular Quotes and Sayings. Retrieved from
http://www.golfian.com/50-popular-normality-quotes-and-sayings/16normality-quotes/
Goldberg, S. (1986). Probability: An Introduction. New York: Dover.
Keynes, J. M. (1921). A Treatise on Probability. London: Macmillan.
Mamhot, M., Mamhot, A., & Adanza, J. (2013). Statistics for General
Education. Purelybooks Trading & Publishing Corp.
Math Warehouse. Normal Distribution Curve and Graph. Retrieved from
https://www.mathwarehouse.com/statistics/normal-distribution-curve-andgraph.php
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Exploring Mathematics in the Modern World
Mises, R. (1964). Mathematical Theory of Probability and Statistics. New
York: Academic Press.
Nalangan, L. and Casinillo, M. (2009). Laboratory Manual in Statistics 1
(Elementary Statistics). Rex Book Store, Inc.
Narag, E. (2010). Basic Statistics with Calculator and Computer Application.
Rex Book Store, Inc.
Pagala, R. (2011). Statistics (Revised Edition). Mindshapers Co., Inc.
Pinterest. Retrieved from
https://www.pinterest.ph/pin/359865826459701508/?lp=true
Socratic Statistics. How do I calculate and interpret a Z-score? Retrieved
from: https://socratic.org/questions/how-do-i-calculate-and-interpret-a-z-score
Star Trek (2018). Measures of Position. Retrieved from
https://stattrek.com/descriptive-statistics/measures-ofposition.aspx?Tutorial=AP
Star Trek (Boxplots). Retrieved from
https://stattrek.com/statistics/charts/boxplot.aspx?Tutorial=AP
Star Trek (Teach Yourself Statistics). Retrieved from:
https://stattrek.com/descriptive-statistics/measures-ofposition.aspx?Tutorial=AP
Statistics How to (Probability and Statistics). Retrieved from
https://www.statisticshowto.datasciencecentral.com/probability-andstatistics/descriptive-statistics/box-plot/
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Unit 5: Geometric Designs (8 hours)
Introduction
More than the practical value, the aesthetic appeal of a geometric figure can
stimulate interest and motivation which leads students to subconsciously
embrace mathematical investigations. As a result, they do not engage only
themselves into mathematical explorations but also to understand society’s
history and culture as well as social systems.
Geometric designs abound in nature and environment, either in 2-dimensional
or even higher dimension forms. They may appear naturally, like the colourful
imprints in the wings of butterflies or the near perfect conical shape of the Mt.
Mayon, or occur as man’s handiwork, like the intricate designs in Islamic
textiles or the pyramids of the ancient Egyptians.
This unit covers several geometric concepts particularly on designs,
transformation and some of its applications.
Learning Outcomes
Upon the completion of this unit, you are expected to:
a. apply geometric concepts, especially isometries in describing and creating
designs; and
b. contribute to the enrichment of the Filipino culture and arts using geometry
Activating Prior Learning
Directions: Write True if the statement is true, otherwise write False.
_____1. A geometric shape can be replicated at a finite number of copies
_____2. Every geometric figure can be transformed into another figure.
_____3. A triangle can be transformed into a hexagon
_____4. Transformation can be done by stretching a geometric figure
_____5. Moving a figure from a fixed distance changes the form of the figure.
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Presentation of Content
At the end of this topics, you should be able to demonstrate the following:
1.
2.
3.
4.
Recognize and Analyze Geometric Shapes
Determine the type of Transformations
Identify Patterns and Diagrams
Determine and Construct Designs, Arts & Culture using
Geometric Designs
A. Recognizing and Analyzing Geometric Shapes
A geometric shape is defined as a geometric information that still remains there
even if scale, orientation, location and reflection are displaced from a particular
geometrical object. We can say that if we move the shape, enlarge it, reflect it
or rotate it, then also the shape remains the same, i.e. it does not change into
another.
The geometric shapes (generally seen in everyday life) are of two types

two dimensional
three dimensional.
Generally, the two-dimensional geometric shapes are represented by the lines
joining the set of points called vertices in a bounded form.
These are known as polygons including triangles, quadrilaterals etc.
Some two-dimensional shapes formed by bounded curves, for example - circle
and ellipse.
Most of the three-dimensional geometric shapes are represented by the lines
joining a set of points as well as two dimensional surfaces containing those
lines. For example - cubes, cuboids, pyramids etc. Few three-dimensional
shapes are formed by bounded curved surfaces, such as sphere and ellipsoid.
B. Transformations
Transformation maps an initial image, called preimage, onto a final image,
called an image.
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Isometry
It is a transformation in which the resulting image is congruent to the pre-image.
It is also called as RIGID TRANSFORMATION
Which of the these transformations are isometries?
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Four types of Rigid Transformation
Reflection
Transformations using Coordinate Geometry
TRANSLATION
The example shows how each vertex moves the same distance in the same
direction.
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Each vertex slides 9 units to the right.
ROTATION Observe the transformation that turns every point of a pre-image
through a specified angle and direction about a fixed point
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REFLECTION The example shows that reflection along the x axis change
the sign of the y-coordinate.
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DILATION Notice how EVERY coordinate of the original triangle has been
multiplied by the scale factor (x2).
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C. Patterns and Diagrams
A pattern is said to be a design having a certain discriminant regularity. A
pattern that is drawn using geometric shapes which are typically repeating.
A wallpaper is an ideal example of geometrical pattern. These patterns are
commonly seen in art as well as in nature. Natural geometrical patterns may
include waves, cracks, spirals, foams etc. Actually, there is a mathematical
structure underlying in each geometrical pattern.
C.1. SYMMETRIC PATTERNS
Note: A plane figure has symmetry if there is a non-trivial transformation that
maps the figure onto itself. A trivial transformation refers to the identity
transformation
Types of Symmetry:
1.Line symmetry/mirror symmetry/reflection symmetry
An object is reflected across a line, like looking in a mirror.
A square has four line symmetries.
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2. Rotational Symmetry
Occurs when a figure is rotated less than 360o about a point so that the image
and the pre-image are indistinguishable
A square has 4 rotational symmetries: 0o, 90o, 180o, 270o
C.2. TESSELLATION
r) with
shapes so that there is no overlapping and no gaps.
polygons that are tessellated is 360o .
-regular
Regular tessellation is made from congruent regular polygons joined sideto-side.
Can you guess which of the polygons can be used for regular tessellation?
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Among the regular polygons, only equilateral triangle, square and regular
hexagon can be used for Regular tessellation.
A semi- regular tessellation uses two or more different regular polygons with
sides of the same length in such a way that all vertices are identical. The
numerical notation shown for these semi-regular tessellations represents the
regular polygon arrangement about each vertex.
An example of semi-regular tessellation (4,8,8)
A Non-regular tessellation does not use regular polygons and also called
demiregular or polymorph. An example of these type are the following.
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Dutch graphic artist M. C. Escher (1898-1972) is known for his creative
use of tessellations in his work. What transformations can you see in this
picture?
The birds and fish have been translated here.
What transformations can you see in this Escher print?
Some birds have been translated and some have been rotated.
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WALLPAPER PATTERNS
Wallpaper patterns are patterns that cover the plane and can be mapped
onto itself by translations in more than one direction.
A Sample for tessellation
Make a tessellation using a rotation
a. draw an equilateral triangle on a blank piece of paper and cut it out
b. draw trapezoid inside the right side of the triangle
c. rotate the trapezoid so you can copy the change on the side as indicated
d. repeat this pattern on a tessellation of equilateral triangles.
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D. DESIGNS, ARTS AND CULTURE
Different patterns can be used to create different geometric designs. When
these designs are used correctly, the designs can be visually effective and
functional.
Patterns can be used to enhance artistic skills and enrich one’s culture
Geometric designs can be seen in the following:
Application
The following is a product of a transformation of the triangle using reflection
and rotation.
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This figure is a product of translation.
Assessment
1.
List all symmetries present in the following:
a. Isosceles triangle
b. Equilateral triangle
c. Rectangle
d. Regular hexagon
e. Regular pentagon
2. Using each of the figures below, draw a wall paper by using:
a.
b.
c.
d.
reflection
Translation
Rotation
Any combination of the above transformation
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3.
There are eight semi-regular tessellations in all. Challenge:
a. Draw tessellation (3, 4, 6, 4)
b. Find out the other 6 semi-regular tessellations
4. On a ¼ sheet of illustration board, create your own tessellation.
8 Semi- regular tessellations
4, 8, 8
3, 4, 6, 4
Reflection
1. What do you think would be the nature of mathematics without patterns and
geometric designs?
References
Geometry: Shapes, Patterns, and Designs by Vistro –Yu
Mathematical Excursions by R. Aufmann et al.
Essential Mathematics for the Modern World by Nocon and Nocon
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Unit 6. Codes
Introduction
In this unit you will learn some
coding schemes that are used to
assign identification numbers or
bar codes, use check digits for
error detection and ZIP code errors
and analyze encrypted data using
cryptography.
Codes were already around since
ancient times. A code is a symbolic
way to represent information. It is
a word or a short phrase that
symbolically assigns a summative, salient, essence capturing, and suggestive
attribute for a portion of language- based or visual data (Saldana, 2013). Readers
may also use other words or phrases to code since in quantitative analysis since
coding is a not a precise science.
Historically, codes were already used by
humans during the ancient times.
Hieroglyphics or “sacred writings”
were codes used by ancient Egyptians in
their writing system. For instance,
Roman numerals were developed to
easily determine the prices of
commodities and services rendered by
the Romans. It was used throughout
Europe until 1600s.
Identification numbers are used to identify individual items, specific products,
people, accounts, or documents. These numbers are useful for easy recognition
and detection of materials and for tracking and inventory of products or
documents (Kirtland, 2001).
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Learning Outcomes
Upon the completion of this unit, you are expected to:
a. Use coding schemes to encode and decode different types of information
for identification, privacy, and security purposes; and
b. Exemplify honesty and integrity when using codes for security purposes
Activating Prior Learning
Students would ask to bring different types of codes found in any type of
materials (e.g. books, specific products, or documents). The materials that they
will bring will be shared to shared classmates for familiarity purposes.
Topic 1: Coding
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
use check digits for error identification;
detect ZIP code errors;
discuss the process of disguising data; and
analyze encrypted data
Presentation of Content
Check Digits
A check digit is used to verify errors on identification numbers. A check digit
is a single digit number that is generated using the other characters from the
identification number. Different identification numbers use different check
digit schemes.
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The Universal Product Code (UPC)
The check digit of UPC is usually found on the far right of the UPC. UPC is
the barcode which is the identification number of a retail item such as a
grocery product. It consists of 12 digits, the first 11 digits or characters specify
the source of the item and the product number. The 12th character, is a module
10 check digit.
Example1
Consider the UPC 88083230854 and find its check digit.
Solution:
Steps
UPC without
check digit
Step1.
Multiply
number in
each position
by 3 or 1 as
indicated
Results of
step 1
Step 2. Add
the result to
get the sum
Step 3.
Subtract the
resulting sum
from the
number that is
a multiple of
10 nearest to
it.
UPC with
check number
A
J
8
3
3
3
24
9
B
C
K L
8
0
1
1
8
D
E
0
F
8
5
1
1
8
3
3
0
H
I
3
4
3
3
8
0
G
9
24
5
2
1
2
12
101
110 – 101 = 9 – check digit
(note: 110 is the nearest multiple of 10 to 101)
8
3
8
0
0
8
141
8
5
3
4
2
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Example 2
Consider the number found at the bottom of Nissin Cup Beef Noodles as
follows:
0 48000 00217 8
Determine how a barcode scanner would detect the error if the number
indicated above was entered into the computer as 0 68000 00217 8 (the
second number was changed from 4 to 6).
Solution:
Steps
UPC without
check digit 8
Step1.
Multiply
number in
each position
by 3 or 1 as
indicated
Results of
step 1
Step 2. Add
the result to
get the sum
Step 3.
Subtract the
resulting sum
from the
number that is
a multiple of
10 nearest to
it.
A
J
0
0
3
3
0
0
B
C
K L
4
0
1
1
D
E
8
0
2
1
1
1
3
3
4
0
24
6
0
1
F
G
0
7
3
3
0
21
H
I
0
1
0
56
60– 56 = 4 – check digit
Since the result is not equal to the given check digit 8, an
error is detected.
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Barcode
A barcode is a set of vertical bars (i.e. long and short) and spaces which provide
an indispensable tool for tracking a variety of data from pricing to inventory.
In most establishment, cashiers
make use of automated cash
registers. Barcoding is an efficient
way of translating data accurately
and used for automated data
collection. It eliminated the
occurrence of human error since
through the use of a bar scanner,
transmitting data is faster and more
reliable. Barcodes were used on June
26, 1974 with the 10-pack Wrigley’s
Juicy Fruit gum.
The simplest barcode is the Postnet
(Postal Numeric Encoding Technique) Code by the US Postal Service which is
commonly seen in business reply envelopes to assist in directing mail.
The ZIP+4 code which is the ZIP code of the US Postal Service which stated
in 1983, contains 53 long and short vertical bars (Stewart, 1995). The long
bars at the beginning and end of the ZIP+4 code serve as guard bars of the
remaining 50 bars. The 50 bars are divided into 5 blocks and each block
contain 2 long bars and 3 short bars. Each block represents a single digit and
the 10th digit is used for error correction.
Example, if the ZIP+4 code is denoted by 𝐴1 𝐴2 𝐴3 𝐴4 𝐴5 𝐴6 𝐴7 𝐴8 𝐴9 𝐴10 , and
the check digit 𝐴10 has a property that the sum 𝐴1 +𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 +
𝐴6 + 𝐴7 + 𝐴8 + 𝐴9 + 𝐴10 ends with 0, i.e. the sum of the digits is divisible by
10. An error is easily detected in a ZIP+4 code if a block of vertical bars does
not contain exactly 2 long bars and 3 short bars. And because the location of
the wrong block can be detected, the check digit is used to correct such error.
Example 3
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Determine the ZIP+4 code and check digit for the following barcode:
1.
2.
3.
4.
Step
Separate
the guard
bars at
the
beginning
and end
of the
ZIP+4
code.
Divide
the
remainin
g bars in
blocks of
5 and
label
Identify
the
7
ZIP+4
code
Verify
the code
Solution
3
1
2
4
2
3
1
9
8
3+2+2+3+1+9+8+7+1+4 = 40
- Number ending with 0.
Assessment
1. Suppose that packaging of a box number of kitchen tools was
damaged such that the 12th digit of a 12-digit UPC is no longer
readable but the remaining 11 digits were 23015691357. Determine
the correct UPC.
2. Suppose a postal money order identification number and check
digit 98898889954 was erroneously copied as 968988889954. Will
the error be detected? Explain.
3. Determine the ZIP+4 code and check digit for the following
Postnet barcodes:
a.
b.
Topic 2: Cryptography
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Learning Objectives
Upon the completion of this topic, you are expected to:
a. encrypt and decode messages using cryptography; and
b. learn methods of ensuring privacy, security, and authenticity of important
data.
Presentation of Content
Cryptography
The needs of some individuals and organizations, especially the military leaders
and diplomats, for protection of classified information lead to inventing ways
of making text so that the secret file would remain secret. Today, cryptology
has wide application. It does not only benefit the military but also the
government system, business and law firms, investigation, communications
(like telephone lines or by radio which are subject to interception), ATM cards,
credit cards, computer password, electronic, and commerce. This necessity
gradually created a mathematical discipline called Cryptology.
The term cryptology is derived from the Greek words kryptos (κρυπτός) which
means "hidden" and logos (γράφω) which means "words". So, cryptology
simply means “hidden words. It is often used to refer to the study of secrets.
The process of encoding messages through secret codes is called encryption.
Caesar Cipher
The Caesar cipher is the first cryptosystem used by Julius Caesar in sending
messages to his troops. The process is to assign each letter of the English
Alphabet by numbers from 0 to 25 that is A corresponds to 0, B corresponds 1,
C corresponds 2 and so on. For example, to encrypt a message, replace a letter
of the message in position k by a letter in position (k+13). And so, A will be
replaced by N (0+13), B will be replaced by O (1+13), and so on. See the table
below
Plaintext
A
B
C
D
E
F
G
H
I
J
K
L
M
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
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Ciphertext
13
14
15
16
17
18
19
20
21
22
23
24
25
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9
10
11
12
A
B
C
D
E
F
G
H
I
J
K
L
M
How messages are sent?
First, the sender composes a message or information which will be intends to
transmit to a receiver, called a plaintext.
Second, the message is converted coded form called encryption.
Third, the coded message is sent to the receiver. The coded message is called
ciphertext.
Finally, the message is converted back to plaintext. The process of decoding is
called decryption.
Application
Example 1
Encrypt the message I LOVE CRYPTOLOGY.
Solution:
Message
Replace the letters of the message
(plaintext) into ciphertext as shown
in the table below.
I LOVE CRYPTOLOGY
V YBIR PELCGBYBTL
If we are to decrypt the message, simply replace the letters of the encrypted
message with the letters in the plaintext.
Example 2
Decrypt the message “QB ABG BCRA GUVF SVYR”.
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Solution:
Message
Replace the letters of the message
(ciphertext) into plaintext as shown
in the table below.
QB ABG BCRA GUVF SVYR
DO NOT OPEN THIS FILE
Assessment
1. Use Caesar Cipher to encrypt the following messages:
a. FLEE AT ONCE WE ARE DISCOVERED
b. EXPLOSIVES OUTSIDE
c. CALCULUS IS FUN
d. MATHEMATICS IS INTERESTING
2. Decrypt the following messages using Caesar cipher.
a. ORYHIRUFRXQWUB
b. VYBIRPFH
c. FRZRFGENYOERNX
3. Research on the following and present them in class:
a. Hamming code
b. Substitution cipher
c. Pigpen cipher
d. Atbash cipher
Summary
1. Summary
A check digit is used to verify errors on identification numbers. A check digit
is a single digit number that is generated using the other characters from the
identification number.
The check digit of UPC is usually found on the far right of the UPC. It
consists of 12 digits, the first 11 digits or characters specify the source of the
item and the product number. The 12th character, is a module 10 check digit.
A barcode is a set of vertical bars (long and short) and spaces which provide
an indispensable tool for tracking a variety of data from pricing to inventory.
Cryptology simply means hidden words. It is often used to refer to the study of
secrets. The process of encoding messages through secret codes is called
encryption. The message or information intends to transmit to a receiver called
plaintext. The process of decoding the message is called decryption. The result
of encryption performed on plaintext is called ciphertext.
The most commonly used cipher is the Caesar cipher introduced by Julius
Caesar.
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Reflection
A. Barcodes greatly reduces errors, allowing easy inventory checks, and
protects against customers who change the price labels. It is designed
to be read by a special scanner in order to input quickly into a
computer. How important is this in your daily living?
B. You learned how to encrypt and decrypt messages using cryptology.
How significant is this in your daily living?
