Download MEASURES OF CENTRAL TENDENCY (average)

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
Division: Change the second term to
its reciprocal, and then follow the rule
for multiplication.
Multiply by the reciprocal.
4 1

5 2
Estimation
1. Is
(7/8 is close to 8/8 which is almost one
whole)
2. Is
4 2 8
3
  1
5 1 5
5
1
5
2
5
7
closer to 0, ½, or 1?
8
3
5
4
5
5
5
3
closer to 0, ½, or 1?
8
(3/8 is close to 4/8 which is equal to ½)
0
½
1
Converting Fractions, Decimals, &
Percents
3 ÷ ¼ = 12 pieces
Subtraction: Rules are the same as
addition, except once you have a common
denominator, subtract.
3x
Fractions to decimal – divide numerator by
the denominator.
3 
4
4
=
1
3 ÷ 4 = 0.75
Decimal to percent – multiply the decimal
by 100 and write percent sign.
Sometimes you will need to borrow:
0.75 × 100 = 75%
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
Comparing Fractions
1. Write the fractions across from each
other.
2. Cross multiply
3×8 = 24
3
4
4×5=20
?
5
8
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.
24 is greater than 20 so,
1 3
1   3  2  1.5
2 2
3
5
>
4
8
1.5×100 = 150%
(The side with the greater product is
the larger fraction.)
Percent to fraction – drop percent sign and
write the number over 100. Simplify, if
possible.
25% =
If mixed numbers, change to improper
fraction then multiply. Change to mixed
fraction if answer is improper fraction.
25
25  25 1


100 100  25 4
Percent to decimal – drop the percent sign
and divide by 100.
1 1 7 5 35
11
2 1   
2
3 4 3 4 12
12
25% 
25
 25  100  0.25
100
Or drop the percent sign, and move the
decimal two places to the LEFT.
1
25%  25  0.25
Ordering Decimals
Dividing Decimals
0.03, 3.033, 0.1033, 0.0034
1.
1. Write numbers in column, line up
decimals.
0.0300
3.0330
0.1033
0.0034
2.
3.
4.
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)
5.
6.
“House” goes over 1st or top
number.
make sure divisor is a whole
number
(move decimal when necessary)
what you do to the outside,
you do to the inside, or
what you do to the top,
you do to the bottom
Rewrite the problem after
moving decimal
Place decimal in correct place
above division sign
Ex. 2.46 ÷ 0.2 →
0.2
2.46
Perfect Squares and Square
Roots
1×1 = 12 =
1
2
4
2
9
2
16
2
25
2
6×6 = 6 =
36
7×7 = 72 =
49
2×2 = 2 =
3×3 = 3 =
4×4 = 4 =
5×5 = 5 =
2
64
2
81
8×8 = 8 =
9×9 = 9 =
10×10 = 102 = 100
Decimal Operations
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Adding and Subtracting Decimals
Line up the decimal points, and then perform
the operation.
Ex. 204.56
+ 13.17
217.73
307.765
- 26.08_
281.685
Multiplying Decimals
Ignore the decimal point at first and multiply
like whole numbers. Then insert the
decimal point so that the number of decimal
places is the same as the total number of
decimal places in the numbers being
multiplied.
Ex.
3.7 x 2.63
2.63 (2 decimal places)
x 3.7 (1 decimal place)
1841
+ 7890
9.731 (total of 3 decimal places)
11×11 = 11 = 121
12.3
24
.6
2
12×12 = 122 = 144
13×13 = 132 = 169
-2
04
- 4
06
- 6
0
14×14 = 142 = 196
15×15 = 152 = 225
16×16 = 162 = 256
17×17 = 172 = 289
PRIME NUMBERS to 100
2,
3, 5, 7
11, 13, 17, 19
23, 29
31, 37
41, 43, 47
53, 59
71, 73, 79
61, 67
83, 89
97
5
-
4 -3 -2 -1
0 1 2 3
19×19 = 192 = 361
20×20 = 202 = 400
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
Properties of Exponents
For any nonzero number, a and any integermand n,
Zero Exponent ao = 1
Examples: 5o = 1, 342o = 1
Any number to the zero power = 1.
Comparing Integers
-
18×18 = 182 = 324
1 =
4 =
9 =
16 =
25 =
36 =
49 =
64 =
81 =
4
5
-
3< 5
10 > -18
2
Ballpark Comparisons
The symbol,

