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.!" !


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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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

! !"#$%!&'!"#$%!()*$&+,!-,./0.!" !


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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5

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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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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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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8

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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9

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.

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10

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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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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12

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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13

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

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14

!#+,%-#."%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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15

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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16

/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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17

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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18


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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19

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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20


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

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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.

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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

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23

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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24

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

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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

=

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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

%

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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.!" !


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.!" !


29


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.!" !


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.!" !


31


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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.!" !


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


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.!" !


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.!" !


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.!" !


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.

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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.!" !


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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.!" !


56

5.3 Algebraic Representation of Perimeters


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.


are 11M, 9M and 3M.


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

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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


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.!" !


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.!" !


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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? ______

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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 _______.

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61

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?

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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:

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63

–3 meets

+5

This can be written as an Integer Addition:

—3 + +5 = +2

Everybody

ready?

+ + + + +

— — —

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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

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65

4 —4 + —4 = —8

5 +10 + +10 = +20

6 +1 + +1 +—3 + —4 = —5

7 —3 + +11 = ?

8 —30 + +30 = ?

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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

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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.

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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

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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

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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.

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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

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72

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

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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

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74


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?

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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)

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76


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 (–).

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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.!" !


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)

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79

19 ("2)( 13) 20 ("1)( "1)

21 ("6)( +6) 22 ("22)( "3)

23 (9)( 4) 24 ("100)("")

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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.

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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.!" !


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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……….

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