Download MEASURES OF CENTRAL TENDENCY (average)

Math 8 SOL Study Guide
OPERATIONS with
FRACTIONS
Simplifying: divide the numerator and
denominator by the same number.
16 16  4 4


20 20  4 5
Addition: If the addends do not have the
same denominator, find a common
denominator, then change the addends to
equivalent fractions.
(Add the numerators, denominators stay
the same. If necessary, simplify the
sum.)
1 2 1 5 2  2 5 4
9
 

  
2 5 2  5 5  2 10 10 10
Mixed numbers:
2
1
2 2
1
4
1
5
3 2 3
2 3 2 5
4
8
4 2
8
8
8
8
Subtraction: Rules are the same as
addition, except once you have a
common denominator, subtract.
Sometimes you will need to borrow:
1
3
1
2
52  4 2  3
3
3
3
3
Multiplication: multiply numerator by
numerator and denominator by
denominator, simplify if necessary.
3 2 62
3
 

4 5 20  2 10
If mixed numbers, change to
improper fraction then multiply.
Change to mixed fraction if answer is
improper fraction.
CONVERTING FRACTIONS,
DECIMALS, & PERCENTS
ORDERING DECIMALS
0.03, 3.033, 0.1033, 0.0034
Fractions to decimal- divide numerator
by the denominator.
3  3 ÷ 4 = 0.75
4
Decimal to percent – multiply the
decimal by 100 and write percent sign.
0.75 × 100 = 75%
OR move decimal 2 places to the RIGHT
then write percent sign.
0. 75  75%
Fraction to percent – divide numerator
by denominator, multiply by 100, then
write percent sign.
1 3
1   3  2  1.5
2 2
1.5×100 = 150%
1. Write numbers in column, line up
decimals.
0.0300
3.0330
0.1033
0.0034
1. Count how many numbers are after
the decimal.
2. Make all decimals have the same
amount of numbers by writing in
zeros.
3. Compare each place value and write
in order.
0.0034, 0.03, 0.1033, 3.033
(least to greatest)
COMPARING FRACTIONS
1. Write the fractions across from each
other.
2. Cross multiply
Percent to fraction – drop percent sign
and write the number over 100.
Simplify, if possible.
25% =
25
25  25 1


100 100  25 4
3×8 = 24
3
4
Or drop the percent sign, and move the
decimal two places to the LEFT.
25%  25  0.25
1 1 7 5 35
1
2 1   
2
3 4 3 5 15
3
3
5
>
4
8
1. The side with the greater
product is the larger fraction.
ABSOLUTE VALUE
The absolute value of a number is the
distance of the number from zero on the
number line regardless of the direction.
Examples:
Division: Change the second term to its
reciprocal, and then follow the rule for
multiplication. 4  1
5 2
21 = 21
 14 = -14
Multiply by the reciprocal.
4 2 8
3
  1
5 1 5
5
1
5
8
24 is greater than 20 so,
Percent to decimal – drop the percent
sign and divide by 100.
25
25% 
 25  100  0.25
100
?
4×5=20
7 = 7
Real Number System
n – 3 + 2(n+2) = 13