means “is approximately (not exactly)”
1 inch  2.5 cm
1 foot ≈ 30 cm
1 meter ≈ 40 inches, a little longer than a yard
1 mile is slightly farther than 1.5 km
1 kilometer is slightly farther than ½ mile
1 ounce  28 grams
1 nickel has a mass of about 5 grams
1 kilogram is a little more than 2 pounds
1 quart is a little less than 1 liter
Water freezes at 0°C and 32°F.
Water boils at 100°C and 212°F.
Normal body temperature is about 37°C and 98°F.
Room temperature is about 20°C and 70°F.
ORDER OF OPERATIONS
The order you perform addition,
subtraction, division and
multiplication in an equation matters!
RATIO
Grouping Symbols or Parentheses ( )
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)
 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
In the name R I C H A R D, the ratio of vowels to consonant
(part to part) is
As a fraction
2
5
with a colon
with the word “to”
2:5
2 to 5
(GEM DAS)
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
2:7
2 to 7
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Divisibility Rules: A number is divisible by:







2 if the ones digit is divisible by 2 (even).
3 if the sum of the digits is divisible by 3
4 if the number formed by the last two digits
is divisible by 4
5 if the ones digit is 0 or 5
6 if the number is divisible by both 2 and 3
9 if the sum of the digits is divisible by 9
10 if the ones digit is 0
NAME OF
PROPERTY
Identity Property
of Multiplication
Identity Property
of Addition
Inverse Property
of Multiplication
Multiplicative
Property of Zero
Distributive
Property
Commutative
Property
3
ARITHMETIC ALGEBRA
65 × 1 = 65
1a = a
15 + 0 = 15
a + 0 = 15
4×
1
=1
4
a b
 1
b a
345 × 0 = 0
0a = 0
2(3 + 5) =
2∙3+2∙5
x(y+z) =
x∙y+x∙z
4+5=9
5+4=9
3 × 6 = 18
6 × 3 = 18
a+b=b+a
or
a×b=b×a
GRAPHS
All graphs have titles
All parts of graphs are labeled
Line graph – change over time
Bar graph – compares data
PROBABILITY
MEASURES OF CENTRAL
TENDENCY (average)
Sample space:
Mean – the sum of all the numbers divided
by the total amount of numbers
Flip a coin, roll a number cube
Tree diagram:
number
Coin
cube
1
2
3
H
4
5
6
T
Circle graph – compares part to whole
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
outcome
H1
H2
H3
H4
H5
H6
1
2
3
4
5
6
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
T1
T2
T3
T4
T5
T6
Sample space:
Coin
heads
1
H1
2
H2
3
H3
4
H4
5
H5
6
H6
Outlier: a value in the data set that is much
higher or much lower than the rest of the
data set. (Example, if 22 were added to the
data above, it would be much lower than the
rest of the data set so it would be called an
outlier.)
IMPORTANT HINT to finding the
measures of central tendency.
tails
T1
T2
T3
T4
T5
T6
ALWAYS order the numbers from the least
to the greatest!
Example 1
Quiz scores: 6, 10, 10, 9, 10, 7, 8
P(head, even number) =
Stem and leaf
Ordered: 6, 7, 8, 9, 10 ,10, 10
3
1
or or 25%or 0.25
12 4
Mode: 10 (there are 3 10s)
Median: 9 (number in the middle)
 8.57or8.6
Mean: 6  7  8  9 7 10  10  10  60
7
Range: 10 – 6 = 4
P(tails, 5) =
1
or 8.3% or 0.083
12
Data is ordered from least to greatest
Test Scores: 95, 100, 90, 95, 85, 85
Independent Event: when the
outcome of one event has no effect on
the outcome of the other. For example,
rolling one dice and flipping a coin.
Dependent Event: when the outcome
of one event is influenced by the
outcome of the other. For example,
drawing two kings from a deck of
cards without replacing the first king
drawn. ( Not replacing the king makes
this a dependent event.)
0
4
Example 2:
X
X
X
X
X
X
X
X
1
2
3
4
5
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
6
Range: 100 – 85 = 15
Mean as a Balance Point: Mean can be
defined as the point on a number line where
the data distribution is balanced. This means
that the sum of the distances from the mean
of all the points above the mean is equal to
the sum of the distances of all the data
points below the mean.
ALGEBRAIC TERMS
Equation – a mathematical sentence with an equal sign:
5a + 2 = 12
134 = 3x – 12
Expression – a mathematical phrase:
5a + 2
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.
45x
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.
5a + 2 – 3b = 12
Term: 3 (5a, 2, and 3b)
