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Order of Operations 1-1
Patterns and Variables 1-2
Translating Phrases to Algebraic Expressions 1-3
Properties 1-4/2-4
Simplifying Algebraic Expressions 2-5
Integers 3-1
Multiplying and Dividing Integers 3-8/3-9
Factors and Multiples 5-1
Adding and Subtracting Mixed Numbers 6-8
Multiplying Rational Numbers 7-1
Reciprocals and Dividing Rational Numbers 7-2
Scientific Notation 7-7
Solving Two Step Equations 8-1
Functions 8-4
More Simplifying to Solve Equations 8-6
Solving Equations with variables on Both Sides 8-7
Coordinate Plane 9-1
Graphing Linear Equations 9-2
Graphing Systems of Equations 9-9
Scatterplots 9-10
Graphing on the Number Line 9-11
Graphing Inequalities on the Coordinate Plane 9-12
Ratio, Rate, Proportion 10-1/10-3
Percent, Decimals, and Fractions 10-4/10-7
Finding a Percent of a Number 11-1
Finding the Percent One 11-2
Percent of Increase or Decrease 11-5
Calculating Simple Interest 11-8
Circles 12-8
Congruent Figures 12-10
Basic Geometric Figures 12-1
Angle Measure 12-3
Parallel and Perpendicular Lines12-4
Triangles 12-5
Polygons 12-6
Area of Rectangles and Parallelograms 13-1
Area of Triangles and Trapezoids 13-2
Area of Circles 13-3
Volume 13-6/13-8
Surface Area 13-9
The Basic Counting Principle 14-1
Permutations and Combinations 14-2
Probability 14-3
Frequency Tables 14-6
Square Roots 15-1
Aproximating Square Roots 15-2
Solving Equations: Using Square Roots
Special Triangles 15-7 Special Triangles
Trigometric Ratios 15-9

Order of Operations 1-1
1)  Parenthesis
2)  Powers
3)  Divisions / Multiplication > (left to right)
4)  Subtraction / Addition > (left to right)
Algebraic expression involving multiplication and division
9xm= 9*m= 9(m)= 9m
7xaxb= 7*a*b= 7(a)(b)= 7ab
y/3= y/3
ex.1) Evaluate m(7+w); m=9
9(7+3); w=3
9(10)
90

ex.2) Evaluate 8+ab/40; a=8
8+8*10
40
8+80
40
8+2
10

ex.3) Evaluate 3y+9 ; y=9
2y
27+9
18
36
18
2

Patterns and Variables 1-2

Translating Phrases to Algebraic Expressions 1-3
sum +
difference -
product *
quotient /
Translate to a numerical expression
ex.1) the difference of 10 and 5 = 10-5
ex.1) 4 less than 10 = 10-4
ex.3) the product of 8 and 7 = 8*7
ex.4) 45 divided by 3 = 45/3
ex.5) a number increased by 7 = h+7
ex.6) 5 less than a number = n-5
ex.7) 35 increased by twice a number = 35+2*d
ex.8) 1 less than a number divided by 10 = d/10-1

Properties 1-4 / 2-4
Commutative:

a+b = b+a
a*b = b*a
ex.1) 5*7 = 7*5
ex.2) m+7 = 7+m

Associative:

ex.1) (98+47)+53 = 98+(47+53)
98+100
198
ex.2) 7*(5x) = (7*5)x
35x

Identity:

a+0 = a
a*1 = a

Distributive:

a*(b+c) = a*b+a*c
ex.1) a*(3+5) = 9*3+9*5
27+45
72
ex.2) 6(r+s) = 6r+6*5
6r+30
ex.3) (m+7)4 = 4m+28
ex.4) 86*48+86*52

Simplifying Algebraic Expressions 2-5
Simplify - replace with a simpler equivalent expression

ex.1) Simplify  3(2n)
(3*2)n
6n
ex.2) (x+7)+6
x+(7+6)
x+13

Term - parts of an expression separated by + or -

4m+2n-3
terms

Like terms - terms with the same variable 2n and 3n are like terms,
4m and 2m are unlike terms

ex.3) 2x+3x
(2+3)x
5x

ex.4) 25g+15g
(25+15)g
40g
ex.5) 2m+3m+m
6m
ex.6) 4e+3f+e
5e+3f

Integers 3-1
Integers {.-3,-2,-1,0,1,2,3,.}
Numbers such as 4 and -4 are opposites.
The opposite of -4 is written -(-4) = 4.
Absolute Value - distance from zero

ex.1) | 7 | = 7
ex.2) |-4 | = 4
ex.3) | 0 | = 0
ex.4) Find m if | m | = 8.  m = -8,8

