UNIVERSITY OF MINDANAO
College of Arts and Sciences Education
General Education - Mathematics
Physically Distanced but Academically Engaged
Self-Instructional Manual (SIM) for Self-Directed Learning (SDL)
Course/Subject: GE 4 – Mathematics in the Modern World
(Week 1 – 3)
Name of Teacher: Prof. Jocelyn G. Ubas
SIM Prepared by: Prof. Ronnie O. Alejan
THIS SIM/SDL MANUAL IS A DRAFT VERSION ONLY.
THIS IS INTENDED ONLY FOR THE USE OF THE
STUDENTS WHO ARE OFFICIALLY ENROLLED
IN THE COURSE/SUBJECT.
THIS IS NOT FOR REPRODUCTION, COMMERCIAL, AND
DISTRIBUTION OUTSIDE OF ITS INTENDED USE.
EXPECT REVISIONS OF THE MANUAL.
College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Course Information: see/download course syllabus in the Blackboard LMS
CC’s Voice:
Welcome to the course GE 4: Mathematics in the Modern World.
Mathematics is all around you and in everything you do. You
may be a cook or a policeman, a nurse or a businessman, a
computer programmer or a forester, an engineer or an architect, a
musician or a magician, and everyone needs mathematics in day-today life. As a cradle of all creations, the world cannot move an inch
without mathematics. So, it is also important for you to know and
appreciate how mathematics affects and influences your life.
CO
For you to appreciate the uses of mathematics in everyday life,
you need to understand its nature, its language, symbols, and
arguments made about mathematics and mathematical concepts,
which is the ultimate course outcome (CO) of this subject.
This course deals with the nature of mathematics, appreciation
of its practical, intellectual, and aesthetic dimensions, and application
of mathematical tools in daily life.
You will be introduced to the nature of mathematics as an
exploration of patterns in nature and the environment and as an
application of inductive and deductive reasoning. You are
encouraged to go beyond the typical understanding of mathematics,
which is just as merely a set of formulas, but as a source of aesthetics
in patterns of nature.
Likewise, mathematics provides a tool for understanding and
dealing with various aspects of present-day living, such as managing
personal finances, making social choices, appreciating geometric
designs, understanding geometric designs, understanding codes
used in data transmission and security, and dividing limited
resources fairly. These aspects will provide opportunities for actually
doing mathematics in a broad range of exercises that bring out the
various dimensions of mathematics as a way of knowing, and test
your understanding and capacity.
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Week 1-3: Unit Learning Outcomes (ULO):
At the end of the unit, you are expected to
a. Articulate the importance of Mathematics in one’s life;
b. Translate English phrases to Mathematical expressions using the
language, symbols, and conventions of Mathematics;
c. Use different types of reasoning to justify statements and
arguments made about Mathematics; and
d. Apply Polya’s four steps in solving problems.
Big Picture in Focus
ULO-a.
Articulate the importance of Mathematics in one’s life.
Metalanguage
In this section, the essential terms relevant to the study of the nature of
Mathematics and to demonstrate ULO-a will be operationally defined to establish a
common frame of reference as to how the texts work. You will encounter these terms
as we go through the study of the nature of mathematics. Please refer to these
definitions in case you will encounter difficulty in understanding some concepts.
1. Meaning and definitions of Mathematics:
•
Mathematics is a systematized, organized, and exact branch of science.
•
Mathematic deals with quantitative facts, relationships, as well as with
problems involving space and form.
•
Mathematics is a logical study of shape, arrangement, and quantity.
•
Mathematics is not to be considered only as ‘number work’ or
‘computation,’ but it is more about forming generalizations, seeing
relationships, and developing logical thinking & reasoning.
•
Mathematics should be shown as a way of thinking, an art or form of
beauty, and as a human achievement.
•
Mathematics helps in solving problems of life that need numeration and
calculation.
•
Mathematics provides an opportunity for the intellectual gymnastic of the
man’s inherent powers.
•
Mathematics is an exact science and involves high cognitive abilities and
powers.
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
2. 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.
3. Number pattern is a pattern or sequence in a series of numbers. This pattern
generally establishes a common relationship between all numbers.
4. Fibonacci sequence is a set of numbers that starts with a one or a zero,
followed by a one, and proceeds based on the rule that each number (called
a Fibonacci number) is equal to the sum of the preceding two numbers.
5. The Golden ratio (the symbol is the Greek letter "phi," Φ) is a special number
approximately equal to 1.618. It appears many times in geometry, art,
architecture, and other areas. We find the golden ratio when we divide a line
into two parts so that: the long part divided by the short part is also equal to
the whole length divided by the long part.
Essential Knowledge
To perform the aforesaid big picture (unit learning outcomes) for the first three
(3) weeks of the course, you need to fully understand the following essential
knowledge that will be laid down in the succeeding pages. Please note that you are
not limited to refer to these resources exclusively. Thus, you are expected to utilize
other books, research articles, and other resources that are available in the
university’s library e.g., ebrary, search.proquest.com, etc.
1. The Nature of Mathematics. The nature of Mathematics can be made explicit
by analyzing the chief characteristics of Mathematics as outlined by Sophia
(2018).
1.1 Mathematics is a science of discovery
It is the discovery of relationships and the expression of those
relationships in symbolic form – in words, in numbers, in letters, by diagrams,
or by graphs. Today, it is a discovery technique, which is making spectacular
progress. They are being applied in two fields: in pure number relationships
and everyday problems like money, weights, and measures.
1.2 Mathematics is an intellectual game
Mathematics can be treated as an intellectual game with its own rules
and abstract concepts. From these viewpoints, Mathematics is mainly a matter
of puzzles, paradoxes, and problem-solving – a sort of healthy mental exercise.
1.3 Mathematics deals with the art of drawing conclusions
One of the important functions of the school is to familiarize students
with a mode of thought, which helps them in drawing the right conclusions and
inferences. In Mathematics, the conclusions are certain and definite. Hence,
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
the learner can check whether or not he has drawn the correct conclusions,
permit the learner to begin with simple and very easy conclusions, gradually
move over to more difficult and complex ones.
