Step by Step MAC Color
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Transcript of Step by Step MAC Color
!
Basics
This is a work in progress……….
Student Name:
Teacher:
New and Improved.
Now includes
Integers
by
1 + 1 = 2
Step
Step
!
TABLE OF CONTENTS
Chap. Topic Page
1 Powers, Factors & PEMDAS
1.1 Powers and Exponents 1
1.2 Prime Factorizations (Factor Trees) 6
1.3 Order of Operations (P-E-M-D-A-S) 9
2 Algebraic Expressions
2.1 Understanding Algebraic Expressions 12
2.2 Evaluating Algebraic Expressions 13
Self Test 15
2.3 Translating Verbal Phrases ! Algebraic Expression 18
2.4 The Division Bar 26
2.5 The Division Bar as a Grouping Symbol 21
2.6 Simplifying Algebraic Expressions With Powers 26
Self Quiz 28
3 Like Terms"
3.1 “Like” Things 29
3.2 Understanding Like Terms 30
3.3 Combining Like Terms 32
3.4 The Commutative & Associative Addition Properties 35
3.5 Using the Commutative & Associative Addition Properties 36
4 Simplifying Algebraic Expressions
4.1 Combining Like Terms Amongst Unlike Terms 37
4.2 The Distributive Property (aka Removing the Parentheses) 40
4.3 Simplifying When There Is More Than One Set of Parentheses 43
Self Test 45
5 Evaluating Formulas
5.1 Basic Formula Evaluation 48
5.2 The Circle and ! 51
5.3 Algebraic Representation of Perimeters 57
5.4 Algebraic Representation of Areas 59
6 The Integers
6.1 The Counting Numbers & the Whole Numbers 59
6.2 Understanding The Integers 60
6.3 The “Poof” Effect (aka Adding Integers) 62
6.4 Integer Addition “Strings” 71
6.5 Combining Like Terms Using Integer Addition Strings 73
6.6 Integer Multiplication 75
6.7 Integer Multiplication – “The Rules” 77
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
1
1 – Powers, Factors & Order of Operations
Read & Study box !"#$%&'(()*+,
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
Mathematicians simplify the writing of 5 multiplied by itself 13 times as:
513
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”
715 715 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.”
# exponent base !
Repeated
multiplication
Powers! Great. I like
writing it this way!
Tell me more.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
2
Exercise box: !" #$
Instructions: Write the power indicated. 1 Nine to the 4th power 94 2 Six to the 14th power
3 two to the 11th power 4 Eleven to the 10th power
5 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 53
21 95 22 210 23 21 24 980 25 a3 26 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 36 28 10 x 10 x 10 29 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 30 1 x 1 x 1 x 1 x 1
Instructions: Write the power as a multiplication and then multiply.
31 33 32 52 33 103
34 24
3 x 3 x 3
9 x 3
27
35 18 36 34 37 09
38 (!)5
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!Copyright: 2009 by Barry Hauptman
3
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
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!Copyright: 2009 by Barry Hauptman
4
Exercise box: !" #$
Instructions: Rewrite each expression using exponents
1 2 x 2 x 2 x 2 x 2 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 25 x 78
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 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 5 x 5 x 5 5 19 x 19 x 23 x 23 x 23 x 87 6 5 x 2 x 3 x 5 x 3 x 3 x 2 x 2 x 5 x 5 x 2 x 2 7 13 x 13 x 11 x 2 x 13 x 13 x 2 x 2 x 2
Instructions: Write each expression as a multiplication without the exponents.
8 32 x 45 3 x 3 x 4 x 4 x 4 x 4 x 4
9 102 x 226 10 93 x 145 11 22 x 73 x 115 12 15 x 34 x 132
Instructions: Evaluate each after rewriting without the exponents
13 52 x 22 5 x 5 x 2 x 2
25 x 4
?
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!Copyright: 2009 by Barry Hauptman
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14 22 x 32 2 x 2 x 3 x 3
15 12 x 72
16 102 x 32
17 23 x 32 2 x 2 x 2 x 3 x 3
4 x 2 x 9 ? x ? ?
18 22 x 33
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!Copyright: 2009 by Barry Hauptman
6
Read & Study box !"#$%&'(()*+, 1.2 Prime Factorizations (Factor Trees)
15 = 5 x 3 shows 15 in factored form.
NOTE: The number 1 is always a factor of any number 15 = 15 x 1.
Exercises: Complete the table. DO NOT USE 1 AS A FACTOR.
Number Factored Form Factors
1. 10 5 x 2 5 and 2
2. 6
3. 9
4. 21
5. 50
6. 11
12 =
x
x
In factored form 12 = 2 x 2 x 3. Using exponents it can be written.
12 = 22 x 3
Exercises: Complete the table Number Factored Form (three factors) Using exponents
7. 18 3 x 3 x 5 32 x 5
Find the factors of 15. No ones, please!
Find three
factors of 12. No
one’s please!
No ones,
please!
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!Copyright: 2009 by Barry Hauptman
7
Read & Study box !"#$%&'(()*+, 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. Parentheses or grouping symbols
2. Exponents (powers)
3*. Multiplications/Divisions, from left to right
4*. Additions/Subtractions, from left to right
Instructions: Find the value of each.
Example box: Exercise box: A 3 + 7 x 9 Original expression 1 5 + 2 x 8
3 + 7 x 9
Multiply 1
st.
3 + 63
Then Add
66 Answer
B 18 – 6 + Original expression
2 12 + 7 "
18 – 6 +
Divide 1st
18 – 6 + 4
Then from left Subtract.
12 + 4
Then Add
16 Answer
C 44 - 2 ! (15 - 3) Original expression 3 20 - 2 " (11 " 8)
44 - 2 ! (15 - 3)
Operations in ( ) 1st.
44 - 2 ! 12
Then Multiply
44 " 24
Then Subtract
20 Answer
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!Copyright: 2009 by Barry Hauptman
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D (22 + 3) ÷ (9 – 4) Original expression
4 (14 + 1) ÷ (8 " 5)
(22 + 3) ÷ (9 " 4)
Operations in ( )
(start at left)
25 ÷ (9 " 4)
25 ÷ 5
Operations in ( )
Then Divide
5
Answer
E 7 + 32 Original expression
5 6 + 52
7 + 32
Power (exponent)
7 + 9
Then Add
16
Answer
F 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?
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!Copyright: 2009 by Barry Hauptman
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Writing box 1. Explain why the following computation is incorrect .
2 x 52 + 23
102 + 23
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 !"#$%&'%
16 + 10 !"#$%('%
%
26 !"#$%)'%
%
There might
be more
than one
error here!