Order from least to greatest
ex.5) -1,-6,4 = -6,-1,4
ex.6) -2,|-3 |, -1,| 4 |
3         4
-2,-1,|-3 |,| 4 |

Think about money (+ : \$ you have)
(- : \$ you spent)
ex.1) 3+-2 = 1
ex.2) -3+2 = -1
ex.3) -2+-3 = -5
To subtract, we add the opposite of the second number.

|------|   |------|   |------|
| 9-7  | = | 5-1  | = | 2-3  | =
| 9+-7 | = | 5+-1 | = | 2+-3 | =
|------|   |------|   |------|

NOTE: -5 = -5
ex.4) 3-2 = 1
ex.5) -3-2 = -5
ex.6) 3- -2 = 5
ex.7) -3- -2 = -1

Multiplying and Dividing Integers 3-8/3-9
If the signs are the same the answer is positive.  If the signs are

ex.1) 4*-5 = -20
ex.2) -4*5 = -20
ex.3) -4*-5 = 20
ex.4) (-5)3 = -125
ex.5) 42/-7 = -6
ex.6) -42/7 = -6
ex.7) -42/-7 = 6
If n = + then n5 = +
If n = - then n5 = -
If n = + then n8 = +
If n = - then n8 = -

Factors and Multiples 5-1

factors        product
7*4      =      28
factors of 8 = 1,2,4,8
factors of 84 = 1,2,3,4,6,7,12,14,21,28,42,84
Divisibility - one number is divisible by another if their quotient
is a whole number (no remainder)
ex.1) Is 11 a factor of 143?
ex.2) Is 9 a factor of 326?

Divisibility rules
2 - even
3 - add digits, test 3
4 - test last 2 digits
5 - end in 0 or 5

Multiples - counting by a number
ex3.) List the multiples of 5
0,5,10,15.
ex.4) List the first 3 nonzero multiples of 11\
11,22,33.

ex.1) 2/5 + 1/3 = 6/15 + 5/15 = 11/15
ex.2) 3/10 + 6/15 = 9/30 + 12/30 = 21/30 = 7/10
ex.3) -2/5 - 3/6 = -12/30 + -15/30 = -27/30 = -9/10
ex.4) 5/6 - -3/4 = 10/12 + 9/12  = 19/12

Reducing Fractional Expressions

ex.1) 3mb/7mk = 3b/7/k
ex.2) 4ef/12f = 1e/3
ex.3) 15ac2/25a2c   15acc/25aac   15c/25a   3c/5a

ex.1) The west wall measures 60 9/16" and 217 7/8". Find the
total length?

ex.2) 3M stock rose from 50 _ to 51 5/8. What was the increase?

ex.3) GM stock fell from 52 1/8 to 50 3/4

Multiplying Rational Numbers 7-1

Multiply / Divide rules
+ + = +
+ - = -
- + = -
- - = +

ex.1) 2/5 * 5/12 = 10/60
ex.2) -6/4 * -3/5 = 9/10
ex.3) 1/2 of 3/4 = 3/8
ex.4) 2 2/3 (1 1/5) = 16/5
ex.5) 3 1/12 (-2) = 37/6
ex.6) (5/8)2 = 5/8 * 5/8 = 25/64

Reciprocals and Dividing Rational Numbers7-2
- two numbers are reciprocals if their product is 1
ex.1) 2/3 in the reciprocal of 3/2 because 3/2 * 2/3 = 1
ex.2) what is the reciprocal of -2000m/3, _______
how many halves are in 3?
3/ 1/2 = 6
2 1/2 / 1/4 = 10
ex.3) 2/3 / 1/6 = -4
ex.4) -5/24 / 5 = -32/15
ex.5) 8/ (-3 _) = -1/24
ex.6) -3 1/8 / -2 1/12 = 3/2 (1 «)

Review: Exponents are used to show how many times the some factor is
repeated.
exponent
43 = 4 x 4 x 4 = 64
base

ex.1) 82 = 8*8 = 64
ex.2) 24 = 2*2*2*2 = 16
ex.3) (-.03)2 = -.03 x -.03 = .0009
Rule: To multiply two powers with the same base, we add the
exponents.
ex.1) 32 * 34 = 36
ex.2) 103 * 102 = 105
ex.3) 24 * 2 = 25

Try the following: 45/42 = 43

This suggests a rule for simplifying expressions in this form.
Rule: To divide two powers with the same base, we subtract the
exponents.

ex.1) 25/23 = 22                       ex.4) x6/x2 = x4
ex.2) (-3)7/ (-3)4                     ex.5) m5/m
ex.3) 104/10                           ex.6) 34/34

you can also use the rule given above to simplify the expression
52/54

52/54 = 1/52                                52/54 = 5-2

This shows that 1/52 is the same as 5-2

ex.1) 3-2 = 1/32 = 1/9      ex.3) (-2)-4 = 1/-2-4 = 1/16

ex.2) 5-3 = 1/53 = 1/125    ex.4) 4-1 = 1/-4

Scientific Notation 7-7
Review
103 = 1000                10-3 = 1/1000 = .001

105 = 100000              10-5 = 1/1000000 = .00001

106 = 1000000             10-6 = 1/10000000 = .000001

Scientific notation is used to simplify work with very small or very
large numbers.