X
3
Y
6
-
-
2
-
1
0
1
2
3
-
-
4
2
0
2
4
6
Associative
Property
Distributive
Property
2:5
o
In the name R I C H A R D, the ratio of vowels to all the letters
in the name (part to whole) is
2
7
2:7
0a = 0
4+5=9
5+4=9
3 × 6 = 18
6 × 3 = 18
(3+4) + 5 = 3 + (4+5)
or
(2 × 4) × 5 = 2 × (4×5)
2(3 + 5) =
2∙3+2∙5
a+b=b+a
or
a×b=b×a
(a+b) + c = a+ (b+ c)
or
(a × b) × c = a × (b×c)
x(y+z) =
x∙y+x∙z
5 + (-5) = 0∙
y + (-y) = 0
o  isnumber
100 ofnumber
2 to 7
a + 0 = 15
345 × 0 = 0
Any problems that are or can
be stated with percent and
the words “is” or “of” can be
solved with the following
formula:
2 to 5
1a = a
4
“IS and OF”
Percent Problems
with the word “to”
ALGEBRA
a b
 1
b a
Additive Inverse
In the name R I C H A R D, the ratio of vowels to consonant
(part to part) is:
2
5
1
1
● 3n = 12 ●
Multiplicative Inverse
3
3
Multiplicative
Property of Zero
Commutative
Property
RATIOS
Ratios are express 3 ways: Part to part
Part to whole
Whole to whole
Written in 3 different ways:
Example 1: R I C H A R D
As a fraction with a colon
7.
NAME OF
ARITHMETIC
PROPERTY
Identity Property
65 × 1 = 65
of Multiplication
Identity Property
15 + 0 = 15
of Addition
Inverse Property
1
4× =1
of Multiplication
SLOPE- rise over run
Y (output)
Range
Dependent
n – 3 + 2n + 4 = 13 Distributive
n +2n + (-3) + 4 = 13 Commutative
3n + 1= 13
3n +1 + (-1)= 13 + (-1) Additive Inverse
3n + 0 = 12 Identity Property of Addition
3n = 12
8. 1n = 4 Identity Property of Multiplication
9. n = 4
Natural or Counting numbers: {1,2,3,4…}
Whole numbers: {0,1,2,3,4…}
Integers: {…-2, -1, 0, 1,2,3,4…}
Rational Numbers: All the Natural numbers, Whole numbers, Integers,
and numbers that can be expressed as a fraction.
Irrational numbers: Real numbers that CANNOT be expressed as a
fraction
X (input)
Domain
Independent
1.
2.
3.
4.
5.
6.
Percent of Change =
Change (new-original)
Original
* If result is a positive
number then it is a percent
increase.
* If they result is a negative
number then it is a percent
decrease.
Calculating Interest: I = prt
Interest earned = principal × rate × time
Principal (amount started with), Rate (as a decimal),
Time (in years)
2
GRAPHS
Stem and leaf
Sample space:
Coin
heads
1
H1
2
H2
3
H3
4
H4
5
H5
6
H6
All graphs have titles
All parts of graphs are labeled
Line graph – change over time
Data is ordered from least to greatest
Bar graph – compares data
Scatterplot-graph that displays data
from two related sets as ordered
pairs.
tails
T1
T2
T3
T4
T5
T6
P(head, even number) =
3
1
or or 25%or 0.25
12 4
P(tails, 5) =
1
or 8.3% or 0.083
12
Fundamental Counting Principle
Histogram – a bar graph in which the heights
of the bars give the frequency of the data.
There are no spaces between bars.
PROBABILITY
3 ● 2 ● 3 = 18 possible combinations
Sample space:
Flip a coin, roll a number cube
Circle graph – compares part to whole
Tree diagram:
number
Coin
cube
1
2
3
H
4
5
6
T
Theoretical Probability: Expected
probability of an event
outcome
H1
H2
H3
H4
H5
H6
1
2
3
4
5
6
T1
T2
T3
T4
T5
T6
3
To calculate Theoretical Probability:
Number of favorable outcomes
Total number of possible outcomes