References
Baltazar, E.C, Ragasa, C., and Evanelista, J. (2018). Mathematics in the
Modern World. C &E Publishing Company, Inc.
Mading, et.al. (2007). Several Substitution Ciphers in Cryptology.
Unpublished Seminar Paper. Cagayan State University, College of Arts and
Sciences.
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UNIT 7: Linear Programming (8 hours)
Introduction
A Linear programming is a method that seeks the optimum solution which is
either maximum or minimum solution. The word “programming” means
producing a plan or procedure that determines the solution to a problem. It
requires that all the mathematical functions in the model be linear functions
which means that it is limited in a two-dimensional set of axes
Graphical Solution Method is a two-dimensional geometric analysis of Linear
Programming problems with two decision variables.
The Theory of Linear Programming states that the optimal solution will lie at a
corner point of the feasible region
Learning Outcomes
Upon the completion of this unit, you are expected to:
a. sketch the graph of system of inequalities;
b. determine the region bounded by the system of inequalities and the
coordinates of the corner points; and
c. appreciate the application of linear programming in decision making.
Activating Prior Learning
Graphing Linear Inequalities
1. Draw the graph of the equation obtained for the given inequality by replacing
the inequality sign with an equal sign .
 Use a dashed or dotted line if the problem involves a strict
inequality, < or >.
 Otherwise, use a solid line to indicate that the line itself
constitutes part of the solution.
2. Pick a test point lying in one of the half-planes determined by the line
sketched in step 1 and substitute the values of x and y into the given inequality.
 Use the origin whenever possible.
3. If the inequality is satisfied, the graph of the inequality includes the halfplane containing the test point.
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 Otherwise, the solution includes the half-plane not containing
the test point
Graphing of System of Inequalities
The solution set of a system of linear inequalities in two variables x and y is
the set of all points (x, y) that satisfy each inequality of the system.
The graphical solution of such a system may be obtained by graphing the
solution set for each inequality independently and then determining the region
in common with each solution set.
Example
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Solving Linear Programming Model Graphically
A linear programming problem in two unknowns x and y is one in which we
determine the maximum and minimum value of a linear expression.
𝑃=𝑎𝑖𝑥+𝑏𝑖𝑦 (for maximization)
C=𝑎𝑖𝑥+𝑏𝑖𝑦 (for minimization)
are called the objective function, subject to number of linear constraints of the
form
𝑎𝑖𝑥+𝑏𝑖𝑦≤ 𝑐𝑖
or
𝑎𝑖𝑥+𝑏𝑖𝑦≥ 𝑐𝑖
or
𝑎𝑖𝑥+𝑏𝑖𝑦= 𝑐𝑖.
An objective function is an expression, which shows the relationship between
the variables in the problem and the firm’s goal.
There are two types of constraints: structural and non-negativity. The structural
constraint is a limit on the availability of resources; it is also referred as explicit
constraint. Non-negativity constraint is the constraint that restricts all the
variable to zero and positive solution; it is also referred as implicit constraint.
Let’s take the linear programming below.
Maximize: P = 2400x + 3200y
Objective Function
Subject to:
3x + 2y ≤ 18
2x + 4y ≤ 20
Structural constraints
x≤5
x ≥ 0, y ≥ 0
Non-negativity constraints
OPTIMAL VALUE
The highest (for maximization problem) or lowest value (for minimization
problem) of the objective function is referred to Optimal value. The optimal
solution is a combination of decision amounts that yields the best possible value
of the objective function and satisfies all the constraints.
FEASIBLE REGION
The feasible region is the set of combinations of values for the decision variables
that satisfy the non-negativity conditions and all the constraints simultaneously
that is the allowable decisions.
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Extreme point is the corner of the feasible region; it is the location of the
maximum and minimum point of the feasible region.
THE EXTREME POINT THEOREM
The linear objective function will have its optimum solutions at the extreme
points (corner points) of the feasible region whenever the feasible region is
bounded.
Fundamental Theorem of Linear Programming Problem
There are two things we need to consider in solving linear programing problem
such as
 If Linear Programming (LP) problem has optimal solution, there is
always at least one extreme point (corner point) solution of the feasible
region.
 A Linear Programming (LP) problem with bounded nonempty feasible
regions always contain optimal solutions
Maximization Problem
Solving Linear Programming Maximization Problem by GRAPHICAL
METHOD
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Steps:
1. Graph the linear inequalities and determine the feasible region
2. Determine the coordinates of the extreme points( corner points)
3. Substitute the coordinates of the extreme points to the objective function and
identify the highest (for maximization problem) or lowest (for minimization
problem) result.
4. Substitute the coordinates of the extreme points to the objective function and
identify the highest (for maximization problem) or lowest (for minimization
problem) result.
Example 1.
A local boutique produced two designs of gowns A and B and has the following
materials available: 18 𝑚2 of cotton, 20 𝑚2 of silk, and 5 𝑚2 of wool. Design A
requires the following: 3 𝑚2 of cotton, 2 𝑚2 of silk, and 1 𝑚2 of wool. Design
B requires the following: 2 𝑚2 of cotton, 4 𝑚2 of silk. If Design A sells for P 1,
200 and Design B for P 1, 600, how many of each garment should the boutique
produce to obtain the maximum amount of money?
Solution:
Step 1:
Represent the unknown in the problem.
Let x be the number of Design A gowns, and y be the number of Design B
gowns.
Step 2:
Step 3:
Formulate the objective function and constraints by restating the information in
mathematical form (LP model)
Objective Function
Maximize: P = 1, 200x + 1, 600y
The constraints are:
3x + 2y ≤ 18
2x + 4y ≤ 20
x≤5
Structural constraints
x ≥ 0, y ≥ 0
Non-negativity constraints
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Step 4:
Step 5:
Locate the Feasible region and trace the extreme points of the graph
Step 6:
Substitute the coordinates of the extreme points on the feasible region to the
objective function
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Objective function: P = 1200x + 1600y
Step 7. Formulate the decision.
Since the coordinate (4,3) will give the highest value of P 9, 600, the decision
is to create 4 design A gowns and 3 design B gowns in order to maximize the
sales.
Minimization Problems
Example:
A pharmacist produces a drug from two ingredients. Each ingredient contains
the same three antibiotics in different proportions. Each ingredient A produced
results P80 in cost; each ingredient B results P50 in cost. The production of
the antibiotics is dependent on the availability of limited resources. The
resource requirement for the production are as follows:
The company wants to determine the quantity of ingredient A and B that must
go in to drug in order to meet the antibiotics’ minimum requirements at the
minimum cost.
Solution:
Step 1.
Represent the unknown in the problem.
Let x be the quantity of ingredient A and y be the number of quantity in
ingredient B
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Step 2
Tabulate the data about the facts (if necessary)
Step 3
Formulate the objective function and constraints by restating the information in
mathematical form (LP model)
Objective Function
Minimize: C = 80x + 50y
The constraints are:
3x + y ≥6
x + y ≥4
2x + 6y≥12
Structural constraints
x ≥ 0, y ≥ 0
Non-negativity constraints
Step 4: Plot the constraints of the LP problem on a graph.
Step 5: Determine the feasible region and trace the extreme points of the graph
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Step 6. Substitutes the coordinate at the extreme points of the feasible region in
the objective function
Step 7: Formulate the decision
Since the coordinate (1,3) gives the lowest value of P230. The decision is to mix
1 unit of Ingredient A and 3 units of Ingredient B in order to minimize the cost
Assessment
A. Graph the inequality
1. 2x + y ≤ 5
2. 3x+4y ≥ 10
3. x ≥ 0
y≤3
y≥x
4. x ≥ 0
y ≥0
2x + 3y ≤ 15
3x + y ≤ 13
5. Solve the following maximization problem
\
Maximize
P = 30x + 25 y
Subject to :
x≥0
y≥0
x + 2y ≤ 6
2x + y ≤ 6
6. Solve the following minimization problem
Minimize
P = 40x + 75y
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Subject to
x≥0
y≥0
x + y ≤ 20
x + 2y ≤ 25
Solve the LP graphically.
7. Rina needs at least 48 units of protein, 60 units of carbohydrates, and 50
units of fat each month. From
each kilogram of food A, she receives 2 of
protein, 4 of carbohydrates, and 5 units of fat. Food B contains 3 units of
protein, 3 units of carbohydrates, and 2 units of fat per kilogram. If food A cost
P110 e food B costs P90 per kilogram, how many kilograms of each food should
Rina buy each month to keep costs at a minimum?
8. A table manufacturer produces tables in two types regular and deluxe. It
costs P300 to make each regular table, which sells for P550. It costs P480 to
make each deluxe table and each sells for P750. The daily production capacity
is 125 tables and the daily cost cannot exceed P60,000. How many tables for
each type should be made per day to maximize the profit?
Reflection
1. What part of the topic would contribute in maximizing your potential?
2. What do you think are the factors or constraints and objective for your life to
maximize your true potential?
References
Essential Mathematics (For the Modern World) by Nokon & Nokon 2016
Lectures and Powerpoint Presentations Prof Dinah Vidad, Joseph and Dr.
Catherine Vistro-Yu
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Unit 8: The Mathematics of Finance
(12 hours)
Introduction
Finance is indispensable component of our daily needs. Many things can be
made possible because of finance. The financial capability of an individual, a
company or an agency sometimes determine the success of some endeavor that
involves finances. A business firms, for example, rely most on the financial
aspects of the business, to maintain its resources such as people, materials,
equipment and labor. In some ways, these firms may resort to borrowing
money from a lending institutions in order to augment the firm’s financial needs
at a certain rate of interest.
On the other hand, banks, government agencies, and even private institutions
and lenders earn money in the form of interest from the money they lend to
borrowers.
For some personal conditions, we can even borrow money using some high
technology systems such as the use of credit cards.
This unit covers nine different topics in dealing with finances:
the
computation of simple and compound interest, credit cards, consumer loans,
bonds and mutual funds, loan amortization and home ownership. Each aspect
is provided with clear and concrete examples for your easy understanding.
Learning Outcomes
a.
b.
c.
d.
Upon the completion of this unit, you are expected to:
Define simple interest, compound interest, loans and loan amortizations,
credit cards, bonds and mutual funds, and home ownership;
Solve problems involving simple and compound interest, loan and loan
amortization, credit cards, bonds and mutual funds and home ownership;
Relate the knowledge of the different topics discussed into practical life
situations; and
Appreciate Mathematics as a practical arts and science for everyday life.
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Activating Prior Learning
Can you answer these?
1. How many days are there in a year? ______
2. How many days does February have during leap year?
_______
3. How many days does each month of the year have?
4. Supposed I loaned a total amount of P25,200 from a small lending firm.
After 6 months, I repaid the firm a total of P26,000.
a.
b.
c.
How much did the lending firm earn from my loan?
_______________
What do you call the amount earned by the lending firm?
____________
Which one is called:
- principal? ____________________
- rate?___________________.
- term of the loan? _____________________.
- interest? ___________________.
5. I have P10,000. Yesterday, I deposited the money in the bank for a period
of 1 year for a rate of 3% per annum.
a. How much money did I receive, including the amount I deposited, at
the end of the period? ___________________.
6. Mr. Garcia borrowed a P50,000 from a lending firm, payable in 2 years at
2% simple interest.
a. How much will Mr. Garcia pay the agency at the end of the term?
b. How much interest was earned?
c. If you add the interest and the principal to obtain a new principal,
how much would it be?
d. If you compute the simple interest from the new principal, how much
would it be?
e. If you add the interest in number 4 to the amount in number 2 to
obtain another new principal, what is the new principal?
f. If you compute the interest from the new principal in Number 5, how
much would it be?
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Topic 1: Simple Interest (3 hours)
One of the best ways to secure one’s money is to deposit it to the bank. For
some period of time, the depositor gets something in return for the amount
deposited. The person who deposited the money is called the depositor,
while the amount the depositor gets in return is called interest. The sum of
money invested in the bank is called the principal.
Sometimes, when people need to secure fund for a limited time for some
reasons, they resort to borrowing from any private or government lending
agencies or institutions, or even to a private individual at a certain rate of
interest. The person, agency or institution who lends money is called the
lender while the person who borrows money is called the debtor or maker.
The sum of money paid in return for the use of the money is also called the
interest.
The capital or sum of money lend or invested is called the principal, while the
fractional part of principal that is paid on the loan is called the rate of interest,
and it is usually expressed in percent. The specific time or duration (years,
months, days) for which the money is borrowed is called the term of the loan.
The interest is calculated based on the interest rate and the term of the loan.
On the other hand, the sum of the principal and the interest is called the
maturity value also called the final amount, while the date to which the term of
the loan ends is called the maturity date.
Loaned money covers some charges such as processing fee, insurance, and
other legitimate fees. These charges are deducted from the principal loan,
The amount
received by the debtor or maker after deducting the legitimate fees is called
the present value or proceed of the loan.
This part of the unit dwells mainly on simple interest and expose you to some
real-life examples for your better appreciation of simple interest.
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
recognize terms related to simple interest;
calculate simple interest on ordinary and exact time, using actual and
approximate time;
solve practical problems involving simple interest; and
appreciate the importance of simple interest into real-life situations
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Presentation of Content
Simple interest (I) is defined as the product of the principal (P), rate (r) and
time (t).
In formula:
Interest = Principal x rate x time
I=Pxrxt
where:
I
P
R
t
- simple interest
- principal amount
- rate in percent
- time in years
From the original formula, other formulas can be derived such as:
① When Principal is unknown: P =
② When the rate is unknown:
r=
③ When the time is unknown:
t=
I
rt
I
Pt
I
Pr
In computing the simple interest, we always arrive at a final amount (F).
final amount is the sum of the principal and the interest. In formula:
The
Final Amount (F) = Principal (P) + Interest (I)
F
= P + I;
Then from the original formula, we derive other formulas:
I = F - P, and
P =F-I
Before you proceed, be sure to learn more bout the following concepts of simple
interest:
Term of the loan
The term of the loan is the period during which the borrower had used the
money. The term or time may be expressed in days, months or years. In
computing the simple interest, the time (t) should always be expressed in
years. If the time is expressed in days or months, convert it into years,
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A. Time in days:
Divide the number of days by 360
Example: When t = 120 days
Time in years
= 120÷360
= 0.33 year
B. Time in months:
Divide the number of months by 360
Example When t = 7 months
Time in years
= 7 months÷12
= 0.58
year
C. Time in months and days:
Convert both months and days in years by dividing the number of moths
by 12, and the number of days by 360.
Example: When t = 9 months and 23 days
Time in years = (9 months÷12) + (23 days÷360)
= 0.750 + 0.064
= 0.814 year
Note that in business, the interest rate is expressed in percent (%). In
computing the interest, it is necessary to convert the rate into fraction or
decimal equivalent.
Example:
A money is invested at a rate of 6%.
Convert 6% into fraction which is 6/100.
To change it into decimal, simply remove the percent (%) sign and move two
places to the left. Therefore, 6% changed into decimal is 0.06.
Computation of Simple Interest
Having understood the different concepts on simple interest, you are now
ready to compute the simple interest.
Example 1:
1. Find the interest and amount on P55,500 at 5% simple interest for 4 years.
Solution:
Given: P = P55,500
r= 5% (0.05)
Find:
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a. Interest (I)
b. Amount (F)
Exploring Mathematics in the Modern World
T= 4 years
a. Find the interest
I =Pxrxt
= P55,500 x .05 x 4
= P11,100- the interest for 4 years
b.
Final Amount
F =P+I
= P55,500 + 11,100
= P66,600
The invested amount will become P66,600 after 4 years.
Example 2. Find the final amount and interest on P150,000 invested at
6 1/2% simple interest for 4 years and 10 months.
Solution:
Given: P
r
t
= P150,000
= 6 1/2% (.065%)
= 4 years and 10 months.
Find:
a. Interest (I)
b. Final Amount (F)
1. First, convert the time into years:
Since time = 4 years and 10 months, then
t = 4 years + (10 months÷12)
= 4+ 0.833
=4.833 years
2. Solve for the interest:
I=Pxrxt
= P150,000 x .065 x 4.833
= P47,121.75
3. Solve the Final Amount:
F=P+I
= P150,000 + P47,121.75
= 197,121.75
The investor will get back a total of P197,121.75 for investing his/her money
for a period of 4 years and 10 months.
Application
A. Pair-Sharing. Look for your pair. Fill in the boxes with the correct figure
if possible. You may use calculators or any computing gadgets available.
I
P
r
t
3.5%
4 years
P15,500
P5,350
F
P22,300
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P25,300
P4,350
4 1/2%
P23,450
4 years, 3
months
5 years, 2
months
B. Group Activity: Group yourselves into 6 members each. Depict a situation
where the knowledge on interest is emphasized and be able to present a simple
role play on this. The following settings may help you conceptualize your
presentation:
1. A scene in a bank where one withdraws his deposits
2. A “Bombay” style of money lending
3. A lending agency where the manager explains the terms and conditions
of the
loan
4. An appliance center that sells appliances (TV, Refrigerator, etc.) on
installment basis.
C. Problem Set: Solve the following problems. Show your complete
solutions. Use a blank sheet enclosed for your answers.
1. Find the Final Amount and interest on P20,000 for 65 days at 5.25% simple
interest.
2. A businessman charges his client P2,750 on a loan of P15,800 for 2 years and
3 months. What rate is effected on the loan?
3. Nina borrowed P50,000 on April 21, 2018 and repays the loan on
April 21, 2019 with an interest rate of 5-1/2%. Find the amount she paid.
4. Three months after borrowing money, John Mark pays an interest of
P2,700. How much did he borrow if the interest rate is 5.25%?
5. Pamela loans P20,000 at 4.75% simple interest.
to get P2,500 interest?
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How long will it take her
Exploring Mathematics in the Modern World
Topic 2: Computing the Simple Interest using Ordinary
and Exact Time (Ordinary and Exact Interest) - 1.5 hours
As seen earlier, there are some instances when the term of the loan is in given
number of days. It is always, of course, the interest of every lending agencies
to have their money loaned earn higher interest.
On the other hand, amount invested in these agencies tend to, as much as
possible give out smaller interest to the investors. This system is what is called
the “Banker’s Rule”. The rule says that for a certain lender, be it agency or
private individuals, the tendency is for them to engage into a situation where
they get bigger interest. For this reason, two methods of computing the simple
interest are involved. In this topic, you will learn how the above situation is
applied into some practical circumstances.
Learning Objectives
a.
b.
c.
Upon the completion of this topic, you are expected to:
compute for the ordinary and exact interest;
identify the advantages/disadvantages of each type of interest.
appreciate the importance of each type of computing the interest in real
life situation.
Presentation of Content
Answers to above questions are very important as you go along this topic,
and as you experience computing for the simple interest using the ordinary and
exact time.
Ordinary interest is computed using the exact time, while exact interest is
computed using the exact time.
Ordinary Interest
To find the ordinary interest, use the formula for simple interest. In this
case, the time expressed in days is divided by 360. This is because each
month of the year is assumed to have 30 days.