3.45      X       103

A decimal between 1 and 10                a power of 10

Write in standard form.

ex.1) 5.8 x 103 = 5800       ex.2) 6.556 x 102 = 655.6
ex.3) 1.8 x 10-4 = .00018    ex.4) 4 x 10-2 = .04

Write in scientific notation.

ex.1) 4567 = 4.567 x 103       ex.2) 1,234,000 = 1.234 x 106
ex.3) 234,000 = 2.34 x 105     ex.4) 50,000,000 = 5.0 x 107
ex.5) 0.000345 = 3.45 x 10-4   ex.6) 0.0206 = 2.06 x 10-2
ex.7) 0.000008 = 8.0 x 10-6    ex.8) 0.2004 = 2.004 x -10

Why are these not in scientific notation?

ex.1)(10 is less than 12.5) 12.5 x 104     ex.2) 2 x 4-5 (the
base must be 10)

Give two reasons for using scientific notation.
1.)  easier for large numbers
2.)  nicer for comparing

Solving Two Step Equations 8-1

SOLVE
review:     m-17 = 24       3m = 5

m+5 = 11         m/4 = 9

Rule:  Solving equations with combined operations
A: Identify the order the operations were applied to the variable.
B: Undo the operations in reverse order.

Functions 8-4

Definition: A function is a special relationship between two
variables.
Domain- number put into the function (input).

Function notation: f(x)
ex.) f(x) = 3 x +4
then f(5) = 3(5)+4
= 15+4
= 19

More Simplifying to Solve Equations 8-6
Like terms- have the same variable and exponent
ex 3m+2m = 5m
Unlike terms- cannot be simplified
ex 3m+2k
3m2+2m

ex.1) 9k+5 (k+7) = -49

ex.2) 2m+3(m-7) = 44

ex.3) 3(5n)+14+6n = 21

ex.4) -2(7c)-12+5c = 51

Solving Equations with Variables on Both Sides 8-7
1: Get the variable on one side
2: Solve as always
ex.1) -4(2m-5) = -4m+10
-8m+20 = -4m+10
+8m         +8m
20 = 4m+10
-10          -10
10 = 4m
4     4
5/2 = m

ex.2) 2(m+2) = 2m+4

ex.3) 2(m-2) = 2(m+1)

Coordinate Plane 9-1

Graphing Linear Equations 9-2

Graphing Systems of Equations 9-9

Scatterplots 9-10

Graphing on the Number Line 9-11

Graphing Inequalities on the Coordinate Plane 9-12

Ratio, Rate, Proportion 10-1/10-3

Ratio- a comparison of one number to another.
ex.) 3/5 or 3.5 or 3 to 5
Rate- a ratio that involves two different units.
ex.) 216miles/4hours = 54m/h
Proportion- An equation starting that two ratios are equal.

We can use cross product to solve proportions.
ex.1) 2/3 = 24/n     ex.2) 9/12 = 3/n     ex.3) 9/10 = n/22

ex.4) The lions in the zoo eat 40kg.
Of food every 7 days. How many
kg. do they eat in 30 days?

Kg./days     40/7 = k/30

1200 = 7k
7    7

171 = k

Percent, Decimals, and Fractions 10-4/10-7

Percent means hundredths or "out of 100"
Change to decimal.
1.)  43% = 43/100 = .43
2.)  90% = .9
3.)  2% =
4.)  34.7% = .347
5.)  10 «% =

Change to percent.
1.)  .27 = 27%
2.)  .06
3.)  .7
4.)  .065
5.)  .7/8 = 87.5%

Finding a Percent of a Number 11-1
Estimating Percents

ex.1) 63%\$ of 61
« of 60 = 30
ex.2) 27% of 79
¬ of 80 = 20

Using Proportions
ex.3) 90% of 25
90/100 = n/25  2250 = 100n
100    100
22.5 = n

Using Decimals
ex.4) 6.5% of \$80
.065 x 80 = 5.20

ex.5) 15% of \$25
.15 x 25 = 3.75

Using a calculator
ex.6) 22% of 150
33

Finding the Percent One 11-2
Number is of Another
"what percent" … n/100
of _ …

ex.1) 17 is what percent of 25?
n   =  17
100     25

25n   =   1700
25         25
n = 68%

ex.2) What percent is 8 out of 13?