2 x 2 x 2
equals 6???
C’mon.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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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 !"#$%&'%
%
2 ! 5 + 9 !"#$%('%
%
10 + 9 !"#$%)'%
%
19 !"#$%*'%
%
Notebook Exercises:
Instructions: Find the value of each.
1 7 + 9 x 3 2 3 x 2 + 11 3 4 x 5 – 11
4 10 ÷ 5 – 1 5 28 – 5 x 2 6 4 x 2 + 5 x 2 7 24 – 2 ! (15 - 5) 8 (11 + 8) " 20 + 12 9 11 + 7 " 8 / 2 10 5 + 32 11 52 – 25 12 32 + 22 13 23 +(20 ÷ 2) 14 33 +(10 " 5) 15 2 ! 3 + 5 ! 1 + 6 ! 4 16 7 ! 3 + (5 – 3)2
17 22 + 32 + 42 " 52 18 12
" 13 + 14 " 15
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!Copyright: 2009 by Barry Hauptman
11
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, #
The following are examples of constants: 3, 5.2, # , - 7, $ Note: The Greek letter $ is an exception. $ represents a famous constant.
Multiplication Multiplication with constants and variables can be shown in several ways.
Multiplication shown with: Expression Meaning No operation symbol 5Q 5 times Q Parenthesis without operation symbol 3(a) 3 multiplied a Raised dot G! M G times M Power (exponent) y
3 y times y times y
Exercise box: !" #$ 1. Instructions:
Put a circle around the five (5) variables and a box around the six (6) constants.
m, z, 4, 1.7, N, #, $ , $, 3 #, -L , 0.00009
2. Instructions: Complete the table.
Multiplication shown with: Expression Meaning No operation symbol 13y ? Parenthesis without operation symbol # (L) ? Raised dot 3 ! x ? Power Q
5 ?
No operation symbol ? 8 times Z
Parenthesis without operation symbol ? K multiplied by X
Raised dot ? ! times N
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!Copyright: 2009 by Barry Hauptman
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Read & Study box !"#$%&'(()*+, 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)
Multiplications first. Then add. 12 + 24
Answer ! 36
]
Instructions: Evaluate each algebraic expression
Example box: Exercise box: A 5a Original expression 1 3q
a = 6 Variable value q = 4 5(6) Substitute 30 Answer
B ! x Original expression 2 " x x = 24 Variable value x = 12 "(24) Substitute
12 Answer
C M + 15 Original expression 3 J + 9 M = 9 Variable value J = 6 9 + 15 Substitute
24 Answer
D 43 " z Original expression 4 13 " v z = 11 Variable value v = 13
43 " 11 Substitute
32 Answer
E Original expression
5
b = 6 Variable value R = 10
Substitute
4 Answer
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!Copyright: 2009 by Barry Hauptman
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F c3 Original expression 6 d2 c = 2 Variable value d = 7
23 Substitute
2 x 2 x 2 Expand the power
8 Answer
G 3a + x2 Original expression 7 3a + x3 a = 5, x = 7 Variable value a = 1 x = 2 3(5) + 72
Substitute
3(5) + 49
15 + 49 PEMDAS
64 Answer
H x + y + z Original expression 8 q + r + s
x = 1, y = 9, z = 2 Variable value q = 18, r = 3, s = 7
1 + 9 + 2 Substitute
10 + 2 PEMDAS
12 Answer
I abc Original expression 9 xyz
a = 2, b = 10, c = 3 Variable value x = 4, y = 5, z = 6 2 ! 10 ! 3 Substitute
20 ! 3 PEMDAS
60 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
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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!#+,%-#."%Name: _____________________ Teacher:___________
1 For the power 104 the base is ____ and the exponent is ____
2 411 can be read “___ to the ____ power”
3 72 can be read “7 to the 2nd power” or “7 _____________”
3 53 can be read “5 to the 3rd power” or “5_____________”
4 Write 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 as a power. _______
5 Write 9 to the seventh power in base/exponent form _______
6 Write 56 as a repeated multiplication.________________
7 In the expression 114x2 the variable is ______.
8 In algebra 11M means 11 ______ M.
9 Write 100 times g algebraically. ___________
10 Find the value of 9y, if y = 3.
11 Find the value of !x, if x = 10.
Use the variable values to evaluate each expression
12 c + 9, if c = 15 13 45 – z, if z = 40
14 9a + b, if a = 2 and b = 20 15 3x – 4y, if x = 5 and y = 3
16 A2, if A = 4 17 d3
, if d = 2
18 A2 + v3, A = 10 and v = 2 19 5b2
, if b = 3
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!Copyright: 2009 by Barry Hauptman
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20
Write without exponents:
112 x 133
21
Write without exponents:
32 x 24 x 53
22 Write using exponents: 23 Write using exponents:
3 x 3 x 5 x 5 x 5 7 x 7 x 11 x 2 x 2 x 7 x 7
24 Use a factor tree to find the prime
factorization of 32. 25 Use a factor tree to find the
prime factorization of 100.
26 Use a factor tree to find the prime
factorization of 54. 27 Use a factor tree to find the
prime factorization of 120.
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!Copyright: 2009 by Barry Hauptman
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/0.1#2.%
1 Base = 10, exponent = 4
2 Four to the 11th power.
3 Seven squared
3 Five cubed
4 310
5 97
6 5 x 5 x 5 x 5 x 5 x 5
7 X
8 11 times M
9 100g
10 27
11 5
12 24 13 5
14 38 15 3
16 16 17 8
18 108 19 45
20 11 x 11 x 13 x 13 x 13
21 3 x 3 x 2 x 2 x 2 x 2 x 5 x 5 x 5
22 32 x 53 23 22 x 74 x 11
24 25
25 22 x 52
26 2 x 33 27 23 x 3 x 5
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!Copyright: 2009 by Barry Hauptman
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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:
A number increased by 15 is translated into " n + 15.
Four times a number is translated into " 4a
A number decreased by 7. " x – 7
The square of a number. " A2
The quotient of a number and 100. " R ÷ 100
The product of a number and ! . " !B
Nine less than a number. " Z – 9
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. "
Exercise box: !" #$ Instructions: Next to each word write the appropriate symbol from the following list.
+
"
X
÷
( 2 )
1. Less 2. times 3. more
4. Square 5. divide 6. add
7. Decreased 8. product 9. multiply
10. Increase 11. minus 12. twice
13. Sum 14. difference 15. 2nd
power
16. Subtract 17. double 18. quotient
Nine less than?
Nine minus?
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!Copyright: 2009 by Barry Hauptman
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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. "
Instructions: Translate each verbal phrase into an algebraic expression.
