• studied mathematics of fractals could create

Chapter 1
plants growth patterns
Nature of Mathematics

What is Mathematics?

• a study of relationship among numbers,

quantities, and shapes
• a systematic process that provides
• opportunity to solve both simple and
• complex problems in real world contexts
• viewed as a science which involves logical
reasoning, drawing conclusions from assumed
premises, and strategic reasoning based on
accepted rules, laws, or probabilities

Patterns and Numbers in Nature

PATTERN

• Patterns are studied because they are

everywhere; people just need to learn to notice
them.
• Patterns are regular, repeating, or recurring
forms of designs.
• It seeks to discover relationships and
connections between seemingly unrelated bits

of information.

Historical Highlights of Patterns

Plato, Pythagoras, Empedocles

• studied patterns to explain ORDER in NATURE

Joseph Plateau

• studied surface of soap

Ernst Haeckel

• studied patterns of symmetry on organisms

D’arcy Thompson

• studied growth patterns in plants

Alan Turing

• studied patterns of spots and stripes

Aristid Lindenmayer and Benoit Mandelbrot

Patterns and
Numbers in
Nature

Logic Patterns
One kind of logic patterns deals with the
characteristics of
various objects while another deals with order.
It is usually the first to be observed.

Word Patterns

Sequences or patterns within words that are

made up of both
consonants and vowels.

Examples:
KNIFE: KNIVES
Number Patterns LIFE: LIVES

Number pattern is a formal description of the Word Patterns

relationships
among different quantities. Examples:

The coolness of the night (a)

Refreshes my skin (b)

The stars shine so bright (a)

Causing me to grin (b)

Number Patterns
Tiger’s Stripes and
An amazing pattern that
Hyena’s Spots
creates a pyramid or
triangle shape out of the
Patterns are also exhibited in the
binomial coefficients.
external appearances of animals.

It is named after a French

According to a theory by Alan
mathematician Blaise
Turing, chemical reactions and
Pascal.
diffusion processes in cells
determine these growth patterns.

Flowers

Flowers with five petals are said to

be the most common.
There is a definite pattern of
clockwise and counterclockwise arcs It comes from the Greek
or spirals extending outward from tesseres, which means
the center of the flower. "four."

It allows the seeds to occupy the Foam

flower head to maximize the access
of light and nutrients. Patterns in nature
that are formed
Types of Patterns from repeating
spheres.
Spiral
The foam is
These are patterns that approximately
occur naturally in plants self-similar on
and natural systems. smaller and smaller
scales; in other
It reflects the order and words, foam is
growth in nature, and have fractal.
been studied by
mathematicians and used Symmetry
as symbols and
architectural forms. • Bilateral Symmerty
In bilateral symmetry, similar anatomical parts
Fractals are arranged
on opposite sides of a median axis so that only
Best described as a one plane can
non-linear pattern that divide the individual into essentially identical
infinitely repeats in halves
different sizes.

The uniformity of a
fractal is the repeating
shape, although the form
may appear in varried
sizes.
Vitruvian Man
Tessellation
Leonardo da Vinci

Composed of repeating
Showing the proportion and symmetry of the
patterns of the same shape
human body.
without any overlaps or
gaps.
Rotation Symmerty

To form or arrange small

The property a shape has when it look the same
squares in a checkered or
after some rotation by a partial turn.
mosaic pattern.
- Drexel
University
The smallest that a figure can be rotated while
still preserving the original formation is called
the angle of
rotation.

Sequence

A sequence is an ordered list of numbers, called

terms, that may have
repeated values. The arrangement of
these terms is set by a definite rule.

Example: 1, 10, 100, 1000, ...

Number Patterns in Humans
Fibonacci Sequence
Population Growth
The man behind this
Patterns can also be observe in human
sequence is an Italian
movement. Old records of world population can
mathematician, Leonardo
help us predict future
Pisano Bigollo, whose
world population.
nickname was “Fibonacci.”
 a grammar consisting of rules on the use of
these symbols.

a range of meaning that can be communicated

with these
symbols.

Elements of the Mathematical Language

Like other languages, mathematics has nouns,

pronouns, verbs, and sentences.

