College for Research and Technology of Cabanatuan
Burgos Avenue, Cabanatuan City 3100
Tel. 463-2735 e-mail: crt.cabanatuan@gmail.com
LESSON 1
MATHEMATICS IN OUR WORLD
How does mathematics exist?
TYPES OF PATTERNS IN NATURE
1. SYMMETRY – an object is said to have symmetry when it remains unchanged after
transformations such as rotations and scaling are applied into it.
2. FRACTALS – are never ending patterns that are self-similar across different scales.
3. SPIRALS – curved patterns made by series of circular shapes revolving around a central point.
4. CHAOS – simple patterns created from complicated underlying behavior.
SYMMETRIES OF OUR SURROUNDINGS
1. REFLECTION SYMMETRY – mirror symmetry or line symmetry. It is made with a line going
through an object which divides it into two pieces which are mirror images of each other.
2. ROTATIONAL SYMMETRY – also called radial symmetry. It is exhibited by objects when their
similar parts are regularly arranged around a central axis and the pattern looks the same after a
certain amount of rotation.
3. TRANSLATIONAL SYMMETRY – it is exhibited by objects which do not change its size and even if
it moved to another location. Ex: moving the stem to another location does not change the
patterns of the leaves
SHAPES IN NATURE
1. CRYSTALS – are solid materials having a compositions enclosed and arranged in symmetrical
plane surfaces, intersecting at definite angles. EX: snowflakes, Ice, Diamonds, Table salts
2. ROCK FORMATIONS – most of the stone and rocks which we usually see everyday are of
irregular and various shapes.
3. ANIMAL KINGDOM – millions of kinds of animal species we have in the world are various
shapes, most of which are irregular.
THE FIBONACCI NUMBERS
“If a single pair of rabbits will be placed in controlled area and is allowed to live and multiply, how many
pairs of rabbits will be produced in a year considering that in every month, each pair bears a new pair
which becomes productive from the second month and so on?”
Who Was Fibonacci?
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
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European mathematician 1175-1250
Real name Leonardo of Pisa.
Author of Liber abaci or Book of the Abacus.
Remembered today because of Edouard Lucas
One of the first people to introduce the decimal number system into Europe
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What Is the Fibonacci Sequence?
Generalized sequence of first two positive integers and the next number is the sum of the
previous two, i.e. 1,1,2,3,5,8,13,21,…
Why Is It Significant?
Has intrigued mathematicians for centuries. Shows up unexpectedly in architecture, science and
nature (sunflowers & pineapples).
Has useful applications with computer programming, sorting of data, generation of random
numbers, etc.
Existence of Fibonacci numbers in art and nature
Examples:
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

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
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MONALISA painting
Window designs
Human body
Floor Tiles
Churches
Sunflower
Pineapple
General rule:
An = An-1 + An-2
Where:
An is the Fibonacci number
An-1 is the number before An; and
An-1 is the number before An-1
Note that:
0+1=1
1+1=2
1+2=3
2+3=5
3+5=8
5 + 8 = 13
Fibonacci Sequence and the Golden Ratio

A remarkable property of the sequence is that the ratio between two numbers in the sequence
eventually approaches the “Golden Ratio” as a limit.
1/1=1, 2/1=2, 3/2=1.5, 5/3=1.6667, 8/5=1.6, 13/8=1.625, 21/13=1.6154
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Famous Irrational number phi—1.61803398…
Only positive # that becomes its own reciprocal by subtracting 1
Used extensively by Ancient Greeks in architecture
Ratio also shows up in many famous paintings
Da Vinci studied extensively the ratio and body proportions.
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SUMMARY
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
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Fibonacci Sequence has been around for hundreds of years.
Golden Ratio has been used for thousands of years.
Both concepts are still in use today and continue to interest us.
LESSON 2
MATHEMATICAL LANGUAGE AND SYMBOLS
LANGUAGE OF MATHEMATICS – structure used by the mathematicians to communicate mathematically.