1.4 Mathematics is a tool subject
Mathematics has its integrity, its beauty, its structure, and many other
features that relate to Mathematics as an end in it. However, many conceive
Mathematics as a very useful means to other ends, a powerful and incisive tool
of wide applicability.
1.5 Mathematics involves an intuitive method
Intuition is to anticipate what will happen next and what to do about it. It
implies the act of grasping the meaning, significance, or structure of a problem
without explicit reliance on the analytic mode of thought. It is a form of
mathematical activity that depends on the confidence in the applicability of the
process rather than upon the importance of the right answers all the time.
1.6 Mathematics is the science of precision & accuracy
Mathematics is known as an exact science because of its precision. It is
perhaps the only subject that can claim certainty of results. In Mathematics, the
results are either right or wrong or accepted or rejected. There is no midway
possible between rights and wrong. Mathematic can decide whether or not its
conclusions are right. Even when there is a new emphasis on approximation,
mathematical results can have any degree of accuracy required.
1.7 Mathematics is the subject of logical sequence
The study of Mathematics begins with few well – known uncomplicated
definitions and postulates and proceeds step by step to quite elaborate steps.
Mathematics learning always progresses from simple to complex and from
concrete to abstract. It is a subject in which the dependence on earlier
knowledge is particularly great. Algebra depends on Arithmetic, the Calculus
depends on Algebra, Dynamic depends on the Calculus, Analytical Geometry
depends on Algebra and Elementary Geometry, and so on. Thus gradation and
sequence can be observed among topics in any selected branch of
Mathematics.
1.8 Mathematics requires the application of rules and concepts to new
situations
The study of Mathematics requires the learners to apply the skill
acquired to new situations. The students can always verify the validity of
mathematical rules and relationships by applying them to novel situations.
Concept and principle become more functional and meaningful only when they
are related to actual practical applications. Such a practice will make the
learning of Mathematics more meaningful and significant.
1.9 Mathematics deals with generalization and classification
Mathematics provides ample exercise in combining different results
under one head, in making schematic arrangements and classifications. When
the student evolves his definitions, concepts, and theorems, he is making
generalizations. The generalizations and classification of Mathematics are very
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
simple and obvious in comparison with those of other domains of thought and
activity.
1.10 Mathematics is abstract science
Mathematical concepts are abstract in the sense that they cannot be
seen or felt in the physical world. Mathematical concepts cannot be learned
through experiences with concrete objects. Some concepts can be learned
only through their definitions, and they may not have concrete counterparts to
be abstracted from. The concept of prime numbers, the concept of probability,
the concept of a function, the concept of limits, concept of continuous functions,
to list few are all abstract in the sense that they can be learned only through
their definitions and it is not possible to provide concrete objects to correspond
to such concepts. Even when concretization is possible they are only
representation of the concepts and not physical object themselves.
1.11 Mathematics is study of structures
The dictionary meaning of ‘structure’ is, ‘the formation, arrangement
and articulation of parts in anything built up by nature or art’. Therefore, a
mathematical structure should be some sort of arrangement, formation, or
result of putting together of parts. A mathematical structure is a mathematical
system with one or more explicitly recognized (mathematical) properties. We
may create a structure from mathematical systems by making specific
recognition of one or more of the commutative, associative or distribution
properties that the system may have.
1.12 Mathematics is a science of logical reasoning
It goes without saying that logic is an important factor in mathematics.
It governs the pattern of deductive proof through which mathematics is
developed. Of course, logic was used in mathematics centuries ago.
2. Patterns in nature. The patterns found in nature have fascinated scientists for
many years. Humans have looked at the stars to find patterns – called
constellations. Each day we experience a sunset and a sunrise – patterns
caused by the Earth’s rotation around the Sun, which we call time. Patterns
help us organize information and make sense of the world around us.
A pattern exists when a set of numbers, colors, shapes, or sound are
repeated over and over again. Patterns can be found everywhere: including in
animals, plants, and even the solar system!
Some specific patterns are as follows:
2.1 A fractal is a detailed pattern that looks similar at any scale and repeats
itself over time. A fractal's pattern gets more complex as you observe it at larger
scales. Examples of fractals in nature are snowflakes, trees branching,
lightning, and ferns.
2.2 A spiral is a curved pattern that focuses on a center point and a series
of circular shapes that revolved around it.
Examples of spirals are pinecones, pineapples, and hurricanes. The
reason for why plants use a spiral form of the leaf is because they are
constantly trying to grow but stay secure. A spiral shape causes plants to
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
condense themselves and not take up as much space, causing it to be stronger
and more durable against the elements.
2.3 A Voronoi pattern provides clues to nature’s tendency to favor efficiency:
the nearest neighbor, shortest path, and tightest fit. Each cell in a Voronoi
pattern has a seed point. Everything inside a cell is closer to it than to any
other seed. The lines between cells are always halfway between neighboring
seeds. Other examples of Voronoi patterns are the skin of a giraffe, corn on
the cob, honeycombs, foam bubbles, the cells in a leaf, and a head of garlic.
There are so many reasons why understanding patterns in nature is
important. People have built cities and created art based on the patterns they
see. We have used patterns, like the alphabet and sign language to help us
communicate with one another. But since our world is always changing, so do
patterns. Next time you go outside, look around – what are some of the patterns
you see?
To see examples of the different types of patterns in nature, visit
https://www.youtube.com/watch?v=4Mfl2QbSMYY
3. Number patterns. By studying patterns in math, humans become aware of
patterns in our world. Observing patterns allows individuals to develop their
ability to predict future behavior of natural organisms and phenomena. Civil
engineers can use their observations of traffic patterns to construct safer
cities. Meteorologists use patterns to predict thunderstorms, tornadoes, and
hurricanes. Seismologists use patterns to forecast earthquakes and
landslides. Mathematical patterns are useful in all areas of science. Some
of these are as follows:
3.1 Arithmetic Sequence. A sequence is group of numbers that follow a
pattern based on a specific rule. An arithmetic sequence involves a sequence
of numbers to which the same amount has been added or subtracted. The
amount that is added or subtracted is known as the common difference.
For example, in the sequence “1, 4, 7, 10, 13…” each number has been
added to 3 in order to derive the succeeding number. The common difference
for this sequence is 3.