It is designed in such a way that one can write

about numbers, sets, functions, etc. as well as
the process undergone by these elements (like
adding,multiplying, grouping, and evaluating).

Noun. Could be constant, such as numbers or

expression with numbers.

Verb. Could be a equal sign or inequal symbol.

Pronoun. Could be variable like x or y.

EXAMPLE Sentence. Formed by putting together these

elements [sub note: always with equal sign]

CHAPTER 2 VARIABLES
Can be used as a symbol any placeholder for any
Language of Mathematics mathematical object.

Language is “a systematic means of allows us to give a temporary name to what you

communicating by the use of sounds or are seeking so that you can perform a concrete
conventional symbols” (Chen, 2010. p. computation for its possible values.
353).

These definitions describe language in terms of

the following components.
 Kinds of Mathematical Statements

Universal Statement
– a certain property is true for all elements in a
set.

All positive numbers are greater than zero.

Conditional Statement
– if one thing is true, then some other thing also
has to be true.

If 378 is divisible by 18, then 378 is also divisible

by 6.

Existential Statement
– if there is at least one thing for which the
property is true.
Two Ways to Present a Set
There is a prime number that is even.
Roster Method
Language of Sets The elements in a given set are listed or
enumerated. Example A = { 1, 2, 3, 4, 5 }
• collection of elements

• introduced by Georg Cantor in 1879

Notation Symbols and Set Properties

The different symbols used to represent set
builder notation are as follows:
SET BUILDER NOTATION
describing a set by listing its elements or
demonstrating its properties that its members
must satisfy.

CARDINALITY OF SET
The cardinality of set is the number of elements
in a set written as “| A |”

EQUAL SET
The sets are equal if they have the same
elements. Example: { 1, 2, 3, 4, 5 } = { 5, 4, 3, 2, 1
}

Two sets are not equal if they do not have the

same elements. Example: { 1, 2, 3, 4 } ≠ { 5, 4, 3,
2, 1 }

EQUIVALENT SET
Two sets are equivalent if they have the same
number of elements. Example: A = { 1, 2, 3, 4, 5
}
B = { 5, 6, 7, 8, 9 }

JOINT SET
Two sets are joint if they have at least one same
element. Example; A = { 1, 2, 3, 4, 5 }
B = { 5, 6, 7, 8, 9 }

DISJOINT SET
Two sets are disjoint if they have no common
element. Example: A = { 1, 2, 3, 4, 5 }
B = { 6, 7, 8, 9, 10 }

SUBSETS
If A and B are sets, then A is called a subset of B,
written A⊆B, if, and only if, every element of A is
also an element of B.
 Example:
U = { 2, 4, 6, 8, 10, 12 }
A = { 4, 6, 8 }
A’ = { 2, 1O, 12 }

Language of Relation and Functions

RELATION
A relation is a rule that relates values from a set
of values (called the domain) to a second set of
values (called the range).

CARTESIAN PRODUCT x – values are input, independent variable,

The product of two sets denoted by A × B. domain

y – values are output, dependent variable, range

RELATION
A relation is also a set of ordered pair (x, y).

FUNCTION
A function is a relation where each element in
the domain is related to only one value in the
range by some rule.
A function is a set of ordered pair (x, y) such
that no two ordered pairs have the same x-value
but different y-values.
Operations on Set

UNION OF SETS
The union of two sets A and B is A ∪ B.

Example:
A = { 1, 2, 3 }
B = { 4, 5, 6 }
A ∪ B = { 1, 2, 3, 4, 5, 6 }

INTERSECTION OF SETS VERTICAL LINE TEST

The intersection of two sets A and B is A ∩ B. A graph represents a function if and only if each
vertical line intersects the graph at most once.
Example:
A = { 1, 2, 3 } FUNCTION NOTATION
B = { 2, 3, 5 } A equation that is a function may be expressed
A ∩ B = { 2, 3 } using function notation.

