Lesson 1: Mathematics in Our World 

1.1 Overview: What is mathematics? 

There is no generally accepted definition for Mathematics and point of views

on  Mathematics may differ from one person to another. But one thing is for
sure,  Mathematics is everywhere and in everything. We use it in our daily lives
and  its application varies from simple to complex.  

In this chapter, we’ll be discussing Mathematics in Nature through the study

of  patterns and numbers and its importance in our world. 

1.2 Patterns and Numbers in Nature 

What is a Pattern? 

We usually think of it as anything that repeats again and again. A pattern is

an  arrangement which helps observers anticipates what they might see or
what  happens next. A pattern also shows what may have come before. A
pattern  organizes information so that it becomes more useful. The human
mind is  programmed to make sense of data or to bring order where there is
disorder. It  seeks to discover relationships and connections between seemingly
unrelated  bits of information. In doing so, it sees patterns. 

Patterns and Numbers in Nature 

Patterns in nature are visible regularities of form found in the natural world. 
These patterns recur in different contexts and can sometimes be modeled 
mathematically. Natural patterns include symmetries, fractals, spirals, 
meanders, waves, foams, tessellations, cracks, and stripes. Studying patterns 
allows one to watch, guess, create, and discover. The present mathematics is 
considerably more than arithmetic, algebra, and geometry. The method of 
doing it has advanced from simply performing computations or derivations
into  observing patterns, testing guesses, and evaluating results.

TYPES OF PATTERNS

 Symmetry
 Trees or Fractals
 Spirals
 Meanders (Chaos or Flow)
  Waves or Dunes
 Bubbles or Foam
 Tessellations (Arrays, Crystals or Tilings)
 Cracks
 Spots and Stripes

Symmetry 

Mathematically, symmetry means that one shape becomes exactly like another 
when you move it in some way: turn, flip or slide. For two objects to be 
symmetrical, they must be the same size and shape, with one object having a 
different orientation from the first.  

It has four (4) main types: 

a) Bilateral or Reflection or Line Symmetry 

b) Radial or Rotation Symmetry 
c) Translational Symmetry 
d) Glide Reflection Symmetry 

Bilateral or Reflection or Line Symmetry is the simplest kind of symmetry.

It  is one of the most common kinds of symmetry that we see in the natural
world.  It can also be called mirror symmetry because an object with this
symmetry  looks unchanged if a mirror passes through its middle. In other
words, the  objects have a left side and a right side that are mirror images of
each other. If  a shape can be folded in half so that one half fits exactly on top
of the other,  then we say that the shapes are symmetric. The fold is called a
line of  symmetry because it divides the shape into two equal parts. Bilateral 
symmetric objects have at least one line or axis of symmetry. The lines of 
symmetry may be in any direction. 

If a figure is rotated around a centre point and it still appears exactly as it did  
before the rotation, it is said to have Radial or Rotational Symmetry. A 
number of shapes like squares, circles, regular hexagon, etc. have rotational 
symmetry. 

If a something has undergone a movement, a shift or a slide, in a specified 

direction through a specified distance without any rotation or reflection, it is 
said to have Translational Symmetry. 
The symmetry that a figure has if it can be made to fit exactly onto the original 
when it is translated a given distance at a given direction and then reflected  
over a line is called Glide Reflection Symmetry. 

Classification of Symmetry on Plane Figures 

a) Rosette Patterns 
b) Frieze or Border Patterns 
c) Wallpaper Patterns
Rosette Patterns consist of taking motif or an element and rotating and/or 
reflecting that element. 

Frieze or Border Pattern is a pattern in which a basic motif repeats itself

over  and over in one direction. It extends to the left and right in a way that
the  pattern can be mapped onto itself by a horizontal translation. They are
divided  into seven groups. 
Mathematician John Conway created names that relate to footsteps for each of the frieze groups.

a) The first frieze group, F1, contains only translation symmetries. F1 is  also
called a HOP. 

b) The second frieze group, F2, contains translation and glide reflection 
symmetries. F2 is also called a STEP. 

c) The third frieze group, F3, contains translation and vertical reflection 
symmetries. F3 is also called a SIDLE. 

d) The fourth frieze group, F4, contains translation and rotation (by a half
turn) symmetries. F4 is called a SPINNING HOP.

e) The fifth frieze group, F5, contains translation, glide reflection and  rotation
(by a half-turn) symmetries. F5 is also called a SPINNING  SIDLE. 

f) The sixth frieze group, F6, contains translation and horizontal reflection 
symmetries. F6 is also called a JUMP. 
g) The seventh frieze group, F7, contains all symmetries (translation, 
horizontal & vertical reflection, and rotation). F7 is also called a  SPINNING
JUMP. 