Mathematical Operation Symbols and their word expressions
CHARACTERISTICS OF MATHEMATICAL LANGUAGE
1. PRECISE – make an exact and accurate expressions
2. Concise – say things briefly
3. Powerful – express complex thoughts with a comparative event
ENGLISH, NOUN AND SENTENCE
NOUN – a word used to identify the class of people, places or things
SENTENCE – used to state complete thoughts. (one noun and one verb)
Example:
Janice loves mathematics
(Janice and mathematics are nouns and love is verb)
NOUN VERSUS SENTENCE
NOUN
James
Manila
Mango
SENTENCE
James is the name of her brother
The capital of the Philippines is Manila
The national fruit of the Philippines is Mango.
Mathematical operation symbols and their word expressions
OPERATION
WORD EXPRESSION
Add, added to, plus, the sum of, more than, the
total of, increased by, going up by, bigger by
Subtraction (-)
Subtract, subtract from, less, minus, less than,
decreased by, diminished by, take away, reduced
by, the difference between
Multiplication (x)
Multiply, times, the product of, multiply by, times
as much as
Division (÷) or (/)
Divide, divided by, the quotient of, the ratio of,
equal amounts of, per
Equation (=)
Equals, is equal to, exactly as, equivalent to, as
similar to
Mathematical expression is a name given to mathematical object of interest.
Addition (+)
Example:
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The value of the expression 1 + 2 x 3 is 7, because the expression is evaluated by first multiplying 2 and 3
and then adding 1 to the result.
Mathematical sentence expresses a complete thought.
Example:
1.
2.
3.
4.
MATHEMATICAL EXPRESSION
6
7+6
5x6
(2 + 3) – 1
MATHEMATICAL SENTENCE
6 is an even number
7 + 6 = 13
5 x 6 = 30
(2 + 3) – 1= 4
WRITING EXPRESSION AND EXPRESSION
Numbers and/or variables that are connected by operation/s is called expression. An equation usually
has words like “equals”, “is equal to” or is.
Example:
Ten a number is fourteen.
Steps in translating a mathematical phrase:
Step 1. Identify the variables and/or constants
Constant – ten and a number is fourteen
Variable – ten and a number is fourteen
Step 2. Determine the connectors.
Connectors – ten and a number is fourteen.
Step 3. Perform the translation of the phrase into symbols.
Ten
and
a number
is
fourteen
10
+
x
=
14
Examples:
1. Sentence/phrase to Algebraic Expression
Phrase
1. Twice as old as Vic is forty
2. Joe’s age divided by five all less than three is one
3. Sonia’s age multiplied by two is seventy – eight
4. The ratio of sixteen and the number of workers is 4
5. Jaime’s age less two is twenty-four
Algebraic Expression
2v = 40
𝑥
−3=1
𝑠
2s = 78
16
=4
𝑥
J – 2 = 24
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2. Algebraic expression to English expression
1.
2.
3.
4.
5.
Algebraic expression
6 + a = 40
5b = 40
𝑐
= 40
5
2m – 4 = 10
6x + 4 = 40
English expression
Six more than a number is forty
Product of five and a number is 40
The quotient of a number and five is forty
Twice a number less four is forty
Four more than the product of six and a number is forty
Some symbols that commonly used in mathematics.
Symbol
“>” inequality sign
Word expression
….is greater than…..
“<” inequality sign
…..is less than…..
“≠” inequality sign
….is not equal to….
“≈” approximate equal
sign
“∑” summation sign
….is approximately
equal to….
The summation of….
“!” exclamation
….factorial
“√” square root
“%” percent
Three dots
The square root of….
…percent…
Therefore…
“ϵ” epsilon
Subset
“U"
Intersection
….is an element of….
…is a subset of….
…union….
….intersection….
LESSON 3
Use
Indicates value on left is
greater than the value
on right
Indicates value on left is
smaller than the value
on right
Indicates that two value
are different
Indicates two values are
close to each other
Sum of many or
infinitely many values
Product of all positive
integral up to a certain
value
Algebraic expressions
Proportion
Logical statement of
mathematical proof
Sets
Sets
Sets
Sets
Sample expression
3>2
2<3
x≠y
x+y≈z
4! = 24
√4
2.5 %
aϵA
A B
AUB
A B
PROBLEM SOLVING AND REASONING
1. INDUCTIVE REASONING – type of reasoning that comes up to a conclusion by examining specific
examples. It can also use in many life situations.
EXAMPLE:
a. Use inductive reasoning to predict the next number in the lists.
