Title: Tools of Algebra: Variables and Expressions; Exponents and PEMDAS; Working with Integers; Applying the Distributive Property; and Identifying Properties of Real Numbers
1Tools of AlgebraVariables and Expressions
Exponents and PEMDAS Working with Integers
Applying the Distributive Property and
Identifying Properties of Real Numbers
 Compiled and adapted by Lauren McCluskey
2Credits
 Algebra I by Monica and Bob Yuskaitis
 Interesting Integers by Monica and Bob
Yuskaitis  Multiplying Integers
 Dividing Integers
 Order of Operations
 Properties by D. Fisher
 Coordinate Plane by Christine Berg
 Prentice Hall Algebra I
3Algebra I
4Variable
 Variable A variable is a letter or symbol that
represents a number (unknown quantity).  8 n 12
5Expression
 Algebraic expression a group of numbers,
symbols, and variables that express an operation
or a series of operations.  m 8
 r 3
6Evaluate
 Evaluate an algebraic expression To find the
value of an algebraic expression by substituting
numbers for variables.  m 8 m 2 2 8 10
 r 3 r 5 5 3 2
7Simplify
 Simplify Combine like terms and complete all
operations  m 8 m
 2 m 8
 (2m x 2) 8n
 4m 8n
8Words That Lead to Addition
 Sum
 More than
 Increased
 Plus
 Altogether
9Words That Lead to Subtraction
 Decreased
 Less
 Difference
 Minus
 How many more
10Write Algebraic Expressionsfor These Word Phrases
n 10
 Ten more than a number
 A number decrease by 5
 6 less than a number
 A number increased by 8
 The sum of a number 9
 4 more than a number
w  5
x  6
n 8
n 9
y 4
11Types of Equations
 Equations may be
 True (when the expressions on both sides of the
equal sign are equivalent)  False (when the expressions on both sides of
the equal sign are not equivalent)  Open sentences (when they contain one or more
variables.
12Complete This Table
n 2n  3 y
5 2  3
10 2  3
21 2  3
32 2  3
7
17
39
61
13Exponents
 Exponents influence only that which they touch
directly.  For example
 40  d2 cd 3
 (for c 2 and d 5)
 40  (5 5) (2 5) 3
 40  25 (10 3)
 40  25 30
 70  25
 50

Now you try one 40 (d)2 cd 3
14Check your answer
 40 (5 5) (5 2) 3
 40 25 (10 3)
 40 25 30
 95
 Note In this case you multiply
 (5) (5) because of the parentheses.
15Try one more
 40  d2 cd 3
 (for c 2 and d 5)
16Check your answer
 40  (5)(5) (5) (2) 3
 40  (25) (10) 3
 40 (25) (30)
 40 (55)
 15
17Exploring Real Numbers
 Natural Numbers may also be known as counting
numbers 1, 2, 3  Whole Numbers include zero 0, 1, 2,
 Integers include negative numbers
 2, 1, 0, 1, 2,
18Exploring Real Numbers
 Rational Numbers can be written as either
terminating or repeating decimals  Irrational Numbers do not terminate or repeat
when written as decimals  Real Numbers include all rational and irrational
numbers
19Order of Operations
 A standard way to simplify mathematical
expressions and equations.
20Purpose
 Avoids Confusion
 Gives Consistency
For example 8 3 4 11 4 44 Or does
it equal 8 3 4 8 12 20
21 Order of operations are a set of rules that
mathematicians have agreed to follow to avoid
mass CONFUSION when simplifying mathematical
expressions or equations. Without these simple,
but important rules, learning mathematics would  be maddening.
22The Rules
 1) Simplify within Grouping Symbols
 ( ), , ,
 2) Simplify Exponents
 Raise to Powers
 3) Complete Multiplication and Division from Left
to Right  4) Complete Addition and Subtraction from Left to
Right
23Back to Our Example
For example 8 3 4 11 4 44 Or does
it equal 8 3 4 8 12 20
Using order of operations, we do the
multiplication first. So whats our answer?
20
24How can we remember it?
 Parenthesis  Please
 Exponents  Excuse
 Multiplication  My
 Division  Dear
 Addition  Aunt
 Subtraction  Sally
 OR PEMDAS
25Adding / Subtracting Real Numbers
 Inverse property
 For every real number n, there is an additive
inverse n such that  n (n) 0.
 We use the inverse property to solve equations.