Wallpaper Patterns is a pattern with translation symmetry in two directions.

It  is, therefore, essentially an arrangement of friezes stacked upon one another
to  fill the entire plane. Any particular wallpaper pattern is made up of a 
combination of the following symmetries; reflection, rotation and glide 
reflection.

Fractals

Fractals are infinitely self-similar, iterated mathematical constructs having

fractal dimensions. Infinite iteration is not possible in nature so all 'fractal'
patterns are only approximate.

branches of trees, animal circulatory systems, snowflakes, lightning and

electricity, plants and leaves, geographic terrain and river systems, clouds,
crystals.
Spirals

Spirals are common in plants and in some animals, notably molluscs. For
example, in the nautilus, a cephalopod mollusc, each chamber of its shell is an
approximate copy of the next one, scaled by a constant factor and arranged in
a logarithmic spiral. Given a modern understanding of fractals, a growth spiral
can be seen as a special case of self-similarity.

Meanders or chaos or flow

Meanders are sinuous bends in rivers or other channels, which form as a fluid,
most often water, flows around bends. As soon as the path is slightly curved,
the size and curvature of each loop increases as helical flow drags material like
sand and gravel across the river to the inside of the bend. The outside of the
loop is left clean and unprotected, so erosion accelerates, further increasing the
meandering in a powerful positive feedback loop.

Waves or dunes
Waves are disturbances that carry energy as they move. As waves in water or
wind pass over sand, they create patterns of ripples. When winds blow over
large bodies of sand, they create dunes. Dunes may form a range of patterns
including crescents, very long straight lines, stars, domes, parabolas, and
longitudinal or Seif ('sword') shapes.

Bubbles or foam
A soap bubble forms a sphere, a surface with minimal area — the smallest
possible surface area for the volume enclosed. Two bubbles together form a
more complex shape: the outer surfaces of both bubbles are spherical; these
surfaces are joined by a third spherical surface as the smaller bubble bulges
slightly into the larger one.

Tesselations 

A tessellation or tiling is a repeating pattern of figures that covers a plane with 

no gaps or overlaps. It is just like a wallpaper group in which patterns are 
created by repeating a shape to fill the plane. 
Cracks

Cracks are linear openings that form in materials to relieve stress. When an elastic material
stretches or shrinks uniformly, it eventually reaches its breaking strength and then fails suddenly
in all directions, creating cracks with 120 degree joints, so three cracks meet at a node.
Conversely, when an inelastic material fails, straight cracks form to relieve the stress. Thus the
pattern of cracks indicates whether the material is elastic or not.

Spots and stripes

Leopards and ladybirds are spotted; angelfish and zebras are striped. These patterns have an
evolutionary explanation: they have functions which increase the chances that the offspring of
the patterned animal will survive to reproduce.

SYNTHESIS

 Patterns in nature are not just there to be admired. They are vital clues to the rules that
govern natural processes and by using Mathematics, the great secret has been discovered.

 Mathematics definitely a useful way to think about nature, to understand how they
happen, to understand why they happed, to organize the underlying patterns and the regularities
in the most satisfying way to predict how nature will behave to control nature for our own ends
and to make practical use of what we have about our world.

 As Galileo said, the book of nature is written in the Language of Mathematics

1.3 Fibonacci Sequence 

We start with 1 and another 1. Add them, we get 2. Add 1 and 2, we get 3. Add 
2 and 3, we get 5. Add 3 and 5, we get 8. If we continue repeating the process,  
we obtain the sequence 1, 1, 2, 3, 5, 8, 13,.. which is known as the Fibonacci  
sequence. 