Example box: Exercise box: A The sum of two numbers. 1 The sum of a number and 11.
a + b
B A number decreased by 100 2 Three decreased by a number
N " 100
C The product of three numbers 3 The product of two numbers.
abc
D A number divided by 5.
4 Ten divided by a number.
E Nine more than a number. 5 Twenty more than a number.
k + 9
F Seven less than a number 6 Eleven less than a number.
A - 7
G One divided by
a number squared. 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.
1 n2
x + y
z2
x 5
a + b 8
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!Copyright: 2009 by Barry Hauptman
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Notebook Exercises:
Instructions: Write each verbal phrase into an algebraic expression.
1 Five times a number. 2 The product (x) of a number and 13. 3 A number divided by 11. 4 The difference between a number and 2. 5 The square of a number. 6 The sum of three numbers. 7 Twice a number. 8 A number minus 6. 9 A number increased by 100. 10 Seven more than a number. 11 The quotient (÷) of two numbers. 12 Five times the square of a number. 13 Twice a number divided by 4. 14 Five less than a number. 15 The product of four numbers. 16 The sum of three numbers divided by 11. 17 A number times 6. 18 The square of a number. 19 A number minus 33. 20 A number plus 33. 21 The sum of two numbers, divided by the square of a number. 22 The product of 5 and a number. divided by the difference between a number and 3.
23 The sum of two numbers, divided by the sum of the square of a number and 8.
Exercise box: !" #$
Instructions: Translate each algebraic expression into a verbal phrase.
1 4n Four times a number or product of 4 and a number.
2 x + 5 A number increased by 5 or Five more than a number. 3 7 " b 4 10x 5 X2 6
7 abc
8 a + b + c 9 2x 10 M "7 11
12
13 12 + X2
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!Copyright: 2009 by Barry Hauptman
20
Read & Study box !"#$%&'(()*+,
2.5 The Division Bar as a Grouping Symbol
The division bar (_____
) represents a grouping. An operation above or below a division
bar should be performed first as if it were in parentheses.
Example A: Steps
!
15"6
3 x 5
Original expression
x 5 Perform the grouping operation above the division bar first. 15 – 6 = 9
3 x 5 Perform the division on the left next.
9 ÷ 3 = 3
15 Multiply last to find the answer
Example B:
Steps
x = 16, y = 9, z = 5
Original expression with variable values.
Substitute the variable values.
Perform the grouping (operation above the division bar) first.
Exponent (power) next.
1 Divide last to find the answer.
x + y
z2
16 + 9
52
25
52
25
25
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
21
Exercise 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 = Multiply last.
Problem #2:
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 = Divide last.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
22
Exercise box: !" #$ Evaluate each by substituting the given values and using PEMDAS.
1 m = 12, n = 8, p = 4
2 a = 10, b = 2, c = 3
3 x = 2, y = 10, z = 5
4 r = 15, s = 2, t = 9
5 w = 12, z = 8
6 a = 3, b = 4, c = 5
7 a = 4, b = 9, d = 3
8 t = 2, u = 3, v = 4
w = 10, z = 20
m + n p
a – b2 c
xy + z z
w + z w – z
a + 3(b – d2)
2
a2 + b2 c2
r + t s3
tu + (w – v)2
z + 1
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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Writing box
Examine the following expression:
After the variables are substituted the expression equals 1. We know the
value of the variables to be 2, 8 and 12. Unfortunately, someone mixed up the assignments. We do not know which is 2, which is 8 or which is 12.
Instructions: In words, explain how you would go about solving this mix up to find
the values of M, N and P. Use examples in your explanation.
M – N 2P
Which equals 2? Which equals 8? Which equals 12?
What’s M? What’s N?
What’s P? Who knows?????
= 1
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!Copyright: 2009 by Barry Hauptman
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Writing box
Examine the following expression:
After the variables are substituted, the expression equals 1. We know the value of the variables to be 2, 4 and 6. Unfortunately, someone mixed up the assignments. We do not know which is 2, which is 4 or which is 6.
Instructions: In space provided below, explain how you would go about solving this
mix up to find the values of a, b and c. Use examples in your explanation.
4a + b3
2c2
Which equal 2 Which equals 4? Which equals 6?
What’s a? What’s b?
What’s c? Who knows?????
= 1
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
25
Read & Study box !"#$%&'(()*+, 2.6 Simplifying Algebraic Expressions With Powers
Study these: Do these: Expression Simplification Expression Simplification
a 7!x 7x 1 9!a
b 7!x!x 7x2 2 9!a!a!a
c 7!x!x!x!y!y
7x3y2 3 9!a!a!b!b!b!b
d 5(x)(x)(y)(z)(z)(z) 5x2yz2 4 3(a)(b)(b)(b)(b)(c)
e 11(x)(y)(y)(z)(z) 11xy2z2 5 6(a)(a)(b)(b)(b)(c)
f 22mmmmm 22m5 6 102ccccccccc
g 17PQQRSSSS 17PQ2RS4 7 3xxyyyyz
h wwxxxyyyyyy
w2x3y5 8 ggggggghi
i 92pqrrrstttttt 92pqr3st6 9 44abbbbcdeeeef
j (xy)(xy) x2y2 10 (ab)(ab)(ab)
k (mn)(mn)(pq) m2n2pq 11 (zw)(zw)(zw)(xy)
l (st)(vw)(vw)(st) s2t2 v2w2 12 5(XY)(CD)(XY)
m 2(ab)(ab)(ab)(ab) 2a4b4 13 11(cd)(ef)(cd)(ef)
n
(up)3 = (vt)(vt)(vt)
v3t3
14
(xy)3
=
o
(abd)2 =
(abd)(abd)
a2b2d2
15 (cdefg)2
=
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
26
Instructions: Complete the following simplifications:
Expression Simplification Expression Simplification
1 (xy)3 (xy)(xy)(xy)
=
2 (ab)4 (ab)(ab)(ab)(ab)
= 3 (pqr) 2 (pqr)(pqr)
=
4 (cdef)3 (cdef)(cdef)(cdef)
= 5 (mn)3
6 (xy)5
7 (pqr)2
8 (abcd)3
9 (ab)5(xy)2 (ab)(ab)(ab)(ab)(ab)(xy)(xy) = 10 (de)3(fg)2
11 (mn)4(qr)5
12 5(ab)2(qr)5
13 7(xy)(zw)4
14 (x4y3)2 (x4y3)(x4y3) =
15 (a2b2)3
%
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
27
!#+,%3456%Name: _____________________ Teacher:___________
Instructions: Translate each verbal phrase into an algebraic expression.
