Student Name:
Teacher:
Basics
New and Improved.
Now includes
Integers
Step
by
Step
1+1
=2
This is a work in progress……….
TABLE OF CONTENTS
Chap.
1
1.1
1.2
1.3
2
2.1
2.2
2.3
2.4
2.5
2.6
3
3.1
3.2
3.3
3.4
3.5
4
4.1
4.2
4.3
5
5.1
5.2
5.3
5.4
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Topic
Powers, Factors & Order of Operations
Powers and Exponents
Prime Factorizations (Factor Trees)
Order of Operations (P-E-M-D-A-S)
Algebraic Expressions
Understanding Algebraic Expressions
Evaluating Algebraic Expressions
Self Test
Translating Verbal Phrases  Algebraic Expression
The Division Bar
The Division Bar as a Grouping Symbol
Simplifying Algebraic Expressions With Powers
Self Quiz
Like Terms
“Like” Things
Understanding Like Terms
Combining Like Terms
The Commutative & Associative Addition Properties
Using the Commutative & Associative Addition Properties
Simplifying Algebraic Expressions
Combining Like Terms Amongst Unlike Terms
The Distributive Property (aka Removing the Parentheses)
Simplifying When There Is More Than One Set of Parentheses
Self Test
Evaluating Formulas
Basic Formula Evaluation
The Circle and π
Algebraic Representation of Perimeters
Algebraic Representation of Areas
The Integers
The Counting Numbers & the Whole Numbers
Understanding The Integers
The “Poof” Effect (aka Adding Integers)
Integer Addition “Strings”
Combining Like Terms Using Integer Addition Strings
Integer Multiplication
Integer Multiplication – “The Rules”
Page
1
6
10
14
15
17
20
21
23
28
30
31
32
34
37
38
39
42
45
47
50
53
59
60
62
63
65
74
76
78
80
 Step by Step Algebra Basics 
1 – Powers, Factors & Order of Operations
Read & Study box 
1.1 Powers and Exponents
Using exponents is a power-ful method used to simplify the way we show repeated
multiplication.
For example, instead of writing the repeated multiplication:
5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 53
Repeated
multiplication
Mathematicians simplify the writing of 5 multiplied by itself 13 times as:
13
Powers! Great. I like
writing it this way!
Tell me more.
5
For the power 513 the 5 is call the base number or base and the 13 is called the exponent.
The base number is the number being multiplied and the exponent is the number of times it
is multiplied. The number is “read 5 to the thirteenth power” or “the thirteenth power of 5.”
QUESTION:
How do we write 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 as a power?
Answer:
“7 to the fifteenth power”
base 
7
15
 exponent
715 means multiply 7
by itself 15 times.
It is read, “7 to the
15th power” or the
“15th power of 7.”
The Specials
Squares ( 2 ) and Cubes ( 3 )
- Special Names for Special Powers.
Powers having exponents of 2 and 3 are special since they appear often in mathematics
and in geometric representations. For these reasons they have special names. The
special name for powers with exponent 2 is “squared”. The special name for powers
with an exponent of 3 is “cubed”.
.
92, can be read “9 squared.”
113 can be read “11 cubed.”
Copyright: 2009 by Barry Hauptman
1
 Step by Step Algebra Basics 
Exercise box:
 
Instructions: Write the power indicated.
Nine to the 4th power
4
1
2
3
two to the 11th power
5
9
Six to the 14th power
4
Eleven to the 10th power
The 4th power of five
6
Four to the 3rd power
7
The 10th power of six
8
Eight to the 2nd power
9
Three squared
10
One to the 1000th power
11
Sixty to the 3rd power
12
Five to the 5th power
13
Twenty five cubed
14
Ten to the 100th power
15
One hundred squared
16
Fifteen to the 1st power
17
c to the 5th power
18
D cubed
Instructions: Write how each power is read.
19 72
Seven Squared 20
21
23
25
95
21
a3
22
24
26
53
210
980
xz
Instructions: Write each as a power using a base and an exponent
27 3 x 3 x 3 x 3 x 3 x 3
6 28 10 x 10 x 10
3
29
2x2x2x2x2x2x2x2
30
1x1x1x1x1
Instructions: Write the power as a multiplication and then multiply.
34
31
32
33
33
52
103
24
3x3x3
9x3
27
35
18
36
Copyright: 2009 by Barry Hauptman
34
37
09
38
(½)5
2
 Step by Step Algebra Basics 
Writing box
1. In the space provided below, explain why
the diagram at the right could represent
seven squared plus six squared.
72 + 62
2. In the space provided below, explain
why the diagram at the right could
represent eight squared minus four
squared.
82 – 42
Copyright: 2009 by Barry Hauptman
3
 Step by Step Algebra Basics 
Exercise box:
 
Instructions: Rewrite each expression using exponents
1
2x2x2x2x2x7x7x7x7x7x7x7x7
2
5 x 5 x 5 x 11 x 11
3
17 x 17 x 17 x 17 x 17 x 17 x 37
4
2x2x2x2x2x3x3x3x3x5x5x5
5
19 x 19 x 23 x 23 x 23 x 87
6
5x2x3x5x3x3x2x2x5x5x2x2
7
13 x 13 x 11 x 2 x 13 x 13 x 2 x 2 x 2
25 x 78
Instructions: Write each expression as a multiplication without the exponents.
8
32 x 4 5
9
102 x 226
10
93 x 145
11
22 x 73 x 115
12
15 x 34 x 132
3x3x4x4x4x4x4
Instructions: Evaluate each after rewriting without the exponents
13
52 x 2 2
5x5x2x2
25 x
4
?
Copyright: 2009 by Barry Hauptman
4
 Step by Step Algebra Basics 
14
22 x 3 2
15
12 x 7 2
16
102 x 32
17
23 x 3 2
18
22 x 3 3
Copyright: 2009 by Barry Hauptman
2x2x3x3
2x2x2x3x3
4 x
2
x 9
?
x
?
?
5
 Step by Step Algebra Basics 
Copyright: 2009 by Barry Hauptman
6
 Step by Step Algebra Basics 
Circle the
prime
factors.
Copyright: 2009 by Barry Hauptman
7
 Step by Step Algebra Basics 
Copyright: 2009 by Barry Hauptman
8
 Step by Step Algebra Basics 
Copyright: 2009 by Barry Hauptman
9
 Step by Step Algebra Basics 
Read & Study box 
1.3 Order of Operations (P-E-M-D-A-S)
As you know, math requires you to work the operations in a particular order, called the
order of operations. The order is as follows:
1.
2.
3*.
4*.
Parentheses or grouping symbols
Exponents (powers)
Multiplications/Divisions, from left to right
Additions/Subtractions, from left to right
Instructions: Find the value of each.
E xample box:
A
3+7x9
3+7x9
3 + 63
66
B
18 – 6 +
18 – 6 +
18 – 6 + 4
12 + 4
16
C
44 - 2 ● (15 - 3)
44 - 2 ● (15 - 3)
44 - 2 ● 12
44 − 24
20
Copyright: 2009 by Barry Hauptman
E xercise box:
1
Original expression
5+2x8
st
Multiply 1 .
Then Add
Answer
2
Original expression
Divide 1
12 + 7 −
st
Then from left Subtract.
Then Add
Answer
3
Original expression
Operations in (
20 - 2 ● (11 − 8)
st
) 1 .
Then Multiply
Then Subtract
Answer
10
 Step by Step Algebra Basics 
D
(22 + 3) ÷ (9 – 4)
(22 + 3) ÷ (9 − 4)
÷ (9 − 4)
25
25
E
Operations in (
(start at left)
)
Operations in (
)
÷ 5
Then Divide
5
Answer
7 + 32
7 + 32
7+9
16
F
Original expression
Original expression
4
5
(14 + 1) ÷ (8 − 5)
6 + 52
Power (exponent)
Then Add
Answer
5 + 2 ● (1 + 3)2
6
2 ● (9 – 6)2 + 1
5 + 2 ● (4)2
5 + 2 ● 16
5 + 32
37
Answer
Definition: a mnemonic is a remembering device.
PEMDAS is a mnemonic device used to remember the order of
operations rules.
What is “Please Excuse My Dear
Aunt Sally” a mnemonic device for?
Copyright: 2009 by Barry Hauptman
11
 Step by Step Algebra Basics 
Writing box
1. Explain why the following computation is incorrect .
2 x 52 + 23
There might
be more
than one
error here!
102
+ 23
2x2x2
equals 6???
C’mon.
100 + 6
Answer: 106? (Not!)
2a. Perform the indicate calculation. 5 x 23 + 10
2b. Explain each step in this correct solution for 5 x 23 + 10
5 x 23 + 10
The problem
2 x 8 + 10
Step 1:
16 + 10
Step 2:
26
Step 3:
Copyright: 2009 by Barry Hauptman
12
 Step by Step Algebra Basics 
3a. Perform the indicate calculation. 2 ● 5 + (10 – 7)2
3b. Explain each step in this correct solution for 2 ● 5 + (10 – 7)2
2 ● 5 + (10 – 7)2
The problem
2 ● 5 + 32
Step 1:
2●5 +9
Step 2:
10 + 9
Step 3:
19
Step 4:
Notebook Exercises :
Instructions: Find the value of each.
1
4
7
10
13
16
7+9x3
10 ÷ 5 – 1
24 – 2 ● (15 - 5)
5 + 32
23 +(20 ÷ 2)
7 ● 3 + (5 – 3)2
Copyright: 2009 by Barry Hauptman
2
5
8
11
14
17
3 x 2 + 11
28 – 5 x 2
(11 + 8) ● 20 + 12
52 – 25
33 +(10 ● 5)
22 + 32 + 42 − 52
3
6
9
12
15
18
4 x 5 – 11
4x2+ 5x2
11 + 7 − 8 / 2
32 + 22
2●3+5●1+6●4
12 − 13 + 14 − 15
13
 Step by Step Algebra Basics 
2 – Algebraic Expressions
Read & Study box 
2.1 Understanding Algebraic Expressions
a. Variables are represented by letters and variables change based on the
values given for them.
b. Numbers are called constants. Number values do not change.
The following are examples of variables: a, b, x, y, M, Z, Ω
The following are examples of constants: 3, 5.2, ½ , - 7, Π
Note: The Greek letter Π is an exception. Π represents a famous constant.
Multiplication
Multiplication with constants and variables can be shown in several ways.
Multiplication shown with:
No operation symbol
Parenthesis without operation symbol
Raised dot
Power (exponent)
E xercise box:
Expression
5Q
3(a)
G● M
y3
Meaning
5 times Q
3 multiplied a
G times M
y times y times y
 
