summer math enrichment modules 7 - 13 mastering the

Summer Math Enrichment

Modules 7 - 13

“Mastering the Fundamentals”

Chris Millett

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this book.

Academic Excellence in Mathematics ® is a registered trademark of Chris Millett.

Science, Math, and Technology Center of Excellence ® is a registered trademark of Chris Millett.

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Table of Content Module 7: Number Properties and Principles ..................................................................................................................................... 7

Section 7.1 Number Properties .................................................................................................................................................... 9 Common Number Properties ................................................................................................................................................... 9 Commutative Property of Addition ........................................................................................................................................ 9 Commutative Property of Multiplication ............................................................................................................................... 9 Associative Property of Addition ............................................................................................................................................. 9 Associative Property of Multiplication ................................................................................................................................. 10 Distributive Property.............................................................................................................................................................. 10 Identity Property of Addition ................................................................................................................................................ 10 Identity Property of Multiplication ....................................................................................................................................... 10 Property of Zero ..................................................................................................................................................................... 10 Number Properties – Guided Practice .................................................................................................................................. 11

Section 7.2 Multiplication – The Foundation to Academic Excellence in Mathematics ....................................................... 13 The Importance of Mastering Multiplication ....................................................................................................................... 13 Level 1 (Single-Digit by Single-Digit Multiplication) ........................................................................................................... 13 Level 2 (Single-Digit by Double-Digit and Double-Digit by Single-Digit Multiplication) ................................................ 13 Level 3 (Double-Digit by Double-Digit Multiplication) ....................................................................................................... 14 Multiplication – Guided Practice .......................................................................................................................................... 15

Section 7.3 Multiplication Facts (1 – 100) ................................................................................................................................. 17 Importance of Multiplication Facts ....................................................................................................................................... 17 Multiplication Facts – Guided Practice ................................................................................................................................ 19

Section 7.4 Multiplying With Multiples of 10 ........................................................................................................................... 20 Master Multiplication with Multiples of 10 .......................................................................................................................... 20 Multiplying With Multiples of 10 – Guided Practice ........................................................................................................... 21

Section 7.5 Multiplying With Multiples of 100 ......................................................................................................................... 22 Master Multiplication with Multiples of 100 ........................................................................................................................ 22 Multiplying With Multiples of 100 – Guided Practice ......................................................................................................... 23

Section 7.6 Multiplying With Multiples of 1000 ....................................................................................................................... 24 Master Multiplication with Multiples of 1000 ...................................................................................................................... 24 Multiplying With Multiples of 1000 – Guided Practice ....................................................................................................... 25

Section 7.7 Division Facts (1 – 100) ........................................................................................................................................... 26 Importance of Division Facts ................................................................................................................................................. 26 Division Facts – Guided Practice ........................................................................................................................................... 34

Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10) ...................................................................................... 35 Divisibility – Guided Practice ................................................................................................................................................ 38 Number Principles – Guided Reinforcement ....................................................................................................................... 39

Module 8: Number Theory and Terminology ................................................................................................................................... 49 Section 8.1 Factors and Multiples ............................................................................................................................................. 51

Factor (Definition and Examples) ......................................................................................................................................... 51 Greatest Common Factor (GCF) ........................................................................................................................................... 51 Multiple (Definition and Examples) ...................................................................................................................................... 51 Least Common Multiple (LCM) ............................................................................................................................................ 52 Factors and Multiples – Guided Practice ............................................................................................................................. 53

Section 8.2 Prime Numbers and Composite Numbers ............................................................................................................. 54 Prime Number (Definition and Examples) ........................................................................................................................... 54 Composite Number (Definition and Examples) ................................................................................................................... 55 Key Math Factors ................................................................................................................................................................... 56 Key Multiplication Fact of Composite Numbers (1 – 100) .................................................................................................. 58 Prime Factorization (Definition and Examples) .................................................................................................................. 60 Equivalent Products ............................................................................................................................................................... 61 Prime Numbers and Composite Numbers – Guided Practice ............................................................................................ 62