The Fibonacci numbers appear in nature in various places. These numbers

are  evident at the flower head of a sunflower or daisy. Spirals are also easier to
see  and to count on pineapples and pine cones. Fibonacci numbers are there
on  broccoli florets and flowers and on the arrangement of leaves around stems
on  many plants too. 

Named for the famous mathematician, Leonardo Fibonacci

 His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in
Italy.
  He is also known as Leonardo Bonacci or Leonardo of Pisa.
 The name he is commonly called, Fibonacci, was made up in 1838 by the Franco-Italian
historian Guillaume Libri and is short for Filius Bonacci (‘Son of Bonacci’).
 He was responsible for introducing to Europe the Hindu-Arabic numeration system that
we use today when he published Liber Abaci in 1202.

This is called the Fibonacci Spiral.

Pinecones, Speed Heads, Vegetables and Fruits 

Spiral patterns curving from left and right can be seen at the array of seeds in 
the center of a sunflower. The sum of these spirals when counted will be a 
Fibonacci number. You will get two consecutive Fibonacci numbers if you 
divide the spirals into those pointed left and right. Fibonacci sequence appears 
on these fruits and vegetables. 

Flowers and Branches 

Some plants also exhibit the Fibonacci sequence in their growth points, on the 
places where tree branches form or split. A trunk grows until it produces a 
branch, resulting in two growth points. The main trunk then produces
another  branch, resulting in three growth points and then the trunk and the
first  branch produce two more growth points, bringing the total to five as
illustrated  on the image below.

Honeybees 

The family tree of a honey bee perfectly resembles the Fibonacci sequence. A 
honeybee colony consists of a queen, a few drones and lots of workers. The 
following image below shows how the family tree relates. 

The Human Body 

The human body has many elements that show the Fibonacci numbers and
the  golden ratio. Most of your body parts follow the Fibonacci sequence and
the  proportions and measurements of the human body can also be divided up
in  terms of the golden ratio. 

Geography, Weather and Galaxies 

Fibonacci numbers and the relationships between these numbers are evident 
in spiral galaxies, sea wave curves and in the patterns of stream and
drainages.  Weather patterns, such as hurricanes and whirlpools sometimes
closely  resemble the Golden Spiral. The Milky Way galaxy and some other
galaxies  have spiral patterns. Planets of our solar system and their orbital
periods are  closely related to the golden ratio. 

Golden Ratio 
The Golden Ratio and/or the Golden Spiral can also be observed in music, art, 
and designs. Appearing in many architectural structures, the presence of the 
golden ratio provided a sense of balance and equilibrium. Let’s take a look at a 
couple of examples. 

Golden ratio, also known as the golden section, golden mean, or divine
proportion, in mathematics, the irrational number (1 + Square root of√5)/2,
often denoted by the Greek letter φ or τ, which is approximately equal to 1.618.

It is the ratio of a line segment cut into two pieces of different lengths such that
the ratio of the whole segment to that of the longer segment is equal to the
ratio of the longer segment to the shorter segment.

Architecture 

The Great Pyramid of Giza: The Great Pyramid of Giza built around 2560 BC is 
one of the earliest examples of the use of the golden ratio. The length of each 
side of the base is 756 feet, and the height is 481 feet. So, we can find that the 
ratio of the vase to height is 756=481 = 1:5717. 
Arts 

Mona-Lisa by Leonardo Da Vinci: It is believed that Leonardo, as a 

mathematician tried to incorporate of mathematics into art. This painting 
seems to be made purposefully line up with golden rectangle.
1.4 Mathematics for Our World 

Mathematics is everywhere; whether it is on land, sea or air, online or on the 

front line, mathematics underpins every nook and cranny of modern life. Far 
from a quaint subject to be forgotten upon leaving school, it is the glue that 
holds our world. 

Math helps us understand or make sense of the world - and we use the world 
to understand math. It is therefore important that we learn math contents 
needed to solve complex problems in a complex world; learn the mathematical 
knowledge and skills we need to understand the world and make
contributions  to the global community. 

Applications of Mathematics in Our World 

Mathematics has so many uses of applications. 

 a) Mathematics helps organize patterns and regularities in the world; b)
Mathematics helps predict the behavior of nature and many phenomena; c)
Mathematics helps control nature and occurrences in the world for our 
own good; 
d) Mathematics has applications in many human endeavors. 