3.2 Geometric Sequence. A geometric sequence is a list of numbers that
are multiplied (or divided) by the same amount. The amount by which the
numbers are multiplied is known as the common ratio.
For example, in the sequence “2, 4, 8, 16, 32...” each number is
multiplied by 2. The number 2 is the common ratio for this geometric
sequence.
3.3 Triangular Numbers. The numbers in a sequence are referred to as
terms. The terms of a triangular sequence are related to the number of dots
needed to create a triangle. You would begin forming a triangle with three
dots; one on top and two on bottom. The next row would have three dots,
making a total of six dots. The next row in the triangle would have four dots,
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
making a total of 10 dots. The following row would have five dots, for a total
of 15 dots. Therefore, a triangular sequence begins: “1, 3, 6, 10, 15…”
3.4 Square Numbers. In a square number sequence, the terms are the
squares of their position in the sequence. A square sequence would begin
with “1, 4, 9, 16, 25…”
3.5 Cube Numbers. In a cube number sequence, the terms are the cubes
of their position in the sequence. Therefore, a cube sequence starts with “1,
8, 27, 64, 125…”
3.6 Fibonacci Numbers. In a Fibonacci number sequence, the terms are
found by adding the two previous terms. The Fibonacci sequence begins
thusly, “0, 1, 1, 2, 3, 5, 8, 13…”
The Fibonacci sequence is named for Leonardo Fibonacci, born in
1170 in Pisa, Italy. Fibonacci introduced Hindu-Arabic numerals to Europeans
with the publication of his book “Liber Abaci” in 1202. He also introduced the
Fibonacci sequence, which was already known to Indian mathematicians.
The sequence is important, because it appears in many places in nature,
including: plant leafing patterns, spiral galaxy patterns, and the chambered
nautilus’ measurements.
4. The Golden Ratio. The Golden Ratio is a common mathematical ratio found
in nature, which can be used to create pleasing, organic-looking compositions
in your design projects or artwork. It's also known as the Golden Mean, the
Golden Section, or the Greek letter phi (Φ). Whether you're a graphic designer,
illustrator or digital artist, the Golden Ratio can be used to bring harmony and
structure to your projects.
Closely related to the Fibonacci Sequence, the Golden Ratio describes
the perfectly symmetrical relationship between two proportions. Approximately
equal to a 1:1.61 ratio, the Golden Ratio can be illustrated using a Golden
Rectangle.
This is a rectangle where, if you cut off a square (side length equal to
the shortest side of the rectangle), the rectangle that's left will have the same
proportions as the original rectangle.
So if you remove the right-hand square from the rectangle above, you'll
be left with another, smaller Golden Rectangle. This could continue infinitely.
Similarly, adding a square equal to the length of the longest side of the
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
rectangle gets you increasingly closer to a Golden Rectangle and the Golden
Ratio.
Plotting the relationships in scale provides us with what's known as a
Golden Spiral. This occurs organically in the natural world.
If you're still a little confused, the video below gives a good overview of
the Golden Ratio in use. It also shows you how to construct a Golden Ratio in
Illustrator. This is used to create a Golden Spiral, followed by Golden Circles.
https://www.youtube.com/watch?time_continue=13&v=CSoHCHQ3zJw&featu
re=emb_logo
It is believed that the Golden Ratio has been in use for at least 4,000
years in human art and design. However, it may be even longer than that –
some people argue that the Ancient Egyptians used the principle to build the
pyramids.
In more contemporary times, the Golden Ratio can be observed in
music, art, and design all around you. By applying a similar working
methodology, you can bring the same design sensibilities to your own work.
Let's take a look at a couple of examples to inspire you.
Ancient Greek architecture used the Golden Ratio to determine pleasing
dimensional relationships between the width of a building and its height, the
size of the portico and even the position of the columns supporting the
structure. The final result is a building that feels entirely in proportion. The neoclassical architecture movement reused these principles too.
Leonardo da Vinci, like many other artists throughout the ages, made
extensive use of the Golden Ratio to create pleasing compositions. In The Last
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2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Supper, the figures are arranged in the lower two thirds (the larger of the two
parts of the Golden Ratio), and the position of Jesus is perfectly plotted by
arranging golden rectangles across the canvas.
There are also numerous examples of the Golden Ratio in nature – you
can observe it all around you. Flowers, seashells, pineapples and even
honeycombs all exhibit the same principle ratio in their makeup.
Many features of the “ideal” human face are said to have ratios equal to
phi (Φ). For example, the dimension relationships between the eyes, ears,
mouth, and nose. The ratio of the height of the whole head to that of the head
above the nose is also said to be phi (Φ). Other examples supposedly include
the ratio between the total height of the body and the distance from the head
to the fingertips, and the distances from head to naval and naval to hill. Then
there is a proportion between the forearm and upper arm, and the one between
hand and forearm; all of these are said to follow the rule of golden ratio.
You can refer to the source below to help
you further understand the lesson:
Ondaro et al. (2018). Mathematics in the modern world, e-book. Mutya Publishing
House, Inc.
Chapter 1 – Introduction
http://124.105.95.237/index.php/s/zrWBy3QgxFwDGox
Chapter 1 Lesson 1 - Mathematics in our World
http://124.105.95.237/index.php/s/r93FQfrJGYFq86a
Chapter 1 Lesson 1.1
http://124.105.95.237/index.php/s/jyepqLA4ijrPAKz
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General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Activity 1. Now that you know the most essential concepts in the study of the
nature of Mathematics. Let us try to check your understanding of
these concepts. You are directed to answer the questions from
MMW Practice Set 1, Nos. 1 – 3
on pages 1 to 2.
Activity 1. Getting acquainted with the essential concepts of the nature of
Mathematics, what also matters is you should also be able to apply
the mathematical concepts in real-world applications. You are
expected to answer at least five (5) problems from
MMW Practice Set 1, Nos. 4 – 15
on pages 3 to 8.
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Activity 1. Based from the most essential concepts in the study of the nature of
Mathematics and the learning exercises that you have done, please
feel free to write your arguments or lessons learned below.
1.
2.
3.
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Do you have any question for clarification?
Questions / Issues
Answers
1.
2.
3.
4.
5.
