MATH PRELIMS

FRACTALS

– It is a detailed pattern that looks

similar at any scale and repeats

itself over time. It gets more

complex as it is observed at larger

scales.

Examples: snowflakes, trees

branching, lightning, ferns

SPIRALS

– It is a curved pattern that focuses

on a center point and a series of

circular shapes that revolve around

it.

Examples: pine cones, pineapples,

hurricanes

VORONOI

– It provides clues to nature’s

tendency to favor efficiency: the

nearest neighbor, shortest path,

and tightest fit.

Examples: skin of a giraffe, corn on

the cob, honeycombs, foam

bubbles, the cells in a leaf

TESSELLATION

– It is a design or pattern in which a

shape is used repeatedly to cover a

plane with no gaps, overlaps, or

empty spaces.

Examples: oriental carpets, quilts,

origami, Islamic architecture

1. REGULAR TESSELLATION

– A —- is a pattern

made with only one type of regular

polygon.

SEMI-REGULAR

– If the same combination of regular

polygons meets at each vertex, it is

called a —-

IRREGULAR

– The figures used are irregular

polygons and may be the same or

different types.

Once a basic design has been

constructed, it may be extended

through 3 different transformations:

reflection (flip), rotation (turn), or

translation (slide).

TRANSFORMATION

– changes a figure into another

figure. The new figure formed by a

—-is called the image.

In some —, the figure

retains its size and only its position

is changed.

SYMMETRY

– when different sides of something

are alike. This produces mirror

images with only two sides, like the

two sides of our bodies; they may

be symmetrical on several sides,

like the inside of an apple sliced in

half; or they might be symmetrical

on all sides, like the different faces

of a cube.

REFLECTION

is when an object is

flipped in a mirror line tó give a new

image (which then faces the

opposite direction), as shown in the

figure below.

REFLECTION SYMMETRY

– Basically, if you can fold a shape

in half and it matches up exactly, it

has —-. An easy

way to understand reflectional

symmetry is to think about folding.

The line created by the fold is the

line of symmetry.

ROTATION

is when an object is

moved around fixed point by a set

number of degrees either clockwise

or counter clockwise to give a new

image. In the figure below, the

object has been rotated 90° counter

clockwise to give the new image.

ROTATIONAL SYMMETRY

– When some shapes are rotated,

they create a special situation

called — A shape

has— if, after you

rotate less than one full turn, it is

the same as the original shape.

TRANSLATION

is when an object is

moved by sliding it by-a set number

of units vertically and horizontally-In

the figure below, the object has

been & translated 2 units to the

right and 1 unit down.

DILATION

– a transformation in which a figure

is made larger or smaller with

respect to a point called the center

of dilation. The ratio of the side

lengths of the image to the

corresponding side lengths of the

original figure is the scale factor of

the dilation.

FIBONACCI SEQUENCE

– named after the Italian

mathematician Leonard Pisa, his

nickname Fibonacci.

It is also interesting to note that

the ratios of successive Fibonacci

numbers approach the number Ф

(Phi), also known as the Golden

Ratio.

– Formed by adding the preceding

two numbers.

– Starting with 1, the succeeding

terms in the sequence can be

generated by adding the two

numbers that came before them.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,

144..

– In mathematical terms, the

sequence Fn of Fibonacci numbers

is defined by Fn = Fn-1 + Fn-2 with

seed values F0 = 0 and F1 = 1

GOLDEN RATIO

ratio is a special

number found by dividing a line into

two parts so that the longer part

divided by the smaller part is also

equal to the whole length divided by

the longer part.

– It is also known as the Golden

Section, Golden Mean, Divine

Proportion or Greek letter Phi.

– This is approximately equal to

1.618.

– If you keep dividing consecutive

terms of the Fibonacci sequence it

will eventually get close to the

golden ratio.

LINEAR FUNCTION

– also known as first – degree

function where the involved

variables are all raised to one (1).

– Its graph is a straight line,

continuous on both ends.

Formula:

f(n) = an + b

y = ax + b

Where:

n = x = independent variable

f(n) = y = dependent variable

a and b are constants

QUADRATIC FUNCTION

– also known as second – degree

function where one of the involved

variables is raised to two (2) or

squared and the rest are all raised

to one (1).

– Its graph is a parabolic curve,

continuous on both ends.

Formula:

f(n) = an2 + bn + c

y = ax2 + bx + c

Where:

n = x = independent variable

f(n) = y = dependent variable

a, b and c are constants

I (INTEREST)

The payment for

the use of borrowed money or the

amount earned on invested money.