 By mathematical modeling we see the inputs to events and their most likely outcomes. Knowing
these inputs and seeing their consequences and establishing their relationship defined quantitatively,
we can prepare for calamities or natural disasters, or better yet, we can probably stop them from
happening.

 Control theory is defined as a field of applied mathematics that is relevant to the control of
certain physical processes and systems. As long as human culture has existed, control has meant some
kind of power over the environment and control theory may be viewed as the science of modifying that
environment, in the physical, biological, or even social sense.

CHAPTER 2: LOGIC AND SETS 

2.1 PROPOSITION 

What is a Proposition? 

A Proposition is a declarative sentence that can be objectively identified as either

true  or false but not both. If a proposition is true, then its truth value is true and
is denoted  by T or 1; otherwise, its truth value is false ad is denoted by F or 0. 

Examples: 
Proposition  Not a Proposition

2 + 3 = 5.  What time is it?

Rodrigo Duterte is the President of the Ouch!

Philippines. 

"C" is a vowel.  x+2=4

BACR is an Engineering Course.  Go out and play.

All dogs are white.  How far is it to the next gasoline station?

x + 2 = 2x when x = −2  x + 2 = 2x

What is the Negation of a Proposition? 

The Negation of a Proposition p is the proposition which is false when p is true;
and  true when p is false. The negation of p is denoted by ¬p. 
p  ¬p

T  F

F  T

Examples:
p  ¬p

The integer 7 is It is not the case that the integer 7 is odd.

odd.
The integer 7 is not odd.

The integer 7 is even.

3 is less than 5. It is not the case that 3 is less than 5.

3 is not less than 5.

3 is greater than 5.

Two types of Proposition 

a. Simple Proposition Is a proposition with only one subject and only one 
predicate. 

b. Compound Proposition are formed from combination of simple proposition 

using logical connectives or logical operators. 

Logical Connectives/ Logical Operators Alternative  Symbol

Name 

Negation of p  NOT  ¬p

Conjunction of p and q  AND  p∧q

Disjunction of p and q  OR  p∨q

Negation of   NAND  p↑q

Conjunction of p and q 

Negation of Disjunction of p and q  NOR  p↓q

 Exclusive or of p and q  XOR  p ⊕q

Conditional statement  IMPLICATION  p→q

Biconditional   BI-IMPLICATIONS  p↔q

statement 

The conjunction of p and q is the proposition “p and q”, denoted by p∧q, which is
true  only when both p and q are true.
p  q  p∧q

T  T  T

T  F  F

F  T  F

F  F  F

The disjunction of p and q is the proposition “p or q”, denoted by p∨q, which is

false only when both p and q are false. 
p  q  p∨q

T  T  T

T  F  T

F  T  T

F  F  F

The conditional statement p → q is the proposition “If p, then q” is the

proposition  which is false only when p is true and q is false. The converse, inverse
and contrapositive of p → q are the conditional statements q → p, (¬p) → (¬q), and (¬q) →
(¬p), respectively. 
p  q  p→q

T  T  T

T  F  F

F  T  T

F  F  T

Examples:
 Conditional Statement p If Andres Bonifacio is a Katipunero, then Andres Bonifacio is a
→q Hero.

Converse Statement  If Andres Bonifacio is a Hero, then Andres Bonifacio is a

q→p Katipunero.

Inverse Statement  If Andres Bonifacio is not a Katipunero, then Andres Bonifacio is

(¬p) → (¬q)  not a Hero.

Contrapositive   If Andres Bonifacio is not a Hero, then Andres Bonifacio is not a

Statement  Katipunero.
(¬q) → (¬p)

The biconditional statement p ↔ q to be read as “p if and only if q” is the

proposition  which is true only if both p and q are true or both p and q are false. 
p  q  p↔q

T  T  T

T  F  F

F  T  F

F  F  T

Tautology, Contradiction and Contingency 

A compound proposition that is always true regardless of the truth values of its 
component proposition is called a Tautology. 

A compound proposition that is always false regardless of the truth values of it 
component proposition is called a Contradiction. 

A compound proposition that is neither a tautology nor a contradiction is called a 

Contingency. 

Examples:
Tautology  Madelyn is either 18 years old or not 18 years old.