1. Instructions:
Put a circle around the five (5) variables and a box around the six (6) constants.
m,
z,
4,
1.7,
N,
Ω,
¼,
Π,
3 ½, -L , 0.00009
2. Instructions: Complete the table.
Multiplication shown with:
No operation symbol
Parenthesis without operation symbol
Raised dot
Power
No operation symbol
Parenthesis without operation symbol
Raised dot
Copyright: 2009 by Barry Hauptman
Expression
13y
½ (L)
3●x
Q5
?
?
?
Meaning
?
?
?
?
8 times Z
K multiplied by X
½ times N
14
 Step by Step Algebra Basics 
Read & Study box 
2.2 Evaluating Algebraic Expressions
Evaluate algebraic expressions by substituting (replacing) the value for the variable
and performing the operations.
Example: Evaluate the expression 6M + 3y if M = 2 and y = 8.
Original Expression
6M + 3y
Substitute M  2, y  8
6(2) + 3(8)
12 + 24
Multiplications first. Then add.
Answer  36
]
Instructions: Evaluate each algebraic expression
E xample box:
A
5a
a=6
5(6)
30
B
D
Variable value
M=9
9 + 15
Variable value
3
J+9
J=6
Substitute
Answer
Original expression
4
Variable value
Substitute
13 − v
v = 13
Answer
Original expression
b=6
¼x
x = 12
Answer
Original expression
E
2
Substitute
M + 15
32
q=4
Answer
x = 24
½(24)
43 − z
z = 11
43 − 11
3q
Substitute
Original expression
24
1
Variable value
½x
12
C
Original expression
E xercise box:
Variable value
5
R = 10
Substitute
4
Copyright: 2009 by Barry Hauptman
Answer
15
 Step by Step Algebra Basics 
F
c3
c=2
3
2
2x2x2
8
G
3a + x2
a = 5, x = 7
3(5) + 72
3(5) + 49
15 + 49
64
H
x+y+z
x = 1, y = 9, z = 2
1+9+2
10 + 2
12
I
abc
a = 2, b = 10, c = 3
2 ● 10 ● 3
20 ● 3
60
d2
6
Original expression
d=7
Variable value
Substitute
Expand the power
Answer
3a + x3
7
Original expression
a=1 x=2
Variable value
Substitute
PEMDAS
Answer
q+r+s
8
Original expression
q = 18, r = 3, s = 7
Variable value
Substitute
PEMDAS
Answer
xyz
9
Original expression
x = 4, y = 5, z = 6
Variable value
Substitute
PEMDAS
Answer
Notebook Exercises :
Instructions: Use the variable values to evaluate each algebraic expression.
1
3A, if A = 6
2
4b, if b = 9
3
xy, if x = 2 and y = 7
4
FG, if F = 6 and G = 3
5
x + 12, if x = 1
6
Z + 3, if z = 100
7
R – 3, if R = 20
8
14 – Q, if Q = 14
9
20 ÷ q, if q = 2
10
J ÷ 1,000, if J = 10,000
11
D + E – F, if D = 9, E = 10, F = 1
12
5H + 7G, if H = 3 and G = 1
13
2a + 4b, if a = 5 and b = 10
14
M + 4c , if M = 8 and c = 3
15
3Q + 5G + 10K, if Q = 1, G = 2, K = 3
16
3a + 4b + 5c, if a = 2, b = 2, c = 2
17
r2 , if r = 4
18
p2 , if p = 5
19
k3 , k = 5
20
Z15 , if Z = 1
21
y2+ x2, if y = 5 and x = 12
22
2b + c3 , if b = 9 and c = 3
23
abc, if a = 1, b = 2, c = 3
24
efgh, if e = 10, f = 8, g = 4, h = 0
Copyright: 2009 by Barry Hauptman
16
 Step by Step Algebra Basics 
Self Test
Name: _____________________ Teacher:___________
1
2
3
3
4
5
6
7
8
9
10
11
For the power 104 the base is ____ and the exponent is ____
411 can be read “___ to the ____ power”
72 can be read “7 to the 2nd power” or “7 _____________”
53 can be read “5 to the 3rd power” or “5_____________”
Write 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 as a power. _______
Write 9 to the seventh power in base/exponent form _______
Write 56 as a repeated multiplication.________________
In the expression 114x2 the variable is ______.
In algebra 11M means 11 ______ M.
Write 100 times g algebraically. ___________
Find the value of 9y, if y = 3.
Find the value of ½x, if x = 10.
Use the variable values to evaluate each expression
13 45 – z, if z = 40
12 c + 9, if c = 15
14 9a + b, if a = 2 and b = 20
15
3x – 4y, if x = 5 and y = 3
16 A , if A = 4
17
d3, if d = 2
18 A + v , A = 10 and v = 2
19
5b2, if b = 3
2
2
3
Copyright: 2009 by Barry Hauptman
17
 Step by Step Algebra Basics 
20
Write without exponents:
112 x 133
22 Write using exponents:
21
Write without exponents:
23
Write using exponents:
3x3x5x5x5
32 x 2 4 x 5 3
7 x 7 x 11 x 2 x 2 x 7 x 7
24 Use a factor tree to find the prime
25
Use a factor tree to find the
prime factorization of 100.
26 Use a factor tree to find the prime
27
Use a factor tree to find the
prime factorization of 120.
factorization of 32.
factorization of 54.
Copyright: 2009 by Barry Hauptman
18
 Step by Step Algebra Basics 
Answers
1
2
3
3
4
5
6
7
8
9
10
11
12
14
16
18
20
Base = 10, exponent = 4
Four to the 11th power.
Seven squared
Five cubed
310
97
5x5x5x5x5x5
X
11 times M
100g
27
5
24
38
16
108
13
19
5
3
8
45
11 x 11 x 13 x 13 x 13
21
3 x 3 x 2 x 2 x 2 x 2 x 5x 5 x 5
2
3
22 3 x 5
5
24 2
23
22 x 74 x 11
25
22 x 5 2
3
26 2 x 3
27
23 x 3 x 5
Copyright: 2009 by Barry Hauptman
15
17
19
 Step by Step Algebra Basics 
Read & Study box 
2.3 Translating Verbal Phrases  Algebraic Expressions
Algebra is a branch of mathematics that uses symbols to represent numbers,
quantities and verbal phases. The following examples and exercises deal with the
translation of verbal phases into algebraic expressions.
Examples:
 n + 15.
 4a
A number increased by 15 is translated into
Four times a number is translated into
 x–7
2
The square of a number.  A
The quotient of a number and 100.  R ÷ 100
The product of a number and ½ .  ½B
Nine less than?
Nine less than a number.  Z – 9
Nine minus?
Nine minus a number.  9 – Z
The sum of two numbers.  x + y
The difference between two numbers.  a − b
Twice a number.  2a
A number divided by 3. 
A number decreased by 7.
E xercise box:
 
Instructions: Next to each word write the appropriate symbol from the following list.
+
1.
4.
7.
10.
13.
16.
X
−
Less
Square
Decreased
Increase
Sum
Subtract
Copyright: 2009 by Barry Hauptman
2.
5.
8.
11.
14.
17.
times
divide
product
minus
difference
double
( 2)
÷
3.
6.
9.
12.
15.
18.
more
add
multiply
twice
2nd power
quotient
20
 Step by Step Algebra Basics 
Read & Study box 
2.4 The Division Bar  ––––––
{––––} is a division bar. A division bar shows the division of algebraic expressions.
Examples:
The quotient of a number and 100.