We use the symbol Io to denote ordinary interest.
formula for ordinary interest as:
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Then we have the
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Ordinary Interest (Io) = Principal (P) x rate (r) x time (t), where t is
divided by 360 days. In formula:
𝒕
Io = P r
𝟑𝟔𝟎
Note: If the time is expressed in years or months, be sure to change it into
days to be able to divide it by 360.For instance, the time is 2 years and 3
months years, change the time first into days following the steps in the
previous lessons.
Example:
Find the ordinary interest on P15,500 for 130 days, at 5 3/4% simple
interest.
Solution:
Given: P= P15,500
r= 5 3/4% (.0575)
t= 130 days.
Find:
Ordinary Interest (Io)
Io = Pr
t
( ___
360
= P15,500 x 0.0575 x (130 ÷ 360)
= P321.740
The ordinary interest of the money invested/borrowed for 130 days is
P321.74.
Exact Interest
To find the exact interest, use the formula for simple interest, dividing the time
expressed in days by 365.
The number of days is calculated using the exact number of days each month
of the year has, such that January has 31, February has 28 (except for leap
year), March has 31, and so on.
We use the symbol Ie to denote exact interest.
To calculate the exact interest, the following formula is used:
Exact Interest (Ie) = Principal (P) x rate (r) x time (t), where t is divided by 365
days.
𝒕
Ie = P r
𝟑𝟔𝟓
(
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Let us use the example above to illustrate the computation of exact interest
(Ie).
Example.
Find the exact interest on P15,500 for 130 days, at 5 3/4% simple interest.
Solution:
Given: P= P15,500
r= 5 3/4% (.0575)
t= 130 days.
Find: Exact Interest (Ie)
___
t
Io = Pr 360
= P15,500 x 0.0575 x (130 ÷ 365)
= P317.285
(
t
Using exact time, the interest on P15,500 invested/borrowed for 130 days is
P317.285.
Let’s compare the ordinary and exact interest.
Principal
rate
Time (130 days)
interest earned
Ordinary
P15,500
5 3/4%
0.361 year
P321.740
Exact
P15,500
5 3/4%
0.356 year
P317.285
Remember!
a. Ordinary interest is always larger than the exact interest.
b. In computing the simple interest, when there is no type of interest is
specified, always compute it using ordinary interest.
Application
A. Form a group of three. Solve the following problems on ordinary and
exact interest. Show your complete solution. Use the attached blank sheet of
paper for your answer.
1.
2.
3.
How much interest will Nena receive, if she deposited to the bank a total
amount of P50,000 at 2.5% exact interest for 3 years and 3 months?
Suppose you borrow from a lending agency a total amount of P30,000
payable at the end of 1 year and 6 months at 3% simple interest. Compute
the ordinary and exact interest.
Compare the results in 2 and 3.
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B. Answer the following problems in a 1/2 sheet of paper.
1. A rural bank offers 2.5% interest for agricultural loan, amounting to
P30,000 each for 25 farmers in the municipality of Gonzaga. If the loan is to
be paid in 180 days (1 cropping period), compute the ordinary and exact
interest of the loan.
2. Make a brief discussion of the advantages/disadvantages of both types of
interest, both to the lender and to the borrower.
Congratulations! You’ve just learned on how to compute the simple
interest using ordinary and exact time.
Topic 3: Computing the Simple Interest using Actual and
Approximate Time (Interest between Dates) - 3 hours
In previous lessons, the time for which the interest is computed is given in
years, months and days. For some instances when interest is to be computed
from a certain inclusive date, it is necessary that we determine the number of
inclusive days. We can do this in two methods: Using the actual and
approximate time. In same way, we can compute simple interest in two
methods- using the actual and approximate time.
In this topic, you will understand that the term (time) of the loan plays a very
vital role in the computation of the interest.
Learning Objectives
Upon the completion of this topic, you are expected to:
a. determine the actual and approximate number of days between inclusive
dates;
b. solve more challenging problems on interest involving the four methods:
(Ordinary Interest on actual time, Ordinary interest on approximate time,
Exact interest on actual time, and Exact interest on approximate time); and
c. compare the four methods of computing interest.
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Exploring Mathematics in the Modern World
Presentation of Content
Finding the actual and approximate time
When the time or term of the loan is expressed between dates, it is
necessary to determine the actual and approximate time.
Actual time is the exact number of days in any given month. Approximate
time is when all the months within the year are assumed to contain 30 days
each.
Let us work on the following examples to show how to determine the actual
and approximate time.
Example.
Find the actual and approximate time from April 21, 2018 to October 4, 2018.
Solution:
Month
Approximate Time
Actual Time
April
May
June
July
August
September
October
TOTAL
9 (30-21)
30
30
30
30
30
4
163 days
9 (30-21)
31
30
31
31
30
4
166 days
The use of table of months such as above could be laborious and time
consuming, particularly when the date where interest is computed involves
longer period of time.
For this case, an alternative solution in determining the actual and approximate
time may be used.
To use the alternative solution, the table attached at the end of this module
(Table 1) will be used in determining the actual and approximate time.
Using the alternative solutions, determine the Actual and Approximate time
from April 21, 2018 to October 4, 2018.
Finding the Actual time:
Using table 1:
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October 4 =
April 21
=
Difference =
277 days
111 days
166 days
Finding the Approximate Time
To find the Approximate time using the alternative solution, express the dates
in numerals, such that:
Year
Month
Day
October 4, 2018
=
2018
10
4
April 21, 2018
=
2018
4
21
Since you cannot subtract 21 days from 4 days, borrow 1 month (30 days) to
get 34 days. You only then have 9 months left.
Let us re-write:
Year
Month
Day
October 4, 2018
=
2018
9 (10 - 1)
34 (30 + 4)
April 21, 2018
=
2018
4
21
Difference
=
0
=
5 months
13 days
5 x 30= 150 days + 13 days = 163 days
Note: When a type of interest is not specified in any problem, the ordinary
interest, on actual time will be used. This rule is also called the Banker’s Rule.
Now, let’s compute the simple interest on actual and approximate time.
Example 1.
Find the interest on P50,000 at 4..5% from January 17, 2019 to July 7, 2019,
both using actual and approximate time.
Solutions:
A. First find the number of days between January 17, 2019 to July 7, 2019
using actual and approximate time.
Month
Approximate Time
Actual Time
January 17
February
March
April
May
June
July 7
TOTAL
14
30
30
30
30
30
7
171 days
14 (31-17)
28
31
30
31
30
7
171 days
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Note that the actual and approximate time are the same. This is because of
February which, on actual time has only 28 days (except leap year), while on
approximate time, it is assumed to have 30 days.
B. Compute the simple interest using the actual time.
Since there is no type of interest specified in the problem, then use ordinary
interest (Io) in computing for the simple interest. Let us then divide the time
by 360.
Using the formula for simple interest, I=Prt where t is divided by 360.
Given: P
r
t
Find:
= P50,000
= 4.5% (.045)
= 171 days (171÷360)
= 0.475
Interest
Solution:
I
= Prt
= 50,000 x .045 x 0.475
= P1,068.75
Using actual time, the interest on P50,000 at 4.5% from January 17 to July 7,
2019 is P1,068.75.
C. Compute the Interest using approximate time.
Since approximate time (171 days) is the same with the actual time, you will
then get the same result for the interest using approximate time which is
P1.068.75.
Let us try more challenging situations.
Example 2:
Find the interest on P25,000 using ordinary and exact interest, on actual and
approximate time at 4% from July 2, 2018 to November 27, 2018.
In the above example, interest will be computed using the four methods:
a. Ordinary interest for actual time (Io-actual)
b. Ordinary interest for approximate time (Io-approx)
c. Exact interest for actual time, and (Ie-actual)
d. Exact interest for approximate time (Ie-approx)
Solution:
Given: P = P25,000
r = 4%
t = July 2, 2018 to November 27, 2018
Find: Io-Actual, Io-approx, Ie actual, Ie approx
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A. Find first the actual and approximate time
Approximate time
Actual time
Year
Month
Day
2018
11
27
2018
7
2
------------------------------------------Diffrence = 4
25
4 x 30 = 120 + 25
= 145 days
November 27 = 331
July 2
= 183
----------------------------------Difference
= 148 dyas
Since the actual and approximate time are determined, you are now ready to
compute the interests.
B. Find the Ordinary interest using actual time (148 days).
that the divisor of the time for ordinary interest is 360.
Remember
Io actual = Prt
= 25,000 x .04 x (148÷360)
= P411.11
The ordinary interest on P25,000 using actual time is P411.11
C. Find the Ordinary Interest using approximate time (145 days). Again
the time will be divided by 360 for Ordinary time.
Io approx
= Prt
= 25,000 x .04 x (145÷360)
= 402.78
The ordinary interest on P25,000 using approximate time is P402.78
D. Find the Exact Interest using actual time (148 days). Remember that the
time shall be divided by 365 for exact interest.
Ie-actual = Prt
= P25,000 x .04 x (148÷365)
= P405.48
The exact interest on P25,000 using actual time is P405.48.
E. Find the Exact Interest using Approximate Time (145 days).
Ie- approx
= Prt
= 25,000 x -04 x (145 ÷365)
= P397.26
The exact interest on P25,000 using actual time is P397.26
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Now, let us summarize the computed interests using the four methods:
Ordinary interest
Exact Interest
Actual time
P411.11
P405.48
Approximate time
P402.78
P397.26
Based on above summary, you note that ordinary interest is always lower than
the exact interest.
On the other hand, interest computed using the actual time is always higher than
the interest computed on approximate time. This is because actual time uses the
actual number of days in a month, compared to approximate time, which
allocates 30 days for each month of the year.
Congratulations. You have just learned how simple interest can be computed
using the four methods.
Can you do these? Now, let’s find out if you can do the following exercises.
Solve the given problems and fill in the chart with the correct values. Make
some generalizations on actual and approximate time based from the chart of
dates that you have just filled out.
You can use any method in finding the actual and approximate time.
1. Mirla wants to put up a small farm and poultry supply business to cater the
needs of the local farmers in her town. For this, she needs a total amount of
P180,000. Since she is financially incapable to put up her project, she resorted
to borrowing from a rural bank in the locality, who charges 3.5% from
December 15, 2019 to February 12, 2020. How much interest will she pay the
bank for the specified period using actual time? approximate time?
2. Find the ordinary and exact interest on P53,000 at 4.5% using actual and
approximate time from February 1, 2016 to March 16, 2016 (leap year).
Inclusive Dates
ORDINARY INTEREST
Actual Time Approximate
time
December 15, 2019February 12, 2020
January 1, 2016-March
15, 2015 (leap year)
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EXACT INTEREST
Actual Time Approximate
Time
Exploring Mathematics in the Modern World
Topic 4: Compound Interest
In the previous lesson, one type of interest that you have learned is the
simple interest. This time, you will learn another type of interest- the compound
interest. The compound interest is interest that results from adding the interest
to the principal periodically. When interest is added to the principal and the
sum becomes the new principal for which the interest is computed for a certain
periods of time, the resulting amount is the final amount or compound amount.
The interest computed from the new principal is called the compound interest.
Compound interest is computed between successive time. You call the time
between this successive computations as compounding or conversion period.
The number of conversion periods for the whole year is usually denoted by m,
while the number of conversion periods for the whole term of the loan is denoted
by n. Conversion periods are usually expressed by any convenient length of
time, and this is usually the exact division of the year, like monthly, quarterly,
semiannually and annually. For this case, when the term is converted monthly,
then m=12, when converted quarterly, m=4, when converted semi-annually,
m=2, and when it is converted annually, m=1. For the total conversion periods
n for the whole term of the loan, this is obtained by multiplying the conversion
period and the term of the loan (t x m). For example, the total conversion
period for a loan in 10 years converted:
Monthly is 10 x 12,
Semiannually is 10 x 2,
Quarterly is 10 x 4,
Annually is 10 x 1,
n = 120
n = 20
n = 40 and
n = 10
On the other hand, unlike the simple interest, the interest rate in compound
interest is usually expressed as an annual or yearly rate.
This section provides you a clear picture on how interest is computed, other
than simple interest. Some practical applications are also provided for you to
appreciate the use and importance of this type of interest- the compound
interest.
Learning Objectives
a.
b.
c.
Upon the completion of this topic, you are expected to:
Compute for the compound interest;
explain the advantages/disadvantages of compound interest, both to the
lender and the borrower; and
appreciate the importance of this type of interest in real life situation.
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Presentation of Content
As earlier discussed, compound interest is computed from the new principal
(principal plus the interest).
To find for the compound interest, it is very important to note the following:
A) Conversion periods (m) is given in exact division of the year such as:
m = 1 (annually)
m = 2 (semiannually)
m = 4 (quarterly
m =12 (monthly)
B) Total conversion periods (n) is computed using the formula:
n = time (t) x conversion periods (m)
a) Periodic rate (i) per conversion period is computed using the formula:
interest rate (r)
Conversion period per year (m)
r/m
Now, we are ready to find the compound interest. To compute it, the
following formula is used:
Compound Interest (I) = Compound Amount - Principal
I=F–P
where:
Compound Amount (F) = Principal ( 1 + i)n
F
= P (1+i)n
Where F = compound amount
i = periodic rate
n = total conversion period
Example
Find the compound amount and interest on P25,500 invested at 5% for 5
years, compounded quarterly.
Solution:
Given: P
r
t
m
= P25,500
= 5% (.05)
= 5 years
= 4 (quarterly)
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n
i
=txm
=5x4
= 20
= r/m
= .05/4
= 0.0125
Find:
a) Compound Amount (F)
b) Compound Interest (I)
n
=txm
=5x4
= 20
i
= r/m
= .05 / 4
= 0.0125
= P (1+i)n, substituting the values, we have
= 25,500 (1 + 0.0125)20
= 25,500 ( 1.28)
= 32,640
b) I = F - P
= P32,640 - P25, 500
= P7,140
The amount of P25,500 invested for 5 years at 5% compounded quarterly has
compounded into P32,640. It earned an interest of P7,140.
a) F
Now, let us compare compound interest with simple interest.
problem above, the interest is:
I
=
=
=
Prt
P25,500 (0.05) (5)
P6,375
Using the same
Remember!
Compound interest is
always larger than simple
interest.
Application
Select one of your classmates and work on the following exercises:
1. For some reasons, I decided to pawn my 22K ring. The item was
appraised for P15,000. The pawnshop charges 4% compounded monthly. If I
decided to redeem my ring after 1 year, how much will I give back to the
pawnshop at the end of the term? How much interest did the pawnshop
earn?
2. What sum of money will be required to discharge a loan of P50,000 at
the end of 3 years and 2 months at a rate of 4.5% compounded quarterly?
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Assessment
Solve the following problems. Use one whole sheet of paper for your answers.
1. Find the compound amount and interest on P23,560 for 2 years and 3
months at 6%, m=2.
2. What final amount and interest will be due after 4 years on a loan of
P32,400 at 5 1/2% compounded annually?
3. Find the compound amount and interest on P17,450 for 4 years and 2
months at 4.5% compounded semi--annually?
3. What sum of money will be required to discharge a loan of P50,000 at
the end of 3 years and 2 months at a rate of 4.5% compounded quarterly?
4. You borrowed an amount of P30,000 from a lending firm from April 1, 2017
to August 15, 2017. If the firm charges you 4.5% on the loan, how much will
you repay the firm at the end of the period using actual and approximate time?
5. How much interest will be generated using the the four methods on P32,500
at 4.5% simple interest from August 1, 2017 to December 15, 2017.
Summary
You have just learned the practical applications of simple interest. Simple
interest is computed depending on the rate of interest, the time (days, months
and years) the amount borrowed, lent or invested. For some future instances,
the knowledge on simple interest helps everyone aware on how a money
invested or loaned earn additional amount. It is a practical learning and
useful as well because it makes you aware how your money grow in the bank.
In the part of the borrower, it makes him/her aware of how much money will
be returned to the lender for certain rate, for a period of time.
Usually, simple interest is computed with the time given in years, months and
number of days. This section discussed the computation of simple interest
between inclusive dates.
To find the number of days between dates, there are two methods that can be
used: the actual and approximate time.
Interest on actual time is bigger than on approximate time. This is because the
number of days when the interest is computed using the actual time is more than
using the approximate time. It follows that the longer the money is borrowed,
loaned or invested, the more interest it gain.
Compound interest is interest that results from adding the interst to the
principal periodically. To compute for the compound interest, the following
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Exploring Mathematics in the Modern World
concepts must be known: periodic rate (i), the conversion period (m) which is
given in the exact division of the year such as annually, semiannually, and
quarterly and monthly, and total conversion period (n). Compound interest is
greater than simple interest.
Reflection
1. What new ideas about simple interest did you learn?
2. In what practical situations can you say that Mathematics of Finance is
most useful?
3. How do you articulate the importance of financial mathematics in your
life?
4. In what way does compound interest become advantageous or
disadvantageous to both the borrower and the lender?
References
Capitullo, F.M. and Cruz, C.U. Mathematics of Investment
Earnhart, R. and Adina, E. (2018). Mathematics in the Modern World
(Outcome-Based Module). C&N Publishing, Inc. pp 1-11
Baltazar, E.C., Ragasa, C., and Evangelista, J., (2018) Mathematics in the
Modern World. C&N Publishing, Inc.
Baltazar, E.C., Ragasa, C., and Evangelista, J., (2018) Mathematics in the
Modern World. C&N Publishing, Inc.
Earnhart, R. and Adina, E. (2018). Mathematics in the Modern World
(Outcome-Based Module). C&N Publishing, Inc. pp 1-11
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Unit 9: Apportionment and Voting (8 hours)
Introduction
In this unit, we present various methods of apportionment, its properties and
steps involved in the implementation of each apportionment method. For
instance, in a city that requires representatives need to be apportioned among
the several groups according to their present numbers of population. The way
representatives are apportioned is a method of mathematical investigation in
dividing a whole proportional to its various parts.
In this unit, we hope you will also be interested in the value and mathematics
of voting which cover different weighted voting methods, the fairness criteria
and the Arrow’s theorem. This part of instructional module is designed to
guide you in your learning process in a casual way. Now you’ve got the
chance to study at your own pace. I hope that you would be able to accomplish
the objectives presented in this module.
Learning Outcomes
Upon the completion of this unit, you are expected to:
a. explain the meaning of apportionment;
b. identify types of apportionment and weighted voting systems; and
c. perform operations on apportionment and weighted voting.
Activating Prior Learning
True of False. If you think the statement is correct write TRUE otherwise write
FALSE.
_________a, There is no perfect apportionment method. A perfect
apportionment method is one which satisfies the fairness criteria and has no
paradoxes.
_________b. Suppose that one representative will be added to one of the state,
the state with the largest population should receive the new representative.
_________c. The Philippines is using majority for the voting system and not
plurality.
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Teacher Aides. A total of 25 faculty aides are to be apportioned among seven
classes at a certain university. The enrolment in each of the seven classes is
shown in the following table.