1 A number increased by 5. 2 The product of a number and 9. 3 Eight more than a number. 4 The difference between two numbers. 5 A number squared, divided by 3. 6 One less than a number. 7 The product of three numbers. 8 The product of two numbers, divided by a number cubed.
Instructions: Translate each algebraic expression into a verbal phrase.
9 3N 10 A + B 11 2A – 7 12
Instructions: Evaluate each by substituting the given values and using PEMDAS. 13 a = 8, b = 4, c = 2
14 m = 15, n = 3, q = 4
15 w = 10, z = 5
16 a = 8, b = 2, c = 10
Instructions: Simplify each using powers:
17 2xxxyyy 18 4(a)(a)(a)(b)(c)(c)(c)(c) 19 (mn)(mn)(mn)(mn) 20 7(xyz)(xyz)(abc)(abc)(abc) 21 (pqr)(pqr)(xyz) 22 (uv)(wx)(uv)(wx)(uv) 23 (xy)2 24 (abc)3 25 (gh)2(mn)3 26 13(ab)3(def)5
27 (z5w2)2 28 102(x3y)6
a – b c
m + n2 q
2w w – z
a2 – b2 c
a – b c2
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
28
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 Why?
Like Units 9 mm, 12 mm, 1.4 mm, 1,023 mm All are in mm’s
Like Fractions , , , , ,
Like Signed
Numbers -4, -12, -109.4, -7! , -1,012, -99 Like Fruit 1 apple, 7 apples, 10" apples, 58 apples
Consider the following examples of “UnLike” things
Examples of “UnLike” Things
UnLike Fractions , , ,
UnLike Units 8 m, 8 cm, 1.9 inches, 50 yds,
UnLike Signed Numbers -8, +12
UnLike Fruit 3 oranges, 5 pears, 1" 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
+ = 9 in2 - 5in2 = 4 in2 (-8) + (-11) = (-19)
7 apples + 3 prunes = ?
UnLike + = ?
UnLike
4 m2 - 5 in = ?
UnLike (-2) - (+11) = ?
UnLike
For the following, combine if they are “Like” things. If not, write UnLike.
1 7 apples + 2 apples = 2 77 mangos – 15 bananas =
3 4 ft + 15 ft + 3 ft = 4 10 cm3 – 4 mm2 =
5 (-3) + (-6) = 6 + =
7 5 cm + 9 cm – 10 cm = 8 19 in2 – 11 cm2
– 5 in =
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
29
Read & Study box !"#$%&'(()*+,
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. Don#t even think about combining them!
Examples of Like Terms Examples of UnLike Terms
7M and 3M are like terms. 6B and 3Y are NOT like terms.
10ab and 12ab are like terms. 9de and 8ef are NOT like terms.
y5 and 3y5 are like terms. 2x6 and 2x5 are NOT like terms.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
30
Exercise box: !" #$ Instructions: Determine whether each box contains only like terms. Circle “like” or
“unlike” below each box.
1 2 3
4 5 6
9x
11x
5B
2B
A
7A
11C
6M
-ab2
2ab2
a2b
2 2ab
2
x -3x B #B 3A A2
4Z # 11ab2 5ab
2 4a
2b 3ab
2
b
like
unlike
like
unlike
like
unlike
like
unlike
like
unlike
like
unlike
Instructions: Each problem contains 3 like terms. Write a 4th
like term in the empty box.
7 8 9
10 11 12 b
L
3L
-y
10y
k2
2k2
7de de
-mn
3 2mn
3 -g
7h
3k
5g
7h
3k
#L 3y $k2
3.2de 9mn
3 2g
7h
3k
Instructions: Fill in the missing number.
13 5 apples – 3 apples = ____ apples
14 7 cats + 11 cats + 2 cats = ____ cats
15 3 ziggles + 4 ziggles = ____ ziggles
16 + =
Instructions: Complete the statement.
17 8 yards + 3 yards =
18 100 cm2 – 50 cm
2 =
19 9 tons – 2 tons + 11 tons + 4 tons – a ton =
20 + + =
21 20 cats – 7 figs + 3 bats + 4 mm3 – a tomato + = ___________?????!!!
22 Something is odd about problem 21! Explain on the lines below.
_______
11
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
31
Read & Study box !"#$%&'(()*+,
3.3 Combining Like Terms.
In Algebra, adding or subtracting expressions to form a new simpler expression is called
combining. On the previous page, we saw that 3 ziggles + 4 ziggles = 7 ziggles. We
combined the ziggles to get a simpler expression. In algebra this is called combining
like terms and can be shown as:
3z + 4z
7z
Note: The Multiplicative Identity (aka 1).
1 x 5 = ?
1 x 2,333,789 = ?
1 x A =
Because 1 multiplied by any number always equals the identical number ----
1 is the Multiplicative Identity .
Example of
combining like
terms
Does that mean when I see an X,
it’s the same as 1X?
DUH, of course!
X = 1X
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
32
Exercise 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 – 50ab
2 = _______ (Remember the mm
2s)
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 = ? Answer: 6D
10 6M – 3M2 = ? Answer: “can not combine unlike terms”
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
33
11 9x + 3x =
12 10y – 5y =
13 8y2 + 2y
2 =
14 6M + 5D =
15 8z7 + 8z =
16 15A2B + 3A
2B =
17 3x + 7x + 12x + 4x =
18 40 apples + 50 apples =
19 16 cats – 14 bananas =
20 a horse + a horse =
21 h + h =
22 7 pickles – a pickle =
23 7p – p =
24 a + b + c + d =
25 6M – M =
26 7y2 + 3y
2 + y
2 – 11y
2 =
27 A – A =
28 A – A + A – A + A – A =
srewsnA
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
34
Read & Study box !"#$%&'(()*+, 3.4 The Commutative and Associative Properties of Addition rr a. 5 + 3 = ? b. 3 + 5 = ?
c. Are the results the same for 5 + 3 and 3 + 5?
d. Why are the results the same?
5 + 3 = 3 + 5 is an example of
The Commutative Property
a + b = b + a
Now do these:
e. (4 + 6) + 1 = ? f. 4 + (6 + 1) = ?
g. Are the results the same for (4 + 6) + 1 and 4 + (6 + 1)?
h. Why are the results the same?
(4 + 6) + 1 = 4 + (6 + 1) is an example of
The Associative Property
(a + b) + c = a + (b + c)
So, what
happened? The order of the
numbers changed. But
the result did not!
The Commutative Property
allows you to change the
order in an addition.
And??
Here the
grouping
changed?
Right. The
grouping
changed?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
35
Read & Study box !"#$%&'(()*+, 3.5 Using the Commutative & Associative Properties of
Addition
The Commutative Property states that you can change the order when
adding two numbers to attain the same result.