The difference between two numbers, divided by 33
The product of a number and 3, divided by 8.


The sum of two numbers, divided by the square of a number.

x+y
z2
Instructions: Translate each verbal phrase into an algebraic expression.
E xercise box:
E xample box:
A
The sum of two numbers.
1
The sum of a number and 11.
2
Three decreased by a number
3
The product of two numbers.
Ten divided by a number.
a + b
B
A number decreased by 100
N − 100
C
The product of three numbers
abc
D
A number divided by 5.
x
5
4
E
Nine more than a number.
5
Twenty more than a number.
6
Eleven less than a number.
k + 9
F
Seven less than a number
A - 7
G
One divided by
a number squared.
1
n2
7
The square of a number, divided by 4.
H
Twice a number, divided by 3.
8
Twice a number, divided by M.
I
The sum of two numbers, divided by 8
9
Twelve divided by the sum of 2 numbers.
a + b
8
Copyright: 2009 by Barry Hauptman
21
 Step by Step Algebra Basics 
Notebook Exercises:
Instructions: Write each verbal phrase into an algebraic expression.
1
3
5
7
9
11
13
15
17
19
21
22
23
Five times a number.
2
The product (x) of a number and 13.
A number divided by 11.
4
The difference between a number and 2.
The square of a number.
6
The sum of three numbers.
Twice a number.
8
A number minus 6.
A number increased by 100.
10 Seven more than a number.
The quotient (÷) of two numbers. 12 Five times the square of a number.
Twice a number divided by 4.
14 Five less than a number.
The product of four numbers.
16 The sum of three numbers divided by 11.
A number times 6.
18 The square of a number.
A number minus 33.
20 A number plus 33.
The sum of two numbers, divided by the square of a number.
The product of 5 and a number. divided by the difference between a number and 3.
The sum of two numbers, divided by the sum of the square of a number and 8.
E xercise box:
 
Instructions: Translate each algebraic expression into a verbal phrase.
1
2
3
4
5
4n
x+5
7−b
10x
X2
Four times a number or product of 4 and a number.
A number increased by 5 or Five more than a number.
6
7
abc
8
a+b+c
2x
M −7
9
10
11
12
13
12 + X2
Copyright: 2009 by Barry Hauptman
22
 Step by Step Algebra Basics 
Read & Study box 
2.5 The Division Bar as a Grouping Symbol
_____
The division bar (
) represents a grouping. An operation above or below a division
bar should be performed first as if it were in parentheses.
Example A:
Steps
15 − 6
3
x 5
x5
€
3x5
15
Example B:
x+y
z2
Original expression
Perform the grouping operation above the
division bar first. 15 – 6 = 9
Perform the division on the left next.
9÷ 3=3
Multiply last to find the answer
Steps
x = 16, y = 9, z = 5
16 + 9
52
Substitute the variable values.
25
52
Perform the grouping (operation above the
division bar) first.
25
25
Exponent (power) next.
1
Copyright: 2009 by Barry Hauptman
Original expression with variable values.
Divide last to find the answer.
23
 Step by Step Algebra Basics 
E xercise box:
 
. Instructions: Follow the directions to find the value in each problem.
Problem #1:
Steps
x 3
Original expression
Perform the grouping operation above the
division bar first. 10 – 2.
Perform the division next.
Answer =
Problem #2:
Multiply last.
Steps
a = 20, b = 4, c = 2
Original expression with variable values.
Substitute the variable values.
Perform the grouping (operation above the
division bar) first.
Exponent (power) next.
Answer =
Copyright: 2009 by Barry Hauptman
Divide last.
24
 Step by Step Algebra Basics 
E xercise box:
 