Mathematics
Fractals
Patterns in nature
Spirals
Number patterns
Voronoi
Golden ratio
Fibonacci
number
Fibonacci
sequence
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Big Picture in Focus
ULO-b.
Translate English phrases to mathematical expressions using
the language, symbols, and conventions of mathematics.
Metalanguage
In this section, the essential terms relevant to the study of Mathematical
language and symbols and to demonstrate ULO-b will be operationally defined to
establish a common frame of reference as to how the texts work. You will encounter
these terms as we go through this topic. Please refer to these definitions in case you
will encounter difficulty in understanding some concepts.
1. Mathematical expression and Mathematical sentence. 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 thoughts. The
Mathematical analogue of a `noun' will be called an expression. Thus, an
expression is a name given to a mathematical object of interest. The
Mathematical analogue of a `sentence' will also be called a sentence. A
mathematical sentence, just as an English sentence, must state a complete
thought.
2. Synonyms. This simple idea that numbers have lots of different names is
extremely important in Mathematics. English has the same concept:
synonyms are words that have the same (or nearly the same) meaning.
However, this `same object, different name' idea plays a much more
fundamental role in Mathematics than in English.
3. Verbs. Just as English sentences have verbs, so do Mathematical
sentences. In the Mathematical sentence ‘3 + 4 = 7’, the verb is ‘=‘. If you
read the sentence as ‘three plus four is equal to seven ‘, then it's easy to
`hear' the verb. Indeed, the equal sign ‘=‘ is one of the most popular
Mathematical verbs.
4. Notion of truth. Sentences can be true or false. The notion of truth (i.e., the
property of being true or false) is of fundamental importance in the
Mathematical language.
5. Language conventions. Languages have conventions. In English, for
example, it is conventional to capitalize proper names (like `Carol' and
`Idaho'). This convention makes it easy for a reader to distinguish between a
common noun (like `carol', a Christmas song) and a proper noun (like `Carol').
Mathematics also has its conventions, which help readers distinguish
between different types of Mathematical expressions.
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College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Essential Knowledge
To perform the aforesaid big picture (unit learning outcomes) for the first three
(3) weeks of the course, you need to fully understand the following essential
knowledge that will be laid down in the succeeding pages. Please note that you are
not limited to refer to these resources exclusively. Thus, you are expected to utilize
other books, research articles, and other resources that are available in the
university’s library e.g., ebrary, search.proquest.com, etc.
1. Mathematical language. Mathematics has its own language, its own tools,
and mode of operations. The language for the communication of Mathematical
ideas is largely in terms of symbols and words.
For example. A number is a property of a set that tells how many
elements there are in the set. But a numeral is a name or symbol used to
represent a number. Essentially, to distinguish between a number and a
numeral is to distinguish between a thing and the name of a thing.
It is important that a student understands the distinction between a
number and numeral so that he may realize the difference between actually
operating with numbers and merely manipulating symbols representing those
numbers. This is only one item in regard to precision of language. There are
many others, such as distinguishing between the line and picture of a line, a
point and the dot used to represent the point, etc.
The language of Mathematics makes it easy to express the kinds of
thoughts that mathematicians like to express. It is:
•
precise (able to make very fine distinctions);
•
concise (able to say things briefly);
•
powerful (able to express complex thoughts with relative ease).
The language of Mathematics can be learned, but requires the efforts
needed to learn any foreign language.
2. Mathematical symbols. Without language, we cannot talk about anything.
Mathematical talk consists of making use of Mathematics symbolism. In
Mathematics, we express lengthy statements in a very brief form by using
various symbols. Thus, Mathematics has a peculiar language in which symbols
occupy the most important position. Some of the important and familiar
symbols used in Mathematics are shown in group below:
Symbol
=


>
<


( ),[ ]
Name
equals sign
not equal sign
approximately equal
strict inequality
strict inequality
inequality
inequality
parentheses, brackets
Meaning / Use
equality
inequality
approximation
greater than
less than
greater than or equal to
less than or equal to
calculate expression inside first
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Phone No.: (082)300-5456/305-0647 Local 134
+
–
,x


/
mod
%


{ }



plus sign
minus sign
asterisk, times sign
center dot
division sign/ obelus
division slash
modulo
percent
proportional to
lemniscate
braces
set membership
real numbers set
rational numbers set
addition
subtraction
multiplication
multiplication
division
division
remainder calculation
for every hundred, 1% = 1/100
proportional to
infinity symbol
set
an element of
 = { x  x = a/b, a,b  }
 = { x  - < x < }
For other mathematical symbols, you can visit
https://www.rapidtables.com/math/symbols/Basic_Math_Symbols.html
3. Mathematical expressions. Table below summarizes the analogy of English
nouns to Mathematical expressions.
name given to an
object of interest:
a complete thought:
ENGLISH
Noun
(person, place, thing)
Examples:
Carol, Davao, book
MATHEMATICS
Expression
Examples:
5, 2 + 3, 1/2
Sentence
Examples:
The car is blue.
Maria is cooking.
Sentence
Examples:
3+4=7
3+4=8
Since people frequently need to work with numbers, these are the most
common type of mathematical expression. And, numbers have lots of different
names.
For example, the expressions
5
2+3
10  2
(6 – 2) + 1
1+1+1+1+1
all look different, but are all just different names for the same number.
4. Translating phrases into expressions. Many words and phrases suggest
mathematical operations. The following common words and phrases indicate
addition, subtraction, multiplication, and division.
Addition
plus
the sum of
increased by
Subtraction
minus
the difference of
decreased by
Multiplication
times
the product of
multiplied by
Division
divided by
the quotient of
per
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Phone No.: (082)300-5456/305-0647 Local 134
total
more than
added to
fewer than
less than
subtracted from
of
Verbal phrases can be translated into variable expressions. Some
examples are below.
Verbal phrase
The sum of a number and 9
The difference of a number and 21
The product of 6 and a number
The quotient of 48 and a number
One-third of a number
Variable expression
n+9
n – 21
6n
48/n
(1/3)n
Whenever possible, select a single variable to represent an unknown
quantity. Then express related quantities in terms of the first variable selected.