92 + 4 = 4 + 92 old
order
new
order
The Associative Property states that you can change the grouping when
adding numbers to attain the same result.
(16 + 88) + 3 = 16 + (88 + 3) old
group
new
group
Example:
2 + 5 + 3
Study the different solutions using Commutative & Associative Properties
Solution A 2 + 5 + 3 Solution B 2 + 5 + 3 Solution C 2 + 5 + 3
7 + 3 2 + 8 5 + 5
10 10 10
Exercise 1: 8 + 4 + 7
Use the Commutative & Associative Properties to solve four different ways.
A 8 + 4 + 7 B 8 + 4 + 7 C 8 + 4 + 7 D 8 + 4 + 7 12 + __ 8 + __ 15 + __
Exercise 2: x + 8x + 4x
A x + 8x + 4x B x + 8x + 4x C x + 8x + 4x D x + 8x + 4x 9x + __
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
36
4 – Simplifying Algebraic Expressions
Read & Study box !"#$%&'(()*+, 4.1 Combining Like Terms Amongst Unlike Terms
Sometimes an expression has like terms mixed together with unlike terms. For example,
consider this expression:
33A + 99 + 4A
Answer: 7A + 99
Consider this expression:
3 12 + 5m + 9q - 3q + 113
Answer: 323 + 6q + 5m3
Let!s review this one:
Where did the 23 come from? ______________________________________
Where did the 6q come from? ______________________________________
Why is the 5m rewritten and unchanged? _____________________________
Can we just
combine the
like terms? Sure. What
about 99?
Should we
just leave it?
Combine the
like terms
12 + 11 and
9q – 3q?
Okay. And
do we just
rewrite the
5m?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
37
Exercise 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
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
38
Exercise box: !" #$ 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
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
39
Read & Study box !"#$%&'(()*+, 4.2 The Distributive Property (aka Removing the Parentheses)
You will recall 4( ) means “4 times ( )”
Now consider the following:
64(3A + 2)6
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)
4 ! 3A + 4 !2
Answer: 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.
Does this
mean 4 times
(3A + 2)?
Yes!
Do you know
how you can
remove the
parentheses?
Where did the
( ) go?
Would you
like to look up
my sleeves?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
40
Exercise box: !" #$ 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 – "ab2) = 24x + 36m – 3ab2
6
3(2x + m + 5ab2) = ?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
41
Exercise box: !" #$ 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 + !ab)
7
3(2x + m + 5ab2)
8
6(2 hens + 3 pens – 5 anchors)
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
42
Read & Study box !"#$%&'(()*+, 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 Distribute the 5 & the 2
5A + 20 + 6A – 2 Multiply as indicated.
11A + 18 Combine like terms
Answer
Here#s 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 Distribute the 3 & the 2
3x + 15y + __x – __y Multiply as indicated.
________ Combine like terms
Answer
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
43
Exercise box: !" #$
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
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
44
!#+,%-#."%789#'%:::::::::::::::::::::::::::%
1 For the power 115 the base is ____ and the exponent is ____
2 92 can be read 9 to the 2nd power or _____________
3 153 can be read 15 to the 3rd power or _____________
4 Write 3 x 3 x 3 x 3 x 3 x 3 as a power. _______
5 Write 25 as a multiplication.________________
6 20 - 2 " (11 " 8)
7 (14 + 1) ÷ (8 " 5)
8 6 + 52
Use the variable values to evaluate each expression
9 10
Evaluate 5a, if a = 4
Evaluate xy, if x =3 and y = 10
11 12
Evaluate M + 2N, if M = 6 and N = 3
Evaluate 3Z – 5Q, if Z = 10 and Q = 1
13 14
Evaluate R2, if R = 6.
Evaluate j5, if j = 10.
15 16
Evaluate 3f3, if f = 2.
Evaluate k2 + g3 if k = 7 and g = 3
17 18
If x = 1, y = 17 and z = 3
evaluate
If a = 8 and b = 2 evaluate
x + y
z2
5a – ab
b3
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
45
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.
24 4x2 and 12x2 are like terms.
25 9xy and 9yz are like terms.
Combine Like Terms (if possible).
26 5y + 6y 27 3m + 2m
28 17x – x 29 16r2 – 2r2
30 13b7 + 10b7 31 3ab + 10ab
32 3ab + 10ab – ab 33 b – b + b – b
34 !d + !d 35 9a + 10ab + 13abc – abcd
Simplify by combining only the like terms.
36 4x + 2x + 2A 37 3A – 2A + 14
38 12B + 2X + 4B 39 11p2 + 2x – 2p2
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 !(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
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
46
/7!;<=!%
1 Base = 11, exponent = 5 2 Nine squared
3 Fifteen cubed 4 36
5 2 x 2 x 2 x 2 x 2 6 14
7 5 8 31
9 20 10 30
11 12 12 25
13 36 14 100,000
15 24 16 76
17 2 18 3
19 x + 12 22
20 9a
21 X2
23 3M and 8M are like terms. True
24 4x2 and 12x2 are like terms. True
25 9xy and 9yz are like terms. False
26 5y + 6y = 11y 27 3m + 2m = 5m
28 17x – x = 16x 29 16r2 – 2r2 = 14r2
30 13b7 + 10b7= 23b7 31 3ab + 10ab= 13ab
32 3ab + 10ab – ab = 12ab 33 b – b + b – b = 0
34 !d + !d = 1d or d 35 9a + 10ab + 13abc – abcd
36 4x + 2x + 2A = 6x + 2A 37 3A – 2A + 14 = A + 14
38 12B + 2X + 4B = 16B + 2X 39 11p2 + 2x – 2p2 = 9p2 + 2x
40 100(x + y) = 100x + 100y 41 25(W – Z) = 25W – 25Z
42 3(4a + 5) = 12a + 15 43 7(2x – b) = 14x – 7b
44 !(10M + 12N)
= 5M + 6N 45 7(a – b + 2c + 2)
= 7a – 7b + 14c + 14
46 2(x + 3) + 5(x + 7)
= 7x +41 47 5(2a + 1) + 15
= 10a + 20
48 7(3ab + x) + 2(2ab + 3x) = 25ab + 13x
49 r(3 + t) + 4rt = 3r + 5rt
50 2(3x + 5y) + 5x(2x + 1) + 4(7y – 4) + 9M =
= 16x + 38y + 10x2 – 16 + 9M
x + y
5
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
47
5 –Evaluating Formulas
5.1 Basic Formula Evaluations
Read & Study box !"#$%&'(()*+, N
EXAMPLES
a. Find the area (A) of a parallelogram (in ft2).