from
Prentice hall Algebra I
26Matrices
27Adding Matrices

 5 (3) 2.7 (3.9)
 7 (4) 3 2

5
2.7
3
3.9
3
2
7
4
8
1.2
3
1
28Try It!
4
7/8
3/4
5
3/4
1
0
1/2
29Check your answer
 4 (5) 7/8 (3/4)
 3/4 1/2 0 (1)

9
1/8
1
1 1/4
30Scalar Multiplication (Matrices)

 (0.1)(47) (0.1)(13) (0.1)(7.9)
 (0.1)(0.2) (0.1)(64) (0.1)(0)

47
13
7.9
0.1
0.2
64
0
31Check your answer
4.7
0.79
1.3
0.02
6.4
0
32Try It!
25
35
3/5

 3/5 ____ 3/5 _____
 3/5 ____ 3/5 _____
10/9
15
33Check your answer
 3/5 25 15 3/5 35 21
 3/5 10/9 2/3 3/5 15 9

15
21
2/3
9
34For more practice
 Go to pages 2730 for
 or pages 43 and 45 for (scalar).
35Interesting Integers!
36What You Will Learn
 Some definitions related to integers.
 Rules for adding and subtracting integers.
 A method for proving that a rule is true.
37Definition
 Positive number a number greater than zero.
0
1
2
3
4
5
6
38Definition
 Negative number a number less than zero.
0
1
2
3
4
5
6
1
2
3
4
5
6
39Definition
 Opposite Numbers numbers that are the same
distance from zero in the opposite direction
0
1
2
3
4
5
6
1
2
3
4
5
6
40Definition
 Integers Integers are all the whole numbers and
all of their opposites on the negative number
line including zero.
41Definition
 Absolute Value The size of a number with or
without the negative sign.
The absolute value of 9 or of 9 is 9.
42Negative Numbers Are Used to Measure Temperature
43Negative Numbers Are Used to Measure Under Sea
Level
30
20
10
0
10
20
30
40
50
44Negative Numbers Are Used to Show Debt
Lets say your parents bought a car but had to
get a loan from the bank for 5,000. When
counting all their money they add in 5.000 to
show they still owe the bank.
45Integer Addition Rules
 Rule 1 If the signs are the same, pretend the
signs arent there. Add the numbers and then put
the sign of the addends in front of your answer.  OR
 Think Teams Which team won? How much did they
win by?
9 5 14
9 5 14
46Solve the Problems
 3 5
 (3) (4)
 6 7
 9 9
8
7
13
18
47Integer Addition Rules
 Rule 2 If the signs are different pretend the
signs arent there. Subtract the smaller from the
larger one and put the sign of the one with the
larger absolute value in front of your answer.  OR
 Think Teams Which team won? How much did they
win by?
9 5
Larger abs. value
9  5 4
Answer  4
48Solve These Problems
2
 3 5
 4 7
 (3) (4)
 6 7
 5 9
 9 9
5 3 2
3
7 4 3
1
4 3 1
1
7 6 1
4
9 5 4
0
9 9 0
49One Way to Add Integers Is With a Number Line
When the number is positive, count to the
right. When the number is negative, count to the
left.