Examples. For each relationship, select a variable to represent one
quantity and state what that variable represents. Then express the second
quantity in terms of the variable selected.
1. Rivero scored 7 more points than Paras in a basketball game.
Let r = number of points scored by Paras
Let r + 7 = number of points scored by Rivero
2. Bob and Mark share ₱650 in a work done.
Let a = how much Bob receives
Let 650 – a = amount Mark receives
The useful thing about translation is that it can go both ways. Just like we
can translate words into mathematical expressions, we can translate mathematical
expressions back into words. Consider the following examples.
Mathematical expression
3x – 8
x
+6
2
Translation
8 less than 3x or
3x decreased by 8
The quotient of x and 2, plus 6 or
x divided by 2, plus 6
5. Mathematical sentences. A mathematical sentence is the analogue of an
English sentence; it is a correct arrangement of mathematical symbols that
states a complete thought. It makes sense to ask about the TRUTH of a
sentence: Is it true? Is it false? Is it sometimes true/sometimes false?
The sentence `1 + 2 = 3 ' is read as `one plus two equals three' or `one
plus two is equal to three'. A complete thought is being stated, which in this
case is true.
A question commonly encountered, when presenting the sentence
example `1 + 2 = 3 ', is the following: If `=' is the verb, then what is the `+'? Here
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is the answer. The symbol `+' is a connective; a connective is used to `connect'
objects of a given type to get a `compound' object of the same type.
Here, the numbers 1 and 2 are `connected' to give the new number 1
+ 2. A familiar English connective for nouns is the word `and': `cat' is a noun,
`dog' is a noun, `cat and dog' is a `compound' noun.
Examples. Classify each entry as a mathematical expression (EXP), or a
mathematical sentence (SEN). Classify the truth value of each entry that
is a sentence: (always) true (T); (always) false (F); or sometimes
true/sometimes false (T/F).
1.
2.
3.
4.
5.
1/2
x–1=3
1+2+x
12  3 = 2
1+2+x=x+1+2
EXP
SEN, T/F
EXP
SEN, F
SEN, T
You can refer to the source below to help
you further understand the lesson:
Ondaro et al. (2018). Mathematics in the modern world, e-book. Mutya Publishing
House, Inc.
Chapter 1 Lesson 2 Mathematical Language and Symbols
http://124.105.95.237/index.php/s/AE5CoH7XYboC3c5
Chapter 1 Lesson 2.1
http://124.105.95.237/index.php/s/D5WsXTWAsB4HTcL
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Activity 1. Now that you know the most essential concepts in the study of
mathematical language and symbols. Let us try to check your
understanding of these concepts. You are directed to answer
exercises from
MMW Practice Set 2 – A, B, & C
on pages 9 to 10.
Activity 1. Getting acquainted with the essential concepts of the nature of
Mathematics, what also matters is you should also be able to apply
the mathematical concepts in real-world applications. You are
expected to answer exercises from
MMW Practice Set 2 – D, E, & F
on pages 12 to 14.
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Activity 1. Based from the most essential concepts in the study of mathematical
language and symbols and the learning exercises that you have
done, please feel free to write your arguments or lessons learned
below.
1.
2.
3.
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Do you have any question for clarification?
Questions / Issues
Answers
1.
2.
3.
4.
5.
Mathematical
Symbols
Mathematical
Language
Mathematical
sentence
Mathematical
conventions
Mathematical
expression
Mathematical
translations
Noun
Verb
Variables
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Big Picture in Focus
ULO-c.
Use different types of reasoning to justify statements and
arguments made about Mathematics.
Metalanguage
In this section, the essential terms relevant to the study of Mathematical
reasoning and to demonstrate ULO-c will be operationally defined to establish a
common frame of reference as to how the texts work. You will encounter these terms
as we go through this topic. Please refer to these definitions in case you will
encounter difficulty in understanding some concepts.
1. Reasoning is the process of thinking about something in order to make
a decision.
2. Deductive reasoning is a basic form of valid reasoning, which starts out with
a general statement, or hypothesis, and examines the possibilities to reach a
specific, logical conclusion.
3. Inductive reasoning is the opposite of deductive reasoning. It makes broad
generalizations from specific observations. Basically, there is data, then
conclusions are drawn from the data.
4. Conjecture is a conclusion or a proposition which is suspected to be true
due to preliminary supporting evidence, but for which no proof or disproof has
yet been found.
Essential Knowledge
To perform the aforesaid big picture (unit learning outcomes) for the first three
(3) weeks of the course, you need to fully understand the following essential
knowledge that will be laid down in the succeeding pages. Please note that you are
not limited to refer to these resources exclusively. Thus, you are expected to utilize
other books, research articles, and other resources that are available in the
university’s library e.g., ebrary, search.proquest.com, etc.
1. Deductive vs. Inductive
Deductive and inductive, refer to the process by which someone creates
a conclusion as well as how they believe their conclusion to be true.
Deductive reasoning requires one to start with a few general ideas,
called premises, and apply them to a specific situation. Recognized rules, laws,
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theories, and other widely accepted truths are used to prove that a conclusion
is right.
The concept of deductive reasoning is often expressed visually using a
funnel that narrows a general idea into a specific conclusion.
In practice, the most basic form of deductive reasoning is the syllogism,
where two premises that share some idea support a conclusion. It may be
easier to think of syllogisms as the following theorem:
If A = B and C = A, then B = C.
Deductive Reasoning in
Theory:
Deductive Reasoning in
Theory:
General Ideas
A is B
C is A
Deductive Reasoning in
Practice:
All muscles are made out of
living tissue.
All humans have muscles.
Specific Conclusion
Therefore, B is C.
Therefore, all humans are
made out of living tissue.
Note that the above paragraph states that the premises prove the
conclusion, not justify it.
Deductive reasoning is meant to demonstrate that the conclusion is
absolutely true based on the logic of the premises.
Compare the following syllogisms:
All musical instruments make sounds.
Airplanes make sounds.
Therefore, airplanes are musical
instruments.
All art is an imitation of nature.
Music is art.
Therefore, music is an imitation of nature.
The syllogism on the left contains two objectively true premises, but its
conclusion is false because it is possible for airplanes and instruments to be
totally separate entities while still having the same properties. The syllogism on
the right takes premises that overlap and uses them to prove that a statement
is definitely true.