Formula: A = bh
Substituting for b and h
A = (7 ft)(5 ft)
Answer: 35 ft2
Parallelogram
b. Find the area of the triangle (in inches2).
Formula: A = ! bh
A = ! (10”)(6”)
A = (5”)(6”)
Answer: 30 in2
Triangle
c. Find the area of a trapezoid (in m2)
Formula: A = ! h(b1 +b2)
A = ! (4)(12 + 8)
A = !(4)(20)
A = (2)(20)
Answer: 40 m2
Trapezoid
d. Find the volume of the cube (in m3).
Formula: V = s3
V = (2)3
V = (2)(2)(2)
Answer: 8 m3
Cube
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
48
Exercise box: !" #$ 1. Find the area of the rectangle (in mm2).
A = LW
2. Find the area of the triangle (in ft2).
A = ! bh
3. What is the area of the trapezoid (in cm2)?
A = ! 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
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
49
9. Find the area of the parallelogram in ft2.
A = bh
10. Find the volume of the rectangular prism in cm3.
V = LWH
11. Find the volume of the cube in inches3
V = s3
12. Find the surface area of the cube in m2.
A = 6s2
13. Find the surface area of the rectangular prism in yd2.
A = 2LW + 2LH + 2WH
14. Find the surface area of the rectangular prism km2.
A = 2LW + 2LH + 2WH
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
50
Notebook Exercises: 1. Find the area, in square meters, of a square whose side is 9 m.
Use Formula
A = s2
2. Find the perimeter, in cm, of a rectangle with length of 5 cm and
width is 3 cm. P = 2L + 2W
3. Find the perimeter of a rectangle with length 9 yds and width 8 yds. P = 2L + 2W
4. Find the perimeter, in cm, of a square whose side is 12 cm? P = 4s
5. Find the area, in square miles, of a triangle whose height is 10 miles and
base is 4 miles. A = !bh
6. Find the area, in square feet, a parallelogram whose height is 13 feet
and base is 2 feet.
A = bh
7. The side of cube measures 4 km, find its volume in cubic km. V = s3
8. The dimensions of a rectangular solid are 3 cm, 4 cm and 5 cm.
Find its volume, in cubic cm. V = lwh
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. A=!h(b1+b2)
10. Find the area, in square meters, of a rectangle whose length measures
14 meters and width measures 3 meters. A = lw
Read & Study box !"#$%&'(()*+,
5.2 The Circle and !
(Leave all answers in terms of !)
1. Find the circumference of the circle.
Solution: Substituting for r
C = 2!(4)
C = !(8)
Answer: 8!
2. Find the area of the circle.
Solution:
A = !(92)
A = !(81)
Answer: 81!
3. Find the circumference of the circle.
Solution:
A = !(17)
Answer: 17!
9 m
17 mm
4 in.
C = 2! r
A = ! r2
C = !d
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
51
4. The radius of a sphere is 2 feet. Find the volume.
Solution:
V = 4!(23)
3
V = 4!(8)
3
5. Find the surface area of right circular cylinder.
Solution:
A = 2!(5)(10) + 2!(52)
A = 2!(50) + 2!(25) A = 100! + 50!
Answer: 150!
Notebook Exercises:
(Leave all answers in terms of !.) Use Formula
1. Find the circumference of a circle whose radius is 8 feet. C = 2!r
2. Find the area of a circle whose radius is 9 cm. A = !r2
3. Find the circumference of a circle whose diameter is 7 km. C = !d
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. A = 2!rh + 2!r2
6. Find the circumference of a circle whose radius is 9 miles. C = 2! r
7. Find the area of a circle whose radius is 10 cm. A = ! r2
8. Find the circumference of a circle whose diameter is 29 km. C = !d
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. A = 2!rh + 2!r2
V = 4!r3 3
r = 2 ft. Answer: 32!
3
A = 2!rh + 2!r2
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
52
Exercise box: !" #$ Write the formula:
Answer 1. The area of a rectangle is equal to its length times its width. A = lw 2. The area of a square is equal to its side squared. A = 3. The volume of a cube is equal to its side cubed. 4. The area of a triangle is equal to ! 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 ! times its radius squared. 7. The circumference of a circle is equal to two times ! times the radius. 8. The circumference of a circle is equal to ! multiplied by the diameter. 9. The area of a trapezoid is equal to ! 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 ! times the
radius times height plus 2 times ! 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 ! times its radius
squared times its height.
17. The volume of a right triangular prism is ! 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 = !r2h, V = #whl, A = wh + lw + lh + ls
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
53
Exercise box: !" #$
Study Example: Find the area of a triangle whose base is 12 m
and height is 8 m.
Step 1: Draw a sketch.
Step 2: Write the formula (look back for formula)
Step 3: Substitute in the formula
Step 4: Solve
SOLUTION
Step 1: (Using a ruler sketch the triangle)
Step 2: A = !bh
Step 3: !(12m)(8m)
Step 4: (6m)(8m)
Answer: 48 m2
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.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
54
4 Find the area of a trapezoid if:
height = 4 mm
base 1 = 11 mm
base 2 = 5 mm
5 Find the perimeter of a rectangle whose
length is 15 meters and width is 8 meters.
6 Find the perimeter of a square whose side
is equal to 100 feet.
7 Find the circumference of a circle whose
radius is equal to 8 cm. (in terms of !)
8 Find the circumference of a circle whose
diameter is equal to 8 cm. (in terms of !)
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
55
9 Find the area of a circle whose radius is
equal to 7 feet. (in terms of !)
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.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
56
5.3 Algebraic Representation of Perimeters
Read & Study box !"#$%&'(()*+,
Study Example: Represent the perimeter of the triangle
algebraically whose sides are 2x, x and 5x.
Solution
P = 2x + x + 5x
Answer: 8x
Exercises:
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.
Solution
P = 4(7b)
Answer: 28b
Exercises:
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
Exercises: 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.
P = side 1 + side 2 + side 3
P = 4s 7b
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
57
Study Example: Represent the circumference of a
circle whose radius is 6x.
C = 2"(6x)
Answer: 12"x
Exercises: Represent algebraically. 12. The circumference of a circle whose radius is 11d.
13. The circumference of a circle whose radius is 7J.
14. The circumference of a circle whose radius is K
15. The circumference of a circle whose radius is 12!G
16. The circumference of a circle whose diameter is 10Y. (Divide the
diameter by 2 to find the radius.)
17. The circumference of a circle whose diameter is 8L.
18. The circumference of a circle whose diameter is 5C.
5.4 Algebraic Representation of Areas
Read & Study box !"#$%&'(()*+,
Study Example: Represent the perimeter of the
triangle algebraically whose sides are 2x, x and 5x.