Evaluate each by substituting the given values and using PEMDAS.
1 m+n
m = 12, n = 8, p = 4 2
a = 10, b = 2, c = 3
a – b2
p
c
3
5
7
xy + z
z
w+z
w–z
a + 3(b – d2)
2
Copyright: 2009 by Barry Hauptman
x = 2, y = 10, z = 5
w = 12, z = 8
a = 4, b = 9, d = 3
4
6
r+t
s3
a2 + b2
c2
2
8 tu + (w – v)
z+1
r = 15, s = 2, t = 9
a = 3, b = 4, c = 5
t = 2, u = 3, v = 4
w = 10, z = 20
25
 Step by Step Algebra Basics 
Writing box
Examine the following expression:
Which equals 2?
Which equals 8?
Which equals 12?
M–N =1
2P
What’s M?
What’s N?
What’s P?
Who knows?????
After the variables are substituted the expression equals 1. We know the
value of the variables to be 2, 8 and 12. Unfortunately, someone mixed up
the assignments. We do not know which is 2, which is 8 or which is 12.
Instructions: In words, explain how you would go about solving this mix up to find
the values of M, N and P. Use examples in your explanation.
Copyright: 2009 by Barry Hauptman
26
 Step by Step Algebra Basics 
Writing box
Examine the following expression:
Which equal 2
Which equals 4?
Which equals 6?
4a + b3
=1
2
2c
What’s a?
What’s b?
What’s c?
Who knows?????
After the variables are substituted, the expression equals 1. We know the
value of the variables to be 2, 4 and 6. Unfortunately, someone mixed up the
assignments. We do not know which is 2, which is 4 or which is 6.
Instructions: In space provided below, explain how you would go about solving this
mix up to find the values of a, b and c. Use examples in your explanation.
Copyright: 2009 by Barry Hauptman
27
 Step by Step Algebra Basics 
Read & Study box 
2.6 Simplifying Algebraic Expressions With Powers
Do these:
Study these:
Expression
Simplification
Expression
a
7 ●x
7x
1
9●a
b
7●x●x
7x 2
2
9●a●a●a
c
7●x●x●x●y●y
7x 3 y 2
3
9●a●a●b●b●b●b
d
5(x)(x)(y)(z)(z)(z)
5x 2 yz 2
4
3(a)(b)(b)(b)(b)(c)
e
11(x)(y)(y)(z)(z)
11xy 2 z 2
5
6(a)(a)(b)(b)(b)(c)
f
22mmmmm
22m 5
6
102ccccccccc
g
17PQQRSSSS
17PQ 2 RS 4
7
3xxyyyyz
h
wwxxxyyyyyy
w 2x 3y 5
8
ggggggghi
i
92pqrrrstttttt
92pqr 3 st 6
9
44abbbbcdeeeef
j
(xy)(xy)
x 2y 2
10
(ab)(ab)(ab)
k
(mn)(mn)(pq)
m 2 n 2 pq
11
(zw)(zw)(zw)(xy)
l
(st)(vw)(vw)(st)
s 2t 2 v 2w 2
12
5(XY)(CD)(XY)
2a 4 b 4
13
11(cd)(ef)(cd)(ef)
m
2(ab)(ab)(ab)(ab)
n
Simplification
14
(up)3 =
(vt)(vt)(vt)
(xy)3
v 3t 3
o
=
(cdefg)2
15
(abd)2 =
(abd)(abd)
Copyright: 2009 by Barry Hauptman
a 2b 2d 2
=
28
 Step by Step Algebra Basics 
Instructions: Complete the following simplifications:
Expression
1
(xy)
3
Simplification
(xy)(xy)(xy)
=
(pqr)(pqr)
Expression
2
(ab)
4
3
(pqr) 2
5
(mn)
3
6
(xy)
7
(pqr)2
8
(abcd)3
9
(ab)5(xy)2
10
(de)3(fg)2
11
(mn)4(qr)5
12
5(ab)2(qr)5
13
7(xy)(zw)4
14
(x4y3)2
15
(a2b2)3
4
(cdef)3
=
Simplification
(ab)(ab)(ab)(ab)
=
(cdef)(cdef)(cdef)
=
5
(ab)(ab)(ab)(ab)(ab)(xy)(xy) =
(x4y3)(x4y3) =
Copyright: 2009 by Barry Hauptman
29
 Step by Step Algebra Basics 
Self Quiz
Name: _____________________ Teacher:___________
Instructions: Translate each verbal phrase into an algebraic expression.
1
2
3
4
5
6
7
8
A number increased by 5.
The product of a number and 9.
Eight more than a number.
The difference between two numbers.
A number squared, divided by 3.
One less than a number.
The product of three numbers.
The product of two numbers, divided by a number cubed.
Instructions: Translate each algebraic expression into a verbal phrase.
9
3N
10 A + B
11 2A – 7
12
a–b
c2
Instructions: Evaluate each by substituting the given values and using PEMDAS.
13
a = 8, b = 4, c = 2
14
m = 15, n = 3, q = 4
a–b
m + n2
c
q
15
2w
w–z
w = 10, z = 5
16
Instructions: Simplify each using powers:
17 2xxxyyy
18
19 (mn)(mn)(mn)(mn)
20
21 (pqr)(pqr)(xyz)
22
2
23 (xy)
24
2
3
25 (gh) (mn)
26
5 2 2
27 (z w )
28
Copyright: 2009 by Barry Hauptman
a2 – b2
c
a = 8, b = 2, c = 10
4(a)(a)(a)(b)(c)(c)(c)(c)
7(xyz)(xyz)(abc)(abc)(abc)
(uv)(wx)(uv)(wx)(uv)
(abc)3
13(ab)3(def)5
102(x3y)6
30
 Step by Step Algebra Basics 
3 – Like Terms 
3.1 “Like” Things
It is important to know about and be able to recognize “like” things.
Consider the following examples of “Like” things:
Examples of Like Things
Like Units
9 mm, 12 mm, 1.4 mm, 1,023 mm
Like Fractions
Like Signed
Numbers
Like Fruit
Why?
All are in mm’s
,
,
,
,
,
-4, -12, -109.4, -7½ , -1,012, -99
1 apple, 7 apples, 10¼ apples, 58 apples
Consider the following examples of “U n L i k e ” things
Examples of “U n L i k e ” Things
UnLike Fractions
,
UnLike Units
UnLike Signed Numbers
UnLike Fruit
,
,
8 m, 8 cm, 1.9 inches, 50 yds,
-8, +12
3 oranges, 5 pears, 1¼ peaches
How are Like Things Combined?
Some Examples
Like things can be combined easily. UnLike can not be combined easily
4 figs + 11 figs = 15 figs
+
7 apples + 3 prunes = ?
UnLike
+
9 in2 - 5in2 = 4 in2
=
=?
4 m2 - 5 in =
?
UnLike
UnLike
(-8)
-
+ ( 11) = (
-19)
(-2) - (+11) = ?
UnLike
For the following, combine if they are “Like” things. If not, write UnLike .
1
3
5
7 apples + 2 apples =
4 ft + 15 ft + 3 ft =
7
5 cm + 9 cm – 10 cm =
(-3)
+ (-6) =
Copyright: 2009 by Barry Hauptman
2
4
6
77 mangos – 15 bananas =
10 cm3 – 4 mm2 =
8
19 in2 – 11 cm2 – 5 in =
+
=
31
 Step by Step Algebra Basics 
Read & Study box 
3.2 Understanding Like Terms
What is a term?
In algebra, a term is an expression that is a number, a variable, or the product of a
number and one or more variables.
 The expression 10Q is a term.
 The expression 5A + 9B has two terms.
 The expression 4x + 12 also has two terms.
 The expression x2 – 3y5 + 8z has three terms.
 The expression a + b + c + d + e + f + g + h has eight terms.
 The expression 8xyz2 also has only one term.
How many terms does each of expressions have?
5x + 7y + 8M
Answer:
eerht
6b
Answer:
eno
x2 + L + 3Z – 14 + x2
Answer:
evif
d+e–f+g+h+j+k–m–n
Answer:
enin
ab2 – 3ab2 ?
Answer:
owt
7abcdefghijklmnopqrstuvwxyz?
Answer:
eno
What are Like terms?
“Like terms” have exactly the same variables, and if there are powers, exactly the same
exponents. You will see later that “like terms” can be combined to form a single term.
What are unLike terms?
“Unlike terms” are terms that have different variables or different exponents. The variables
and exponents are not exactly the same. You will see later that “Unlike terms” can not,
should not and must not, be combined. Donʼt even think about combining them!
Examples of Like Terms
Examples of U n L i k e Terms
7M and 3M are like terms.
10ab and 12ab are like terms.
y5 and 3y5 are like terms.
6B and 3Y are NOT like terms.
9de and 8ef are NOT like terms.
2x6 and 2x5 are NOT like terms.
Copyright: 2009 by Barry Hauptman
32
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Circle “like” or “unlike” below each box of four terms.
1
2
3
4
5
6
9x 11x
x -3x
5B 2B
B ½B
A 7A
3A A2
11C 6M
4Z Ω
-ab2 2ab2
11ab2 5ab2
a2b2 2ab2
4a2b 3ab2
like
unlike
like
unlike
like
unlike
like
unlike
like
unlike
like
unlike
b
Instructions: Each problem contains 3 like terms. Write a 4th like term in the empty box.
7
8
9
L 3L
½L
-y 10y
3y
k2 2k2
¼k2
10
11
12
-mn3 2mn3
9mn3
-g7h3k 5g7h3k
2g7h3k
b
7de
de
3.2de
Instructions: Fill in the missing number.
13
5 apples – 3 apples = ____ apples
14
7 cats + 11 cats + 2 cats = ____ cats
15
3 ziggles + 4 ziggles = ____ ziggles
16
+
_______
11
=
Instructions: Complete the statement.
17
8 yards + 3 yards =
18
100 cm2 – 50 cm2 =
19
9 tons
–
2 tons + 11 tons + 4 tons
+
–
a ton =
–
a tomato +
20
+
21
20 cats
22
Something is odd about problem 21! Explain on the lines below.
–
=
7 figs + 3 bats + 4 mm3
Copyright: 2009 by Barry Hauptman
= ___________?????!!!
33
 Step by Step Algebra Basics 
Read & Study box 
3.3 Combining Like Terms.
In Algebra, adding or subtracting expressions to form a new simpler expression is called
combining. On the previous page, we saw that 3 ziggles + 4 ziggles = 7 ziggles. We
combined the ziggles to get a simpler expression. In algebra this is called combining
like terms and can be shown as:
3z + 4z
7z
Example of
combining like
terms
Note: The number 1 is the multiplicative identity,
because 1 multiplied by any number always equals the identical number.
1x5= ?
1 x 2,333,789 = ?
1xA= ?
1 is the Multiplicative Identity.
Does that mean when I see an X, it’s
the same as 1X?
DUH, of course!
X = 1X
Copyright: 2009 by Barry Hauptman
34
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Refer back to previous pages before doing these.
Remember the ziggles!!!
Combine like terms
1 5a – 3a = _______
(Remember the apples!)
2 7c + 11c + 2c = _______ (Remember the cats!)
3
5e + e = _______ (Remember the 1/11ths!)
4
8y + 3y = _______ (Remember the yards!)
5
100ab2 – 50ab2 = _______ (Remember the mm2s)
6
9t – 2t + 11t + 4t = _______ (Remember the tons!)
7
12tfs + tfs + 2tfs = _______ (Remember the Alamo!)
8
20c – 7f + 3b + 4m – t + 9h = ???????!!!!! Why can’t this be done?
Answer:
Instructions: If the terms shown are like terms combine them into a single term.
If the terms are unlike terms, write “can not combine unlike terms.”
9
4D + 2D = ?
Copyright: 2009 by Barry Hauptman
Answer: 6D
35
 Step by Step Algebra Basics 
10
6M – 3M2 = ?
Answer: “can
not combine unlike terms”
11
9x + 3x =
20
a horse + a horse =
12
10y – 5y =
21
h+h=
13
8y2 + 2y2 =
22
7 pickles – a pickle =
14
6M + 5D =
23
7p – p =
15
8z7 + 8z =
24
a+b+c+d=
16
15A2B + 3A2B =
25
6M – M =
17
3x + 7x + 12x + 4x =
26
7y2 + 3y2 + y2 – 11y2 =
18
40 apples + 50 apples =
27
A–A=
19
16 cats – 14 bananas =
28
A–A+A–A+A–A =
srewsnA
Copyright: 2009 by Barry Hauptman
36
 Step by Step Algebra Basics 
Read & Study box 
3.4 The Commutative and Associative Properties of Addition
rr
a. 5 + 3 = ?
b. 3 + 5 = ?
c. Are the results the same for 5 + 3 and 3 + 5?
d. Why are the results the same?
The order of the
numbers changed. But
the result did not!
So, what
happened?
And??
The Commutative Property
allows you to change the
order in an addition.
5 + 3 = 3 + 5 is an example of
The Commutative Property
a+b=b+a
Now do these:
e. (4 + 6) + 1 = ?
f. 4 + (6 + 1) = ?
g. Are the results the same for (4 + 6) + 1 and 4 + (6 + 1)?
h. Why are the results the same?
(4 + 6) + 1 = 4 + (6 + 1) is an example of
Here the
grouping
changed?
Copyright: 2009 by Barry Hauptman
The Associative Property
(a + b) + c = a + (b + c)
Right. The
grouping
changed?
37
 Step by Step Algebra Basics 
Read & Study box 
3.5 Using the Commutative & Associative Properties of
Addition
The Commutative Property states that you can change the order when
adding two numbers to attain the same result.
92 + 4 = 4 + 92
old
order
new
order
The Associative Property states that you can change the grouping when
adding numbers to attain the same result.
(16 + 88) + 3 = 16 + (88 + 3)
old
group
new
group
Example:
2+5+3
Study the different solutions using Commutative & Associative Properties
Solution A
2+5+3
7+3
10
Exercise 1:
Solution B
2+5+3
2+8
10
Solution C
2+5+3
5+5
10
8+4+7
Use the Commutative & Associative Properties to solve three different ways.
A
8+4+7
12 + __
B
?
x + 8x + 4x
9x + __
C
?
Exercise 2:
A
8+4+7
8 + __
8+4+7
15 + __
?
x + 8x + 4x
B
x + 8x + 4x
C
x + 8x + 4x
?
Copyright: 2009 by Barry Hauptman
38
 Step by Step Algebra Basics 
4 – Simplifying Algebraic Expressions
Read & Study box 
4.1 Combining Like Terms Amongst Unlike Terms
Sometimes an expression has like terms mixed together with unlike terms. For example,
consider this expression:
Can we just
combine the
like terms?
33A + 99 + 4A
Sure. What
about 99?
Should we
just leave it?
Answer:
7A + 99
Consider this expression:
3 12 + 5m + 9q - 3q + 11
Combine the
like terms
12 + 11 and
9q – 3q?
Answer:
Okay. And
do we just
rewrite the
5m?
323 + 6q + 5m3
Letʼs review this one:
Where did the 23 come from? ______________________________________
Where did the 6q come from? ______________________________________
Why is the 5m rewritten and unchanged? _____________________________
Copyright: 2009 by Barry Hauptman
39
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Explain each result in the space provided.
1
2B + 9C + 4B = 6B + 9C
2
16x - 5x + 8 = 11x + 8
3
4a2 + 3a2 + 8 + 2 + 9a5 = 7a2 + 10 + 9a5
4
3m3 + 8xy + 3m3 + xy = 6m3 + 9xy
5
4ab3 – 3ab3 + 12a3b7= ab3 + 12a3b7
Copyright: 2009 by Barry Hauptman
40
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Simply by combining Like Terms
1
3A + 16Q + 2A
2
11y – 3y + 88
3
L + 22L + 7a – a
4
10a2 + 10 + 19a2
5
6a – 4a + 9 - 8 + 3xy + 10xy
6
4ab2 + xyz + 7 – 3 + 19ab2
7
9abc + 5xyz + 2 mb2 + 11abc
8
8 keys + 2 pens – 5 keys + 2 pens + a wrench
Copyright: 2009 by Barry Hauptman
41
 Step by Step Algebra Basics 
Read & Study box 
4.2 The Distributive Property (aka Removing the Parentheses)
You will recall 4(
) means “4 times (
)”
Now consider the following:
64(3A + 2)6
Does this
mean 4 times
(3A + 2)?
Yes!
Do you know
how you can
remove the
parentheses?
The Distributive Property can be used to remove the parentheses.
“Distribute” the multiplier 4 to each of the terms inside, 3A and 2.
4(3A + 2)
Where did the
(
) go?
Answer:
4 ● 3A + 4 ●2
Would you
like to look up
my sleeves?
D12A + 8 D
Finish this example by distributing the 3 to remove the (
).
3(6x + y) = 3●6x + 3● __
18x + __
Answer:
D
D
SUMMARY: The distributive property states that when multiplying a number
by an addition or subtraction of two or more numbers multiiply each of the
numbers being added/subtracted by that number and remove the parentheses.
Then write as an addition/subtraction of the resulting numbers.
Copyright: 2009 by Barry Hauptman
42
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Explain each result in the space provided.
1
4(a + b) = 4a + 4b
2
7(2m – 3) = 14m – 21
3
10(x + 3a2) = 10x + 30a2
4
x(7 + y) = 7x + xy
5
12(2x + 3m – ¼ab2) = 24x + 36m – 3ab2
6
3(2x + m + 5ab2) = ?
Copyright: 2009 by Barry Hauptman
43
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Use the “Distributive Property” to remove the parenthesis
Remember: Multiply the number outside by each part of the addition/subtraction inside,
and then remove the parentheses.
1
7(p + q)
2
9(3m – 5)
3
10(x2 + 3ab)
4
a(5 + 2a)
5
2(x + y – z)
6
8h(2x + 4m + ½ab)
7
3(2x + m + 5ab2)
8
6(2 hens + 3 pens – 5 anchors)
Copyright: 2009 by Barry Hauptman
44
 Step by Step Algebra Basics 
Read & Study box 
4.3 Simplifying When There Is More Than One Set of Parentheses.
Consider this expression with two sets of parentheses:
65(A + 4) + 2(3A - 1)6
5 ●A + 5 ●4 + 2 ●3A –2 ●1
5A + 20 + 6A – 2
11A + 18
Distribute the 5 & the 2
Multiply as indicated.
Combine like terms
Answer
Hereʼs another example with two sets of parentheses.
Finish this example by using the distributive property to remove both sets of
parentheses and than combining the like terms.
63(x + 5y) + 2(4x - y)
3 ●x + 3 ●5y + 2 ●4x – 2 ●y
3x + 15y + __x – __y
________
Distribute the 3 & the 2
Multiply as indicated.
Combine like terms
Answer
Copyright: 2009 by Barry Hauptman
45
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Simplify each.
1
7(a + b) + 3(a + b)
= 7a + 7b + 3a + 3b
= 10a + 10b
2 5(x + y) + 2(x + y)
3
5(x + 2) + 4(x – 1)
4
7(4n + 2) + 3(8 – 2n)
5
2(a2 + b2) + 5(a2 + b2)
6
4(P + 2Q) + 5(Q + 1)
7
7(R + S) + 9 – 7S
8
4(2P + Q) + 11 – Q
Copyright: 2009 by Barry Hauptman
46
 Step by Step Algebra Basics 
Self Test
Name: ___________________________
1
2
3
4
5
6
7
8
For the power 115 the base is ____ and the exponent is ____
92 can be read 9 to the 2nd power or _____________
153 can be read 15 to the 3rd power or _____________
Write 3 x 3 x 3 x 3 x 3 x 3 as a power. _______
Write 25 as a multiplication.________________
20 - 2 ● (11 − 8)
(14 + 1) ÷ (8 − 5)
6 + 52
Use the variable values to evaluate each expression
Evaluate 5a, if a = 4
9
10 Evaluate xy,
if x =3 and y = 10
11 Evaluate M + 2N,
if M = 6 and N = 3
12
Evaluate 3Z – 5Q,
if Z = 10 and Q = 1
2
13 Evaluate R , if R = 6.
14
Evaluate j5, if j = 10.
3
15 Evaluate 3f , if f = 2.
16
Evaluate k2 + g3
if k = 7 and g = 3
17 If x = 1, y = 17 and z = 3
x+y
evaluate
18
If a = 8 and b = 2 evaluate
z
Copyright: 2009 by Barry Hauptman
2
5a – ab
b3
47
 Step by Step Algebra Basics 
Write each verbal phrase as an algebraic expression
19 A number increased by 12.
20 Nine times a number
21 A number squared
22 The sum of two numbers, divided by 5
Write either “True” or “False” for each.
23 3M and 8M are like terms.
2
2
24 4x and 12x are like terms.
25 9xy and 9yz are like terms.
Combine Like Terms (if possible).
26 5y + 6y
28 17x – x
7
7
30 13b + 10b
32 3ab + 10ab – ab
34 ½d + ½d
27
29
31
33
35
3m + 2m
16r2 – 2r2
3ab + 10ab
b–b+b–b
9a + 10ab + 13abc – abcd
Simplify by combining only the like terms.
36 4x + 2x + 2A
37 3A – 2A + 14
2
2
38 12B + 2X + 4B
39 11p + 2x – 2p
Use the Distributive Property to remove the parenthesis.
40 100(x + y)
41 25(W – Z)
42 3(4a + 5)
43 7(2x – b)
44 ½(10M + 12N)
45 7(a – b + 2c + 2)
Simplify each.
46 2(x + 3) + 5(x + 7)
47 5(2a + 1) + 15
48 7(3ab + x) + 2(2ab + 3x)
49 r(3 + t) + 4rt
50 2(3x + 5y) + 5x(2x + 1) + 4(7y – 4) + 9M
Copyright: 2009 by Barry Hauptman
48
 Step by Step Algebra Basics 
ANSWERS
1
3
5
7
9
11
13
15
17
19
20
21
23
24
25
26
28
30
32
34
36
38
40
42
44
46
48
50
Base = 11, exponent = 5
2
Fifteen cubed
4
2x2x2x2x2
6
5
8
20
10
12
12
36
14
24
16
2
18
x + 12
22
9a
X2
3M and 8M are like terms. True
4x2 and 12x2 are like terms. True
9xy and 9yz are like terms. False
5y + 6y = 11y
27
17x – x = 16x
29
7
7
7
13b + 10b = 23b
31
3ab + 10ab – ab = 12ab
33
½d + ½d = 1d or d
35
4x + 2x + 2A = 6x + 2A
37
12B + 2X + 4B = 16B + 2X
39
100(x + y) = 100x + 100y
41
3(4a + 5) = 12a + 15
43
½(10M + 12N)
45
= 5M + 6N
2(x + 3) + 5(x + 7)
47
= 7x +41
7(3ab + x) + 2(2ab + 3x)
49
= 25ab + 13x
2(3x + 5y) + 5x(2x + 1) + 4(7y – 4) + 9M
= 16x + 38y + 10x2 – 16 + 9M
Copyright: 2009 by Barry Hauptman
Nine squared
36
14
31
30
25
100,000
76
3
x+y
5
3m + 2m = 5m
16r2 – 2r2 = 14r2
3ab + 10ab= 13ab
b–b+b–b=0
9a + 10ab + 13abc – abcd
3A – 2A + 14 = A + 14
11p2 + 2x – 2p2 = 9p2 + 2x
25(W – Z) = 25W – 25Z
7(2x – b) = 14x – 7b
7(a – b + 2c + 2)
= 7a – 7b + 14c + 14
5(2a + 1) + 15
= 10a + 20
r(3 + t) + 4rt
= 3r + 5rt
=
49
 Step by Step Algebra Basics 
5 –Evaluating Formulas
5.1 Basic Formula Evaluations
Read & Study box 
N
EXAMPLES
a. Find the area (A) of a parallelogram (in ft2).
Formula: A = bh
Parallelogram
Substituting for b and h
A = (7 ft)(5 ft)
Answer: 35 ft2
b. Find the area of the triangle (in inches2).
Triangle
Formula: A = ½ bh
A = ½ (10”)(6”)
A = (5”)(6”)
Answer: 30 in2
c. Find the area of a trapezoid (in m2)
Trapezoid
Formula: A = ½ h(b1 +b2)
A = ½ (4)(12 + 8)
A = ½(4)(20)
A = (2)(20)
Answer: 40 m2
d. Find the volume of the cube (in m3).
Cube
Formula: V = s3
V = (2)3
V = (2)(2)(2)
Answer: 8 m3
Copyright: 2009 by Barry Hauptman
50
 Step by Step Algebra Basics 
E xercise box:
 