Contradiction  Madelyn is both 18 years old and not 18 years old.

Contingency  Madelyn is either 17 years old or 18 years old.

DE MRGAN’S LAW FOR STATEMENTS

For statement P and Q

1. ∼(P ⋁ Q) ≡ P ⋀ ∼Q

2. ∼(P ⋀ Q) ≡ ∼P ⋁ ∼Q

TAUTOLOGY is a statement which is true in every case, (always true) SELF CONTRADICTION is a
statement always false

EXAMPLE 4: is the statement x + 2 = 5 a tautology of self-contradiction

SOLUTION: The statement is not true for all values of X and it is not false for all values of X.

2.2 SET THEORY 

What is a Set? 

A Set is well-defined collection of objects called elements. 

Examples: 

1. The collection of Jordan Shoes is a Set. 

2. The collection of all vowels in the English alphabet is a Set. 
3. The collection of handsome guys is not a set because one cannot objectively 
identify if a given guy is handsome or not because the word “handsome” is 
subjective in nature. 

Upper case letters are usually used to named sets and can be commonly described in 
three (3) ways: a) Listing or Roster Method, b) Set-Builder Notation and c) Descriptive 
Method. 

The Listing Method or Roster Method describes the set by listing all the elements 
between braces and separated by commas (Note: in enumerating the elements of a 
certain set, each element is listed only once and the arrangement of elements in the
list  is immaterial) 

The Set-Builder Notation uses a variable (a symbol, usually a letter that can
represent  different elements of a set), braces, and a vertical bar that is read as “such
that”. This  is usually used when elements are too many to list down.  
The Descriptive Method uses a short verbal statement to describe a

set.

Examples: 

Listing or Roster V = {a, e, i, o, u}

Method 

Set-Builder Notation  V = {x|x is a vowel in the English alphabet}

Descriptive Method  V is the set of all vowels in the English alphabet

Listing or Roster A = {0, 1, 2, 3, 4, 5, 6, 7)

Method 

Set-Builder Notation  A = {x|x is a Whole Number, 0 ≤ X < 8}

Descriptive Method  A is set of Whole Numbers that is less than 8

Special Sets 

N – set of natural numbers {0, 1, 2, 3, …} 

Z – set of integers {…, -2, -1, 0, 1, 2, …} 
Q – set of rational numbers 
R – set of real numbers 
C – set of complex numbers 

What is a Universal Set? 

A Universal Set is the set of all elements under consideration, denoted by capital .
All other sets are subsets of the universal set. 

What is an Element? 

An Element (∈) is any object that belongs to a set. 

Examples: 

V = {a, e, i, o, u} 

a ∈ V o ∈ V 
e ∈ V u ∈ V 
i ∈ V 
What is the Cardinality of Sets? 

The Cardinality of Set is the number of distinct elements of a set. The Cardinality of 
Set is denoted by |S|. 

Examples:
Examples  Cardinality of Set

Set A = {1}  |A| = 1

Set B = {1,2}  |B| = 2

Set C = {1,2,3}  |C| = 3

Set D = {1,2,3,4}  |D| = 4

Set E = |E| = 1
{1,1,1,1,1} 

What is a Subset? 

A subset is a set whose elements are all members of another set. 

The symbol "⊆" means "is a subset of". 

The symbol "⊂" means "is a proper subset of". 

Examples: 

Set A = {1,2,3,4} 

Set B = {1,2,3,4,5,6,7} 

Set C = {6,7,8,9} 

Set U = {1,2,3,4,5,6,7,8,9} 
 Set A is a Subset of Set B (A⊆B) 
Set C is not a Subset of Set B (C⊄B) 
Set A, Set B, and Set C is a Subset of Set U 

What is a Power Set? 

In mathematics, the power set (or powerset) of a set S is the set of all subsets of S,
including the empty set and S itself. 

Examples: 

Set S = {1,2,3} 

ρ(S) = { }, {1}, {2}, {3}, {1,2}, {2,3}, {3,4}, {1,2,3}

What is an Equal Set and Equivalent Set? 

Two sets are Equal if they have exactly the same elements irrelevance of order and 
repetition. It could be anything like numbers, pictures, alphabets, etc. 