Solution
A = !(8x)(9x) = (4x)(9x) = 36(x)(x)
Answer: 36x2
Exercises: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.
C = 2!r 6x
A = !bh 9x
8x
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
58
Study Example: Represent the area of a square whose side is equal to 5K.
A = (5K)2 = (5K)(5K) = 25(K)(K)
Answer: 25K2
Exercises: Represent algebraically. 5. The area of a square whose side is 3b.
6. The area of a square whose side is 7R.
7. The area of a square whose side is 3.5Z
8. The area of a square whose side is Q.
Study Example: Represent algebraically the area of
a rectangle, whose length 5 is and width is 9G.
A = (5)(9G)
Answer: 45G
Exercises: Represent algebraically. 9. The area of a rectangle whose base is 3M and height is 7.
10. The area of a rectangle whose base is 10x and height is 3.
11. The area of a rectangle whose base is 4.5y and height is 2.
Study Example: Represent the area of a circle whose radius is 4b.
A = "(4b)2 = "(4b)(4b) = "(4)(4)(b)(b)
= "(16b2)
Answer: 16"b2
Exercises: Represent algebraically. 12. The area of a circle whose radius is 2L.
13. The area of a circle whose radius is 5h.
14. The area of a circle whose radius is f.
15. The area of a circle whose radius is 3.4G
16. The area of a circle whose diameter is 6Y. (Divide the
diameter by 2 to find the radius.)
17. The area of a circle whose diameter is 10b.
18. The area of a circle whose diameter is 3C.
A = s2
A = lw 9G
5
5K
A = !r2 4b
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
59
6 –The Integers
Read & Study box !"#$%&'(()*+, 6.1 The Counting & the Whole Numbers
The most common number system is the “counting numbers” or “natural numbers” which
are as follows:
The Counting numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, … (and so on).
Add a zero (0) to this system and you get the “whole numbers”.
The Whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, … (and so on)
Writing box
1. What is the difference between the Counting Numbers and the Whole Numbers?
2. What number is a Whole Number that is not a Counting Number? _______
3. What do the three dots (…) at the end of the number systems above mean?
4. What is the smallest Counting Number? _______
5. What is the smallest Whole Number? _______
6. Why is true that there is no largest Counting Number?
7. Is it true that there is no largest Whole Number? ______
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
60
Read & Study box !"#$%&'(()*+, 6.2 Understanding The Integers Find or draw the opposite of each:
up off true ! " # minus + –7 Write the
opposite Down
As you have learned, the opposite of “+” is “–” and, therefore, the opposite of the
negative of a number is the positive of the number. Thus, the opposite of –7 is +7.
The following set of numbers is called The Integers.
… , –6, –5, –4, –3, –2, –1, 0, +1, +2, +3, +4, +5, +6, …
Looked at another way, the Integers can be divided into three parts as follows:
Positive Whole Numbers +1, +2, +3, +4, +5, +6, …
Negative Whole Numbers –1, –2, –3, –4, –5, –6, …
Zero
0
Writing box 1. Explain the type of numbers that have to be added to the Whole Numbers to form the
Integers?
2. Fill in the missing words in the following sentence:
The Integers consist of the __________ whole numbers, the __________ whole
numbers and _______.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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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 … when representing the Integers?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
62
Read & Study box !"#$%&'(()*+, 6.3 The “Poof” Effect (aka Adding Integers)
THE “POOF” EFFECT: Something very interesting happens when a meets a
They meet They “poof” They#re gone
meets
Every time a “+” and “–” meet ……… “POOF” they both disappear.
Kind of mortal enemies, you might say.
–3 meets +5?
+
–
+
–
+
+
– +
+
– – +
See all the “POOFS” and
result on the next page
How many
“poofs” when:
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
63
–3 meets
+5
This can be written as an Integer Addition:
—3 + +5 = +2
Everybody
ready?
+ + + + +
— — —
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
64
Look at this “pre-poofed” example:
+6 + —7 = —1 Why –1?
Explain:
Exercises: Adding Integers
Instructions: Explain each result in the space provided. 1
—2 + +9 = +7
2 —13 + +1 = —12
3 —103 + +103 = 0
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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4 —4 + —4 = —8
5 +10 + +10 = +20
6 +1 + +1 +—3 + —4 = —5
7 —3 + +11 = ?
8 —30 + +30 = ?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
66
Exercises:
Instructions: Add the integers. 1 —2 + +5
2 +1 + —3
3 —1,039 + +1,039
4 —8 + —3
5 +1 + +1
6 +5 + +2 +—3 + —1
7 —30 + +11
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
67
“Why’s my sign?”
Instructions: The numbers have been shaded out and only the signs are shown.
Explain why the result of each Integer addition will have the sign as indicated.
1 — + — = — Explain why the result is negative.
2 + + + = + Explain why the result is positive.
3
+ + — = + or — Explain why the sign can be + or —.
4 + + + = ? What’s my sign? Why? Explain.
5 — + + = ? What’s my sign? Why? Explain.
6 — + — = ? What’s my sign? Why? Explain.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
68
“What’s my sign?”
Instructions: Determine the sign of each Integer Addition and rewrite in the
appropriate column.
Sign of the Addition No sign
+ — 0 1
+8 + —3
+ 2
+8 + +3
3 —8 + —3
— 4
—8 + +3
5 —8 + +8
0 6
—1 + +7
7
+1 + —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
17
—1 + +5 + +7
18
—8 + +2 + +6
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
69
“What’s my SUM?”
Instructions: Determine the sign of each Integer Addition and write the SUM in
the appropriate column.
Sign No sign
+ — 0 1
+8 + —3 +5
2 +8 + +3
3 —8 + —3 —11
4 —8 + +3
5 —8 + +8 0
6 —1 + +7
7 +1 + —7
8
—1 + —7
9
+1 + +7
10
—6 + +6
11
—2 + +9
12
+2 + +9
13
—2 + —9
14
+2 + —9
15
—2 + +2
16
+9 + —9
17
—1 + +5 + +7
18
—8 + +2 + +6
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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Exercises: Instructions: Find the sum for each Integer Addition
1 +8 + —3 = +5 2 +8 + +3 =
3 —8 + —3 =
4 —8 + +3 = 5 —8 + +8 = 6 —1 + +7 =
7 +1 + —7 = 8 —1 + —7 = 9 +1 + +7 =
10 —6 + +6 = 11 —2 + +9 = 12 +2 + +9 =
13 —2 + —9 = 14 +2 + —9 = 15 —2 + +2 =
16 +9 + —9 =
17 —1 + +5 + +7 =
18 —8 + +2 + +6 =
19 —7 + +2 + +3 =
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 = ?