Although deductive arguments rarely come in the exact form of a
syllogism, the same thought process can be used to evaluate their strength
and create counterarguments.
Inductive reasoning uses a set of specific observations to reach an
overarching conclusion; it is the opposite of deductive reasoning.
So, a few particular premises create a pattern which gives way to a
broad idea that is likely true. This is commonly shown using an inverted funnel
(or a pyramid) that starts at the narrow premises and expands into a wider
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conclusion. There is no equivalent to a syllogism in inductive reasoning,
meaning there is no basic standard format.
All forms of inductive reasoning, though, are based on finding a
conclusion that is most likely to fit the premises and is used when making
predictions, creating generalizations, and analyzing cause and effect.
Inductive reasoning in Theory
Inductive Reasoning in Practice
Specific Observations
My neighbor’s cat hisses at me daily.
At the pet store, all the cats hiss at me.
Therefore, all cats probably hate me.
Broad Conclusion
Just as deductive arguments are meant to prove a conclusion, inductive
arguments are meant to predict a conclusion. They do not create a definite
answer for their premises, but they try to show that the conclusion is the most
probable one given the premises. In the above example, there are several
possible factors that could contribute to a cat’s reaction toward the arguer.
Perhaps she wears a deodorant that cats dislike, or maybe she is hostile
toward cats and neglected to mention it. But, considering neither of these
factors are acknowledged in the premises, these are not considered the most
probable conclusions. The most probable conclusion, given the premises that
have been supplied, is that cats hate the arguer.
An inductive argument is either considered weak or strong based on
whether its conclusion is a probable explanation for the premises.
Compare these inductive arguments:
The cost of college has been increasing
over the past several decades.
Therefore, higher taxes on the rich are
probably the best way to help middle
class America thrive.
The past five Marvel movies have been
incredibly successful at the box office.
Therefore, the next Marvel movie will
probably be successful.
Once again, the reasoning on the left is weak while the right is strong.
On the left, the two statements made are likely true on their own, but the first
premise does not predict the second to be true. Since there is no obvious
correlation between the two, the argument is weak. On the right, the premise
identifies a pattern, and the conclusion provides a logical continuation of this
pattern without exaggeration. Thus, the argument is strong.
Mathematically Acceptable Statements
Consider the following Statement:
“The sum of two prime numbers is always even.”
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The given statement can either be true or false since the sum of two
prime numbers can be either be an even number or an odd number. Such
statements are mathematically not acceptable for reasoning as this sentence
is ambiguous. Thus, a sentence is only acceptable mathematically when it
is “Either true or false, but not both at the same time.” Therefore, the basic
entity required for mathematical reasoning is a statement. This is the
mathematical statement definition.
2. Types of Reasoning in Mathematics
In terms of mathematics, reasoning can be of two major types which
are:
1. Inductive Reasoning
2. Deductive Reasoning
The other types of reasoning are intuition, counterfactual thinking,
critical thinking, backwards induction and abductive induction. These are the 7
types of reasoning which are used to make a decision. But, in mathematics,
the inductive and deductive reasoning are mostly used which are discussed
below.
Note: Inductive reasoning is non-rigorous logical reasoning and
statements are generalized. On the other hand, deductive reasoning is rigorous
logical reasoning and the statements are considered true if the assumptions
entering the deduction are true. So, in mathematics, deductive reasoning is
considered to be more important than inductive.
Inductive Reasoning
In the Inductive method of mathematical reasoning, the validity of the
statement is checked by a certain set of rules and then it is generalized. In
other words, in the inductive method of reasoning, the validity of the statement
is checked by certain set of rules and then it is generalized. The principle of
mathematical induction uses the concept of inductive reasoning.
As inductive reasoning is generalized, it is not considered in geometrical
proofs. Here, is an example which will help to understand the inductive
reasoning in Mathematics better.
Example of Inductive Reasoning
Statement: The cost of goods is ₱10 and the cost of labor to manufacture the
item is ₱ 5. The sales price of the item is ₱ 50.
Reasoning: From the above statement, it can be said that the item will provide a
good profit for the stores selling it.
Deductive Reasoning
The principal of deductive reasoning is actually the opposite of the
principle of induction. On the contrary to inductive reasoning, in deductive
reasoning, we apply the rules of a general case to a given statement and make
it true for particular statements. The principle of mathematical induction uses
the concept of deductive reasoning (contrary to its name). The below-given
example will help to understand the concept of deductive reasoning in
mathematics better.
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Example of Deductive Reasoning
Statement: Pythagorean Theorem holds true for any right-angled triangle.
Reasoning: If triangle XYZ is a right triangle, it will follow Pythagorean Theorem.
3. Types of Reasoning Statements
There are three main types of reasoning statements:
• Simple Statements
• Compound Statements
• If-Then Statements
Simple Statements
Simple statements are those which are direct and do not include any
modifier. These statements are easiest to solve and does not require much
reasoning.
An example of a simple statement is
The Sun rises in the east.
In this statement, there is no modifier and thus it can be simply
concluded as true.
Compound Statement
With the help of certain connectives, we can club different statements.
Such statements made up of two or more statements are known as compound
statements. These connectives can be “and”, “or”, etc. With the help of such
statements, the concept of mathematical deduction can be implemented very
easily. For a better understanding consider the following example:
Statement 1:
Statement 2:
Even numbers are divisible by 2.
2 is also an even number.
These two statements can be clubbed together as:
Compound Statement: Even numbers are divisible by 2 and 2 is also an even
number.
Let us now find the statements out of the given compound statement:
Compound Statement: A triangle has three sides and the sum of interior angles
of a triangle is 180°.
The Statements for this compound statement are:
Statement 1:
Statement 2:
A triangle has three sides.
The sum of the interior angles of a triangle is 180°.
These statements which are both related to triangles are mathematically
true. These two statements are connected using “and.”
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If-Then Statements
According to mathematical reasoning, if we encounter an if-then
statement i.e. ‘if a then b’, then by proving that a is true b can be proved to be
true or if we prove that b is false then a is also false.