COMPLEMENT OF SETS The notation f(x) read “f of (x)” represents the

The set of elements found in universal set but variable y (output).
not in set A.
example
y = 2x + 6 can be written as f(x) = 2x + 6 often used in applications that involve
prediction, forecasting, or behavior.
Given the equation f(x) = 2x + 6, evaluate when
x = 3. Inductive Reasoning
reasoning using obsereved patterns.
f(x) = 2x + 6
f(3) = 2(3) + 6 STEPS:
f(3) = 12 1. Observe data
2. Find a pattern
3. Make a conjecture
CHAPTER 3 Use Inductive Reasoning to make
PROBLEM - SOLVING involves three basic Conjecture
functions:
1. Seeking information
2. Generating new knowledge
3. Making decisions

Problem
a statement requiring a solution, usually by
means of mathematical operation/geometric
construction.

Method
ways or techniques used to get which will,
usually involve one or more problem solving
strategies

Answer
can be a number, quantity, or some other entity
that the problem is asking for

Solution
the whole process of solving a problem
including the method of obtaining an answer
and the answer itself
DEDUCTIVE REASONING
Mathematical Reasoning the process of reaching a specific
the ability of a person to analyze conclusion by applying general;
problem,situations, and construct logical assumptions procedures, or principles
arguments to justify the process or hypothesis
involves making logical argument and
INDUCTIVE REASONING conclusions by applying generalizations
uses specific examples to reach general to specific situations
conclusion called “conjecture”.
example:
a logical process in which multiple premises,all All Engineers are very good at Math.
believed true or found true most of the time,are
combined to obtain a specic conclusion. Cedric is an Engineer.
Therefore, Cedric is very good at Math.

3. Carry out the Plan

• Work carefully
• Keep an accurate and
neat record of all your
attempts.
• Realize that some of
your initial plans will
COUNTEREXAMPLES not work and that you
A statement is a true statement may have to devise
provided that it is true in all cases. If you another plan or modify
can find one case for which a statement your existing plan.
is not true, called a counterexample,
then the statement is a false statement. 4. Review the Solution
• Ensure that the solution
Example is consistent with the
facts of the problem.

• Interpret the solution in

the context of the
problem.

PROBLEM SOLVING WITH STRATEGIES • Ask yourself whether

POLYA’S PROBLEM SOLVING there are
STRATEGY generalizations
1. Understand the Problem of the solution that
• Restate the problem in your could apply to other
own words problems.
• Determine what is asked
• Identify the given data,
conditions and
• information

• Identify the unknown

2. Devise a Plan
PROBLEM SOLVING WITH PATTERNS branch of statistics that interprets and draw conclusions
from the data.

MEASURES OF CENTRAL TENDENCY

MEASURES OF CENTRAL TENDENCY

[MEAN OF UNGROUPED DATA]

Arithmetic Sequence Mean

The difference between one term and the next
term is constant. most commonly used measures of central
tendency
In general, we can write arithmetic sequence often referred to as “average”
like this:

For each sequence, if it is arithmetic,

find the common difference.
MEASURES OF CENTRAL TENDENCY
Geometric Sequence
[MEDIAN OF UNGROUPED DATA]
Each term is found by multiplying the previous
term by a constant.

Median
In general, we can write geometric sequence
like this: midpoint of the data set or set of scores

the data must be arranged in order, from least

to greatest or vice versa

For each sequence, if it is geometric,

find the common ratio.

CHAPTER 4
STATISTICS Mode
number / value / observation in a data set which
Descriptive Statistics appears the most number of times
branch of statistics that involves the collection,
organization, summarization, presentation of The mode of a list of numbers is the number
data that occurs most frequently.

Inferential Statistics
MEASURES OF CENTRAL TENDENCY

[GROUPED DATA]

RANGE – GROUPED
measures the distance between largest and the smallest
class boundary

PROCEDURES FOR COMPUTING STANDARD

DEVIATION

VARIANCE
the average of the squared deviations from the mean

MEASURES OF VARIATION
indicate the degree or extent to which numerical values
are spread or clustered about the average value in a
distribution

RANGE - STANDARD DEVIATION

UNGROUPED a measure of how spread out numbers are
rough estimation of dispersion measures the distance
between largest and the smallest values
 MEASURES OF POSITION
QUARTILE (GROUPED DATA)

DECILE (GROUPED DATA)

VARIANCE AND
STANDARD
DEVIATION (GROUPED DATA)

PERCENTILE (GROUPED DATA)

MEASURES OF POSITION
QUARTILE, DECILE,
PERCENTILE
(GROUPED DATA)