Class
Number of Students
38
Mathematics in the Modern World
39
Understanding the Self
35
Purposive Communication
27
Art Appreciation
21
Chemistry
31
Science, Technology and Society
33
Ethics
224
Total
a. Determine the standard divisor. What is the meaning of the standard
divisor in the context of this problem?
b. Use the Hamilton method to determine the number of teacher aides to be
apportioned to each class.
c. Use Jefferson method to determine the number of teacher aids to be
apportioned to each class. Is this apportionment in violation of the quota
rule?
d. How do the apportionment results produced using the Jefferson method
compare with the result produced using the Hamilton method?
Recreation: A company is planning its annual summer retreat and has asked
its employees to rank five different choices of recreation in order of
preference. The result are given in the table below.
Recreations
Picnic in a park
Water skiing at a lake
Amusement park
Riding horse at ranch
Dinner cruise
Number of votes
1
3
2
5
4
10
Rankings
2
1
1
2
5
5
4
3
3
4
18
6
3
4
1
5
2
28
4
3
2
1
5
16
a. Using the plurality voting system, what activity should be planned for the
retreat?
b. Use the plurality with elimination method to determine which activity
should be chosen.
c. Using the Borda count method of voting, which activity should be
planned?
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Topic 1: Apportionment
This topic introduces you to the basic idea of apportionment. What is
apportionment? What are the different methods of apportionment, and its
possible outcomes and behavior according to fairness criteria?
Learning Objectives
Upon the completion of this topic, you are expected to:
a. use each method of apportionment to distribute items fairly to several
groups of population;
b. identify which among the apportionment methods will satisfy the fairness
criteria; and
c. identify if an apportionment problem will result to some paradoxes.
Presentation of Content
Apportionment is the act of distributing by allotting or apportioning;
distribution according to a plan; "the apportionment of seats in the House of
Representatives is based on the relative population of each state"
wordnetweb.princeton.edu/perl/webwn.
Historically, it all started in the United States constitution. “Representatives…
shall be apportioned among the several states…according to their respective
numbers…”
For instance, in most representative governments, political power has most
recently been apportioned among constituencies based on population, but there
is a long history of different approaches. en.wikipedia.org/wiki/Apportionment
_(politics)/. The two most common apportionment plans were put forward by
Alexander Hamilton and Thomas Jefferson. Other methods of apportionments
are Adam’s method, Webster’s method and Huntington-Hill Method.
Here is the definition:
I. Hamilton Method
This method is based on standard divisor and standard quota of the population.
Standard Divisor is the quotient of the total population divided number of
allocated items/person.
𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒊𝒗𝒊𝒔𝒐𝒓 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑡𝑜 𝑎𝑝𝑝𝑜𝑟𝑡𝑖𝑜𝑛
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Exploring Mathematics in the Modern World
For allocating congressional seats to states based on population, the standard
divisor gives the number of people per seat in congress on a national basis.
Standard Quota is the whole number part of the quotient of a population
divided by the standard divisor.
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑔𝑟𝑜𝑢𝑝
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑖𝑣𝑖𝑠𝑜𝑟
Lower Quota – Standard quota, rounded down to the nearest whole number.
Upper Quota – Standard quota, rounded up to the nearest whole number.
𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒒𝒖𝒐𝒕𝒂 =
II. Jefferson Method
This method is based on a modified divisor, Dm, and a modified quota of the
population.
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑔𝑟𝑜𝑢𝑝
𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝑑𝑖𝑣𝑖𝑠𝑜𝑟
How to find the modified divisor? Here are the steps…
𝑴𝒐𝒅𝒊𝒇𝒊𝒆𝒅 𝒒𝒖𝒐𝒕𝒂 =
1.
2.
3.
4.
5.
Pick a Dm that is slightly less than the standard divisor.
Divide each group’s population by Dm.
Round down to the nearest whole number.
Find the sum of the whole number.
Is the sum of the number of items to be apportioned? Yes, STOP, you
found Dm.
6. If NO, change the value of Dm and repeat the steps 2-5.
Increase Dm if the sum is too high; decrease Dm if the sum is too low.
Let’s look for the first example…
Example 1: To illustrate how the Hamilton and Jefferson plans were used
to calculate the number of representatives each state should have, we will
consider the fictitious country of Atlantic, with a population of 20,000 and five
states. The population of each state is given in the table blow.
Population of Atlantic
State
Alto
Bajo
Viejo
Pardo
Sur
Total
Population
11,227
878
3,515
1,562
2,916
20,000
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Exploring Mathematics in the Modern World
a. The Hamilton Plan
Under the Hamilton plan, the total population of the country (20,000) is
divided by the number of representatives (25). This gives the number of
citizens represented by each representative.
For Atlantic, we have
𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒 𝑡𝑜 𝑎𝑝𝑝𝑜𝑟𝑡𝑖𝑜𝑛
𝟐𝟎, 𝟎𝟎𝟎
=
= 𝟖𝟎𝟎
𝟐𝟓
What is the meaning of the number 800 calculated above? It is the number of
citizens represented by each representative.
𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒊𝒗𝒊𝒔𝒐𝒓 =
Now divide the population of each state by the standard divisor and round the
quotient down to a whole number.
For example, both 13.1 and 13.9 would rounded to 13. Each whole number
quotient is called a standard quota.
Standard Quota is the whole number part of the quotient of a population
divided by the standard divisor.
Apportionment Table of Atlantic
State
Population
Quotient
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
(
)
𝑫
11,207
11,201
Alto
≈ 14.009
800
874
874
Bajo
≈ 1.093
800
3516
3,516
Viejo
≈ 4.395
800
1502
1,502
Pardo
≈ 1.878
800
2,917
2,907
Sur
≈ 3.646
800
Total
20,000
Standard
Quota
No. of
Rep
14
14
1
1
4
4
1
2
3
4
23
25
The total number of representatives is 23, not 25 as required by Atlantic’
constitution. When this happens, the Hamilton plan calls for revisiting the
calculation of the quotients and assigning an additional representative to the
state with the largest decimal remainder. This process is continued until the
number of representatives equals the number required by the constitution. We
must add two more representatives, the state with the two highest decimal
remainders are Pardo (1.878) and Sur (3.646). Thus each of these states gets
an additional representative.
b. The Jefferson Plan
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Exploring Mathematics in the Modern World
As we saw with the Hamilton plan, dividing by the standard divisor does not
always yields the correct number of representatives short by two. The
Jefferson plan attempts to overcome this difficulty by using a modified
standard devisor. This number is chosen, by trial and error, so that the sum of
the standard quotas is equal to the total number of representatives. In a specific
apportionment calculation, there may be more than one number that can serve
as the modified standard divisor. In our case Dm is 740. So that we have the
Jefferson apportionment table.
State
Population
Quotient
Standard
quota of
Jefferson
Plan
15
1
4
2
3
25
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
(
)
𝑫𝒎 = 𝟕𝟒𝟎
15.136
1.181
4.751
2.029
3.928
11,201
Alto
874
Bajo
3,516
Viejo
1,502
Pardo
2,907
Sur
Total
20,000
Dm =Modified Standard Divisor
Now, we compare the two methods of apportionment. Can you spot the
difference?
Hamiltonian versus Jefferson Plan
State
Alto
Bajo
Viejo
Pardo
Sur
Total
Population
11,201
874
3,516
1,502
2,907
20,000
Hamilton Plan
14
2
4
2
3
25
Jefferson Plan
15
1
4
2
3
25
Example 2: Apportioning Board Members Using the Hamilton and Jefferson
Methods
Suppose the 18 members on the board of Electric Cooperative are selected
according to the populations of the five cities in the province.
a. Use the Hamilton method to determine the number of board members each
city should have.
b. Use the Jefferson method to determine the number of board members each
city should have.
City
A
B
C
Population
7020
2430
1540
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3720
5290
D
E
Solution:
a. First find the total population of the cities.
7020 + 2430 + 1540 + 3720 + 5290 = 20,000
b. Calculate the standard divisor
20000
𝑆𝐷 =
= 1111.11
18
Here is the apportionment table solution for Hamilton Plan
City
Population
Quotient
Standard
Quota
A
7020
6
B
2430
2
2
C
1540
1
2
D
3720
3
3
E
5290
7020
1111.11
= 6.318
2430
1111.11
= 2.187
1540
1111.11
= 1.386
3720
1111.11
= 3.348
5290
1111.11
= 4.761
Number
of Board
Members
6
4
5
Total
20,000
16
18
Apportionment Solution Table by Jefferson Method
City
Population
Quotient
A
7020
B
2430
C
1540
7020
925
= 7.589
2430
925
= 2.627
1540
925
= 1.665
Number
Quotient
Number
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
of Board
of Board
Members ( 𝑫𝒎 = 𝟗𝟓𝟎 ) Members
7020
7
7
= 7.389
950
2
2430
= 2.558
950
2
1
1540
= 1.621
950
1
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Exploring Mathematics in the Modern World
3720
925
= 4.022
5290
925
= 5.719
D
3720
E
5290
Total
20,000
4
3720
= 3.916
950
3
5
5290
= 5.568
950
5
19
18
This result in the 4th column yields too many board members. Thus, we must
increase the modified standard divisor. By experimenting with different
divisors, we find that 950 gives the correct number of board members.
Now, I believe you are ready to answer the following Activity.
Application
Activity 9.1:
Apportionment
1. Suppose the 20 members of a committee from five Asian countries are
selected according to the populations of the five countries, as shown
Country
Indonesia
Philippines
Thailand
Malaysia
Singapore
Population
269,536,000
108,106,000
69,306,000
32,454,000
5,868,000
a. Use the Hamilton method to determine the number of representatives
each country should have.
b. Use Jefferson method to determine the number of representatives each
country should have.
2. A university is composed of four colleges. The enrollment in each
college is given in the following table.
College
Enrollment
Arts and
Science
1250
Engineering
985
Computing
Sciences
1420
Industrial
Technology
1595
There are 350 new computers to be apportioned among the four colleges
according to their respective enrolments. Use Hamilton’s method and
Jefferson’s method to find each college’s apportionment of computers.
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That was a little difficult, wasn’t it? Well, the purpose of the activity was to
let you understand the two main apportionment method. Please continue
reading the rest of the unit. In particular, other methods of apportionments.
Other Methods of Apportionment
III. Adam’s Method of Apportionment
If the sum of the upper quotas does not equal the correct number of
representatives. Use a modified standard divisor (Dm) that yields the correct
number of representatives by trial and error.
Choose Dm greater than D such that the sum of the upper quotas equals the
number of representatives.
Example 2: Solve Example 1 using Adam’s method. We need 25
representatives.
Standard
State Population
Quotient
Quotient
Number
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
quota
of Rep.
(
)
(
)
𝑫 = 𝟖𝟎𝟎
Alto
Bajo
Viejo
Pardo
Sur
Total
11,201
874
3,516
1,502
2,907
20,000
14.009
1.093
4.395
1.878
3.646
15
2
5
2
4
28
𝑫𝒎 = 𝟖𝟕𝟓
Adam
12.801
0.999
4.018
1.717
3.322
13
1
5
2
4
25
IV. Webster’s Method of Apportionment
This method is a variation of the Jefferson plan and Adam’s method.
Instead of using the lower quota or the upper quota, use the regular rules of
rounding to determine the regular quota (R).
Choose modified standard divisor Dm such that the sum of the regular
quotas equals the number of representatives by trial and error.
Example 3: Solve Example 1 using Webster’s method. We need 25
representatives.
Population
State
Quotient
Standard
No. of Rep.
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
Webster
Method
quota
(
)
𝑫 = 𝟖𝟎𝟎
Alto
Bajo
Viejo
Pardo
Sur
Total
11,201
874
3,516
1,502
2,907
20,000
14.009
1.093
4.395
1.878
3.646
188
15
2
5
2
4
28
14
1
4
2
4
25
Exploring Mathematics in the Modern World
V. Huntington-Hill Number
When there is a need to add one item to one of the several groups in a population,
the additional one item is given to the group with the highest Huntington-Hill
number.
𝑃2
The formula is given by 𝐻 = 𝐴(𝐴+1), where P is the size of the group and A is
the current number of items assigned to this group.
Example 4: Solve Example 1 when there is a need to add one
representative/item so that we have 26 representatives. To which state a new
representative should be assigned using Huntington-Hill method?
State
Population
Quotient
Standard
No. of Rep
𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏
Quota
(
)
𝑫
11,201
874
3,516
1,502
2,907
20,000
Alto
Bajo
Viejo
Pardo
Sur
Total
14.009
1.093
4.395
1.878
3.646
14
1
4
1
3
23
14
1
4
2
4
25
Solution
Using the formula for the Huntington-Hill number we obtain the following.
11,2012
8742
𝐻1 = 14(14+1) = 597,440 , 𝐻2 = 1(1+1) = 381,938 ,
3,5162
1,5022
𝐻3 = 4(4+1) = 618,112.8 , 𝐻4 = 2(2+1) = 376,000.7
2,9072
𝐻5 = 4(4+1) = 422,532.5
The state of Viejo has the greatest Huntington-Hill number. Thus, the additional
representative should be given to the state of Viejo.
Activity 9.2: Other Methods of Apportionment
A. The following table shows the enrollments for the four division of a
college. There are 50 new overhead projectors that are to be apportioned
among the divisions based on the enrollments.
Division
Population
1,280
3,425
1,968
2,936
9609
Health
Business
Engineering
Science
Total
1. Use the Hamilton method to determine the number of projectors to be
apportioned to each division.
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Exploring Mathematics in the Modern World
2. Use the Jefferson method to determine the number of projectors to be
apportioned to each division.
3. Use the Webster method to determine the number of projectors to be
apportioned to each division.
B. The following table shows the thirty sections of math courses are to be
offered in introductory algebra, intermediate algebra, college algebra and literal
arts math. The preregistration figures for the number of students planning to
enrol in their sections are given in the following table.
Course
Intro
Algebra
382
9
Intermediate
Algebra
405
9
College
Algebra
213
5
Liberal
Arts Math
345
7
Enrollment
Number of
Sections
To which course a new section should be assigned using Huntington-Hill
method?
What is Fairness in Apportionment?
Apportionment that satisfies the quota criterion. What is quota criterion?
Quota Criterion-The number of allocations given to a subset of a population is
the standard quota or no one more than the standard quota.
The Hamilton method always satisfies the quota criterion. Both Jefferson and
Webster may violate this criterion because of the way the modified standard
divisor is chosen. Even Huntington-Hill method may also violate quota
criterion.
What are the flaws of Apportionment Methods?
1. Alabama Paradox-An increase in the total number of items to be
apportioned results in the loss of an item for a group.
2. Population Paradox-Group A loses items to Group B, even though the
population of group a grew at a faster rate than that of group B.
3. New–States Paradox-The addition of a new group changes the
apportionments of other methods.
What is Balinski and Young’s Impossibility Theorem?
There is no perfect apportionment method. It is impossible to develop an
apportionment method that can satisfy the quota rule and the same time avoid
all types of paradoxes. Any apportionment method that does not violate the
quota rule must produce paradoxes, and any apportionment method that does
not produce paradoxes must violate the quota rule.
Hamilton’s Method (favors larger subgroups) is the only method that satisfies
the quota rule but produces Alabama paradox, population paradox, and newstates paradox.
Jefferson’s Method (favors larger states) produces no paradoxes but violates
the quota (upper) rule. The dilemma of this method is there is no formula for
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Exploring Mathematics in the Modern World
the modified divisor, Dm; trial and error must be used. Usually, there is more
than one Dm that will work.
Adam’s Method (favors smaller states) is a mirror image of Jefferson’s method
so they have the same advantages and disadvantages?
Webster’s Method (favors smaller states) as the best overall apportionment
method according to experts because it produces no paradoxes with rare
violation of the quota (upper and lower) rule. There is not a formula for the
modified divisor, trial and error must be used.
You have done so much at this point. You are entitled to some rest before you
proceed. Why don’t you take a short break and then come back to finish the
unit module
Topic 2: Introduction to Voting System
We know you are now ready to grasp this concept of voting. The right to vote
is one of the most valued privileges you enjoyed in this life of democracy.
Sometimes, we are puzzled by the fact that those best candidate did not get
elected maybe because of the way our plurality voting system works, or is it
possible to elect someone or pass a proposition that has less than majority
support?
As we proceed through this section, we will look at the definitions of voting,
and some of the voting systems. You may ask yourself what type of voting
system our country is practicing-is it plurality voting or the majority system?
Learning Objectives
Upon the completion of this topic, you are expected to:
a. use each voting methods to determine the winner among candidates or
options; and
b. identify which among the voting methods will satisfy the fairness criteria.
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Exploring Mathematics in the Modern World
Presentation of Content
I. What is voting?
Voting is a tool used by group of people in making a united decision. It can be
presented suitably using election system where one select one particular
candidates out of a set of candidates on the basis of ballots cast by voters.
What is Majority System of Voting?
It is the most common voting system applied on an election with two
candidates only. Majority means that candidate with more than 50% of the
votes. For instance, in an election process consisting of two candidates and
100 voters, the candidate with 51 votes or higher wins the election. Often,
there is not a majority winner.
What is Plurality of Voting?
Each voter votes for one candidate, and the candidate with the most votes or
having the most number of first-place votes is declared winner. The winning
candidate does not have to have a majority of the votes.
Preference ballot – a type of ballot structure used in several electoral systems
in which voters rank a list or group of candidates in order of preference.
Take a glimpse of preference ballots. Have you seen one?
Written numbers, Column marks, Written names and Touch screen.
https://en.wikipedia.org/
wiki/File:Preferential_ballo
t.svg
https://en.wikipedia.org/
wiki/File:Rankballotoval.gi
f
https://en.wikipedia.org/
wiki/File:Rankballotname2.
gif
https://en.wikipedia.org/
wiki/File:Rankballottouch.gif
Methods to determine the outcome of an election from a preference table:
1. Plurality method
2. Borda count Method
3. Plurality-with-elimination method
4. Top-Two Runoff Method
5. Pairwise comparison method
Example 1: There are three candidates running for the Student Government
Association: Alan (A), Bravo (B) and Christian (C).The preference ballots for
the three candidates are shown. Fill in the number of votes in the first row of
the given preference table.
BAC ABC CAB CAB ABC CAB BAC CAB
BAC CBA BAC ABC ABC CAB ABC CBA
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BAC
ABC
CAB
BAC
ABC
ABC
CAB
ABC
CAB
BAC
CBA
BAC
CBA
CAB
CAB
CAB
ABC
CBA
ABC
ABC
BAC
ABC
Rank
First Choice
Alan
Christian
Bravo
Christian
Second Choice
Bravo
Alan
Alan
Bravo
Third Choice
Christian
Bravo
Christian
Alan
13
11
9
5
Number of Voters
a.
b.
c.
d.
How many students voted in the election?
How many students voted Alan as their first choice?
How many people selected candidates in the order BAC?
Who will win the presidency using the Plurality method?