Coefficient: 5, 3 (both numbers are next to variables)
Variable: a, b
Example 1:
Example 3:
Y + 12 = 15
3h = 15
To solve this addition equation subtract 12
from both sides of the equation.
To solve this multiplication problem, divide
both sides by the coefficient, 3.
y + 12 = 15
-12 -12
y
=3
3 h = 15
3
3
h=5
check: 3 + 12 = 15
15 = 15
Check: 3×5 = 15
15 = 15
Example 2:
Example 4:
25 = y – 10
h÷6=4
To solve this subtraction equation, add 10 to
both sides.
To solve this division problem, multiply both
sides by 6.
25 = y – 10
+10
+10
35 = y
h÷6=4
×6 ×6
h = 24
Check: 25 = 35 – 10
25 = 25
Check: 24 ÷ 6 = 4
4 = 4
5
Modeling Equations
=x
TRIANGLES
Area & Perimeter
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
=1
What is the value of
?
Equilateral/ Acute
Isosceles/Right
Scalene/Obtuse
Graphing Inequalities
-
5 -4 -3 -2 -1
0
1 2 3 4 5
= includes that number
Example, n ≥ -3
Does not include the number
Example, n > -3
What is the difference between
arithmetic and geometric sequences?
While both are numerical patterns,
arithmetic sequences are additive and
geometric sequences are multiplicative.
Ex. 1,5, 9, 13….is arithmetic because
you + 4 each time.
Ex. 2, 6, 18, 54…. is geometric because
you x 3 each time.
Absolute Value
The absolute value of a number is the
distance of the number from zero on the
number line regardless of the direction.
Ex.
 7 = 7 and 21 = 21
A= lw (length x width)
P = 2l + 2w OR P = 2 (l+w)
Circles
chord
QUADRILATERALS
Polygon that has 4 sides and 4 angles
Sum of interior angles: 360 °
Square – a rectangle with
4 congruent sides or a
rhombus with 4 right
angles. All angles = 90o, opposite
sides are parallel.
Rectangle – opposite sides
congruent, all angles
congruent. All angles = 90o, opposite
sides parallel
Rhombus – all 4 sides
congruent, opposite angles
congruent, opposite sides
parallel (also a
parallelogram)
Parallelogram – opposite
sides parallel, opposite
angles congruent.
Trapezoid – one pair
of parallel sides.
Kite - a quadrilateral with two pairs of
adjacent congruent sides. One pair of
opposite angles is congruent.
Diameter
Radius
Area = 3.14  radius radius
Circumference = 2  3.14  radius
C= 2πr or C=  d
Pi = 3.14
GEOMETRY:
Types of angles:
Acute angle - angles less than 900
Right angle – angles = 900
Obtuse angle – angles greater than 900 and
less than 1800.
Straight angles – angles = 1800.
Complementary angles –
2 adjacent angles that equal 900.
300 + 600 = 900
“IS and OF” Percent Problems
Supplementary angles –
2 adjacent angles that equal 1800.
Any problems that are or can be stated
with percent and the words “is” or “of”
can be solved with the following formula:
o
o  isnumber
100 ofnumber
900 + 900 = 1800
6
Geometry
SURFACE AREA
Perpendicular lines - special
intersecting lines that form right angles
where they intersect.
Polygons:
A polygon is a closed figure made be
joining line segments, where each line
segment intersects exactly two others.
The three shapes below are examples of
polygons.
The figure below is not a polygon, since it
is not a closed figure:
Heptagon: A seven-sided polygon.
The sum of the angles of a heptagon is
900 degrees.
Regular
Rectangular Prism
Irregular
Octagon: An eight-sided polygon.
The sum of the angles of an octagon is
1080 degrees.
Regular
Irregular
Nonagon: A nine-sided polygon. The
sum of the angles of a nonagon is 1260
degrees.
S.A. = 2lw + 2lh+ 2wh
V=l×w×h
Congruent figures have the same size and
shape. Example below.
The figure below is not a polygon, since it
is not made of line segments:
Regular
Irregular
Decagon: A ten-sided polygon. The sum
of the angles of a decagon is 1440 degrees.
Similar figures have the same shape but not
the same size. Example below.
The figure below is not a polygon, since
its sides do not intersect in exactly two
places each:
Regular
Irregular
POLYGON PREFIXES
A regular polygon is a polygon whose
sides are all the same length, and whose
angles are all the same.
Pentagon: A five-sided polygon. The
sum of the angles of a pentagon is 540
degrees.
Examples:
tri- 3
quad- 4, tetra- 4
penta- 5
hexa – 6
hepta- 7
octo- 8
nona- 9
deca- 10
Regular
Irregular
Hexagon: A six-sided polygon. The sum
of the angles of a hexagon is 720 degrees.
Regular
Irregular
7