If we encounter a statement which says a if and only if b then we can
reason such a statement by showing that a is true and then b is also true and
if b is true then a is definitely true.
Example.
a. 8 is multiple of 64
b. 8 is a factor of 64
Since one of the given statements i.e. a is true, therefore, a or b is true.
4. How to Deduce Mathematical Statements?
For deducing new statements or for making important deductions from
the given statements three techniques are generally used:
1. Negation of the given statement
2. Contradiction Method
3. Counter Statements
Let us take a look at these methods one by one.
Negation of the Given Statement
In this method, we generate new statements from the old ones by the
rejection of the given statement. In other words, we deny the given statement
and express it as a new one. Consider the following example to understand it
better.
Statement 1: Sum of squares of two positive numbers is positive.
Now if we negate this statement then we have,
Statement 2: Sum of squares of two positive numbers is not positive.
Here, by using “not”, we denied the given statement and now the following can
be inferred from the negation of the statement:
There exist two numbers, whose squares do not add up to give a positive number.
This is a “false” statement as squares of two numbers will be positive.
From the above discussion, we conclude that if (1) is a mathematically
acceptable statement then the negation of statement 1 (denoted by statement
2) is also a statement.
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Contradiction Method
In this method, we assume that the given statement is false and then try
to prove the assumption wrong.
Example.
Statement: The derivative of y = 9x2 + sin x with respect to x is 18x + cos x.
For proving the validity of this statement,
Suppose that dy/dx ≠ 18x + cos x. We know that the derivative of
xn is given by n • xn−1. Therefore, the derivative of 9x2 is 18x and the
derivative of sin x is given by cos x.
Also,
d/dx (f (x) + g(x)) = df (x)/dx + dg(x)/dx.
Therefore, d/dx (9x2 + sin x) = 18x + cos x.
Hence, our assumption is wrong, and the statement is a valid
statement.
Counter Statements
Another method for proving validity is to use a counter statement i.e.
giving a statement or an example where the given statement is not valid.
Example.
Statement: If x is a prime number then x is always odd.
To show that the given statement is false we will try to find a counter
statement for this. We know that 2 is a prime number i.e. it is divisible by only
itself and 1. Also, 2 is the smallest even number. Therefore, we can say that 2
is a prime number which is even. Hence, we can say that the statement is not
true for all prime numbers, therefore, the given statement is not valid.
Example Questions Using the Principle of Mathematical Reasoning
Question 1:
Consider the following set of statements and mention which of these are
mathematically accepted statements:
1) The Sun rises in the east.
2) New Delhi is a country.
3) Red rose is more beautiful than a yellow rose.
Solution:
When we read the first statement, we can straightaway say that the first
statement is true, and the second one is false. As far as the third statement is
considered it may depend upon perceptions of different people. Hence, it can
be true for some people and at the same time false for others. But such
ambiguous statements are not acceptable for reasoning in mathematics.
Thus, a sentence is only acceptable mathematically when it is either true
or false but not both at the same time. So, statement 1 and 2 are
mathematically accepted statements while statement 3 is not accepted
mathematically.
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Question 2:
The sum of three positive numbers x, y and z is always negative.
Solution:
This statement is acceptable. It can never be true because all positive
numbers are greater than zero and therefore the sum of positive numbers can
never be negative.
Question 3:
The product of three real numbers x, y and z is always zero.
Solution:
In this given statement we cannot figure out if the statement is true or
false. Such a sentence is not mathematically acceptable for reasoning.
Question 4:
Statement (a): A circle with infinite radius is a line
Statement (b): A circle with zero radii is a point
Solution:
Since (a) is true and (b) is also true then both statements (a) and (b) are
also true. For two given statements (a) or (b) to be true, show that either (a) is
true or prove that (b) is true i.e. if any one of the statements is true then (a) or
(b) is also true.
You can refer to the source below to help
you further understand the lesson:
Ondaro et al. (2018). Mathematics in the modern world, e-book. Mutya Publishing
House, Inc.
Chapter 1 Lesson 3 Problem Solving and Reasoning
http://124.105.95.237/index.php/s/A9aEf4EZrifXfQT
Chapter 1 Lesson 3.1
http://124.105.95.237/index.php/s/HKTA8HMY8giKant
Chapter 1 Lesson 3.2
http://124.105.95.237/index.php/s/mdgZTRKXL2tTYEe
Chapter 1 Lesson 3.3
http://124.105.95.237/index.php/s/LzYi69WT9cDH4nd
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Activity 1. Now that you know the most essential concepts in problem solving,
let us try to check your understanding of these concepts. You are
directed to answer exercises from
MMW Practice Set 3 – A
on page 15.
Activity 1. Getting acquainted with the essential concepts in problem solving,
what also matters is you should also be able to apply the
mathematical concepts in real-world applications. You are expected
to answer at least five (5) exercises each from
MMW Practice Set 3 – B & C
on pages 16 to 17.
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Activity 1. Based from the most essential concepts in problem solving and the
learning exercises that you have done, please feel free to write your
arguments or lessons learned below.
1.
2.
3.
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Do you have any question for clarification?
Questions / Issues
Answers
1.
2.
3.
4.
5.
Reasoning
Inductive reasoning
Deductive reasoning
Counterexample
Conclusion
Conjecture
Reasoning statement
Contradiction
Argument
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Big Picture in Focus
ULO-d.
Apply Polya’s four steps in solving problems.
Metalanguage
In this section, the essential terms relevant to the study of problem solving
and to demonstrate ULO-d will be operationally defined to establish a common
frame of reference as to how the texts work. You will encounter these terms as we
go through the study of the nature of mathematics. Please refer to these definitions
in case you will encounter difficulty in understanding some concepts.
1. Problem-solving refers to mathematical tasks that have the potential to
provide intellectual challenges for enhancing students' mathematical
understanding and development.
2. Logic Problems are puzzles which require people to use deductive
reasoning skills, meaning they need to look at different pieces of information
in order to arrive at an answer. Logic problems are not like regular math
problems. In fact, many types of logic problems contain all of the answers,
you just have to figure them out.
There are many different types of logic problems with some being more
challenging than others. Regardless of what type of logic problem it is, here
are some general tips on how to successfully solve one:
• Use a pencil instead of a pen. Many times, it takes several tries to get to
the correct answer. This will allow you to erase and start over as needed.
