Math Reviewer

Arithmetic

• Sum = addend + addend (or terms)

• Difference = minuend – subtrahend

• Product = multiplicand



multiplier (or factors)

• Quotient = dividend



divisor

• Take note!

a

is

undefined

Arithmetic

• Addition

– Same sign: add the numbers, copy the sign

– Opposite signs: subtract the numbers, copy sign of

larger number

• Subtraction

– Change the operation into addition, then change

the sign of the subtrahend

• Multiplication/division

Properties of arithmetic

• Commutative (



,



) a + b = b + a

• Associative (



,



)

• Distributive (



over



and



over +)

• Identity element (0 for



and 1 for



)

• Inverse elements

– Additive inverses have the sum 0

– Multiplicative inverses have the product 1

Properties of equality

• Reflexive

• Symmetric

• Transitive

• Substitution

a

a



a

b

b

a



_;

_then



c

a

c

b

b

a



_and



_,

_then



.

by

replaced

is

if

retains

the

same

value

involving

If

,

then

any

expression

b

a

a

b

a



Order of operations: PEMDAS

• Parenthesis or grouping

symbols

• Exponents

• Multiplication and

division (whichever

comes first)

• Addition and

subtraction (whichever

comes first)

Laws of exponents

• A power is an

expression involving

repeated

multiplication.

• base: b; power or

exponent: m or n

• Take note!

n

m

m

m

m

m

m

n

m

n

m

mn

n

m

n

m

n

m

a

a

b

a

b

a

a

b

ab

b

b

b

b

b

b

b

b

b

1

since

;

1

)

(

1

hence

;

)

(

0

0



































a

a

a





1

0

₁

Scientific notation

• Write numbers as the product of a number less than

10 and a power of 10.

• The sign of the exponent is positive when the

original figure is less than 1 and negative when more

than 1.

3

2

10

6

.

5

0056

.

0

245



2

_

.

45



10

_



Prime, Composite or Denominate?

• Prime

– Numbers whose only factors are 1 and itself

• Composite

– Numbers with factors other than itself and 1

• Denominate

GCF and LCM

• Greatest common

factor

– Useful in reducing

fractions to lowest or

simplest terms

• Least common multiple

– Useful in operations

involving dissimilar

fractions

Fractions

• Improper fractions to mixed numbers

• Mixed numbers to improper fractions

b

remainder

b

a

quotient

b

a

_

₍

_

₎

b

a

c

b

b

a

c



(

*

)



Fractions

• Equivalent fractions

– Have the same value

– Result from the multiplication and division of the

numerators and denominators of a fraction by the

same number

• Reducing to simple fractions

– Determine the GCF of the numerator and

denominator; divide them by the GCF

Ratio and proportion

• Ratios may be written as fractions or by using

colons

• Proportions are the equality between two

ratios

3

:

2

3

2

_is

_the

_same

_as

6

:

4

::

3

:

2

6

4

3

2

_

_is

_the

_same

_as

Sharing/dividing in a ratio

When sharing/dividing any amount/quantity

(e.g.,

n) in a ratio (e.g., a : b : c):

• Add

• Divide

• Multiply

parts

of

number

c

b

a







part

per

share

parts

of

number

n

_

c

or

b

or

c

or

b

or

a

share

of

a

part

per

share

_*



Rules for ratio and proportion

.

then

,

If

.

then

,

If

.

then

,

If

c

d

a

b

d

c

b

a

d

b

c

a

d

c

b

a

bc

ad

d

c

b

a













Variation

• Direct:

– Variable

y varies at the same rate as variable x

• Inverse:

– As variable

x increases, y decreases and vice-versa

• Joint

– More than one variable is involved in a direct

variation

kx

y



x

k

y



Mathematical sentences

• is – was – will be – has:



• plus – sum – more than – greater than –

farther than – years from now – increased by

– exceeds by:



• decreased by – minus – less than – diminished

by – younger than – years ago – subtracted

from:



Factoring polynomials

• Factor out the common monomial factor

• Difference of two squares

• Perfect square trinomial

• FOIL method and quadratic polynomials

2

2

2

₂

_ab

_b

₍

_a

_b

₎

a









)

)(

(

2

2

_b

_a

_b

_a

_b

a









c

bx

ax

form

standard

a



b

)(

c



d

)



_:

ac

2





(

ad





bc

)



bd

(

Notes in solving word problems

• Consecutive numbers:

• Consecutive

even/odd numbers:

• Mixture (e.g., of coins, vehicles, chemicals):

total amt = number of coins * denomination

number of feet/wheels = animals/vehicle * #

of feet/wheels of each

2

1





_,

_x

x

,

x

4

2





_,

_x

x

,

x

Notes in solving word problems

• Motion:

• Work:

• Age: add when it says “# of years from now”

and subtract when it says “# of years ago”

time

distance

speed



complete

to

person

the

takes

it

days/hours

of

number

day/hour)

1

(in

person

one

by

done

job

a

of

part

1



Notes in solving word problems

• Interest

• Compound interest

or future value

• Continuous compound interest

Prt

I



nt

n

r

P

A



(

1



)

rt

Pe

A



t

r

PV

FV



(

1



)

Notes in solving word problems

• Percent change

• Discount

• Commission

original

original

new

change

%





price

original

rate

Discount





earnings

total

rate

Commission





discount

price

original

price

Discounted





Functions

• A function is a set of ordered pairs such that

for each first component, there is at most one

value of the second component

• Three things that specify a function:

– Domain

– Range

– A rule which assigns to each member of the

domain exactly one member of the range

Functions

• Domain

– set of all first components in each element of the

set

• Range

– set of all the second components in each element

of the set

• When

y is a function of x, the set of values of x

is the domain of the function and the set of

Lines

• Parallel lines do not meet.

• Intersecting lines meet at one point. The sum

of any 2 adjacent angles formed is 180

°.

Kinds of Angles

• Acute: less than 90

_°

• Right: 90

_°

• Obtuse: more than 90

°, but less than 180°

• Straight: 180

_°

Angle relationships

• Supplementary angles

– Sum of measures is 180

°

• Complementary angles

Pythagorean theorem

• Applies to right triangles

• Legs:

a and b

• Hypotenuse:

c

2

2

2

_a

_b

c





Polygons

• Polygons are flat, closed figures consisting of

line segments.

• Exterior angles of a polygon add up to 360

°.

• The sum of the interior angles of a polygon,

with the number of sides

n:

• The sum of the exterior angles of a simple,

closed polygon is 360

_°.

)

2

(

Measurements

• Perimeter: sum of the length of all sides

• Area: size covered by a surface

s

P

_square



4

c

b

a

P

_triangle







w

l

P

_rectangle



2



2

r

nce

Circumfere



₂



2

s

A

_square



_A

_rectangle



_lw

bh

A

_triangle

2

1



2

r

A

_circle





Measurements

• Volume: the space occupied or contained by a

substance or shape

3

s

V

_cube



lwh

or

abc

V

_rectangula

_r

_prism



3

3

4 r

V

_sphere





bh

V

_pyramid

3

1



_V

_cone

_r

2

_h

3

1

_



Statistics, Probability and

Number Series

Measures of central tendency

• Mean

– Add the numbers and divide the sum by the

number of values (which were added).

• Median

– The middle number when numbers are written in

order. (When the number of values is even, the

median is the mean of the two middle values.)

• Mode

Counting techniques

• If two or more actions are performed in a

particular order, such that the first action

produces

a results, the second action

produces

b results, the third action produces c

results, and so on, then:

results

possible

of

number

total

...

*

c

*

b

*

a



Counting techniques: Permutation

• Objects are arranged

in a definite order:

– n different things

taken

r at a time

– n things taken n at a

time in which

q are

alike,

r are alike and

so on

– n distinct objects are

arranges along a

)!

(

!

r

n

n

P

_r

n



_

!...)

!

(

!

r

q

n

P

_r

n



)!

1

(





_n

P

Counting techniques: Combination

• Objects are not arranged in a particular order

– n different things taken r at a time

!

)!

(

!

r

r

n

n

C

_r

n



_

Probability

outcomes

possible

of

number

total

the

happen

can

outcome

an

ways

of

number

outcome

an

of

y

Probabilit



Arithmetic sequences and series

2

)

1

(





n

n

S

_n

• Arithmetic sequence/progression: a sequence

of numbers where successive members of the

sequence have a constant difference (

d).

• Formula for

n-th term

• Formula for sum of first

n values of a

sequence

d

n

a

Geometric sequences and series

• Geometric sequence/progression: a sequence

of numbers where successive members of the

sequence are produced by multiplying them

by a constant multiplier (

r, or common ratio)

• Formula for

n-th term

• Formula for sum of first

n values of a

sequence

1

1

*





n

n

a

r

a

r

a

S

_n



1

(

1



n

)

References and further reading

British Broadcasting Corporation. (2012). GCSE Bitesize: Maths. Retrieved 27

December 2012, from http://www.bbc.co.uk/schools/ gcsebitesize/

maths/

Fisico, M., L. Sia, J. Ho, Z. Maralit, J. Tan, . . . J. Rosello, R. Franco. (1998). 21

st

Century Mathematics Series. Quezon City: Phoenix Publishing House, Inc.

Pierce, Rod. (2012). Math Is Fun. Retrieved 28 December 2012, from

http://www.mathsisfun.com/index.htm

Stapel, E. (2012). The Purplemath Forums. Retrieved 27 December 2012, from

http://www.purplemath.com/ index.htm