1. Find the area of the rectangle (in mm2).
A = LW
2. Find the area of the triangle (in ft2).
A = ½ bh
3. What is the area of the trapezoid (in cm2)?
A = ½ h(b1 + b2)
4. What is the area of the trapezoid (in mm2)
A = LW
5. Find the area of the square (in m2).
A = s2
6. Find the perimeter of the rectangle (in yd).
P = 2L + 2W
7. Find the perimeter of the rectangle (in cm).
P = 2L + 2W
8. Find the perimeter of the square.
P = 4s
Copyright: 2009 by Barry Hauptman
51
 Step by Step Algebra Basics 
9. Find the area of the parallelogram in ft2.
A = bh
10. Find the volume of the rectangular prism in cm3.
V = LWH
11. Find the volume of the cube in inches3
V = s3
12. Find the surface area of the cube in m2.
A = 6s2
13. Find the surface area of the rectangular prism in yd2.
A = 2LW + 2LH + 2WH
14. Find the surface area of the rectangular prism km2.
A = 2LW + 2LH + 2WH
Copyright: 2009 by Barry Hauptman
52
 Step by Step Algebra Basics 
Use Formula
Notebook E xercises:
1. Find the area, in square meters, of a square whose side is 9 m.
2. Find the perimeter, in cm, of a rectangle with length of 5 cm and
width is 3 cm.
3. Find the perimeter of a rectangle with length 9 yds and width 8 yds.
4. Find the perimeter, in cm, of a square whose side is 12 cm?
5. Find the area, in square miles, of a triangle whose height is 10 miles and
base is 4 miles.
6. Find the area, in square feet, a parallelogram whose height is 13 feet
and base is 2 feet.
7. The side of cube measures 4 km, find its volume in cubic km.
8. The dimensions of a rectangular solid are 3 cm, 4 cm and 5 cm.
Find its volume, in cubic cm.
9. The height of a trapezoid is 10 ft and its bases measure 11 ft. and 16 ft.
Find the area of the trapezoid in square feet.
10. Find the area, in square meters, of a rectangle whose length measures
14 meters and width measures 3 meters.
A = s2
P = 2L + 2W
P = 2L + 2W
P = 4s
A = ½bh
A = bh
V = s3
V = lwh
A=½h(b1+b2)
A = lw
Read & Study box 
5.2 The Circle and π
(Leave all answers in terms of
1. Find the circumference of the circle.
Solution: Substituting for r
C = 2π(4)
C = π(8)
π)
4 in.
C = 2π r
Answer: 8π
2. Find the area of the circle.
Solution:
A = π(92)
A = π(81)
A = π r2
9m
Answer: 81π
3. Find the circumference of the circle.
Solution:
A = π(17)
C = πd
17 mm
Answer: 17π
Copyright: 2009 by Barry Hauptman
53
 Step by Step Algebra Basics 
4. The radius of a sphere is 2 feet. Find the volume.
Solution:
V = 4π(23)
3
V = 4π(8)
3
V = 4πr3
3
r = 2 ft.
Answer: 32π
3
5. Find the surface area of right circular cylinder.
Solution:
A = 2π(5)(10) + 2π(52)
A = 2π(50) + 2π(25)
A = 100π + 50π
A = 2πrh + 2πr2
Answer: 150π
Notebook E xercises:
(Leave all answers in terms of π.)
Use Formula
1. Find the circumference of a circle whose radius is 8 feet.
C = 2πr
2. Find the area of a circle whose radius is 9 cm.
A = πr2
C = πd
3. Find the circumference of a circle whose diameter is 7 km.
4. The radius of a sphere is 3 ft. Find the volume.
5. Find the surface area of right circular cylinder whose height is 3 mm
and radius is 2 mm.
6. Find the circumference of a circle whose radius is 9 miles.
A = 2πrh + 2πr2
7. Find the area of a circle whose radius is 10 cm.
C = 2π r
A = π r2
8. Find the circumference of a circle whose diameter is 29 km.
C = πd
9. The radius of a sphere is 10 ft. Find the volume.
10. Find the surface area of right circular cylinder whose height is 5
mm and radius is 3 mm.
Copyright: 2009 by Barry Hauptman
A = 2πrh + 2πr2
54
 Step by Step Algebra Basics 
E xercise box:
 