n   =   8
100     13

ex.3) What percent of 72 is 12?

n   =   12
100      72

ex.4) What % of 72 is 90?

n   =   90
100      72

Percent of Increase of Decrease 11-5

change
original = _ %

ex.1) My weight on January 1st = 140
My weight on February 1st = 144

change
original = 4/140 = .0285 = 3%

ex.2) Jenny runs the mile in 9 minutes
in June. In July she can run it in 8
minutes.

change
original = 1/9 = .111 = 11%

Calculating Simple Interest 11-8

When you borrow money from a bank, credit union, or loan company
you pay for the use of it.  The amount you pay for the use of money
is called interest.

The amount of interest you pay depends upon the principle (amount
borrowed), the rate (percent of interest) charged, and the length of
time the money is kept (time).

Interest = Principle * Rate * Time

I = P * R * T
ex.1) Calculate the interest         ex.2) Principle = \$80,000
on a \$800 loan at 8% interest              Rate = 8% per year
per year if you paid it back in            Time = 30 years
2 years.

ex.3) You use your credit card       ex.4) Principle = \$5000
to buy \$500 worth of clothes.              Rate = 1.5% per month
You have to pay 15% interest               Time = 2 years
per year. How much interest
would you pay after 1 year?
What did the clothes really cost?

ex.5) You put \$1000 in the bank      ex.6) Principle = \$1000
and leave it in for 3 months at            Rate = 8% per year
2.5% interest per year. How much           Time = 3 months
interest do you make? How much
money do you have now?

ex.7) Principle = \$1,000,000
Rate = 8% per year
Time = 3 months

Circles 12-8

Congruent Figures 12-10

Basic Geometric Figures 12-1

Angle Measure 12-3

Parallel and Perpendicular Lines 12-4

Triangles 12-5

Polygons 12-6

Area of Rectangles and Parallelograms 13-1

Area of Triangles and Trapezoids 13-2

Area of Circles 13-3

Volume 13-6/13-8

Surface Area 13-9

The Basic Counting Principle 14-1

To find the total number of choices for an event, multiply the
number of choices for each individual part.

ex.1) How many outcomes are possible if you first toss a coin,
then roll a die?
2 * 6 = 12 outcomes

ex.2) How many different pizza combinations can you make with 2
crusts, 2 sauces, and 4 toppings? (only a one topping pizza)
2 * 2 * 4 = 16

10 * 10 * 10 * 26 * 26 * 26 = 17576000

Permutations and Combinations 14-2

An arrangement of a group of objects in a certain order is
called a permutation.

ex.1) Andy, Bob and Chris are to be seated at 3 desks arranged
in a row. How many ways can the 3 students arrange themselves?
3 * 2 * 1 = 6

ex.2) Steve, Rachel, Teddy and Kristin are on our 4x100 relay.
How many running orders are there for this relay?
4 * 3 * 2 * 1 = 24

ex.3) 20 people are running for student council. How many
different permutations are there?
20 * 19 = 380 (president & vice president)

A selection of objects without regard to order is called a
combination.

ex.4) choose 2 of these 4: Art, Band, computer, Drafting.

ex.5) Almond Joy, Butterfinger, Crunch, Dollar Bar; choose 3 of 4.

Probability 14-3

Probability = ways for an outcome to occur
total outcome
Flipping a coin:
ex.1) P(getting tails) = «

Rolling a Die:

ex.3) P(rolling a 5) = 1/6
ex.4) P(not rolling a 5) =
ex.5) P(rolling a 1 or 5) = 2/6 = 1/3
ex.6) P(rolling a 7) =
ex.7) P(rolling a number <7) = 6/6 = 1

Note: If any outcome is certain to happen then
the probability of that outcome is 1.
If any outcome is impossible then the
probability of that outcome is 0.

ex.8) Find P(winning the daily 3)
1/1000
1/10 * 1/10 * 1/10 = 1/1000

Frequency Tables 14-6

Data is often recorded using a frequency table.

This table shows how many students received each score on a math
quiz.

_
Mean = total points/number of scores = 468/27 = 17.3
Median = middle score = 17
Made = Score that occurs most often = 17
Range = Difference between the largest and smallest score = 20-13 = 7

Square Roots 15-1

Approximating Square Roots 15-2

Solving Equations: Using Square Roots 15-3

Special Triangles 15-7

Trigonometric Ratios 15-9

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