+4 + +1 ++3 + +7 + +8 + —2 + —10 +—1 + —5
+23 + —18
+5
Okay, if you are a
positive, regroup
on the left!
And, if you are a
negative, regroup
on the right!
The Positive Like Signs added up is: The Negative Like Signs added up is:
Wow! All of that
adds up to +5?
Yep. Just regroup,
add the like signs
and then add the
results.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
71
Exercises: Regrouping by Like Signs
Instructions: Find the sum by regrouping and adding the like signs first.
1 +1 + +3 +—12 ++4 + —1 = ?
Positives Negatives
2 —7 + —11 ++5 + +3 +—11 + —9 ++8 = ?
Positives Negatives
Read & Study box !"#$%&'(()*+, 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.
61 – 2 + 3 – 76
Solution:
1 – 2 + 3 – 7 =
4 – 9 = –5
Do we do it
the same way
as before?
Yes. Combine
the like signs
and then
“poof” away.
+4 + -9
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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Exercise box: !" #$
Instructions: Perform the indicated addition.
1 2
8 – 5
3 – 6
3 4
–1 – 3 + 2
–2 – 7
5 6
10 – 3
3 – 10
7 8
–8 – 5 + 13
3 – 10 + 9
9 10
–12 + 12
7 – 7
11 12
–1 + 2 –1 + 2 –1 + 2
– 4 – 4
+3 +
-6
+8 +
-5
-2 +
-7
-1 +
-3 +
+2
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
73
Read & Study box !"#$%&'(()*+, 6.5 Combining Like Terms Using Integer Addition Strings Study Example: Simplify by combining like terms.
3a – 9a =
– 6a
Exercise box: !" #$
Instructions: Simplify each expression by combining like terms.
1 2
–8B – 3B
2a – 5a
3 4
–1e – 3e + 2e
–2q2 + 7q2
5 6
10g5 – 3g5
3hf – 10hf
7 8
–8g – 5g + 13g
3r – 10r + 9r
+3a +
-9a
-8B +
-3B
+2a +
-5a
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
74
Read & Study box !"#$%&'(()*+,
Now consider this expression: "8k + 9k + 53M =6=
Answer: k + 53M6
Explain the result below
The answer is k + 53M because
Exercise box: !" #$
Instructions: Simplify each expression by combining ONLY like terms.
1 2
9B + 34M –2B
3a – 7a + 13b
3 4
–1k + 3k + 2j
–3q2 –11z + 7q2
5 6
10d3 – 3g + 3d3
13hf – 10gh + 2gh
7 3x + 2y + 5y + 7x
8 -5x + 7x -14y – 8y + 3M
How did this
happen?
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
75
Read & Study box !"#$%&'(()*+, 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)
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
76
“What’s my sign?”
Instructions: Complete this table.
Example Rewrite as an addition Answer Sign 1 5(1) (1) + (1) + (1) + (1) + (1) 5 + 2 4(3)
3 (+2)(+3) (+3) + (+3) +6 + 4 (5)("2) ("2) + ("2) + ("2) + ("2) + ("2) "10 5 (4)("1) ("1) + ("1) + ("1) + ("1) "4 " 6 ("1)(4) Same as (4)("1) =
("1) + ("1) + ("1) + ("1) "4 "
7 (+2)(+5) 8 (3)("2) 9 ("2)(3) Same as (3)("2) =
("2) + ("2) + ("2)
10 (4)( "2) 11 (3)(1) 12 ("2)(4) Same as 14 ("1)(6) Same as 15 (2)(11)
Okay, here’s your task? If you know
the signs being multiplied, can you tell
the sign of the answer?
Do you mean, what is the sign of the
result if you multiply, say, + by –?
Exactly. So does, (+)(–)
equal + or –?
Can I look at the
table above?
Sure.
Let’s see. It looks like
(+)(–) equals (–).
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
77
6.8 Integer Multiplication, “The Rules”
Looking at the previous table we see:
(+)(+) = + (")(+)= " (+)(") = "
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 _____________
Nice going.
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.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
78
A positive times a negative is a _____________
A negative times a negative is a _____________
Using the Integer Multiplication Rules
Instructions: Write the reason for each answer in the space provided.
1 ("2)( "3) = +6 Why?
2 5("4) = "20 Why?
3 ("6)(+!) = "3 Why?
4 (+8) (+1) = +8 Why?
Instructions: Find the product for each Integer Multiplication
5 5("2) 6 7("10)
7 ("3)("1) 8 (+3)("1)
9 ("10)(4) 10 ("1)(14)
11 (+7)( +1) 12 (0)( +1)
13 (+8)( "8) 14 ("8)( "8)
15 (6)( "9) 16 15(3)
17 ("1)("4) 18 ("!)(+12)
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
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19 ("2)( 13) 20 ("1)( "1)
21 ("6)( +6) 22 ("22)( "3)
23 (9)( 4) 24 ("100)("")
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
80
“What’s my sign?”
Instructions: Complete this table.
Example Rewrite as an addition Answer Sign 1 5(1) (1) + (1) + (1) + (1) + (1) 5 + 2 4(3)
3 (+2)(+3) (+3) + (+3) +6 + 4 (5)("2) ("2) + ("2) + ("2) + ("2) + ("2) "10 5 (4)("1) ("1) + ("1) + ("1) + ("1) "4 " 6 ("1)(4) Same as (4)("1) =
("1) + ("1) + ("1) + ("1) "4 "
7 (+2)(+5) 8 (3)("2) 9 ("2)(3) Same as (3)("2) =("2) + ("2) + ("2) 10 (4)( "2) 11 (3)(1) 12 ("2)(4) Same as 14 ("1)(6) Same as 15 (2)(11)
Okay, here’s your task? If you know
the signs being multiplied, can you tell
the sign of the answer?
Do you mean, what is the sign of the
result if you multiply, say, + by –?
Exactly. So does, (+)(–)
equal + or –?
Can I look at the
table above?
Sure.
Let’s see. It looks like
(+)(–) equals (–).
Nice going.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
81
6.8 Integer Multiplication, “The Rules”
Looking at the previous table we see:
(+)(+) = + (")(+)= " (+)(") = "
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 _____________
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.
! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !
!Copyright: 2009 by Barry Hauptman
82
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)(+!) = "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 ("!)(+12)
19 ("2)( 13) 20 ("1)( "1)
21 ("6)( +6) 22 ("22)( "3)
23 (9)( 4) 24 ("100)("")
This is a work in progress……….