Solution:
a. Sum of all number of votes: Add the row/column totals for each
preference ballot. 13+11+9+5=38 voters.
b. Getting the sum of all first place votes, we obtain the following table.
Candidate Alan got 13 votes.
Candidate
Rankings
Total First Place
Votes
Alan
1
2
2
3
13
Bravo
2
3
1
2
9
Christian
3
1
3
1
11+5 = 16
Number of Voters
13
11
9
5
38
c. Using the table below, there are 9 voters who selected candidates in the
order Bravo-Alan-Christian.
Rank
First Choice
Alan
Christian
Bravo
Christian
Second Choice
Bravo
Alan
Alan
Bravo
Third Choice
Christian
Bravo
Christian
Alan
Number of Voters
13
11
9
5
d. We can see that candidate Christian has the most number of first place
votes. Thus, Christian wins in this election according to the plurality
method. However, Christian did not get the majority of the votes (at
least 20 votes).
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Exploring Mathematics in the Modern World
Example 2: Determine the Winner Using Plurality Voting
Fifty people were asked to rank their preferences of five varieties of
chocolate candy, using 1 for their favourite and 5 for their least favourite.
This type of ranking of choices is called a preference schedule.
Varieties of Chocolate Candy
Rankings
Caramel Center
5
4
4
4
2
4
Vanilla Center
1
5
5
5
5
5
Almond Center
2
3
2
1
3
3
Tofee Center
4
1
1
3
4
2
Solid Chocolate
3
2
3
2
1
1
Number of Voters
17
11
9
8
3
2
To answer the question, we will make a table showing the number of firstplace votes for each candy.
Caramel Center
First-place votes
0
Vanilla Center
17
Almond Center
8
11+9=20
Tofee Center
3+2=5
Solid Chocolate
We can see that Toffee Center has the most number of first-place votes with a
sum of 20. Thus, the winner is Toffee center using the plurality method.
However, Christian did not get the majority of the votes (at least 20 votes).
II. Borda Count Method of Voting
In this method each candidate is assigned a weight according to the voter’s
preferences.
If there are n candidates or issue in an election, each voter ranks the candidates
or issues by giving n points to the voter’s first choice, n-1 points to the voter’s
second choice, and so on, with the voter’s least favourite choice receiving 1
point. The candidate or issue that receives the most total points is the winner.
Example 3: Using Example 1, who will win the presidency using the Borda
Count method?
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Exploring Mathematics in the Modern World
Solution:
Rank
First Choice
3 pts
Second Choice
2 pts
Third Choice
1 pt
Number of
A
C
B
C
3*13=39
3*11=33
3*9=27
3*5=15
B
A
A
B
2*13=26
2*11=22
2*9=18
2*5=10
C
B
C
A
1*13=13
1*11=11
1*9=9
1*5=5
13
11
9
5
Voters
Following the weights over 38 ballots, we arrived with the Borda counts for
the following candidates. Alan’s count: 39+22+18+5=94 points, Bravo’s count
is 27+26+10+11 = 74 points, and Christian’s count is 33+15+13+9=70 points.
The candidate with the largest Borda count is candidate Alan. The result is
different from Plurality of voting.
Illustration 4:
Applying the Borda count method to the education measures, a measure
receiving a first-place vote receives 3 points (there are three different
measures). Each measure receiving a second-place vote receives 2 points, and
each measure receiving a third-place vote receives 1 point. The calculations
are show below.
Measure A:
Measure B:
15- 1st-place votes:
12-1st-place votes:
15x3 = 45
12x3 = 36
0 - 2nd-place votes:
9-2nd-place votes:
0x2 = 0
9x2 = 18
21 -3rd-place votes:
15-3rd-place votes:
21x1=21
15x1=15
Total: 66
Total: 69
The largest Borda count is Measure C.
Measure C:
9-1st-place votes:
9x3 = 27
27-2nd-place votes:
27x2 = 54
0-3rd-place votes:
0x1=0
Total: 81
III. Plurality with Elimination Method
This method is a variation of the plurality method of voting. Like the Borda
count method, the method of plurality with elimination considers a voter’s
alternate choices. Candidate with the majority (over 50%) of first-place votes
is the winner. Instead of calculating 50%, you can count the number of votes
received.
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Exploring Mathematics in the Modern World
a.
b.
c.
d.
Find the total number of votes.
Divide the total by 2.
Round up.
You must have more votes than the number in (c) to have a
majority.
If no candidate receives a majority, eliminate the candidate with the fewest
first-place votes. If there is a tie for the fewest votes, eliminate all tied
candidates.
Either hold another election or adjust the preference table by moving the
candidates in each column below the eliminated candidate up one place.
Continue this process until a candidate receives a majority of first-place votes.
Example 5: I. The members of a club are going to elect a president. If the 101
members of the club mark their ballots as shown in the table below, who will
be elected president using the Plurality-with-Elimination method?
Candidate
Alma
Brando
Chito
Denver
Number of Voters
2
1
3
4
30
Rankings
2
2
4
3
1
4
3
1
18
12
2
4
3
1
24
3
2
1
4
10
2
1
4
3
7
Solution
For round 1, candidate Alan should be eliminated.
Total First place Votes
0
37
28
36
Alma
Brando
Chito
Denver
Round 2
Candidate
Brando
Chito
Denver
No. of Votes
Rankings
1
2
3
30
3
2
1
24
3
1
2
18
2
3
1
12
196
2
1
3
10
1
3
2
7
Total First Place
Votes
37
28
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Exploring Mathematics in the Modern World
Candidate Chito has the lowest first-place vote and should be eliminated.
Brando has the highest first-place vote. Thus, the winner for this method is
candidate Brando.
IV. The Top-Two Runoff Method
The two candidates with the most number of first-place votes are
removed from the preference list and then are re-ranked for a new preference
list. The one with the highest first-place votes in the new preference list
between these two candidates will be declared the winner. The top-two run off
method satisfies the majority criterion.
Example 6: Use Example 5 to apply the Top-Two runoff method to the
preference list.
Solution:
Counting the total first place votes for each candidate, we obtain the following
table
Candidate
Alma
Brando
Chito
Denver
No. of Votes
Rankings
2
1
2
3
30
2
3
2
1
24
2
3
1
2
18
2
2
3
1
12
Total First Place
Votes
3
2
1
3
10
2
1
3
2
7
0
37
28
36
The top two candidates with the most number of first-place votes are
candidates Brando and Denver. Thus we remove candidates Alma and Chito
from the list. This gives us the following preference list.
Candidate
Rankings
Total First Place
Votes
Brando
Denver
No. of Votes
1
2
30
2
1
24
2
1
18
2
1
12
1
2
10
1
2
7
30+10+7=47
24+18+12=54
Thus, candidate Denver wins this election. Note that in the Plurality method
candidate Brando wins the election.
V. Pairwise Comparison Voting Method
This method of voting is sometimes referred to as “head –to –head” method.
In this method, each candidate is compared one-on-one with each of the other
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Exploring Mathematics in the Modern World
candidates. In an election with n candidates, the number of comparison (C)
that must be made is 𝐶 =
𝑛(𝑛−1)
2
. A candidate receives 1 point for a win, 0.5
points for a tie, and 0 points for a loss. The candidate with the greatest number
of points wins the election.
Example 7. Using the Pairwise Comparison Voting Method, what is the
prefered name of the new foot ball stadium by the alumni and students?
There are four proposals for the name of a new foot ball stadium at a college:
Panther Stadium, after the team mascot; Sanchez Stadium, after a large
university contributor; Mosher Stadium, after a famous alumnus known for
humanitarian work; and Fritz satdium, after the college’s most winning football
coach. The preference schedule cast by alumni and students is shown below.
2
1
3
4
752
Panther Stadium
Sanchez Stadium
Mosher Stadium
Fritz Stadium
Numbr of Ballots
Rankings
3
1
4
2
1
4
2
3
678
599
2
4
3
1
512
4
3
2
1
487
Solution
We will create a table to keep track of each of the head-to-head comparisons.
Before we begin note that a matchup between, say Panther and Sanchez is the
same as the match up between Sanchez and Panther. Therefore, we will shade
the duplicate cells and the cells between the same candidates. This is shown
below.
Versus
Panther
Sanchez
Mosher
Fritz
Panther
Sanchez
Mosher
Fritz
To complete the table, we will place the name of the winner in the cell of each
head-to head match. For instance, for the Panther-Sanchez matchup,
1. Panther was favoured over Sanchez on 678+599+512=1789 ballots.
Sanchez was favoured over Panther on 752+487=1239 ballots.
The winner of this matchup is Panther, so that the name is placed in the Panther
versus sanchez cell. D othis for each of the match ups.
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Exploring Mathematics in the Modern World
Versus
Panther
Sanchez
Mosher
Fritz
Panther
Sanchez
Panther
Mosher
Panther
Sanchez
Fritz
Fritz
Fritz
Fritz
2. Panther was favoured over Mosher on 752+599+512=1863 ballots.
Mosher was favoured over Panther on 678+487=1165 ballots.
The winner of this matchup is Panther.
3. Panther was favoured over Fritz on 752+599=1351 ballots.
Fritz was favoured over Panther on 678+512+487=1677 ballots.
The winner of this matchup is Fritz.
4. Sanchez was favoured over Mosher on 752+599+487=1838 ballots.
Mosher was favoured over Sanchez on 678+512=1190 ballots.
The winner of this matchup is Sanchez.
5. Sanchez was favoured over Fritz on 752+599=1351 ballots.
Fritz was favoured over Sanchez on 678+512+487=1677 ballots.
The winner of this matchup is Fritz.
6. Mosher was favoured over Fritz on 752+678=1430 ballots.
Fritz was favoured over Mosher on 599+512+487=1598 ballots.
The winner of this matchup is Fritz.
From the above table, Fritz has three wins, Panther has two wins, and Mosher
has one win. Using pairwise comparison, Frit Stadium is the winning name.
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Exploring Mathematics in the Modern World
Example 8: Using Example 1, who will win the presidency using the
Pairwise comparison method?
Candidate
Rankings
Alan
1
2
2
3
Bravo
2
3
1
2
Christian
3
1
3
1
Number of Votes
13
11
9
5
Solution
The table below shows the head-to-head comparisons match up between
candidates.
Versus
Alan
Bravo
Christian
Alan
Bravo
Alan
Christian
Alan
Bravo
1. For Alan-Bravo matchup, Alan was favoured over Bravo on 13+11=24 votes,
Bravo was favoured over Alan on 9+5=14 votes. Alan wins the matchup.
2. Alan was favoured over Christian on 13+9=22 votes. Christian was favoured
over Alan on 11+5=14 votes. The winner of this matchup is Alan again.
3. Bravo was favoured over Christian on 13+9=22 votes. Christian was favoured
over Bravo on 11+5=14 votes. Bravo wins this matchup.
From the above table, Alan has two wins and Bravo has one win. Using pairwise
comparison, Alan wins the election.
What is fairness Criteria in voting? Requirements a fair voting system must
meet.
The following are different ways to define fairness in voting options.
1. Majority Criterion-if the winning candidate receives a majority of the
first-place votes in an election, then that candidate should win the election.
2. Monotonicity Criterion -If a candidate wins an election and also in a reelection where the changes that favor the candidate, then that candidate
should win the re-election.
3. Condorcet Criterion-If a candidate is favoured when compared
separately (head-to-head) with every other candidate, then that candidate
should win the election.
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4. Irrelevant Alternatives Criterion-a winning candidate in an election
remains the winner in any recount even if the losing candidate withdraw
from the election.
What is Arrow’s Impossibility Theorem?
This theorem states that it is mathematically impossible to develop a voting
system that satisfies each of the four fairness criteria.
Performance Summary of the Voting Methods
The following table summarizes the performance of each voting method. The
input “yes” means that the method satisfies the said fairness criteria.
Plurality Elimination Borda
Top-
Pairwise
Count
Two
Comparison
Runoff
Majority
yes
yes
no
yes
yes
Monotonicity
yes
no
yes
no
yes
Condorcet
no
no
no
no
Yes
Irrelevant
no
no
no
no
no
Alternatives
Now, I believe you are ready to answer the following Assessment Questions.
Assessment
1. The members of a club are going to elect a president from four nominees
using the Borda count method. If the 100 members of the club mark their
ballots as shown in the table below, who will be elected president?
Candidate
Alma
Brando
Chito
Denver
Number of Voters
2
1
3
4
30
2
4
3
1
24
201
Rankings
2
2
4
3
1
4
3
1
18
12
3
2
1
4
10
2
1
4
3
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2. Using the Plurality with Elimination Voting Method
A university wants to add a new sport to its existing program. To help ensure
that the new sport will have student support, the students of the university are
asked to rank the four sports under consideration. The results are shown in the
following table.
Ranks
Lacrosse
3
2
3
1
1
2
Squash
2
1
4
2
3
1
Rowing
4
3
2
4
4
4
Golf
1
4
1
2
2
3
326
297
287
250
214
197
Number of ballots
3. Suppose that 30 members of a regional planning board must decide where
to build a new airport. The airport consultants to the regional board have
recommended four different sites. The preference schedule for the board
members is shown in the following table. Apply the Top-Two runoff method
to the preference list.
Aparri
Ballesteros
Lallo
Sta-Ana
Number of Ballots
3
2
1
4
12
Ranking
1
3
2
4
11
2
3
4
1
5
3
1
2
4
2
4. Use the Pairwise Comparison Voting Method
A service club is going to sponsor a dinner to raise money for a charity. The
club has decided to serve Italian, Mexican, Thai, Chinese, or Indian food. The
members of the club were surveyed to determine their preferences. The result
are shown in the table below.
Rankings
Italian
2
5
1
4
3
Mexican
1
4
5
2
1
Thai
3
1
4
5
2
Chinese
4
2
3
1
4
Indian
5
3
2
3
5
Number of Ballots
33
30
25
20
18
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5. There are four candidates are running for pesident of the math club: Peter(P),
Sharon (S), Carol (C) and Brent (B). The result of the election are shown in the
following prefernce table:
Number of Votes
First Choice
Second Choice
Third Choice
Fourth Choice
15
B
S
P
C
19
C
P
S
B
23
P
S
B
C
10
P
B
C
S
18
S
C
P
B
15
B
S
C
P
a. How many students voted in theelcetion?
b. How many steudtns voted Brent as their first choice?
c. Who would win the presidency using the Plurality method?
d. Who would win the presidency using the Borda count method?
e. Who would win the presidency using the Plurality with elimination method?
f. Who woud win the presidency using the pairwise comparison method? If a
tie, use the plurality method between the winners o determine the tie breaker.
You have done so much at this point. You are entitled to some rest before
you proceed. Why don’t you take a short break and then come back to finish
the unit module?
Topic 3: Weighted Voting System
Learning Objectives
Upon the completion of this topic, you are expected to:
a. determine the winning coalitions in a weighted voting system;
b. determine the critical voter; and
c. compute the Banzhaf power index and use this value to determine the
voter’s power.
Presentation of Content
Weigthed Voting Systems is a Biased Voting System
A weigthed voting system is one in which some voters have more wieght on the
outcome of an election. It is a voting system where voters are not necessarily
equal in the number of votes they control.
A few examples are:
a. Stockholders/shareholders of a company: the more stock you own,
the more say you have in decision making for the company.
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b. The United Nations Security Council – 15 voting nations: 5
permanent members (Britain, China, France, Russia, United States), 10
nonpermanent members appointed for a 2-year rotation. Permanent members
have more “votes” than non permanent members.
c. The Electoral College—Each state gets a number of “electors” (votes)
equal to the number of Senators plus the number of Representatives in Congress.
California has 55 votes but North Dakota only has 3 votes. Each state is a
voter but states with heavy concentration of population receive a bigger “vote”.
Notation: 𝑊 = {𝑄|𝑤1 , 𝑤2 , … , 𝑤𝑛 }
Where Q is the quota, which his the required number of votes to pass a
resolution and 𝑤𝑖 is the weight of a voter which corresponds to the amount of
votes controlled by a voter.
Example: 𝑊 = {14|8, 6,5,1}.
Quota = 14, total votes =8+6+5+1=20
Player 1(𝑃1 )= controls 8 votes or has a wieght of 8.
Player 2(𝑃2 )= controls 6 votes or has a wieght of 6.
Player 3(𝑃3 )= controls 5 votes or has a wieght of 5.
Player 4(𝑃4 )= controls 1 votes or has a wieght of 1.
Types of Weigthed Voting Systems
1. One Person-One Vote System
Each person has only one vote. Thus, a majority of the votes is required
to pass resolution.
Example: 𝑊 = {𝑄|1, 1,1,1,1,1}
Since the toal number of votes is 7 then Q=4 is the required number of
votes to pass a resolution. Simple majority which is 50% +1.
2. Dictatorship
One particular person has weight that is greater than the quota and sum
of all the wieghts of other voters.
Example: 𝑊 = {10|11,5,3,1,1}
Person 𝑃1 with 11 as wieght of vote will always dictate any voting
outcome. Note that even if we combine the other votes, it will not exceed
the said qouta Q=10.
A dummy is a voter whose wieght does not affect any voting outome.
In our example above 𝑃2 , 𝑃3 , and 𝑃4 are dummies.
Note that when a system has a dictator, all other playes are considered
dummies.
3. Null System
This is a weigthed voting system which cannot pass any resolution
because the sum of all the votes is always less than the quota.
Example 3: 𝑊 = {17|2,1,8,3,2}
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Impossible to win since the sum of votes 𝑃1 + 𝑃2 + 𝑃3 + 𝑃4 =
2+1+8+3+2=16 less than the Quota=17.
4. The Veto Power System
This is one type of weigthed voting system where each voter has a veto
power meaning if one voter does not vote no resolution will be passed.
This tye of voting system will occur when quota is equal to the sum of
all the votes.
If at least one voter in a wiegthed voting system has a veto power, the
sytem is said to be a veto power system.
Example: 𝑊 = {18|4,5,1,2,6}
Here, ∑ 𝑊 = 4+5+1+2+6=18 which is equal to the quota Q=18.
Example: 𝑊 = {12|9,5,4,2}
There is no dictator. if P1 chooses to vote against the motion can the
other player combine weight to meet the qouta? ∑ 𝑊 = 𝑃2 + 𝑃3 + 𝑃4 =
11 too low and cannot pass motion without 𝑃1 . Thus, 𝑃1 has a veto
power.
5. Coalition
A coalition in a voting system is an alliance formed by a group of voters
with a common goal which is either to favor a resolution or a vote
against it.
We define the following terms related to coalition
A winning coalition is a group of voters whose sum of all votes is greater than
or equal to the quota
A losing coalition is group of voters whose sum of all votes is less the quota
A critical voter is a voter who must be present in a winning coalition in order
for it to remaining a winning coalition.
What is the number of possible coalitions given the number of voters?
The number of possible coalition of voters is the number of possible subsets that
can be formed from these voters. This includes the set containing all the voters
and the singleton subsets.
Example 1: Find the critical voters/players in each of the following
coalitions.