Examples:  

Set A = {1,2,3,4} 
Set B = {3,1,4,2} 
Set C = {1.1.2.2.3.3.4.4} 

Set A, Set B and Set C are Equal Sets. 

Two sets are Equivalent if they have the same number of

elements. Examples: 

Set A = {1,2,3} 
Set B = {x, y, z} 

Set A and Set B are Equivalent Sets. 

Sizes of Sets 

a. Null Set – an Empty Set 

Examples: A = { } 
B = {0} 

b. Unit Set – set with only one element 

Examples: C = {1} 
 D = {2} 

c. Infinite Set – set with infinite number of elements 

Examples: E = {1,2,3,4,5,….} 

d. Finite Set – set with definite number of elements 

Examples: F = {1,2,3,4,5}
Operation of Sets 

a. Union 
The Union of Sets A and B is represented by A∪B. The union contains those 
elements that are either in A or in B or in Both. 

Examples: Set A = {1,2,3} 

Set B = {3,4,5} 
A∪B = {1,2,3,4,5} 

b. Intersection 
The Intersection of Sets A and B is represented by A∩B. The intersection
contains  those elements in both A and B. 

Examples: Set A = {1,2,3} 

Set B = {3,4,5} 
A∩B = {3}
c. Complement 
Given the universal set U, the complement of Set A, denoted by Ā, contains all 
the elements in the universal set not present in Set A. 

Examples: Set A = {1,2,3} 

Set B = {3,4,5} 
Set U = {1,2,3,4,5,6,7} 
Ā = {4,5,6,7} 

d. Difference or Relative Complement 

The difference of A and B, denoted by A-B, is the set containing those elements 
that are in A but not in B. The difference of A and B is also called the
complement  of B with respect to A. 

Examples: Set A = {1,2,3} 

Set B = {3,4,5} 
A-B = {1,2}
e. Exclusive-Or or Symmetric Difference 
The symmetric difference of A and B, denoted by A⊕B, is the set containing
those  elements in either A or B, but not in both A and B. 

Examples: Set A = {1,2,3} 

Set B = {3,4,5} 
A⊕B = {1,2,4,5} 

f. Cartesian Product 
The Cartesian Product of A and B, denoted by AxB, is the set of all ordered
pairs  (a,b), where a is an element of Set A and b is an element of Set B. 

Examples: Set A = {1,2,3} 

Set B = {3,4,5} 
AxB = {(1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5)}
CHAPTER 3: PROBLEM SOLVING 

What is a Problem? 

➢ A question that needs a solution. 

➢ It is a question or condition that is difficult to deal with and has not been solved.
➢ It is a situation in which an individual or group is called upon to perform a task for
which  there is no readily accessible algorithm which determines completely the method
of  solution. 

What is Problem Solving? 

➢ Problem solving is the act of defining a problem; determining the cause of the problem; 
identifying, prioritizing, and selecting alternatives for a solution; and implementing a 
solution. 
➢ Problem solving is the methods we use to understand what is happening in our 
environment, identify things we want to change and then figure out the things that need 
to be done to create the desired outcome. 

Understanding Problem Solving 

Before getting too far into the discussion of Problem Solving, it is worth pointing out that we
find  it useful to distinguish between the three words method, answer and solution. 

METHOD + ANSWER = SOLUTION 

By method we mean the means used to get an answer. This will generally involve one or more
Problem Solving Strategies. On the other hand, we use answer to mean a number, quantity
or  some other entity that the problem is asking for. Finally, a solution is the whole process
of  solving a problem, including the method of obtaining an answer and the answer itself. 

Who is George Polya? 

George Pólya (December 13, 1887 – September 7th, 1985) was a Hungarian mathematician.
In  1945, he devised a model for problem solving and published it in his book “How to Solve
it”. 

“Mathematical problem solving is finding a way around a difficulty, around an obstacle, and
finding a solution to a problem that is unknown.” - George Polya
George Polya’s Guideline to Problem Solving 

Polya’s Four Steps 

1. Understand the Problem 

➢ Sometimes the problem lies in understanding the problem. If you are unclear as to what 
needs to be solved, then you are probably going to get the wrong results. In order to show 
an understanding of the problem, you, of course, need to read the problem carefully. 
Sounds simple enough, but some people jump the gun and try to start solving the
problem  before they have read the whole problem. Once the problem is read, you need to
list all  the components and data that are involved. This is where you will be assigning
your  variable. 