Experimental Probability: The
actual number of times an outcome
occurs in an experiment.
To calculate Experimental Probability:
P = total actual outcomes
total events
* If the outcome of one event does NOT
influence the occurrence of the other event, they
are called independent events. (Ex. Choosing
a card from a deck – replacing it back in the
deck- and choosing again.)
* If the outcome of one event has an impact on
the outcome of the other event, the events are
called dependent. (Ex. Choosing 2 cards from a
deck without replacing the first card drawn).
MEASURES OF CENTRAL TENDENCY
(average)
Mean – the sum of all the numbers divided
by the total amount of numbers
Median – the number in the middle of the set
of numbers that are in numerical order.
- If two numbers are in the middle, find
the mean of the two numbers
Mode – most repeated number(s).
- not all data sets have a mode.
- Sometimes you may have more than
one mode.
Range – the greatest number minus the least
number
IMPORTANT HINT to finding the measures
of central tendency.
ALWAYS order the numbers from the least
to the greatest.
Example 1
Quiz scores: 6, 10, 10, 9, 10, 7, 8
ORDER OF OPERATIONS
The order you perform addition,
subtraction, division and
multiplication in an equation
matters!
Parentheses ( ) or Grouping
Exponents yx
Multiply or Divide (divide or
multiply rank equally-solve left to
right)
Add or Subtract (subtract or add rank
equally-solve left to right)
(PEMDAS)
Please Excuse My Dear Aunt Sally!
(GEM DAS)
OPERATIONS WITH
INTEGERS
Addition
 If the signs of the terms are the
same, add.
 If the signs of the terms are
different, then subtract. The
answer takes the sign of the term
with the highest absolute value.
 Change the subtraction sign to
addition, and also change the sign
of the number directly behind the
subtraction to the opposite.
 Then follow the rules for addition.
Mode: 10 (there are 3 10s)
Median: 9 (number in the middle)
Mean: 6  7  8  9  10  10  10  60  8.57or8.6
7
7
Range: 10 – 6 = 4
1×1 = 12 =
1
2
4
2
9
2
4×4 = 4 =
16
5×5 = 52 =
25
2×2 = 2 =
3×3 = 3 =
2
36
2
49
2
64
2
81
6×6 = 6 =
7×7 = 7 =
8×8 = 8 =
9×9 = 9 =
10×10 = 102 = 100
2
11×11 = 11 = 121
12×12 = 122 = 144
13×13 = 132 = 169
14×14 = 142 = 196
15×15 = 152 = 225
16×16 = 162 = 256
Subtraction
 Add the Opposite!
Ordered: 6, 7, 8, 9, 10 ,10, 10
Perfect Squares and Square
Roots
17×17 = 172 = 289
18×18 = 182 = 324
19×19 = 192 = 361
20×20 = 202 = 400
Example 2:
Test Scores: 95, 100, 90, 95, 85, 85
Ordered: 85, 85, 90, 96, 96, 100
Mode: 85, 96 (2 of each number)
Median: 90  96  186  93
2
2
Mean: 85  85  90  96  96  100  552  92
6
Range: 100 – 85 = 15
1 =
4 =
9 =
16 =
25 =
36 =
49 =
64 =
81 =
1
2
3
4
5
6
7
8
9
100 = 10
121 = 11
144 = 12
169 = 13
196 = 14
225 = 15
256 = 16
289 = 17
324 = 18
361 = 19
400 = 20
Multiplication and Division
Scientific Notation, or exponential
Multiply or divide, then apply the
following rules to determine the sign
of the answer. (4 “Groups Of” 2)
notation, is a method of writing numbers in the
form a × 10n , where a is greater than or equal
to 1 and less than 10.