Write the formula:
1. The area of a rectangle is equal to its length times its width.
2. The area of a square is equal to its side squared.
Answer
A = lw
A=
3. The volume of a cube is equal to its side cubed.
4. The area of a triangle is equal to ½ its base times its height.
5. The area of a parallelogram is equal to its base times its height.
6. The area of a circle is equal to its π times its radius squared.
7. The circumference of a circle is equal to two times π times the radius.
8. The circumference of a circle is equal to π multiplied by the diameter.
9. The area of a trapezoid is equal to ½ its height times the sum of its bases.
10. The perimeter of a square is equal to four times its side.
11. The perimeter of a rectangle is equal to twice iength plus twice width.
12. The volume of a rectangular prism is equal to its length multiplied by its
width multiplied by its height.
13. The volume of a right circular cylinder is equal to two times π times the
radius times height plus 2 times π times the radius squared.
14. The surface area of a cube is equal to 6 times its side squared.
15. The surface area of a rectangular prism is equal to 2 times the length
times the width, plus 2 times length times the height, plus two times the
width times the height.
The following formulas have not been shown previously.
16. The volume of a right cirular cylinder is equal to π times its radius
squared times its height.
17. The volume of a right triangular prism is ½ its width times its height
times if length.
18 The surface area of a right triangular prism is equal to width (w) times
height (h) plus length (l) times width (w) plus length (l) times height (h)
plus length (l) times side (s).
Answers to 26, 27 and 28.
V = πr2h, V = ½whl, A = wh + lw + lh + ls
Copyright: 2009 by Barry Hauptman
55
 Step by Step Algebra Basics 
E xercise box:
 