➢ Ask questions, experiment, or otherwise rephrase the question in your own words. 

2. Devise a Plan 
➢ Find the connection between the data and the unknown. Look for patterns relate to a 
previously solved problem or a known formula, or simplify the given information to give 
you an easier problem. 

➢ Once the problem is understood, set the problem aside for a while. Your subconscious 
mind may keep working on it. Moving on to think about other things may help you stay  
relaxed, flexible, and creative rather than becoming tense, frustrated, and forced in your 
efforts to solve the problem. 

3. Carry out the Plan 

➢ Check the steps as you go. 

➢ This is where you solve the equation you came up with in your 'devise a plan' step. The 
equations in this tutorial will all be linear equations. 

➢ Follow through with the plan that you have chosen. If it continues not to work discard it  
and try another approach. Do not be misled, this is how mathematics is done, even by 
professionals. The key is to keep trying until something works. 

4. Look Back 

➢ Examine the solution obtained. In other words, check your answer. 

➢ Basically, check to see if you used all your information and that the answer makes sense. 
If your answer does check out, make sure that you write your final answer with the
correct  labeling.

Example: Susie’s age this year is a multiple of 5. Next year, her age is a multiple of 7.
What  is her present age? 

1. Understand the Problem 

Given: Susie's Age now is a multiple of 5 

Susie's Age next year is a multiple of 7 

2. Devise a Plan 

Consider the list of Multiples of 5: 

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 

3. Carry out the Plan 

Add 1 to each multiple of 5 and look for the number that is a multiple of 7. The
numbers  you will find are 21 and 56. 

6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 

4. Look back 
 Check if 21 and 56 are multiples of 7 

21 ÷ 7 = 3 
56 ÷ 7 = 8 
91 ÷ 7 = 13 

Susie’s Age current age is either 20 or 55 or 90 

What is Reasoning? 

➢ It is our ability to use logical thinking to come up with a decision. 

Two major types of reasoning: 

 a. Inductive Reasoning 
b. Deductive Reasoning

Inductive Reasoning 

➢ It is the process of reasoning that arrives at a general conclusion based on observation of 
specific examples. 

➢ Conjecture is the conclusion formed by using inductive reasoning which may or may not 
be true. 

Examples: 

1 is an odd number 
11 is an odd number 
21 is an odd number 
Therefore, all numbers ending with 1 is odd number. 

a = b 
b = c 
Then a = c 

Essay test is difficult 

Problem solving test is difficult 
Therefore, all tests are difficult 

Deductive Reasoning 

➢ It is the process of reasoning that arrives at a conclusion based on previously accepted 

general statements. 

➢ It is the process of reaching a conclusion by applying general ideas called

“Premises”.

Examples: 
 Premise 1: Lorenze is a man. 
Premise 2: All men are mortal. 
Conclusion: Lorenze is a mortal. 

Premise 1: Noble gases are stable. 

Premise 2: Neon is a noble gas. 
Conclusion: Neon is stable 

Inductive Reasoning versus Deductive Reasoning 

Inductive Reasoning  Deductive Reasoning

Specific Argument -> General General Conclusion -> Specific Argument

Conclusion 

Logically True  Logically True

But realistically may or may not be true  Realistically True

What is Counter Example? 

➢ A counterexample to an argument or a proposition is a situation which shows that the 

argument can have true premises and a false conclusion. 

➢ A counterexample to an argument is a substitution instance of its form where the 

premises are all true and the conclusion is false. Since the validity and invalidity is a 
matter of form and since only an invalid argument can have true premises and a false 
conclusion, a counterexample to an argument proves both that it and its form are invalid. 

Examples: 

The Prime number 2 is a counterexample to the statement, “All prime numbers are

odd”. If God exists, then life has meaning. But there is no God. Therefore, life is

meaningless. 

All carabaos are mammals. 

All carabaos are animals. 
So, all animals are mammals. 

If you are in Congress, you’re either a Senator or a Representative. 

If you are a Senator, you have a 6-year term 
So, if you are in Congress, you have a 6-year term.