Section 8.3 Principles of Numbers ............................................................................................................................................. 63 Positive Numbers .................................................................................................................................................................... 63 Negative Numbers ................................................................................................................................................................... 63

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Zero .......................................................................................................................................................................................... 63 Number Line Representation of Numbers ........................................................................................................................... 63 Principles of Numbers – Guided Practice ............................................................................................................................. 64

Section 8.4 Number Terminology .............................................................................................................................................. 65 Integer ...................................................................................................................................................................................... 65 Consecutive Integers ............................................................................................................................................................... 65 Even Number .......................................................................................................................................................................... 65 Odd Number ........................................................................................................................................................................... 65 Rational Number .................................................................................................................................................................... 65 Irrational Number .................................................................................................................................................................. 65 Remainder ............................................................................................................................................................................... 66 Digits and Place Value ............................................................................................................................................................ 66 Number Terminology – Guided Practice .............................................................................................................................. 67

Section 8.5 Arithmetic Operations on Numbers ...................................................................................................................... 69 Common Arithmetic Operations on Numbers ..................................................................................................................... 69 Standard Arithmetic Symbols ............................................................................................................................................... 69 Addition (Sum) ........................................................................................................................................................................ 69 Subtraction (Difference) ......................................................................................................................................................... 70 Multiplication (Product) ........................................................................................................................................................ 71 Division (Quotient) ................................................................................................................................................................. 72

Module 8 (Number Theory and Terminology) – Review Exercises........................................................................................ 73 Factors and Multiples ............................................................................................................................................................. 73 Prime Numbers and Composite Numbers ............................................................................................................................ 75 Positive & Negative Number Arithmetic .............................................................................................................................. 78

Module 9: Principles of Fractions ..................................................................................................................................................... 79 Section 9.1 Introduction to Fractions ........................................................................................................................................ 81

Introduction to Fractions ....................................................................................................................................................... 81 Types of Fractions .................................................................................................................................................................. 81 Simplest-Form Fraction ......................................................................................................................................................... 81 Equivalent Fractions .............................................................................................................................................................. 82 Reciprocal ................................................................................................................................................................................ 82 Least Common Denominator (LCD) ..................................................................................................................................... 82 Reducing (Simplifying) Fractions .......................................................................................................................................... 83 Unreducing (Unsimplifying) Fractions ................................................................................................................................. 83 Converting Improper Fractions to Mixed Fractions ........................................................................................................... 83 Converting Mixed Fractions to Improper Fractions ........................................................................................................... 83 Comparing Fractions.............................................................................................................................................................. 84 Converting Fractions to have a Common Denominator ..................................................................................................... 84 Introduction to Fractions – Guided Practice ........................................................................................................................ 85

Section 9.2 Fraction Arithmetic ................................................................................................................................................. 87 Adding and Subtracting Fractions with Like Denominators .............................................................................................. 87 Adding and Subtracting Fractions with Unlike Denominators .......................................................................................... 87 Adding Mixed Fractions ......................................................................................................................................................... 88 Subtracting Mixed Fractions ................................................................................................................................................. 89 Multiplying Fractions ............................................................................................................................................................. 90 Multiplying Mixed Fractions ................................................................................................................................................. 90 Dividing Fractions by Fractions ............................................................................................................................................ 90 Dividing Fractions by Whole Numbers ................................................................................................................................ 91 Dividing Whole Numbers by Fractions................................................................................................................................. 91 Dividing Mixed Fractions ....................................................................................................................................................... 92 Fraction Arithmetic – Guided Practice................................................................................................................................. 93

Section 9.3 Advanced Fraction Principles ................................................................................................................................ 94 Fractions (Tenths, Hundredths, Thousandths, and Beyond) .............................................................................................. 94 Introduction to Complex Fractions ....................................................................................................................................... 95 Solving Complex Fractions .................................................................................................................................................... 95 Introduction to Word Problems with Fractions................................................................................................................... 95 Solving Word Problems with Fractions ................................................................................................................................ 96 Advanced Fraction Principles – Guided Practice ................................................................................................................ 97