Study Example:
Find the area of a triangle whose base is 12 m
and height is 8 m.
Step
Step
Step
Step
1:
2:
3:
4:
SOLUTION
Step 1: (Using a ruler sketch the triangle)
Draw a sketch.
Write the formula (look back for formula)
Substitute in the formula
Solve
Step 2:
A = ½bh
Step 3:
½(12m)(8m)
Step 4:
(6m)(8m)
Answer: 48 m 2
1
Find the area of a triangle whose base is
equal to 16 mm and height is 9 mm.
2
Find the area of a parallelogram whose
base is 10 in and height is 6 in.
3
Find the area of a rectangle whose base is
10 yards and height is 3 yards.
Copyright: 2009 by Barry Hauptman
56
 Step by Step Algebra Basics 
4
Find the area of a trapezoid if:
height = 4 mm
base 1 = 11 mm
base 2 = 5 mm
5
Find the perimeter of a rectangle whose
length is 15 meters and width is 8 meters.
6
Find the perimeter of a square whose side
is equal to 100 feet.
7
Find the circumference of a circle whose
radius is equal to 8 cm. (in terms of π)
8
Find the circumference of a circle whose
diameter is equal to 8 cm. (in terms of π)
Copyright: 2009 by Barry Hauptman
57
 Step by Step Algebra Basics 
9
Find the area of a circle whose radius is
equal to 7 feet. (in terms of π)
10
Find the volume of a cube whose side is
equal to 3 m.
11
Find the surface area of a cube whose side
is equal to 4 mm.
12
Find the surface area of a rectangular prism
whose length is 4 inches, width is 10 inches
and height is 3 inches.
Copyright: 2009 by Barry Hauptman
58
 Step by Step Algebra Basics 
5.3 Algebraic Representation of Perimeters
Read & Study box 
Study Example: Represent the perimeter of the triangle
algebraically whose sides are 2x, x and 5x.
Solution
P = 2x + x + 5x
P = side 1 + side 2 + side 3
Answer: 8x
E xercises:
1. Represent the perimeter of a triangle, algebraically, whose sides
are 4Q, 9Q and Q.
2. Represent the perimeter of a triangle, algebraically, whose sides
are 11M, 9M and 3M.
3. Represent the perimeter of a triangle, algebraically, whose sides
are 12R, 6R and 2R.
4. Represent the perimeter of the triangle, algebraically, whose
sides are Z, Z and Z.
Study Example: Represent the perimeter of a square,
algebraically, whose side is 7b.
7b
P = 4s
Solution
P = 4(7b)
Answer: 28b
E xercises:
5. Represent the perimeter of a square whose side is 3b.
6. Represent the perimeter of a square whose side is 2R.
7. Represent the perimeter of a square whose side is 3.5Z
8. Represent the perimeter of a square whose side is Q.
Study Example: Represent the perimeter of a rectangle
whose length is 4a and width is a.
P = 2(4a) +2(a)
= 8a + 2a
Answer: 10a
E xercises:
Represent algebraically.
9. The perimeter of a rectangle whose length is 5M and width is M.
10. The perimeter of a rectangle whose length is 4D and width is D.
11. The perimeter of a rectangle whose length is 1.2Y, width is 6Y.
Copyright: 2009 by Barry Hauptman
59
 Step by Step Algebra Basics 
Study Example: Represent the circumference of a
circle whose radius is 6x.
6x
E xercises:
C = 2πr
C = 2π(6x)
Answer: 12πx
Represent algebraically.
12. The circumference of a circle whose radius is 11d.
13. The circumference of a circle whose radius is 7J.
14. The circumference of a circle whose radius is K
15. The circumference of a circle whose radius is 12½G
16. The circumference of a circle whose diameter is 10Y. (Divide the
diameter by 2 to find the radius.)
17. The circumference of a circle whose diameter is 8L.
18. The circumference of a circle whose diameter is 5C.
5.4 Algebraic Representation of Areas
Read & Study box 
Study Example: Represent the perimeter of the
triangle algebraically whose sides are 2x, x and 5x.
A = ½bh
9x
8x
Solution
A = ½(8x)(9x)
= (4x)(9x)
= 36(x)(x)
Answer: 36x 2
E xercises: Represent algebraically.
1. The area of a triangle whose height is 10y and base is 3y.
2. The area of a triangle whose height is 4z and base is 10z.
3. The area of a triangle whose height is 12M and base is 20M.
4. The area of a triangle whose height is 6Z and base is Z.
Copyright: 2009 by Barry Hauptman
60
 Step by Step Algebra Basics 
Study Example:
Represent the area of a square whose side is equal to 5K.
A = s2
5K
A = (5K)2
= (5K)(5K)
= 25(K)(K)
Answer: 25K 2
E xercises:
5.
6.
7.
8.
Represent algebraically.
The area of a square whose side is 3b.
The area of a square whose side is 7R.
The area of a square whose side is 3.5Z
The area of a square whose side is Q.
Study Example: Represent algebraically the area of
a rectangle, whose length 5 is and width is 9G.
A = (5)(9G)
A = lw
9G
Answer: 45G
5
E xercises:
Represent algebraically.
9. The area of a rectangle whose base is 3M and height is 7.
10. The area of a rectangle whose base is 10x and height is 3.
11. The area of a rectangle whose base is 4.5y and height is 2.
Study Example:
Represent the area of a circle whose radius is 4b.
4b
2
A = πr
A = π(4b)2
= π(4b)(4b)
= π(4)(4)(b)(b)
= π(16b2)
Answer: 16πb 2
E xercises:
Represent algebraically.
12. The area of a circle whose radius is 2L.
13. The area of a circle whose radius is 5h.
14. The area of a circle whose radius is f.
15. The area of a circle whose radius is 3.4G
16. The area of a circle whose diameter is 6Y. (Divide the
diameter by 2 to find the radius.)
17. The area of a circle whose diameter is 10b.
18. The area of a circle whose diameter is 3C.
Copyright: 2009 by Barry Hauptman
61
 Step by Step Algebra Basics 
6 –The Integers
Read & Study box 
6.1 The Counting & the Whole Numbers
The most common number system is the “counting numbers” or “natural numbers” which
are as follows:
The Counting numbers are 1, 2, 3, 4, 5, 6, 7, …
Add a zero (0) to this system and you get the “whole numbers”.
The Whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, …
Writing box
1. What is the difference between the Counting Numbers and the Whole Numbers?
2. What number is a Whole Number that is not a Counting Number? _______
3. What do the three dots (…) at the end of the number systems above mean?
4. What is the smallest Counting Number? _______
5. What is the smallest Whole Number? _______
6. Why is true that there is no largest Counting Number?
7. Is it true that there is no largest Whole Number? ______
Copyright: 2009 by Barry Hauptman
62
 Step by Step Algebra Basics 
Read & Study box 
6.2 Understanding The Integers
Find or draw the opposite of each:
Write the
opposite
up
off

true


minus
+
–
7
Down
As you have learned, the opposite of “+” is “–” and, therefore, the opposite of the
negative of a number is the positive of the number. Thus, the opposite of –7 is +7.
The following set of numbers is called The
Integers.
… , –6, –5, –4, –3, –2, –1, 0, + 1, + 2, + 3, + 4, + 5, + 6, …
Looked at another way, the Integers can be divided into three parts as follows:
Positive Whole Numbers
Negative Whole Numbers
+
1, +2, +3, +4, +5, +6, …
–
1, –2, –3, –4, –5, –6, …
Zero
0
Writing box
1. Explain the type of numbers that have to be added to the Whole Numbers to form the
Integers?
2. Fill in the missing words in the following sentence:
The Integers consist of the __________ whole numbers, the __________ whole
numbers and _______.
Copyright: 2009 by Barry Hauptman
63
 Step by Step Algebra Basics 
3. Why are these two ways of showing the Integers both correct?
…, – 5, – 4, – 3, – 2, – 1, 0, + 1, + 2, + 3, + 4, + 5, …
…, – 3, – 2, – 1, 0, + 1, + 2, + 3, …
4. The following is an incorrect way of illustrating the Integers. Why is it incorrect?
–
6, – 5, – 4, – 3, – 2, – 1, 0, + 1, + 2, + 3, + 4, + 5, + 6
5. Why is it important to add the
Copyright: 2009 by Barry Hauptman
… when representing the Integers?
64
 Step by Step Algebra Basics 
Read & Study box 
6.3 The “Poof” Effect (aka Adding Integers)
THE “POOF” EFFECT:
Something very interesting happens when a
They meet
+
meets
+
–
meets a
They “poof”
Theyʼre gone
–
Every time a “+” and “–” meet ……… “POOF” they both disappear.
Kind of mortal enemies, you might say.
How many
“poofs” when:
–
3
–
–
+
meets
–
+
+
5?
+
+
+
See all the “POOFS” and
result on the next page
Copyright: 2009 by Barry Hauptman
65
 Step by Step Algebra Basics 
Everybody
ready?
–
3
+
meets
+
— — —
+
5
+
+
+
This can be written as an Integer Addition:
—
Copyright: 2009 by Barry Hauptman
+
+
3+ 5= 2
66
 Step by Step Algebra Basics 
Look at this “pre-poofed” example:
+
6 + —7 = —1
Why –1?
Explain:
E xercises: Adding Integers
Instructions: Explain each result in the space provided.
1
2
3
—
2 + +9
—
= +7
13 + +1
—
= —12
103 + +103
Copyright: 2009 by Barry Hauptman
= 0
67
 Step by Step Algebra Basics 
4
—
5
+
6
+
7
—
8
—
4 + —4 =
10 + +10
—
8
= +20
1 + +1 +— 3 + — 4
3 + +11
30 + +30
Copyright: 2009 by Barry Hauptman
= —5
=?
=?
68
 Step by Step Algebra Basics 
E xercises:
Instructions: Add the integers.
—
1
2 + +5
2
+
1 + —3
3
—
1,039 + +1,039
4
—
8 + —3
5
6
7
+
1 + +1
+
—
5 + + 2 +— 3 + — 1
30 + +11
Copyright: 2009 by Barry Hauptman
69
 Step by Step Algebra Basics 
“Why’s my sign?”
Instructions: The numbers have been shaded out and only the signs are shown.
Explain why the result of each Integer addition will have the sign as indicated.
1
—
+—
=
—
2
+
++
=
+
+
+—
=
+ or —
4
+
++
=
5
—
++
=
6
—
+—
=
3
Copyright: 2009 by Barry Hauptman
Explain why the result is negative.
Explain why the result is positive.
?
?
?
Explain why the sign can be + or —.
What’s my sign? Why? Explain.
What’s my sign? Why? Explain.
What’s my sign? Why? Explain.
70
 Step by Step Algebra Basics 
“What’s my sign?”
Instructions: Determine the sign of each Integer Addition and rewrite in the
appropriate column.
Sign of the Addition
+
1
+
8 + —3
2
+
8 + +3
3
—
8 + —3
4
—
8 + +3
5
—
8 + +8
6
—
1 + +7
7
+
8
—
1 + —7
9
+
1 + +7
10
—
6 + +6
11
—
2 + +9
12
+
2 + +9
13
—
2 + —9
14
+
2 + —9
16
+
9 + —9
—
No sign
0
+
—
0
1 + —7
17
—
1 + +5 + +7
18
—
8 + +2 + +6
Copyright: 2009 by Barry Hauptman
71
 Step by Step Algebra Basics 
“What’s my SUM?”
Instructions: Determine the sign of each Integer Addition and write the SUM in
the appropriate column.
Sign
+
1
+
8 + —3
2
+
8 + +3
3
—
8 + —3
4
—
8 + +3
5
—
8 + +8
6
—
1 + +7
7
+
8
—
1 + —7
9
+
1 + +7
10
—
6 + +6
11
—
2 + +9
12
+
2 + +9
13
—
2 + —9
14
+
15
—
16
+
+
No sign
—
0
5
—
11
0
1 + —7
2 + —9
2 + +2
9 + —9
17
—
1 + +5 + +7
18
—
8 + +2 + +6
Copyright: 2009 by Barry Hauptman
72
 Step by Step Algebra Basics 
E xercises:
Instructions: Find the sum for each Integer Addition
1 +
2 +8 + +3 =
8 + —3 = + 5
4
7
10
13
16
17
18
19
—
8 + +3 =
+
1 + —7 =
—
6 + +6 =
—
2 + —9 =
+
9 + —9 =
—
1 + +5 + +7
—
8 + +2 + +6
—
7 + +2 + +3
5
8
11
14
—
8 + +8 =
—
1 + —7 =
—
2 + +9 =
+
2 + —9 =
3 —8
+ —3 =
6 —1
9
12
15
+ +7 =
+
1 + +7 =
+
2 + +9 =
—
2 + +2 =
=
=
=
Study example: Regrouping by Like Signs
Instructions: Find the sum by adding up the like signs first.
+
4 + +1 +—2 + —10 ++3 + +7 + —1+ +8 +—5 = ?
Okay, if you are a
positive, regroup
on the left!
+
4 + +1 ++3 + +7 + +8
And, if you are a
negative, regroup
on the right!
+
—
The Positive Like Signs added up is:
+
The Negative Like Signs added up is:
23
Wow! All of that
adds up to +5?
Copyright: 2009 by Barry Hauptman
2 + —10 +—1 + —5
+
+
5
—
18
Yep. Just regroup,
add the like signs
and then add the
results.
73
 Step by Step Algebra Basics 
E xercises:
Regrouping by Like Signs
Instructions: Find the sum by regrouping and adding the like signs first.
1
+
1 + +3 +—12 ++4 + —1 = ?
Positives
2
—
Negatives
7 + —11 ++5 + +3 +—11 + —9 ++8 = ?
Positives
Negatives
Read & Study box 
6.4 Integer Addition “Strings”
Consider this expression:
+
1 + —2 + 3 + —76
In Algebra, this expression is a called an “addition string” of integers (signed
numbers). It can be written without the RAISED positive and negative signs like
this.
Do we do it
the same way
as before?
61 – 2 + 3 – 76
Yes. Combine
the like signs
and then
“poof” away.
Solution:
1–2+3–7 =
+
4 + -9
Copyright: 2009 by Barry Hauptman
4
–
9=
–5
74
 Step by Step Algebra Basics 
E xercise box:
 