50Adding on a Number Line
3 5
2

51Adding Integers Is With a Number Line
6 4
2

52Adding Integers Is With a Number Line
3 7
4

53Integer Subtraction Rule
Subtracting a negative number is the same as
adding its opposite. Change the signs and add.
2 (7) is the same as 2 (7) 2 7 9!
54Integer Subtraction Rule
Subtracting a negative number is the same as
adding its opposite. Change the signs and add.
2 (7) is the same as 2 (7) 2 7 9!
55Here are some more examples.
12 (8) 12 (8) 12 8 20
3 (11) 3 (11) 3 11 8
56Check Your Answers
1. 8 (12) 8 12 20 2. 22 (30)
22 30 52 3. 17 (3) 17 3 14 4.
52 5 52 (5) 57
57Multiplying / Dividing Real Numbers
Multiplying and dividing positive and negative
numbers is easy when you remember the
rules positive positive positive
negative negative negative  
 positive negative negative  
58MULTIPLYING INTEGERS
59Problem 1
(3)( 5)
60Problem 2
(5)(3)
61Problem 3
(2)(10)
62Problem 4
(3)(8)
63Problem 5
(6)(8)
64Problem 6
(3)(9)
65Problem 7
(7)(3)
66Problem 8
(9)(0)
67Problem 9
(9)(7)
68Problem 10
(16)(10)
69Problem 11
(9)(5)
70Problem 12
(4)(9)
71Problem 13
(5)(1)
72Problem 14
(10)(4)
73Problem 15
(15)(2)
74Problem 16
(5)(11)
75Problem 17
(10)(4)
76Problem 18
(4)(7)
77Problem 19
(12)(5)
78Problem 20
(8)(4)
79Check your answers
 15 11) 45
 2) 15 12) 36
 3) 20 13) 5
 4) 24 14) 40
 5) 48 15) 30
 6) 27 16) 55
 7) 21 17) 40
 8) 0 18) 28
 9) 63 19) 60
 10) 160 20) 32
80DIVIDING INTEGERS
Remember
1. IF THE SIGNS ARE THE SAME THE
ANSWER IS POSITIVE
2. IF THE SIGNS ARE
DIFFERENT THE ANSWER
IS NEGATIVE
81Problem 1
(6)( 3)
82Problem 2
(6) (3)
83Problem 3
(10) (2)
84Problem 4
(16) (2)
85Problem 5
(12) (6)
86Problem 6
(9) (3)
87Problem 7
(8) (4)
88Problem 8
(9) (0)
89Problem 9
(21) (7)
90Problem 10
(16) (8)
91Problem 11
(10) (5)
92Problem 12
(12) (6)
93Problem 13
(5) (1)
94Problem 14
(10) (5)
95Problem 15
(15) (3)
96Problem 16
(22) (11)
97Problem 17
(16) (4)
98Problem 18
(14) (7)
99Problem 19
(12) (6)
100Problem 20
(8) (4)
101Check your answers
 2 11) 2
 2 12) 2
 5 13) 5
 8 14) 2
 2 15) 5
 3 16) 2
 2 17) 4
 0 18) 2
 3 19) 2
 2 20) 2
102The Distributive Property
 You can use the Distributive Property
 to multiply a sum or difference
 by a number.
 You can also use the Distributive Property
 to simplify algebraic expressions by
 removing the parentheses.

103A good way to remember how to apply the
Distributive Property is to visualize 2
rectangles
31 3
3 x 3x
3
So 3( x 1) (3 x) (3 1) 3x 3
1
x
104Try It!
a) 2/3(6y 9) b) 0.25(6q 32) c) (8 3r)
5/16 d) 4.5(b 3)
105Check your answers
 4y 6
 1.5q 8
 2 1/2 15/16 r
 4.5b 13.5
106Properties
by D. Fisher
107Which Property?
(2 1) 4 2 (1 4)
Associative Property of Addition
108Which Property?
3 7 7 3
Commutative Property of Addition
109.
Which Property?
8 0 8
Identity Property of Addition
110Which Property?
6 4 4 6
Commutative Property of Multiplication
111Which Property?
2(5) 5(2)
Commutative Property of Multiplication
112Which Property?
3(2 5) 32 35
Distributive Property
113Which Property?
6(78) (67)8
Associative Property of Multiplication
114Properties Using Variables
115Which Property?
6(3 2n) 18 12n
Distributive Property
116Which Property?
2x 3 3 2x
Commutative Property of Addition
117Which Property?
ab ba
Commutative Property of Multiplication
118Which Property?
a 0 a
Identity Property of Addition
119Which Property?
a(bc) (ab)c
Associative Property of Multiplication
120Which Property?
a1 a
Identity Property of Multiplication
121Which Property?
a b b a
Commutative Property of Addition
122Which Property?
a(b c) ab ac
Distributive Property
123Which Property?
a (b c) (a b) c
Associative Property of Addition
124The Coordinate Plane
By Christine Berg Edited ByVTHamilton
125Definition
Coordinate Plane
 The plane formed when 2 perpendicular number
lines intersect at their zero points
126Coordinate Plane
 The perpendicular number lines form a grid on the
plane
127(No Transcript)
128Xaxis
 The horizontal number line
 Positive to the right
 Negative to the left
129Yaxis
 The vertical
 number line
 Positive upward
 Negative downward
130Origin
 Where the x and y axes intersect at their zero
points
131(No Transcript)
132Quadrants
 The x and y axes divide the coordinate plane into
4 parts called quadrants
133II
I
IV
III
134Ordered Pair
 A pair of numbers (x , y) assigned to a point
on the coordinate plane
135Ordered Pair
Xcoordinate
Ycoordinate
136Plotting a Point
 Step 1
 Begin at the Origin
137Plotting a Point
 Step 2
 Locate x on the xaxis
138Plotting a Point
 Step 3
 Move up or down to the value of y
139Plotting a Point
 Step 4
 Draw a dot and label the point