P (PRINCIPAL)

– The amount

borrowed/invested.

r (Rate of Interest)

A fractional

part of the principal that is paid on

loan or investment.

t (time)

– The number of years for

which the money is borrowed or

invested.

F (Final amount or Maturity

Level)

The sum of the principal

and interest earned within the time.

COMPOUND INTEREST

the interest calculated on the

initial principal and the accumulated

interest from previous periods.

WHAT IS LANGUGE?

– Language is a powerful tool for

communication and expression,

essential for sharing ideas,

knowledge, and emotions, thereby

shaping human experience and

culture (Crystal, 2005).

"What do you think makes

Math different from other subjects?"

– it uses special symbols and

language to express ideas very

clearly and precisely.

– has its own vocabulary and ways

of writing

– makes math unique and

universal.

MATHEMATICS EXPRESSION

– Does not express a complete

thought (phrase)

MATHEMATICAL SENTENCE

– States complete thought (with

mathematical symbol =, ≠, >, <, ≤

or ≥

SETS

– a group or collection of objects or

things.

– each object of a set is called an

element or a member of a set.

Example:

A set of silverware{spoon, fork,

knife}

In Mathematics, we also have set.

Example:

A set of counting numbers from

1 to 10

{1,2,3,4,5,6,7,8,9,10,}

ROSTER METHOD

we use this method in naming a

set by using any capital letter and

list the elements inside the bracket.

roster

- not all elements were

written, uses three dots then the

last elements.

Example 1: The set of counting

numbers which are even from 1 to

15.

Answer: E = {2,4,6,8,10}

This is read as “E is the set of

counting numbers which are even

from 1 to 15.”

SET-BUILDER NOTATION

Instead of listing the elements of

the set inside the bracket, we

describe the elements of the set.

Example 2:

V = {a,e,i,o,u}, or the set of vowels

of the English Alphabet.

Answer: V = { x / x is a vowel of the

English alphabet }

This is read as “V is the set of all x

such that x is a vowel of the English

Alphabet”

Example 3: Describe the set of

whole numbers less than 30 in two

ways:

Roster Method

D = {0,1,2,3 ...29}

Set-Builder Notation

D = {x / x is a whole number less

than 30}

ELEMENTS OF THE SET

The objects belonging to a set

are called

∈

- we use this symbol when an

element belongs to a set

∉

- we use this symbol when an

element doesn’t belong to a set.

CARDINALITY OF A SET

– The number of elements in a

given set.

– answers the question “how

many” is a cardinal number.

The cardinal number of A can be

represented by n(A).

Example 1: If M = {month with 31

days} then the set is M = {January,

March, July, August, October,

December}. The cardinal number is

7.

written as n(M) = 7

EMPTY SET

a set with NO ELEMENTS.

– represented by the symbol { } or

Ø.

– The cardinality of an empty set is

0.

– Written as { } or Ø.

FINITE SET

– a set having countable number of

elements.

– known as countable set and

cardinality can be determined.

Example: A set of even numbers

from 1 to 20

V = {2,4,6,8,10,12,14,16,18,20}

Cardinality: n(V) = 10

INFINITE SET

– a set with an unlimited number of

elements.

– cardinality cannot be determined.

Transcribed by: Ms. Escoses BAPS

world Modern

Example: S =  Set of all stars in

the universe 

Cardinality: n(S) = cannot be

determined.

Example 2: S = Set of whole

numbers

Cardinality: n(S) = cannot be

determined.

VENN DIAGRAM

Very useful in showing the

relationship between sets.

– consists of a rectangle

representing the universal set and

circles inside the rectangle

represents the sets being

considered in discussion.

UNIVERSAL SET

– a set of all elements under

consideration, represented by the

symbol U.

UNION OF SETS

– The union of set A and set B,

denoted by A U B, is the set whose

elements are contained in either set

A or set B, or both.

U - union symbol

The Venn Diagram shows the

shaded part of set A and B. The

shaded part represents the union

of two sets.

INTERSECTION OF SETS

– The intersection of set A and set

B, denoted by A ∩ B is a set whose

elements are contained in both set

A and set B.

∩ - Intersection Symbol

DISJOINT SET

– When two or more sets have no

common elements, then the

intersection of the given sets is

empty. We say that these sets are

—sets.

COMPLEMENT OF A SET

– A —-of a set A, is

denoted by A’, is the set of all

elements in the universal set that

are not in A.