Instructions: Perform the indicated addition.
1
2
3–6
8–5
3
–1 – 3 + 2
+
–2 – 7
3 + -6
-
2 + -7
4
+
8 + -5
-
1 + -3 + + 2
5
3 – 10
6
10 – 3
7
3 – 10 + 9
8
–8 – 5 + 13
9
7–7
10
–12 + 12
11
–4–4
12
–1 + 2 –1 + 2 –1 + 2
Copyright: 2009 by Barry Hauptman
75
 Step by Step Algebra Basics 
Read & Study box 
6.5 Combining Like Terms Using Integer Addition Strings
Study Example: Simplify by combining like terms.
3a – 9a =
+
3a + -9a
– 6a
E xercise box:
 
Instructions: Simplify each expression by combining like terms.
1
2a – 5a
2
+
-
2a + 5a
–8B – 3B
3
–2q2 + 7q2
4
–1e – 3e + 2e
5
3hf – 10hf
6
10g5 – 3g5
7
3r – 10r + 9r
8
–8g – 5g + 13g
Copyright: 2009 by Barry Hauptman
-
8B + -3B
76
 Step by Step Algebra Basics 
Read & Study box 
Now consider this expression:
How did this
happen?
−8k + 9k + 53M =6=
Answer:
k + 53M6
Explain the result below
The answer is k + 53M because
E xercise box:
 
Instructions: Simplify each expression by combining ONLY like terms.
1
3a – 7a + 13b
2
9B + 34M –2B
3
–3q2 –11z + 7q2
4
–1k + 3k + 2j
5
13hf – 10gh + 2gh
6
10d3 – 3g + 3d3
7
3x + 2y + 5y + 7x
8
-5x + 7x -14y – 8y + 3M
Copyright: 2009 by Barry Hauptman
77
 Step by Step Algebra Basics 
Read & Study box 
6.6 Integer Multiplication
When multiplying two integers the following four combinations of signs are possible:
(+)(+)
(+)(−)
(−)(+)
(−)(−)

the appropriate combination of signs
(+)(+)
(−)(−)
(+)(−)
(−)(+)
4(3)
(+2)(−1)



(5)(−3)
(−7)(−10)

(−6)( +9)

Complete This: (Place a  in the appropriate box.)
(+)(+)
1
5(−2)
2
(−3)(−1)
3
(−10)(4)
4
(+7)( +1)
5
(+8)( −8)
6
(6)( −9)
7
(−1)(−4)
8
(−2)( 13)
9
(9)( 4)
Copyright: 2009 by Barry Hauptman
(−)(−)
(+)(−)
(−)(+)
78
 Step by Step Algebra Basics 
“What’s my sign?”
Instructions: Complete this table.
1
2
3
Example Rewrite as an addition
(1) + (1) + (1) + (1) + (1)
5(1)
Answer Sign
4(3)
(+2)(+3) (+3) + (+3)
4
(5)(−2) (−2) + (−2) + (−2) + (−2) + (−2)
5
(4)(−1) (−1) + (−1) + (−1) + (−1)
6
(−1)(4)
7
(+2)(+5)
(3)(−2)
(−2)(3)
(4)(−2)
(3)(1)
(−2)(4)
(−1)(6)
(2)(11)
8
9
10
11
12
14
15
(4)(−1) =
(−1) + (−1) + (−1) + (−1)
Sam e as
Sam e as
5
+
+6
−10
−4
−4
+
−
−
(3)(−2) = (−2) + (−2) + (−2)
Sam e as
Sam e as
Okay, here’s your task? If you know
the signs being multiplied, can you tell
the sign of the answer?
Do you mean, what is the sign of the
result if you multiply, say, + by –?
Exactly. So does, (+)(–)
equal + or –?
Sure.
Can I look at the
table above?
Let’s see. It looks like
(+)(–) equals (–).
Nice going.
Copyright: 2009 by Barry Hauptman
79
 Step by Step Algebra Basics 
6.7 Integer Multiplication, “The Rules”
Looking at the previous table we see:
(+)(+)
=
+
(−)(+)=
−
(+)(−) =
−
Wait a second,
something’s missing!
What’s missing?
(—)(—) is not there?
WHAT’S GOING ON?
You’re right! Here’s why. It’s too difficult
too explain now; just remember that
negative x negative is a positive.
Are you sure? That
doesn’t sound right!
Positive.
Get it? I’m positive,
(—)(—) = a positive.
Here are all the rules.
All of The Integer Multiplication Rules
(+)(+) = +
(−)(+)= −
(+)(−) = −
(−)(−) = +
Complete the following fill-ins:
A positive times a positive is a ______________
A negative times a positive is a _____________
A positive times a negative is a _____________
A negative times a negative is a _____________
Copyright: 2009 by Barry Hauptman
80
 Step by Step Algebra Basics 
Using the Integer Multiplication Rules
Instructions: Write the reason for each answer in the space provided.
1
(−2)(−3) = +6 Why?
2
5(−4) = −20 Why?
3
(−6)(+½) = −3 Why?
4
(+8) (+1) = +8
Why?
Instructions: Find the product for each Integer Multiplication
5
5(−2)
6
7(−10)
7
(−3)(−1)
8
(+3)(−1)
9
(−10)(4)
10
(−1)(14)
11
(+7)(+1)
12
(0)(+1)
13
(+8)(−8)
14
(−8)( −8)
15
(6)(−9)
16
15(3)
17
(−1)(−4)
18
(−½)(+12)
19
(−2)(13)
20
(−1)(−1)
21
(−6)(+6)
22
(−22)(−3)
23
(9)(4)
24
(−100)(−¼)
This is a work in progress……….
Copyright: 2009 by Barry Hauptman
81