Compute the voting power of each voters. The weigthed voting system for a
company owned by 4 people is given by
𝑊 = [𝟏𝟓: 𝟏𝟑, 𝟗, 𝟓, 𝟐]
a. Find all the winning coalitions.
b. For each winning coalition, determine the critical voters/players.
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Solution.
13+9+5+2
We have, 𝑄 =
= 14.5 ≈ 15
2
Owners
𝑷𝟏
𝑷𝟐
𝑷𝟑
𝑷𝟒
Shares
13
9
5
2
The total possible coalitions are 24 − 1 = 15
The winning coalitions (WC) are those combinations whose total votes is
greater than the quota 15. These are{𝑃1 𝑃2 },{𝑃1 𝑃3 }, {𝑃1 𝑃4 },{𝑃1 𝑃2 𝑃3 },{𝑃1 𝑃2 𝑃4 },
{𝑃1 𝑃3 𝑃4 }, {𝑃2 𝑃3 𝑃4 } and {𝑃1 𝑃2 𝑃3 𝑃4 } .
Number
Coalitions
Total
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
𝑃1
𝑃2
𝑃3
𝑃4
𝑃1 𝑃2
𝑃1 𝑃3
𝑃1 𝑃4
𝑃2 𝑃3
𝑃2 𝑃4
𝑃3 𝑃4
𝑃1 𝑃2 𝑃3
𝑃1 𝑃2 𝑃4
𝑃1 𝑃3 𝑃4
𝑃2 𝑃3 𝑃4
𝑃1 𝑃2 𝑃3 𝑃4
Quota
13
9
5
2
22
18
15
14
11
7
27
24
20
16
29
15
Winning
Coalition
Critical
Voters/Players
yes
yes
yes
𝑃1 & 𝑃2
𝑃1 & 𝑃3
𝑃1 & 𝑃4
yes
yes
yes
yes
yes
𝑃1
𝑃1
𝑃1
𝑃2 , 𝑃3 & 𝑃4
none
Banzhaf Power Index
How a voting power is measured in a weighted voting system.
A player’s power is proportional to the number of coalitions for which that
player/voter is critical. The more often a player is critical, the more power he
holds.
The voting power of a player is measured by Banzhaf power index and it is
defined as follows:
𝐵=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎 𝑣𝑜𝑡𝑒𝑟 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑙𝑎𝑦𝑒𝑟
𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑙𝑎𝑦𝑒𝑟𝑠 𝑖𝑛 𝑎𝑙𝑙 𝑤𝑖𝑛𝑛𝑖𝑛𝑔 𝑐𝑜𝑎𝑙𝑖𝑡𝑖𝑜𝑛𝑠
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Example 2: W = {5: 3,2,2} which player has the most power?
Solution:
From a Banzaf point of view:
Look at all winning coalitions and find the critical players in each
Number
Coalitions
Total
1
2
3
5
6
8
11
𝑃1
𝑃2
𝑃3
𝑃1 𝑃2
𝑃1 𝑃3
𝑃2 𝑃3
𝑃1 𝑃2 𝑃3
Quota
3
2
2
5
4
4
7
5
𝐵=
Winning
Coalition
Critical
Voters/Players
yes
yes
𝑃1 & 𝑃2
𝑃1 & 𝑃3
yes
𝑃1
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎 𝑣𝑜𝑡𝑒𝑟 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑙𝑎𝑦𝑒𝑟
𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑙𝑎𝑦𝑒𝑟𝑠 𝑖𝑛 𝑎𝑙𝑙 𝑤𝑖𝑛𝑛𝑖𝑛𝑔 𝑐𝑜𝑎𝑙𝑖𝑡𝑖𝑜𝑛𝑠
3
1
1
𝐵(𝑃1 ) = 5 = 60%, 𝐵(𝑃2 ) = 5 = 20% and 𝐵(𝑃3 ) = 5 = 20%.
The most power voter is P1.
Congratulations!
You really worked hard, I can tell. I hope that you will continue to progress
and be encouraged by the additional knoweldge that you are gaining from the
modules.
Application
A. Four partners decide to start a business: 𝑃1 buys 8 shares, 𝑃2 buys 7
shares, 𝑃3 buys 3 shares and 𝑃4 buys 2 shares.
a. The quota is set two-thirs of the total number of votes.
Describe as a weighted voting system.
b. The partnership above decides to make a quota of 19 votes.
B. A committee has 4 members (P1, P2, P3, P4). P1 has twice as many
votes as P2. P2 has twice as many votes as P3. P3 and P4 have the
same number of votes. The quota is 49.
Describe the weighted
voting system using the notation [q: w1, w2, w3, w4] given the
definitions of quota below. (Hint: write the weighted voting system
as [49: 4x, 2x, x, x] and then solve for x.
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a) The quota is a simple majority
b) The quota is more than three-fourths
Assessment
A. Given the weighted voting system:
𝑊 = [16: 8,6,4,4,1] state the following:
a. The number of players:
b. The weight of P5.
c. The total number of votes
d. The minimum % of the quota.
B. Determine which players, if any, are:
a) [15: 16, 8, 4, 1]
[24: 16, 8, 4, 1]
dictators, veto power, dummies
b)
[18: 16, 8, 4, 1]
c)
C. Compute the voting power of each voters. The wiegthed voting system
for a company owned by 4 people is given by
𝑊 = [601: 425,250,175,350]
a. Find all the winning coalitions.
b. For each winning coalition, determine the criticla
voters/players.
Solution.
D. The weigthed voting system for a company owned by 4 people is given
by
𝑊 = [𝟏𝟓: 𝟏𝟑, 𝟗, 𝟓, 𝟐] Find the Banzhaf Power index of each players.
E. Find the Banzhaf Power index for the weighted voting system:
[51: 30, 25, 25, 20]
F. Find the Banzhaf Power index for the weighted voting system:
[4: 3, 2, 1]
Summary
You just have learned the methods of apportionment, voting system and
weighted voting system. Congratulations! You did a lot in this unit. Recall that
we began by defining important terms such apportionment, voting systems and
weights of voting. We summarized the two main apportionment methods:
Hamilton Plan and Jefferson Method, followed by others methods such as:
Adam’s method, Webster’s method and Huntington-Hill Method with the
fairness criteria in apportionment. Next thing we discussed five methods of
voting such as: Plurality method, Borda count Method, Plurality-withelimination method, Top-Two Runoff Method and Pairwise comparison
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method with the fairness voting criteria. Lastly, you have learned types of
weighted voting systems, critical voters and the Banzhaf Power index of each
voters. With all these you are now quite ready to move on to the next unit.
Reflection
References
Aufmann, R. et. al. Mathematical Excursion Chapter 4.
Earnhart, R.T & Adina, E.M.(2018). Mathematics in the Modern World. C &E
Publishing, Inc.
MGF 1107 CH 14 Notes-Apportionment Denson
file:///C:/Users/User/Desktop/MAthematics%20in%20the%20modern
%20world/mgf%201107%20ch%2014%20apportionment.pdf
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Unit 10: Logic (12 hours)
Introduction
The term “logic” is used quite a lot, but not always in its technical sense. Logic
is the science or study of how to evaluate arguments and reasoning. Logic is
what allows us to distinguish correct reasoning from poor reasoning. Logic is
important because it helps us reason correctly — without correct reasoning, we
don’t have a viable means for knowing the truth or arriving at sound beliefs.
When it comes to evaluating arguments, there are specific principles and criteria
which should be used. If we use those principles and criteria, then we are using
logic, if we aren’t using those principles and criteria, then we are not justified
in claiming to use logic or be logical. This is important because sometimes
people don’t realize that what sounds reasonable isn’t necessarily logical in the
strict sense of the word.
The principles and criteria of mathematical logic specify methods of reasoning
mathematical statements. Logical reasoning provides the theoretical base for
many areas of mathematics and consequently computer science. It has many
practical applications in computer science like design of computing machines,
artificial intelligence, definition of data structures for programming languages
etc.
Designed to make logic interesting—without sacrificing content or rigor—this
unit introduces basic concepts about propositional logic, explains the
symbolization of English sentences and develops proofs using truth-table and
truth techniques for evaluating arguments.
Learning Outcomes
a.
b.
c.
d.
e.
Upon the completion of this unit, you are expected to:
Define a proposition;
Determine the types of propositions;
Find the truth value of a proposition;
Construct the truth table of a proposition; and
Demonstrate an ability to prove statements and arguments expressed in
symbolic logic.
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Activating Prior Learning
Directions: Below are English sentences. Determine if the sentence is true or
false or neither. Write T if true, F if false and N if neither on blank before the
number.
_____1. Tuguegarao is the capital city of Cagayan.
_____2. Do you need some help?
_____3. I am excited!
_____4. Congruent angles have different measures.
_____5. If 3x + 2 = 8, then x = 2.
Topic 1. Proposition and Types of Propositions
Learning Objectives
a.
b.
c.
d.
Upon the completion of this topic, you are expected to:
identify if a sentence is a proposition or not;
write the negation, conjunction, disjunction, conditional and biconditional
proposition of the given simple propositions.
write the converse, inverse and contrapositive statement of a conditional
proposition; and
translate English sentences into the language of propositional logic or viseversa.
Presentation of Content
English sentences are either true or false or neither. Consider the sentences
below.
Five is a prime number.
5 + 2 < -10
How are you?
Proposition
The first sentence is true, the second one is false while the third sentence is
neither true nor false. A proposition is a statement that is either true or false but
not both. Therefore, the first two sentences are examples of propositions
because the first one is true and the second sentence is false. The third sentence
is not a proposition because it is not a declarative sentence. A proposition is
often a declarative sentence.
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Activity 1. Put a check mark (√ ) on the blank if the sentence is a proposition
and a cross mark (X) if it is not a proposition.
_____1. How many odd numbers are there from 1 to 20?
_____2. A 45 degree angle is acute.
_____3. Two is the only even number that is prime.
_____4. If x = 2, then 𝑥 2 > 4.
_____5. Solve problem x + 3 = 5.
In logic, proposition are usually denoted by letters such as P, Q, R, S and T.
For a common notation we use capital letters because small letters are used for
variables that represent numbers. Thus English sentences as propositions are
translated using capital letters in propositional logic.
Example.
Let
P = Five is a prime number.
Q = 5 + 2 < - 10
Types of Propositions
We will be now concerned to learn more about five types of complex
propositions (negations, conjunctions, disjunctions, conditionals and
biconditionals), which are constructed from simpler propositions by means of
the five connectives.
1. Negation; ∼
Definition. A negation is a compound proposition formed by the denial of the
given proposition.
Example
Let
P = Five is a prime number.
∼P = Five is not a prime number.
Q = Two is not an odd number.
∼Q = Two is an odd number.
Note: The symbol “∼” is used to denote the word “not” written before the
proposition.
Other ways to express negations are:
It is not true that…
It is not the case that…
2. Conjunction; ∧
Definition. A conjunction is a compound proposition made up of two simple
propositions connected by the word “and” or “but”.
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Example
Let
P = Two is an even number.
Q = Two is a composite number.
P ∧ Q = Two is both even and a composite number.
P ∧ ∼Q = Two is even but not a composite number.
∼P ∧∼Q = Two is not even and not a composite number.
= Neither two is even nor a composite number
∼( P∧ Q ) = It is not true that two is both even and a composite
number.
Note:
P∧Q
P ∧ ∼Q
∼P ∧∼Q
These propositions are conjunctions, the
first proposition is a conjunction of two
simple propositions, the second proposition
is a conjunction of a simple proposition
and a negation, while the third is a
conjunction of two negations.
∼( P∧ Q )
This is a negation. A negation of a
conjunction.
The symbol “∧” is used to denote the connective word “and” or “but” written
between the propositions. The connective word “but” is preferably used when
one proposition is positive and the other proposition is negative.
There are in fact a lot of ways in which conjunctions may be expressed in
English, among others are:
both ... and ...
... as well as ...
... however ...
... though ...
... although ...
... even though ...
... nevertheless …
... still …
... but still …
... also …
... while …
... despite the fact that …
... moreover …
... in addition …
3. Disjunction; ∨
Definition. A disjunction is a compound proposition made up of two simple
propositions connected by the word “ or “.
Example
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P = Two is an even number.
Q = Two is a composite number.
R = Two is greater than zero.
P ∨ Q = Two is even or a composite number.
∼P
∨ Q = Two is
not even or a composite number.
P ∨ (∼Q ∧∼R) = Two is even or neither composite nor greater than zero.
Note: The symbol “∨” is used to denote the connective word “or” written
between the propositions.
Disjunctions are not as abundant as conjunctions in English. While some are
not so obvious, most are indicated by the occurrence of the following phrases:
... or ...
either ... or ...
… or else …
4. Conditional Proposition; →
Definition. It is a compound proposition which takes the form “if – then”
statement.
Example
P = Two is an even number.
Q = Two is a composite number.
R = Two is greater than zero.
P → Q = If two is even, then it is a composite number. ( P is called the
antecedent or
hypothesis while Q is called the consequence or
conclusion)
( P ∧ Q )→ ∼R = If two is both even and a composite number, then it is not
greater than zero. (the proposition is a conditional proposition whose
hypothesis is the conjunction P ∧ Q and the conclusion is a negation; ∼R )
Note: The symbol “→” is used to denote the connective word “if-then”
written between the propositions.
Some other common ways of expressing the conditional P
 if P, then Q;

P implies Q;
 if P, Q;
 P only if Q;
 P is sufficient for Q;
 Q if P;
 Q whenever P;
 Q is necessary for P.
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5. Biconditional Proposition; ↔
Definition. It is a compound proposition which take the form of “if and only
if” statement.
Example
P = Two is an even number.
Q = Two is a composite number.
R = Two is greater than zero.
P ↔ R = Two is even if and only if it is greater than zero.
∼( P ∨ Q) ↔ ∼R = It is not true that two is even or composite if and only if it
is not greater than zero.
Note: The symbol “↔” is used to denote the connective word “if and
only if” written between the propositions.
The biconditionals are indicated by the occurrence of the following phrases:
... if and only if ...
... if but only if...
… when and only when …
… just in case …
… iff …
… exactly if …
Activity 2. Translate the following propositions in English sentences using the
given propositions below.
Classify the proposition as to negation, conjunction, disjunction,
conditional or biconditional.
P = I cheat.
R = I will study hard.
S = I will fail.
T = I am good.
Proposition
Type of
Proposition
Translation
1. T → ∼ S
2. R ∨ ( P ∧ S )
3. ∼S ↔ (∼P ∧ R )
4. (∼R ∨ ∼S ) → T
5. (∼ R ∧
( ∼T ∨ S )
P)→
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Activity 3. Translate the English sentences to symbols using the
propositions below.
P = A student misses lecture.
Q = A student studies.
R = A student fails.
__________1. A student studies or misses lecture.
__________2. It is not the case that a student studies or does not miss lecture.
__________3. A student fails if and only if the student misses lectures and
does not study.
__________4. If a student studies and does not miss lecture, then the student
does not fail.
__________5. A students studies and if the student does not fail, then the
student does not miss lectures.
CONVERSE, INVERSE and CONTRAPOSITIVE
There are some important related implications following from P
namely:
1. The proposition Q → P is called the converse.
2. The inverse of P → Q is ∼P → ∼Q;
3. The contrapositive of P → Q is ∼Q → ∼P
→ Q,
Activity 4. Write the converse, inverse and contrapositive statements of each
conditional statement (use English sentence and symbols).
1. If a polygon has four sides, then it is a quadrilateral. P →Q
Converse: ______________________________________________________
Inverse: ________________________________________________________
Contrapositive: __________________________________________________
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2. If 𝑥 2 = 81, then x ≠ 9.
P →∼Q
Converse: ______________________________________________________
Inverse: ________________________________________________________
Contrapositive: __________________________________________________
Application
I. Complete the table by writing the equivalent symbol of the English
sentence or write the English sentence corresponding to the given symbol
using the following propositions below.
P = All odd numbers are prime.
Q = All even numbers are composite.
R = All prime numbers are greater than one.
Symbol
P ∨∼Q
English sentence
It is not true that if all odd numbers are prime then
not all prime numbers are greater than one.
∼P ∧ R
If all prime numbers are greater than one then it is
not true that not all odd numbers are prime or all
even numbers are composite.
( R ∧Q ) ↔P
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II. Fill-up the table by writing the converse, inverse and contrapositive of the
conditional propositions in symbols.
Proposition
1. ∼P →Q
2. P →∼Q
3. (P ∧ Q ) →S
4. ∼P →( Q ∨∼P
)
Converse
Inverse
Contrapositive
Topic 2. Truth Value and Truth Table
Learning Objectives
a.
b.
c.
d.
e.
Upon the completion of this topic, you are expected to:
give the rules in determining the truth value of a proposition;
evaluate the truth value of a proposition;
construct the truth table of a proposition;
identify if a proposition is a tautology, contradiction or contingency; and
determine if two propositions are logically equivalent or not logically
equivalent.
Presentation of Content
TRUTH VALUE
Defnition: The truth value of a proposition is its truth or falsity, that is either T
or F.
Example:
Let
P = Ten is an even number.
Q = Ten is greater than fifteen.
Truth values:
Now, using the truth values of P and Q,
What is the truth value of ∼P? of ∼Q?
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What is the truth value of P ∧ Q, P ∨Q, P →Q and P↔ Q?
TRUTH TABLE
Definition. It is a table showing the complete list of all the possible truth
values of the proposition.
Here are the rules in determining the truth values of negation, conjunction,
disjunction, conditional and bi-conditional propositions.
Rules in Determining the Truth Value of a Proposition
1. NEGATION
Rule:
If P is true, then ∼P is false.
If P is false, then ∼P is true.
Example:
a. Let P = Ten is an even number.
The truth value of P if true (T) since ten is divisible by 2 then is an even
number.
Now, negating P, that is, “∼P = Ten is not an even number.” becomes false.
b. Let P = Ten is greater than fifteen.
The truth value of P is false (F), and when it is negated, that is, “∼P = Ten is
not greater than fifteen.” the truth value is true (T).
This can be summarized using the truth table below.
The truth table shows all the possible truth values of proposition P, that is, P
can either be true (T) of false (F). So that if P is true (T), ∼P is false (F). If P
is false (F), ∼P is true (T).
2. CONJUNCTION
Rule: P ∧ Q is true if both P and Q are true,
otherwise false.
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To construct the truth table for the statement P ∧ Q containing two
propositions P and Q, first we must consider only four possible assignments of
the truth values of P and Q then build two columns with all possible values of
P and Q (i.e., (T;T);(T;F);(F;T);(F;F)). This is shown in the table below.
The truth table shows that P ∧ Q is true when both P and Q are true, other
combinations of the truth value of P and Q such as (T;F), (F;T), (F;
F),respectively, P ∧ Q is false (F) because not both propositions are true.
3. DISJUNCTION
Rule: P ∨ Q is true if either P is true or Q is true
or both P and Q are true,
otherwise false.