• Make sure you read the entire problem multiple times and write down all
of the things that you know are true. Search for clues that might be hidden
and make sure you understand the problem. If you get stuck in the middle
of the problem, go back and re-read. Something that didn't seem helpful
before, might make more sense in the middle of the problem.
• Sometimes, drawing the problem out, creating a grid, or using real-life
objects helps the problem make more sense.
• Don't get impatient. Logic problems can take a while to solve, reading and
re-reading to figure out the clues. It isn't about getting to the correct
answer right away, it is about the critical thinking skills you will develop.
Look at it as a fun challenge for your brain!
3. Problem-solving using Pattern is a strategy in which students look for
patterns in the data in order to solve the problem. Students look for items or
numbers that are repeated or a series of events that repeats.
4. Recreational mathematics is mathematics done for recreation or as a
hobby and intended to be fun. Typically it involves games or puzzles that
relate to mathematics, although the term can cover other material. Typically,
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recreational mathematics involves general logical and lateral thinking skills,
as opposed to advanced mathematical concepts, so that the average person
is at least able to understand and appreciate a recreational problem and its
solution. Recreational puzzles can also increase people's appreciation of
mathematics as a whole.
Essential Knowledge
To perform the aforesaid big picture (unit learning outcomes) for the first three
(3) weeks of the course, you need to fully understand the following essential
knowledge that will be laid down in the succeeding pages. Please note that you are
not limited to refer to these resources exclusively. Thus, you are expected to utilize
other books, research articles, and other resources that are available in the
university’s library e.g., ebrary, search.proquest.com, etc.
Polya’s Four Steps In Problem-Solving.
Whether you like it or not, whether you are going to be a mother, father,
teacher, computer programmer, scientist, researcher, business owner, coach,
mathematician, manager, doctor, lawyer, banker (the list can go on and
on), problem solving is everywhere. Some people think that you either can do
it or you can't. Contrary to that belief, it can be a learned trade. Even the best
athletes and musicians had some coaching along the way and lots of
practice. That's what it also takes to be good at problem solving.
George Polya, known as the father of modern problem solving, did extensive
studies and wrote numerous mathematical papers and three books about
problem solving. He created his famous four-step process for problem
solving, which is used all over to aid people in problem solving:
Step 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.
Step 2: Devise a plan (translate).
When you devise a plan (translate), you come up with a way to solve
the problem. Setting up an equation, drawing a diagram, and making a chart
are all ways that you can go about solving your problem.
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Step 3: Carry out the plan (solve).
The next step, carry out the plan (solve), is big. This is where you solve the
equation you came up with in your 'devise a plan' step.
Step 4: Look back (check and interpret).
You may be familiar with the expression 'don't look back'. In problem
solving it is good to look back (check and interpret). 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 1.
Twice the difference of a number and 1 is 4 more than that number. Find the
number.
Step 1: Understand the problem.
Make sure that you read the question carefully several times. Since we are
looking for a number, we will let
x = a number
Step 2: Devise a plan (translate).
Twice the difference of a number and 1 is 4 more than that number
2 (x – 1) = x + 4
Step 3: Carry out the plan (solve)
2(x + 1) = x + 4
2x – 2 = x + 4
2x – 2 – x = x + 4 – x
x–2=4
x–2+2=4+2
x=6
- Remove ( ) using distribution property
- Get all x terms on one side
- Get all constant terms on right side
Step 4: Look back (check and interpret).
If you take twice the difference of 6 and 1, that is the same as 4 more than 6,
so this does check.
Final Answer: The number is 6.
Example 2.
One number is 3 less than another number. If the sum of the two numbers is
177, find each number.
Step 1: Understand the problem.
Make sure that you read the question carefully several times. We are looking
for two numbers, and since we can write the one number in terms of another
number, we will let
x = another number; and
one number is 3 less than another number is
x – 3 = one number
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Step 2: Devise a plan (translate).
The sum of two numbers is 177
x + (x – 3) = 177
Step 3: Carry out the plan (solve).
x + (x – 3) = 177
2x – 3 = 177
2x – 3 + 3 = 177 + 3
2x = 180
x = 90
- Remove ( ) using distribution property
- Get all constant terms on right side
- Divide the equation by 2
Step 4: Look back (check and interpret).
If we add 90 and 87 (a number 3 less than 90) we do get 177.
Final Answer: One number is 90. Another number is 87.
You can refer to the source below to help
you further understand the lesson:
Ondaro et al. (2018). Mathematics in the modern world, e-book. Mutya Publishing
House, Inc.
Chapter 1 Lesson 3 Problem Solving and Reasoning
http://124.105.95.237/index.php/s/A9aEf4EZrifXfQT
Chapter 1 Lesson 3.1
http://124.105.95.237/index.php/s/HKTA8HMY8giKant
Chapter 1 Lesson 3.2
http://124.105.95.237/index.php/s/mdgZTRKXL2tTYEe
Chapter 1 Lesson 3.3
http://124.105.95.237/index.php/s/LzYi69WT9cDH4nd
36
College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Activity 1. Now that you know the most essential concepts in Polya’s Four-Step
in problem solving, let us try to check your understanding of these
concepts. You are directed to answer exercises from
MMW Practice Set 3 – F
on page 24.
Activity 1. Getting acquainted with the essential concepts in problem solving,
what also matters is you should also be able to apply the
mathematical concepts in solving problems. You are expected to
answer at least five (5) exercises each from
MMW Practice Set 3 – D, E, & G
on pages 18 to 29.
37
College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Activity 1. Based from the most essential concepts in problem solving and the
learning exercises that you have done, please feel free to write your
arguments or lessons learned below.
1.
2.
3.
38
College of Arts and Sciences Education
General Education - Mathematics
2nd Floor, DPT Building, Matina Campus, Davao City
Phone No.: (082)300-5456/305-0647 Local 134
Do you have any question for clarification?
Questions / Issues
Answers
1.
2.
3.
4.
5.
Polya’s four steps
Problem-solving
George Polya
Understanding the
problem
Translate
Carry out the plan
Look back
Recreational
problems
Logic problems
39