1, 3, 6, 11,,?
Solution:
The first two numbers differ by 2, the second and the third by 3, the third and fourth by 2 again. It
appears that when two numbers differ by 2, the next difference would be 3, followed again by 2, then
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by 3. Since the difference between 8 and 11 is 3, we predict the next number to 11 to be a number 2
more than 11, which is 13.
b. If n is an integer, then the absolute value of n is greater than 0.
Solution:
Since 0 is an integer, we let n = 0. Now |n| = |0| = 0 > n. We have found a counter example. Thus, the
statement “If n is an integer, then the absolute value of n is greater than 0 is a false statement.
2. DEDUCTIVE REASONING – It is the process by which one makes conclusion based on previously
accepted general assumptions, procedures and principles.
Example:
PREMISE – There are 23 books on the bookshelf and 16 on the lower shelf. There are no books on the
bookshelf.
CONCLUSION – therefore, there are 39 books in the bookshelf.
POLYA’S PROBLEM-SOLVING STRATEGY
POLYA’S 4 STEP PROBLEM-SOLVING STRATEGY
1. Understand the problem – you must have clear understanding of the problem and have ready
answers to questions.
2. Devise a plan – some of procedures include in making a list of known information, making a list
of information that is needed, sketching a diagram, making an organized list that shows all
possibilities, making a table or chart, working backwards, trying to solve a similar but simpler
problem.
3. Carry out the plan – you need to work carefully and keep an accurate and neat record of all you
attempts.
4. Review the solution
Example
1. Katrina sells eggs by piece. On the first day, she sold a half more than half the number of eggs
for sale. On the second day, she sold a half more than half the remaining number of eggs on the
first day. On the third day, she again sold a half more than a half the remaining number of eggs
from the second day. Only a dozen eggs were left for the fourth day. How many eggs did she
have originally?
Understand the problem. We need to determine the number of eggs before Katrina started selling them
Devise a plan. Algebraic solution is okay but working backwards is more preferable because we knew
already the end result.
Carry out the plan. Since only 12 eggs were left, equivalent to half less than half the number before she
started selling eggs on the third day. It follows that half the number is 12.5 which means that there were
25 eggs left on the second day. There were 25.5 = 51 eggs just before she started selling eggs on the
second day, and so before she started selling eggs on the first day, there were originally 51.5 x 2 = 103
eggs.
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Review the solution. To check our solution, we start 103 eggs and proceed through each day of sale.
Day 1: 103 – (51.5 + 0.5) = 103 – 52 = 51
Day 2: 51 – (25.5 + 0.5) = 51 – 26 = 25
Day 3: 25 – (12.5 + 0.5) = 25 – 13 = 12
2. A basketball team won two out of their last four games. In how many different orders could they
have two wins and two losses in four games?
 WWLL
 WLLW
 LWWL
 LWLW
 LLWW
 WLWL
3. How many rectangles do we have? 18 rectangles
LESSON 4
DATA MANAGEMENT
DIVISIONS OF STATISTICS
1. DESCRIPTIVE – Statistical procedure concerned with describing the characteristics and
properties of a group of persons, places or things that was based on verifiable facts.
Example:
How many students are interested to take statistics online?
A basketball player wants to find his average shots for the past 10 games.
2. INFERENTIAL – statistical procedure used to draw inferences for the population on the basis of
information obtained from the sample using the technique of descriptive statistics.
Example:
Is there significant difference in the academic performance of male and female students in statistics?
A politician wants to estimate his chance of winning in the upcoming senatorial election.
DATA COLLECTION AND PRESENTATION OF STATISTICAL DATA
METHODS FOR COLLECTING DATA
1. DIRECT OR INTERVIEW METHOD – person to person exchange of data from interviewer to
interviewee.
2. INDIRECT OR QUESTIONNAIRE METHOD – responses are written and given more time to
answer. Questionnaires are given to the respondents.
3. REGISTRATION METHOD – The respondents provide information in the compliance with certain
laws, policies, rules, regulations, decrees, or standard practices.
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4. EXPERIMENTAL METHOD – the researcher wants to control the factors affecting the variable
being studied to find out cause and effect relationships.
5. OBSERVATION METHOD – utilized to gather data regarding attitudes, behavior and cultural
pattern of the samples under investigation.