++=+
--=+
-+=+-=-
+
–
+
–
–
–
+
6
-
Example: 63,970 = 6.397 × 104
Example: 5.842 × 10-5 = 0.00005842
Properties of Exponents:
“Zero Pairs”
For any non-zero number, a and any integers m
and n, any number to the zero power = 1.
Zero Exponent: a0 = 1 Ex. 50 = 1. 3420 = 1
-
5+3=?
Negative exponents:
1
1
Example: 6 = 6 2 = 36
-2
4
ALGEBRAIC TERMS
Equation – a mathematical sentence with an equal sign:
5a + 2 = 12
134 = 3x – 12
Expression – a mathematical phrase:
5a + 2
Variable – a symbol, usually a letter used to represent a
number
Coefficient – the number next to the variable
Term – parts of an expression or equation separated by a
“+” or “–” sign.
SOLVING ALGEBRAIC EQUATIONS
HINTS
1. Solve for the variable (you want the variable
to stand alone)
2. Do the opposite. If the equation is:

Addition, then subtract

Subtraction, then add

Multiplication, then divide

Division, then multiply
3. What you do to one side of the equation,
you do to the other.
Example 1:
45x
5a + 2 – 3b = 12
Term: 3 (5a, 2, and 3b)
Coefficient: 5, 3 (both numbers are next to variables)
Variable: a, b
Inequality – a mathematical sentence that contains the
symbols <, >, ≤, or ≥. EX. 3x + 4 > 22
Example 3:
3h – 9 = 15
z + 12 = 15
To solve this addition equation subtract 12
from both sides of the equation.
To solve this two-step equation, first add nine
to both sides, and then divide both sides by
the coefficient, 3.
z + 12 = 15
-12 -12
z
=3
3h - 9 = 15
+ 9 +9
3h + 0 = 24
3
3
h=8
check: 3 + 12 = 15
15 = 15
Example 2:
Check: 3(8) – 9 = 15
Example 4:
25 = y – 10
-
5n + 2 < 12
-2 < -2
-5n + 0 < 10
To solve this subtraction equation, add 10 to
both sides.
25 = y – 10
+10
+10
35 = y
1 1
● 5n < - ● 10
5
5
n > -2
-
5
Check: 25 = 35 – 10
5
-
4
-
3
-
2
-
1
0
1
2
3
4
TRIANGLES
Sum of interior angles = 180 °
Classified by sides:
Equilateral: All three sides are
congruent
Isosceles: Two congruent sides
Scalene: All three sides are different
lengths
Classified by angles;
Right: One interior angle is 90
Acute: triangle with three acute angles
Obtuse: Triangle with one obtuse angle
Equilateral/ Acute
Isosceles/Right
Scalene/Obtuse
Vertical angles—angles on opposite
sides of the vertex where two lines
cross. These pairs of angles are always
equal.
Circles
Diameter
a° = b°
Radius
Area = 3.14  radius radius
Adjacent angles- angles that share a
vertex and a line.
Circumference = 2  3.14  radius
C= 2πr or C =  d
Pi = 3.14
Types of angles
QUADRILATERALS
Polygon that has 4 sides and 4 angles
Sum of interior angles: 360 °
Parallelogram – opposite
sides congruent, opposite
angles congruent, 2 pairs of parallel
sides (opposite sides parallel)
tri- 3
Acute angle - angles less than 900
quad- 4, tetra- 4
penta- 5
Right angle – angles = 900
hexa – 6
hepta- 7
Rhombus – all 4 sides
congruent, opposite angles
congruent, opposite sides
parallel (also a
parallelogram).
Obtuse angle – angles greater than 900
and less than 1800.
Rectangle – opposite sides
congruent, all angles congruent.
All angles = 90o, opposite sides parallel
Complementary angles –
2 adjacent angles that equal 900.
Square – a rectangle with 4 congruent
sides or a rhombus with 4 right angles.
All angles = 90o, opposite sides are
parallel
Trapezoid –
one pair of
parallel sides.
POLYGON PREFIXES
Straight angles – angles = 1800.
300 + 600 =
90
0
Supplementary angles –
2 adjacent angles that equal 1800.
900 + 900 =
1800
Kite – a quadrilateral with two pairs of
adjacent congruent sides. One pair of
opposite angles is congruent.
6
octo- 8
nona- 9
deca- 10
Pythagorean Theorem
TRANSFORMATIONS
Dilation
Rotation
A dilation is a transformation that
produces an image that is the same
shape as the original, but is a different
size. A dilation stretches or shrinks the
original figure.
VOLUME & SURFACE AREA
Q: How does changing one dimension
of a rectangular prism affect the volume
of the prism?
A: There is a direct relationship
between the volume and increasing one
dimension. For example if the length
doubles, then the volume will double.
Rectangular prism
A rotation is a transformation that turns a
figure about a fixed point called the center of
rotation. An object and its rotation are the
same shape and size, but the figures may
be turned in different directions.
SOLIDS
V=l×w×h
S.A. = 2lw + 2lh + 2wh
Cylinder
Reflection
In mathematics, the reflection of an object is
called its image. If the original object was labeled
with letters, such as polygon ABCD, the image
may be labeled with the same letters followed by a
prime symbol, A'B'C'D'. The line (where a mirror
may be placed) is called the line of reflection.
The distance from a point to the line of reflection
is the same as the distance from the point's image
to the line of reflection.
 Pyramids & Cylinders are named by
the shape of their bases.
 Pyramids – has one base and a
vertex across from the base.
 Cylinders – has two bases opposite
of each other.
 Other– sphere, cone, cylinders have
curved faces
 NETS – 2 dimensional pattern that
can
be folded up to create a 3-d solid
V = πr2h
S.A. = 2πr2 + 2πrh
Cone
V=
Triangular prism
A reflection can be thought of as a "flipping" of an
object over the line of reflection.
1 2
πr h
3
S.A. = πr2 + πrl
Pyramid
Translation
cube
Pentagonal
pyramid
A translation moves an object
without changing its size or shape
and without turning it or flipping
7
1
Bh
3
1
S.A. = lp + 2B
2
V=