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Module 9 (Principle of Fractions) – Review Exercises .......................................................................................................... 101 Fraction Fundamentals ........................................................................................................................................................ 101 Fraction Arithmetic .............................................................................................................................................................. 103 Advanced Fraction Principles.............................................................................................................................................. 109

Module 10: Principles of Decimals ................................................................................................................................................ 113 Section 10.1 Introduction to Decimals .................................................................................................................................... 115

Decimal .................................................................................................................................................................................. 115 Decimal Place Value ............................................................................................................................................................. 115 Comparing Decimals ............................................................................................................................................................ 116 Ordering Decimals ................................................................................................................................................................ 116

Introduction to Decimals – Guided Reinforcement ............................................................................................................... 117 Section 10.2 Rounding Decimal Numbers .............................................................................................................................. 118

What is Decimal Rounding? ................................................................................................................................................ 118 Principles of Rounding Decimal Numbers ......................................................................................................................... 118 Examples of Rounding Decimal Numbers .......................................................................................................................... 119 Rounding Decimal Numbers – Guided Practice ................................................................................................................ 120

Section 10.3 Decimal Arithmetic ............................................................................................................................................. 121 Common Arithmetic Operations Involving Decimals ....................................................................................................... 121 Decimal Addition .................................................................................................................................................................. 121 Decimal Subtraction ............................................................................................................................................................. 122 Decimal Addition and Subtraction – Guided Practice ...................................................................................................... 123 Introduction to Decimal Multiplication .............................................................................................................................. 124 Multiplying a Number with a Decimal by a Number without a Decimal ........................................................................ 124 Multiplying a Number with a Decimal by another Number with a Decimal ................................................................... 125 Introduction to Decimal Division ........................................................................................................................................ 126 Dividing Numbers without a Decimal Point (Resulting in an Answer with a Decimal Point) ....................................... 126 Dividing a Number without a Decimal Point into a Number with a Decimal Point ....................................................... 127 Dividing a Number with a Decimal Point into a Number without a Decimal Point ....................................................... 128 Dividing a Number with a Decimal Point into another Number with a Decimal Point .................................................. 129

Module 10 (Principle of Decimals) – Review Exercises ......................................................................................................... 131 Decimal Fundamentals ......................................................................................................................................................... 131 Decimal Arithmetic ............................................................................................................................................................... 133 Supplemental Decimal Arithmetic ...................................................................................................................................... 135

Module 11: Principles of Percentages ............................................................................................................................................ 139 Section 11.1 Introduction to Percentages................................................................................................................................ 141

Percentage (Definition & Examples) ................................................................................................................................... 141 Converting from Percentage to Fraction ............................................................................................................................ 141 Converting from Fraction to Percentage ............................................................................................................................ 141 Converting from Decimal to Percentage ............................................................................................................................. 141 Converting from Percentage to Decimal............................................................................................................................. 141

Section 11.2 Calculating Percentages ...................................................................................................................................... 142 What is a Certain Percent of a Certain Value? .................................................................................................................. 142 What Percent is a Value of another Certain Value? .......................................................................................................... 142 What Value is a Certain Percent of a Certain Value? ....................................................................................................... 142 What is a Certain Percent Increase of a Value? ................................................................................................................ 143 What is a Certain Percent Decrease of a Value?................................................................................................................ 143 What is a Certain Percent More than a Value? ................................................................................................................. 144 What is a Certain Percent Less than a Value? ................................................................................................................... 144

Section 11.3 Percentage Equivalents ....................................................................................................................................... 145 Important Fraction, Decimal, Percentage Equivalents ..................................................................................................... 145

Section 11.4 Word Problems with a Percent .......................................................................................................................... 146 Solving Word Problems with Containing Percentages ...................................................................................................... 146 Percentage – Guided Practice .............................................................................................................................................. 147