The truth table below shows that the first three combinations of the truth
values of P and Q (T;T), (T;F), (F;T), respectively, P ∨ Q is true since at
least one of the propositions P and Q is true, last combination of the truth
value of P and Q which is (F; F), P ∨ Q is false (F) because none of the
propositions is true.
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4. CONDITIONAL
Rule: P → Q is false if P (antecedent) is true and
Q (consequence) is false, otherwise true.
The truth table below shows that the combinations of the truth values of P and
Q (T;T), (F;T), (F;F), respectively, P→ Q is true because the conditional
statement is false only when the antecedent (P) is true and the consequence
(Q) is false, the combination of the truth value of P and Q which is (T; F), P
→ Q is false (F) because antecedent (P) is true and the consequence (Q) is
false.
5. BICONDITIONAL
Rule: P ↔ Q is true if P and Q have the same
truth values, that is P and Q are both true or P and
Q are both false, otherwise false.
The truth table below shows that the combinations of the truth values of P and
Q (T;T), (F;F), P↔ Q is true because the propositions P and Q have the
same truth values. The combination of the truth value of P and Q which is (T;
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F), (F;T), P ↔ Q is false (F) because the propositions P and Q have
different truth values.
Activity 1. Determine the truth values of the following propositions using the
truth values of the simple propositions below. Write your answer on the blank
after the proposition. The first item is done for you.
P:T
Q:F
R:F
S:T
1. P ∧∼R : ____T_____
Solution:
First, determine the truth value of ∼R, this is shown in the table.
P
T
∼R
T
R
F
∼R is true(T) since R is false (F).
After ∼R, evaluate the truth value of the conjunction of P and ∼R (P ∧∼R).
P
R
∼R
T
F
T
P ∧∼R is true (T) because both P and ∼R are true.
P ∧∼R
T
2. ∼S ∨∼Q : ________
3. ( P ∧ Q )→∼S : _______
4. Q ↔ ∼( P ∨ R ) : _______
5. (R ∨ S) → ( P ∧∼Q) : ______
Activity 2. Construct the truth table of the following propositions below. The
first two items are done for you.
1. ∼P →∼Q
First , list the four combinations of the truth values of P and Q.
P
T
T
F
F
Q
T
F
T
F
Second, evaluate the truth values of ∼P and ∼Q.
P
T
T
F
F
∼P
F
F
T
T
Q
T
F
T
F
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∼Q
F
T
F
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Last step is evaluate the truth value of ∼P →∼Q. In this conditional
statement, the antecedent is ∼P and the consequence is ∼Q. So, applying the
rule for conditional proposition, we can now write the complete truth table of
∼P →∼Q.
P
T
T
F
F
∼P
F
F
T
T
Q
T
F
T
F
∼Q
F
T
F
T
∼P →∼Q
T
T
F
T
2. ∼P ↔ ( Q ∨ S )
First, write down all possible combinations of the truth values of P, Q and S.
There are 8 combinations of their truth values as shown below. Observe the
pattern on how the truth values are listed.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
S
T
F
T
F
T
F
T
F
Second, negate P (∼P ) and evaluate ( Q ∨ S ).
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
∼P
F
F
F
F
T
T
T
T
S
T
F
T
F
T
F
T
F
Q∨S
T
T
T
F
T
T
T
F
Finally, evaluate the truth value of ∼P ↔ ( Q ∨ S ).
P
T
T
T
T
Q
T
T
F
F
S
T
F
T
F
∼P
F
F
F
F
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Q∨S
T
T
T
F
∼P ↔ ( Q ∨ S )
F
F
F
T
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F
F
F
F
3.
4.
T
T
F
F
T
F
T
F
T
T
T
T
T
T
T
F
T
T
T
F
∼( P →Q )
∼(( P ∨ Q ) ↔ ∼S )
TAUTOLOGY, CONTRADICTION AND CONTINGENCY
TAUTOLOGY
Definition. It is a proposition which is always true for every assignment of the
truth values of its simple components.
Example:
(( P →Q ) ∧ ∼Q ) →∼P
The truth table of (( P →Q ) ∧ ∼Q ) →∼P is reflected below.
P
Q
T
T
F
F
T
F
T
F
P
→Q
T
F
T
T
( P →Q )
∧∼Q
F
F
F
T
∼Q
F
T
F
T
∼P
(( P →Q ) ∧∼Q)→∼P
F
F
T
T
T
T
T
T
Since the truth value of (( P →Q ) ∧ ∼Q ) →∼P is always true as seen in the
last column of the truth table considering all combinations of the truth values
of P and Q, therefore (( P →Q ) ∧ ∼Q ) →∼P is called a tautology.
CONTRADICTION
Definition. It is a proposition which is always false for every assignment of
truth values of its simple propositions.
Example: P ∧ ( ∼P ∧ Q )
Below is the truth table of P ∧ ( ∼P ∧ Q ).
P
T
T
F
F
Q
T
F
T
F
∼P
F
F
T
T
∼P ∧ Q
F
F
T
F
P ∧ ( ∼P ∧ Q )
F
F
F
F
Since the truth value of P ∧ ( ∼P ∧ Q ) is always is false as seen in the last
column of the truth table considering all combinations of the truth values of P
and Q, therefore P ∧ ( ∼P ∧ Q ) is called a contradiction.
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Exploring Mathematics in the Modern World
CONTIGENCY
Definition. It is a proposition that is neither a tautology nor a contradiction.
Example: ∼( ∼P V Q )
P
Q
∼P
∼P ∨ Q
∼( ∼P V Q )
T
T
F
T
F
T
F
F
F
T
F
T
T
T
F
F
F
T
T
F
Lastly, based from the truth table of ∼( ∼P V Q ), the proposition is neither a
tautology nor a contradiction ( not always true or not always false), ∼( ∼P V
Q ) is then called a contingency.
LOGICAL EQUIVALENCE; ≅
Definition: Two propositions are equivalent if they have the same truth values
no matter what truth values their constituent propositions are.
Example:
Show whether or not the propositions ∼( P ∨ Q ) and ∼P ∧∼Q are equivalent.
First construct the truth table of each proposition.
a. ∼( P ∨ Q )
P
T
T
F
F
P∨Q
T
T
T
F
Q
T
F
T
F
∼( P ∨ Q )
F
F
F
T
b. ∼P ∧∼Q
P
T
T
F
F
Q
T
F
T
F
∼P
F
F
T
T
∼Q
F
T
F
T
∼P ∧∼Q
F
F
F
T
Compare the corresponding truth values of ∼( P ∨ Q ) and ∼P ∧∼Q and you
will see that the values are the same for the same assignment of the truth
values of P and Q. Therefore the propositions ∼( P ∨ Q ) and ∼P ∧∼Q are
logically equivalent written as: ∼( P ∨ Q ) ≅ ∼P ∧∼Q.
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Exploring Mathematics in the Modern World
Now try the following items:
Activity 3.
not.
Prove whether or not the propositions are logically equivalent or
1. P →Q and ∼P ∨ Q
2. P ↔Q and ( P → Q ) ∧ ( Q →P )
Application
I. Using the truth values of P, Q, R and S below, evaluate the truth value of
each of the following propositions. Write your answer on the blank before
the number.
P :F
R:T
Q:T
S:F
_____4. P →( R ∨∼S )
_____5. ( P ∧ Q ) ↔( T ∨ S )
_____1. Q ∧ ∼Q
_____2. ∼( R → S )
_____3. ∼S ∨ ( Q ∧ R )
II. Construct the truth table of the proposition. Determine if the proposition is
a tautology, contradiction or contingency. Write your answer on the blank
before the number.
__________1. P ∧∼P
P
T
F
__________2. P∨∼Q
P
T
T
F
F
Q
T
F
T
T
__________3. (∼P ∨Q) ↔ (P →Q)
P
T
T
F
F
Q
T
F
T
T
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Topic 3. Arguments and Validity
Learning Objectives
Upon the completion of this topic, you are expected to:
a. define an argument; and
b. determine if an argument is valid or invalid using truth table.
Presentation of Content
ARGUMENT
Definition. An argument is a list of statements called premises followed by a
statement called the conclusion.
𝑃_1 Premise
𝑃_2 Premise
𝑃_3 Premise
.
.
.
𝑃_𝑟 Premise
_______________________
∴ C Conclusion
The argument is said to be valid if the statement ( 𝑃1 ∧ 𝑃2 ∧ 𝑃3 ∧ … 𝑃𝑟 ) →
𝐶 is a tautology.
To prove whether an argument is valid or invalid you may follow these steps:
a. Form the conditional statement in the form of ( 𝑃1 ∧ 𝑃2 ∧ 𝑃3 ∧ … 𝑃𝑟 ) →
𝐶 where the antecedent is the conjunction of the premises and the
consequence is the conclusion based from the definition.
b. Construct the truth table of the conditional statement.
c. Identify from the truth table if the conditional statement is a tautology or
not.
Example: Determine the validity of the argument below using the truth table.
P→Q
Q→R
------------P→R
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Exploring Mathematics in the Modern World
In the argument, the premises are P → Q and Q → R, while the conclusion is
P → R.
Now following the steps:
a. Form the conditional statement.
[ (P → Q ) ∧ (Q → R ) ] → (P → R)
b. Construct the truth table. Apply your knowledge in constructing the truth
table.
P
Q
R
P→
Q
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
T
F
F
T
T
T
T
Q
→
R
T
F
T
T
T
F
T
T
(P → Q ) ∧
(Q → R )
T
F
F
F
T
F
T
T
P → [ (P → Q ) ∧ (Q → R )
R
] → (P → R)
T
F
T
F
T
T
T
T
T
T
T
T
T
T
T
T
c. Based from the table, the proposition is a tautology. Therefore the given
argument is valid.
Now, for your activity, try the following arguments.
Application
Show if the argument is valid or invalid.
1.
∼P → Q
P
------------∼Q
2.
P∧Q→R
∼R
-----------∼P ∨ ∼Q
3.
P→Q
R→Q
∼P ∧ R
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Exploring Mathematics in the Modern World
Q
Assessment
I.
1.
For each item below, do the following.
a. Classify the proposition as to negation, conjunction, disjunction,
conditional or biconditional.
b. Translate the propositions to English sentence using the following
simple propositions.
P = Fifteen is greater than twelve.
Q = Fifteen is not a multiple of five.
R = Fifteen is not a composite number.
Item
P ∧∼ Q
Type of Proposition
Translation
2. ∼Q ∨∼R
3. R→∼ P
4. ∼( Q ∧ R )
5. P ↔ ( ∼Q ∨ ∼R )
II. Construct the truth table of each of the following propositions then
determine whether the proposition is a tautology, contradiction or a
contingency.
1.
(P→ Q) ∨ (Q →P): ________________
P
Q
T
T
T
F
F
T
F
F
2.
( P ∨ Q ) ∧ ( ∼P ∧ ∼ Q ): _________________
P
Q
T
T
T
F
F
T
F
F
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Exploring Mathematics in the Modern World
3.
∼(P ∧ Q ) ↔( ∼P ∨∼Q ): ________________
P
T
T
F
F
Q
T
F
T
F
III. Show whether the argument is valid or invalid.
1.
P
P →Q
-------------Q
2.
P∨Q
∼P
-------------Q
3.
P →Q
Q
------------P
Summary
Proposition - is a statement that is either true or false but not both.
Types of Proposition
Negation
Conjunction
Disjunction
Conditional
Biconditional
Logical connector
not
and or but
or
if-then
if and only if
Related Conditionals
Formed By
Conditional
given hypothesis and conclusion
Converse
interchange hypothesis and conclusion
of the conditional
Inverse
negate both hypothesis and conclusion
of the conditional
Contrapositive
negate both hypothesis and conclusion
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Symbol
P→Q
Q→P
∼P→∼Q
∼Q → ∼P
Exploring Mathematics in the Modern World
of the converse
Rules for the Truth Value of a Proposition
Types of Proposition
Rule
Negation
If P is true, then ∼P is false. If P is false, then ∼P is true.
Conjunction
P ∧ Q is true if both P and Q are true, otherwise false.
Disjunction
P ∨ Q is true if either P is true or Q is true or both P and
Q are true, otherwise false
Conditional
P ↔ Q is true if P and Q have the same truth values,
that is P and Q are both true or P and Q are both false,
otherwise false
Biconditional
P ↔ Q is true if P and Q have the same truth values, that
is P and Q are both true or P and Q are both false,
otherwise false.
TAUTOLOGY is a proposition which is always true for every assignment of the
truth values of its simple components.
CONTRADICTION is a proposition which is always false for every assignment
of the truth values of its simple propositions.
CONTIGENCY is a proposition that is neither a tautology nor a contradiction.
LOGICAL EQUIVALENCE Two propositions are equivalent if they have the
same truth values no matter what truth values their constituent propositions are.
ARGUMENT is a list of statements called premises followed by a statement
called the conclusion.
𝑃_1 Premise
𝑃_2 Premise
𝑃_3 Premise
.
.
.
𝑃_𝑟 Premise
_______________________
∴ C Conclusion
VALID ARGUMENT
The argument is said to be valid if the statement (𝑃1 ∧
𝑃2 ∧ 𝑃3 ∧ … 𝑃𝑟 ) → 𝐶 is a tautology.
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Exploring Mathematics in the Modern World
Reflection
1. Create a slogan that will promote logic as an important tool in everyday life.
2. If you are a proposition, what would it be? Why? Give an example and tell
the truth value.
3. What value in life did you develop in learning logic? Why?
References
Books:
JOHNSON BAUGH, RICHARD: Discrete Mathematics, 7th Edition, 2009
U. DAEPP, P. GORKIN, Reading, Writing, and Proving. A Closer Look at
Mathematics (Undergraduate Texts in Mathematics), Springer, Bucknell
University, 2011.
Online:
https://www.cs.purdue.edu/homes/spa/courses/cs182/mod1.pdf
https://www.geeksforgeeks.org/mathematical-logic-introductionpropositional-logic-set-2/
http://kpaprzycka.wdfiles.com/local--files/logic/W02.pdf
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Unit 11. Mathematics of Graphs
Introduction
In this unit, you will learn how to analyze and solve various problems such as
how to determine the least expensive route of travel on vacation, scheduling
task to avoid overlapping of meetings or conferences, finding the most
appropriate routes in order to save time and effort.
Basically, the methods used to study and solve these problems can be traced
back to an old recreational puzzle. In the early eighteenth century, the Pregel
River in city called Konigsberg, located in modern-day Russia, surrounded an
island before splitting in two. There are seven bridges crossed the river and
connected with four different land areas (see the map below).
During that time, many citizens living in the area attempted to take a stroll that
would lead them to cross each bridge and return them to the starting point
without traversing the same bridge twice. Sad to say, no one of them can do it,
no matter where they choose to start with. You may also try for yourself using
paper and pencil and you will see that it not easy.
In 1736, Leonard Euler (1707-1783) a Swiss mathematician proved that it is
impossible to traverse each of the bridges of Konigsberg exactly once and
return to the starting point. His analysis of the challenge laid the groundwork
for a branch of mathematics known as graph theory. In this chapter, we will
investigate how Euler approached the problem of the seven bridges of
Konigsberg in the next topic.
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Exploring Mathematics in the Modern World
Learning Outcomes
Upon the completion of this unit, you are expected to:
a. Use mathematical concepts and tools in networks and graphs; and
b. Support the use of mathematics in various aspects and endeavors in life
Activating Prior Learning
Students would think and ask of all various connections we experience in our
lives such as friends are connected on social media, cities are connected by
bridges, computers are connected in the internet, etc. Students would share
their ideas and experiences regarding various connections.
Topic 1. Graph Coloring
Learning Objectives
Upon the completion of this topic, you are expected to:
a. identify if the given graph is 4-colorable or not,
b. color the map using the fewest number of colors, and
c. apply graph coloring in scheduling and sudoku problem
Presentation of Content
Definition. A graph is 4-colorable if no two vertices connected by an edge
share the same color and the least number of different colors required is four.
Four Color Theorem
Every planar graph is 4- colorable.
Example 1. The graph shown below requires five colors if we wish to color it
such that no edge joins two vertices of the same color. Does this
contradict the four color theorem?
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Exploring Mathematics in the Modern World
Solution:
No. This graph is K5, and therefore it is not a planar graph, so the four color
theorem does
not apply.
Example 2.
A fictional map below shows the boundaries of countries on a
rectangular continent.
Represent the map as a graph, and find a coloring of
the graph using fewest possible number of colors. Then color the map
according to the graph coloring.
Solution;
First, draw a vertex in each country and then connect two vertices with an
edge if the corresponding countries are neighbors. (see figure 2). Now, try to
color the vertices of the resulting graph so that no edge connects two vertices
of the same color. We know we will need at least two colors, so one strategy is
simply to pick a starting vertex, give it a color, and then assign colors to the
connected vertices one by one. Try to reuse the same colors, and use a new
color only when there is no other option. For this graph we need four colors.
To see why we need 4 colors, notice that one vertex colored green in the
second figure below connects to a ring of 5 vertices. Three different colors are
required to color the 5-vertex ring, and the green connects to all these, so it
requires a fourth color.
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Exploring Mathematics in the Modern World
g
Application
Example 3. A Scheduling Application of Graph Coloring
Eight different school clubs wants to schedule a meeting on the last day of the
semester.
Some club members, however, belong to more than one of these
clubs, so clubs that share members cannot meet at the same time. How many
different time slots are required so
that all members can attend all
meetings? Clubs that have a member in common are
indicated with an “X”
in the table below.
Ski club
Stud
gov
Debate
Society
Honor
society
Stud
newspaper
Community
Outreach
Campus
Democrats
Campus
rep
Ski
club
___
Student
gov
X
Debate
club
Honor
society
X
Student
newspaper
X
___
X
X
X
X
___
X
X
X
___
X
X
X
___
X
X
X
X
___
X
X
X
___
X
X
X
X
X
X
Community
Outreach
X
X
236
Campus
Democrats
X
Campus
rep
X
X
X
___
Exploring Mathematics in the Modern World
Solution:
We can represent the given information by a graph. Each club is represented
by a vertex, and an edge connects two vertices if the corresponding clubs have
at least one common member.
SC
CR
SG
CD
DC
HS
CO
SN
Two clubs that are connected by an edge cannot meet simultaneously. If we let
a color correspond to a time slot, then we need to find a coloring of the graph
that uses the fewest possible number of colors. The graph is not 2-colorable,
because we can find circuits of odd length. However, by trial and error, we can
find a 3-coloring. One example is shown below. Thus, the chromatic number
of the graph is 3, so we need 3 different time slots.
SC
CR
SG
CD
DC
HS
CO
SN
Each color corresponds to a time slot, so one scheduling is
First time: ski club, debate club, student newspaper
Second Time slot: student government, community outreach
Third slot: honor society, campus Democrats, campus Republicans
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Exploring Mathematics in the Modern World
Assessment
1. Draw graph that represents the information given in the table
below involving professors and the subjects that are assign to
them in a semester.