WAYS IN PRESENTING DATA
1. TEXTUAL PRESENTATIONS – data are presented in paragraphs or sentences.
Example:
Nominally, the peso improved by 1.4 percent as of April 14, 2003 compared to its level
in 2002, followed by the Thai baht which gained 0.86 percent; Indonesian rupiah, 0.68 percent;
and Taiwan dollar, 0.2 percent.
Other currencies, on the other hand, depreciated during the same period. The
Singapore dollar fell 2.33 percent. The South Korean won slid 2.14 percent while the Japanese
yen dropped 0.61 percent.
2. TABULAR PRESENTATION – it use rows and columns and data are presented systematically and
ordered.
Example:
3. GRAPHICAL PRESENTATION
a. SCATTER GRAPH – graph used to present measurements or values that are thought tobe
related.
b. LINE CHART/GRAPH – useful for showing trends over a period of time.
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c. PIE CHART/GRAPH – circular graph that are useful in showing how a total quality is
distributed among a group of categories.
d. COLUMN AND BAR GRAPH – it is applicable only for grouped data. Used for discrete,
grouped data of ordinal or nominal scale.
MEASURES OF CENTRAL TENDENCY
- It is a numerical descriptive measure in which indicate or locate the center of a distribution of a set of
data.
1. MEAN or ARITHMETIC MEAN – commonly known as average. It is represented by
EXAMPLE:
The numbers of students at six different classrooms are 25, 27, 29, 24, 35 and 28. Find
the mean.
M = Tx/n
M = 25 + 27+ 29+ 24+ 35 +28/6
M = 168/6
M = 28
2. MEDIAN – the middle position of arranged values from lowest to highest. It is the middlemost
score.
EXAMPLE 1:
Given 3, 4, 7, 8, 10, 12 and 14 = 8 is the median because it is in the middle term.
EXAMPLE 2:
Given the following scores: 80, 81, 82, 83, 84, and 85, compute for the median.
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80, 81, 82, 83, 84, 85 = 82 and 83 is the middlemost so we will get the average of this.
Md = 82 + 83 = 165
Md = 165 ÷ 2 = 82.5
therefore, 82.5 is the median
EXAMPLE 3:
Solve for the median of the following raw scores: 6,7,8,9,9,10,12,14,15
Md =
𝟗 + 𝟗 + 𝟏𝟎
𝟑
Md = 9.33
therefore, 9.33 is the median
3. MODE – most frequently observed value that occurs.
EXAMPLE:
Find the mode of the ff. data: 63, 100, 84, 88, 73, 86, 97,95, 97
Arranging the values from least to greatest makes it easier to find the mode
63, 73, 84, 86, 88, 95, 97, 97, 100
Find the number that appears more or most frequently
The value 97 appears twice.
All other numbers appear just once. Therefore, 97 is the MODE
A set of data can have:
 No mode
 One mode
 More than one mode
9, 11, 16, 6, 7, 17, 18 – NO MODE
9, 11, 16, 8, 16 – HAS ONE MODE
18, 7, 10, 7, 18 – MORE THAN ONE MODE
4. WEIGHTED MEAN – is a mean calculated by giving values in a data set more influence according
to some attribute of the data.
Mw =
𝐓(𝐰𝐟)
Where:
𝐓𝐟
T(wf) = total or sum of weight times frequency
Tf = N = total or sum of the frequencies
EXAMPLE:
Ely’s wants to determine if he passed the subject. Given the following data, did Ely pass the subject?
Grading system
Quizzes
Recitation
Assignment
Seatwork
Term exam
TOTAL
Ely’s grade :
=
=
Weighted percentage
25% = .25
15% = .15
10% = .10
20% = .2
30% = .3
100%
Ely’s score
74
90
85
65
73
(74 x .𝟐𝟓) + (90 x .𝟏𝟓) + (85 x .𝟏𝟎) + (65 x .𝟐) + (73 x .𝟑)
𝟏𝟎𝟎%
𝟏𝟖.𝟓 + 𝟏𝟑.𝟓 + 𝟖.𝟓 + 𝟏𝟑 + 𝟐𝟏.𝟓
= 75.4%
𝟏𝟎𝟎%
Therefore, Ely passed the subject with this grade
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