Algebra Statistics Trigonometry Calculus
Ben
√
√
√
Ariel
√
√
√
√
Angelo
√
√
√
Shaly
√
2. Five classes in an elementary school have arranged a tour at a
zoo where the students get to feed the animals
Class 1 wants to feed the elephants, giraffes, and hippos.
Class 2 wants to feed the monkeys, rhinos, and elephants.
Class 3 wants to feed the monkeys, deer, and sea lions.
Class 4 wants to feed the parrots, giraffes, and the polar bears.
Class 5 wants to feed the sea lions, hippos and the polar bears.
If the zoo allows animals to be fed only once a day by one class
of students, can the tour be accomplished in two days? (assume
that each class will visit the zoo inly on one day). If not, how
many days will be required?
Summary
Representing maps as Graphs. Draw a vertex in each region of the map.
Connect 2 vertices if the corresponding regions share a common border.
The Four-Color Theorem. Every planar graph is 4-colorable. (In some
cases less than 4 colors may be required. Also, if the graph is not planar,
more than 4 colors may be necessary).
Applications of Graph Coloring. Determining the chromatic number of a
graph and finding a corresponding coloring of the graph can solve some
practical applications such as scheduling meeting or events.
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Exploring Mathematics in the Modern World
Reflection
Graph coloring is significant most especially in our daily scheduling activities.
How do you also apply graphs in your daily life?
References
Aufmann, R., et. Al., (2018). Mathematics in the Modern World. Rex Book
Store, Inc. (RSBI). ISBN 978-971-23-9357-0
Baltazar, E.C, Ragasa, C., and Evanelista, J. (2018). Mathematics in the
Modern World. C &E Publishing Company, Inc.
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Unit 12: Mathematical Systems
Introduction
The study of mathematical system has been driven purely by its inherent beauty
and by human curiosity. Moreover, with the advancement of human knowledge,
mathematical system has found important applications in the real world.
Modular arithmetic is a type of number system where numbers are represented
by the remainder after division by the modulo number. It is a form of arithmetic
dealing with the remainders after integers are divided by a fixed "modulus" m.
Basically, it is a kind of integer arithmetic that reduces all numbers to ones that
belongs to a fixed set [0 ... n-1]. We can say two integers, a and b, are congruent
mod m (where m is a natural number) if both numbers divided by m produce
the same remainder. In other words, m must evenly divide their difference, a b.
Modular arithmetic is the arithmetic of congruences, sometimes known
informally as "clock arithmetic." In modular arithmetic, numbers "wrap around"
upon reaching a given fixed quantity, which is known as the modulus (which
would be 12 in the case of hours on a clock, or 60 in the case of minutes or
seconds on a clock).
The Swiss mathematician Leonhard Euler pioneered the modern approach to
congruence in 1750, when he explicitly introduced the idea of congruence
modulo a number N. Modular arithmetic was further advanced by Carl Friedrich
Gauss in his book published in 1801.
In this module, we will explore modular arithmetic in greater detail and learn
how to recognize congruence classes. Furthermore, we will also solve realworld problems that operate in cyclical process to illustrate the importance of
modular arithmetic.
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Exploring Mathematics in the Modern World
Learning Outcomes
Upon the completion of this unit, you are expected to:
a. Explain modular arithmetic
b. Perform operations using modular arithmetic; and
c. Apply modular arithmetic in solving real-life problems.
Activating Prior Learning
Modulus Time
Look around you. Do you see that clock on the wall?
It’s modulus 12.
Why? Because the standard method for telling time is to split the day into two
12 hour segments. Instead of counting up to 24, we count to 12 twice.
In fact, circular counting is a fundamental representation of modular
arithmetic.
Furthermore when you convert between military time and standard time,
you’re performing modular arithmetic. For example, we know that 18:00 is the
same as 6:00 pm because when we divide 18 by 12, we’re left with 6 as a
remainder.
So, let me posit this: what time would it be right now in a universe that used
modulus 8 in their time system?
As I’m writing this it’s 8 pm. In a universe that uses modulus 8, the time
would be 6 o’clock.
(solution: 8 pm is the 20nd hour of the day. So, we take 20 mod 8. Eight of
course goes into 20 twice with a remainder of 4. That means 20 mod 8 = 4.)
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Exploring Mathematics in the Modern World
Topic 1: Modular Arithmetic and Its Application
Presentation of Content
When we divide two integers we will have an equation that looks like the
following:
𝑎
= 𝑞 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 𝑟
𝑏
a is the dividend
b is the divisor
q is the quotient
r is the remainder
Sometimes, we are only interested in what the remainder is when we divide a
by b. For these cases there is an operator called the modulo operator
(abbreviated as mod). Using the same a, b, q, and r as above, we would have: a
( mod b) = r. We would say this as a modulo b is equal to r where b is referred
to as the modulus.
For example:
15
= 3 remainder 3
4
15 (mod 4) = 3
Visualize modulus with clocks
Observe what happens when we increment numbers by one and then divide
them by 4.
0
4
1
4
2
4
3
4
4
4
5
4
6
4
7
4
8
4
= 0 remainder 0
= 0 remainder 1
= 0 remainder 2
= 0 remainder 3
= 1 remainder 0
= 1 remainder 1
= 0 remainder 2
= 1 remainder 3
= 2 remainder 0
The remainders start at 0 and increases by 1 each time, until the number
reaches one less than the number we are dividing by. After that, the
sequence repeats.
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Exploring Mathematics in the Modern World
Equivalence Classes
Now, you are probably thinking that modular arithmetic is useless because you
keep getting the same answers over and over again.
You’re right! In fact that’s the beauty of modular arithmetic. It gives us a new
way to relate numbers to one another.
Check this out.
Let’s represent modulus 4 with the following circle diagram.
0
1
3
2
Recall that when you divide by 4, you have 4 possible remainders: 0, 1, 2, and
3.
Let’s calculate 0, 1, 2 and 3 mod 4:
0 (mod 4)
1 (mod 4)
2 (mod 4)
3 (mod 4)
≡
≡
≡
≡
0
1
2
3
Place the numbers in their respective sections of the modulus 4 diagram.
0
3
1
0
1
3
2
And continue calculating:
4 (mod 4) ≡ 0
5 (mod 4) ≡ 1
6 (mod 4) ≡ 2
7 (mod 4) ≡ 3
8 (mod 4) ≡ 0
9 (mod 4) ≡ 1
10 (mod 4) ≡ 2
11 (mod 4) ≡ 3
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Exploring Mathematics in the Modern World
Add these numbers to the diagram:
0
1
1, 5, 9
0, 4,
8
3
3, 7,
11
2, 6,
10
2
Each of these sections represents an equivalence class.
Modulo Operations
• The notion of modular arithmetic is related to that of the remainder in division.
The operation of finding the remainder is sometimes referred to as the modulo
operation.
• We define Zn as the set of integers from 0,1,2,…,n-1 modulo n, i.e.
Zn = {0, 1, 2, …, n-1}
Note that Zn has exactly n nonnegative integers. In particular, we define Zn
with the following set of positive integers:
• Z6 = {0, 1, 2, 3, 4, 5}, modulo 6
• Z7 = {0, 1, 2, 3, 4, 5, 6}, modulo 7
• Z10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, modulo 10
In Zn, modulo is simply the remainder r when an integer 𝑎 ∈ ℤ is simply
divided by n and has a remainder r < n.
Perform the following operations:
A. In Z8
1. 3 + 7
2. 14 + 20
3. 106 + 204 `
4. 15 ● 4
5. 18 ● 15
B. In Z15
1. 10 + 15
2. 23 + 54
3. 101 + 79
4. 13 ● 5
5. 23 ● 12
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Exploring Mathematics in the Modern World
Constructing Modulo Tables
Zn is closed under the binary operations of addition and multiplication of
integers modulo n.
Let us construct the addition and multiplication table for Z5.
In Z5 = {0, 1, 2, 3, 4}
+
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
0
2
2
3
4
0
1
3
3
4
0
1
2
4
4
0
1
2
3
●
0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
1
3
3
0
3
1
4
2
4
0
4
3
2
1
Application
A. Clock Arithmetic
In a 12 hour clock, determine the time for each of the following:
a. 50 hours after 6:00 a.m.
b. 17 hours before 10:00 p.m.
This problem works in modulo 12 operation.
a. 50 + 6 ≡ 56 ≡ 8 (mod 12). So the answer is 8:00 a.m..
b. 10 – 17 ≡ -7 ≡ 5 ( mod 12) So, the time is 5:00 a.m..
B. Week-day Arithmetic
Another application of modular arithmetic involves day-of-the-week
arithmetic. If we associate each day of the week with a number, as shown
below,
Monday = 1
Tuesday = 2
Wednesday = 3
Thursday = 4
Friday = 5
Saturday = 6
Sunday = 7
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Exploring Mathematics in the Modern World
then 5 days after Wednesday is Monday; 15 days after Tuesday is Wednesday.
Symbolically we write
3 + 5 ≡ 1 (mod 7)
2 + 15 ≡ 3 (mod 7)
and
Similarly, If today is Tuesday, what day of the week will it be 95 days from
now?
95 ≡ 4 (mod 7)
Thus, counting 4 days after Tuesday is Saturday.
B. Credit Cards
A credit card is an electronic, plastic card issued by a financial institution that
allows an individual borrow money at the point of sale to complete a purchase.
Credit cards are very important part of life because they give you the ability to
pay items and necessities when you do not have the cash or money.
A valid credit card number (also known as Primary Account Number – PAN)
has several fields and each of them has a meaning. For the technically inclined,
this number complies to the ISO/IEC 7812 numbering standard. A credit card
number contains a six-digit issuer identification number (IIN), an individual
account identification number, and a single digit checksum.
Credit card numbers are normally 13 to 16 digits long. The first digit of the
issuer identification number is the major industry identifier (MII). It identifies
the industry where the card will be most used in.
The table below shows the identification prefixes used by four popular card
issuers.
Card Issuer
Prefix
Number of Digits
Master Card
51 to 55
16
Visa
4
13 or 16
American Express
34 to 37
15
Discover
6011
16
Mod 10 Algorithm
The mod 10 algorithm is a checksum (detection of errors) formula which is the
common name for the Luhn algorithm. This formula has been in use to validate
a lot of identification numbers besides credit cards since its development by
scientist Hans Peter Luhn from IBM.
The Luhn Formula:
1. Drop the last digit from the number. The last digit is what we want to
check against .
2. Reverse the numbers.
3. Muliply the digits in odd positions (1, 3, 5, etc.) by 2 and subtract 9 to
all any result higher than 9.
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4. Add all the numbers together.
5. The check digit (the last number of the card) is the amount that you need
to add to get a multiple of 10 (modulo 10)
Luhn Example:
Original
Number
Drop the
last digit
Reverse
the digits
Multiply
odd digits
by 2
Subtract
0 to
numbers
over 9
Add all
the
numbers
4
5 5
6 7
3 7
5 8
6 8
9 9
8 5 5 Total
4
5 5
6 7
3 7
5 8
6 8
9 9
8 5
5
8 9
9 8
6 8
5 7
3 7
6 5
5 4
10 8 18 9 16 6 16 5 14 3 14 6 10 5 8
1
8 9
9 7
6 7
5 5
3 5
6 1
5 8
1
8 9
9 7
6 7
5 5
3 5
6 1
5 8
85
85 (modulo 10) = 5 ( last digit of the card)
C. The ISBN
An ISBN is an International Standard Book Number. ISBNs were 10 digits in
length up to the end of December 2006, but since 1 January 2007 they now
always consist of 13 digits. ISBNs are calculated using a specific mathematical
formula and include a check digit to validate the number.
Each ISBN consists of 5 elements with each section being separated by spaces
or hyphens. Three of the five elements may be of varying length:





Prefix element – currently this can only be either 978 or 979. It is always 3
digits in length
Registration group element – this identifies the particular country,
geographical region, or language area participating in the ISBN system. This
element may be between 1 and 5 digits in length
Registrant element - this identifies the particular publisher or imprint. This
may be up to 7 digits in length
Publication element – this identifies the particular edition and format of a
specific title. This may be up to 6 digits in length
Check digit – this is always the final single digit that mathematically validates
the rest of the number. It is calculated using a Modulus 10 system with
alternate weights of 1 and 3.
An ISBN is essentially a product identifier used by publishers, booksellers,
libraries, internet retailers and other supply chain participants for ordering,
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listing, sales records and stock control purposes. The ISBN identifies the
registrant as well as the specific title, edition and format.
Formula for the ISBN Check Digit
d13 = 10 – (d1 + 3d2 + d3 +3d4 + d5 + 3d6 + d7 + 3d8 + d9 + 3d10 + d11 + 3d12) mod
10
If d13 = 10, then the check digit is 0.
For example:
Determine if the ISBN code 9781861972712 is valid or not.
Solution:
d13 = 10 – [9 + 3(7) + 8 +3(1) + 8 + 3(6) + 1 + 3(9) + 7 + 3(2) + 7 + 3(1)] mod
10
d13 = 10 – [118] mod 10
d13 = 10 -8
d13 = 2
The ISBN code is valid.
D. UPC
A UPC, short for universal product code, is a type of code printed on retail
product packaging to aid in identifying a particular item. It consists of two parts
– the machine-readable barcode, which is a series of unique black bars, and the
unique 12-digit number beneath it.
The purpose of UPCs is to make it easy to identify product features, such as the
brand name, item, size, and color, when an item is scanned at checkout. In fact,
that’s why they were created in the first place – to speed up the checkout process
at grocery stores. UPCs are also helpful in tracking inventory within a store or
warehouse.
Parts of a UPC
After paying a fee to join, GS1 assigns a 6-digit manufacturer identification
number, which becomes the first six digits in the UPC on all the company’s
products. That number identifies the particular manufacturer of the item.
The next five digits of the UPC is called an item number. It refers to the actual
product itself. Within each company is a person responsible for issuing item
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numbers, to ensure that the same number isn’t used more than once and that old
numbers referring to discontinued products are phased out.
Many consumer products have several variations, based on, for example, size,
flavor, or color. Each variety requires its own item number. So a box of 24 oneinch nails has a different item number than a box of 24 two-inch nails, or a box
of 50 one-inch nails.
The last digit in the 12-digit UPC is called the check digit. It is the product of
several calculations – adding and multiplying several digits in the code – to
confirm to the checkout scanner that the UPC is valid. If the check digit code is
incorrect, the UPC won’t scan properly.
Formula for the UPC Check Digit
The UPC is a 12-digit number that satisfies a congruence equation that is similar
to the one for ISBNs. The last digit is the check digit. If we label the 12 digits
of the UPC as d1, d2, ... , d12, we can write a formula for the UPC check digit
d12.
d12 = 10 – (3d1 + d2 + 3d3 +d4 + 3d5 + d6 + 3d7 + d8 + 3d9 + d10 + 3d11) mod 10
If d12 = 10, then the check digit is 0.
Example:
Is 1 – 32342 – 65933 – 9 a valid UPC?
Solution:
d12 = 10 – (3d1 + d2 + 3d3 +d4 + 3d5 + d6 + 3d7 + d8 + 3d9 + d10 + 3d11) mod 10
d12 = 10 – [3(1) + 3 + 3(2) + 3 + 3(4) + 2 + 3(6) + 5 + 3(9) + 3 + 3(3)] mod 10
d12 = 10 – [91] mod 10
d12 = 10 – 1
d12 = 9
Therefore, the UPCis valid.
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Assessment
Solve the following problems:
1. In a 12-hour clock, determine the time
a. 6 hours after 10 o’clock
b. 13 hours before 9 o’clock
c. 26 hours after 5:00 a.m.
d. 47 hours before 8:00 p.m.
2. What day of the week is 26 days from now if today is a Monday?
3. Today is Sunday. What day will it be 450 days from now ?
4. Construct the modulo tables for addition and multiplication for m = 8.
5. Perform the following operations based on the indicate modulo.
a. In Z4, find 5 + 2
b. In Z6, find 4 ● 7
c. In Z8, find - 4
d. In Z5, find 5−3
e. In Z10, find 9 - 4
f. In Z9, find 3 + 17
g. In Z7, find 32 - 8
6. Determine which credit card number is valid and which is not valid.
a. 4024007187744080
b. 4929997923363698
c. 4485191778625256
d. 4539315377710946
e. 4241659513432318
7. Determine if the ISBN 97897198 of the book Mathematics in the Modern
World by Earnhart, R and Adina E. is valid or not.
8. The UPC of Philippine product Argentina beef loaf 170g is 7 – 48485 –
80009 – 𝒙𝟏𝟐. Identify the missing check digit.
9. Select three different products found in your home with UPC. Show that the
UPCs of the products are valid.
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Summary
Modular arithmetic is a system of arithmetic for integers, which considers
the remainder. In modular arithmetic, numbers "wrap around" upon reaching a
given fixed quantity (this given quantity is known as the modulus) to leave a
remainder. A number x (mod N) is the equivalent of asking for the remainder
of x when divided by N.



If m is a positive integer and m | a – b, then we say that a is congruent
to b modulo m and in symbol, a ≡ b (mod m). If a is not congruent to b
modulo m, then we write a ≢ b (mod m).
Every integer is congruent modulo m to exactly one of the integers
0,1,2,….,m-1.
Two integers a and b are said to be congruent (or in the same
equivalence class) modulo N if they have the same remainder upon
division by N. In such a case, we say that a ≡ b (mod N).
Modular arithmetic is an extremely flexible problem solving tool in the
validation of various identification numbers such as credit card numbers, ISBN,
UPC and many others.


The Luhn formula (also known as the modulos 10) is used to generate
and/or validatae and verify the accuracy of credit-card numbers.
Formula for the ISBN Check Digit
d13 = 10 – (d1 + 3d2 + d3 +3d4 + d5 + 3d6 + d7 + 3d8 + d9 + 3d10 + d11 + 3d12)
mod 10
If d13 = 10, then the check digit is 0.
 Formula for the UPC Check Digit
d12 = 10 – (3d1 + d2 + 3d3 +d4 + 3d5 + d6 + 3d7 + d8 + 3d9 + d10 + 3d11) mod 10
If d12 = 10, then the check digit is 0.
Reflection
What is your realization of the significance of modular arithmetic in checking
validity of identification numbers?
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References
Aufman, R.N., et.al (2010) Mathematical Excursions (Second Edition).
Brooks/Cole Cengage Learning
Lockwood, J. et.al. (20190 Mathematics in the Modern World. Rex Book
Store
Baltazar, E., et.al. (2018) Mathematics in the Modern World. C & E
Publishing, Inc.
Earhart, R. T. & Adina, E. M., (2018) Mathematics in the Modern World. C &
E Publishing, Inc.
http://mathworld.wolfram.com/ModularArithmetic.html
https://www.khanacademy.org/computing/computerscience/cryptography/modarithmetic/a/what-is-modular-arithmetic
https://www.isbn-international.org/content/what-isbn
https://www.shopify.com/encyclopedia/universal-product-code-upc
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