Module 11 (Principle of Percentages) – Review Exercises .................................................................................................... 149 Percentage Fundamentals .................................................................................................................................................... 149 Percentage Calculations ....................................................................................................................................................... 151

Module 12: Advanced Arithmetic Operations ................................................................................................................................ 155 Section 12.1 Ratios .................................................................................................................................................................... 157

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Ratio (Definition) .................................................................................................................................................................. 157 Ratio (Expressions) ............................................................................................................................................................... 157 Ratio Examples ..................................................................................................................................................................... 157 Ratio – Guided Practice ....................................................................................................................................................... 158

Section 12.2 Proportion ............................................................................................................................................................ 159 Proportion (Definition & Example) .................................................................................................................................... 159 Proportion – Guided Practice .............................................................................................................................................. 160

Section 12.3 Other Advanced Arithmetic Operations ........................................................................................................... 161 Principles of Exponents ........................................................................................................................................................ 161 Square Root ........................................................................................................................................................................... 162 Order of Operations ............................................................................................................................................................. 164 Arithmetic Word Problem Fundamentals .......................................................................................................................... 164 Sequence ................................................................................................................................................................................ 165 Set ........................................................................................................................................................................................... 166 Advanced Arithmetic Operations – Guided Practice ........................................................................................................ 167

Section 12.4 Advanced Counting Principles ........................................................................................................................... 169 Multiplication Principle of Counting .................................................................................................................................. 169 Advanced Counting Principles ............................................................................................................................................ 169 Combination .......................................................................................................................................................................... 170 Permutation ........................................................................................................................................................................... 171 Advanced Counting Principles – Guided Practice ............................................................................................................. 172

Module 12 (Advanced Arithmetic Operations) – Review Exercises ..................................................................................... 173 Ratio ....................................................................................................................................................................................... 173 Proportion ............................................................................................................................................................................. 174 Advanced Arithmetic Operations ........................................................................................................................................ 175 Advanced Counting Principles ............................................................................................................................................ 182

Module 13: Principles of Roots and Radicals ................................................................................................................................. 185 Section 13.1 Introduction to Radicals ..................................................................................................................................... 187

What is a Radical? ................................................................................................................................................................ 187 Radical Terminology ............................................................................................................................................................ 187 Learn the Following to Master Radicals ............................................................................................................................. 187

Section 13.2 Rules of Radicals ................................................................................................................................................. 188 Product Rule of Radicals ...................................................................................................................................................... 188 Quotient Rule of Radicals .................................................................................................................................................... 188

Section 13.3 Radical Manipulation .......................................................................................................................................... 189 Simplifying Radicals ............................................................................................................................................................. 189 Radicals and Rational Exponents ........................................................................................................................................ 189 Positive versus Negative Rational Exponents ..................................................................................................................... 189 Radical Manipulation – Guided Practice ........................................................................................................................... 190

Section 13.4 Radical Arithmetic .............................................................................................................................................. 192 Adding Radicals .................................................................................................................................................................... 192 Subtracting Radicals ............................................................................................................................................................ 192 Multiplying Radicals ............................................................................................................................................................ 193 Special Rule for Multiplying Radicals ................................................................................................................................ 194 Multiplying Radicals with Additional or Subtraction ....................................................................................................... 194 Radical Arithmetic – Guided Practice ................................................................................................................................ 195 Dividing Radicals .................................................................................................................................................................. 197 Conjugate Pair ...................................................................................................................................................................... 197 Multiplying Conjugates ........................................................................................................................................................ 197 Rationalizing the Denominator............................................................................................................................................ 197 Dividing Radicals – Guided Practice................................................................................................................................... 198

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Module 7: Number Properties and Principles

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Section 7.1 Number Properties

Common Number Properties

➢ Commutative Property of Addition

➢ Commutative Property of Multiplication

➢ Associative Property of Addition

➢ Associative Property of Multiplication

➢ Distributive Property

➢ Identity Property of Addition

➢ Identity Property of Multiplication

➢ Property of Zero

Commutative Property of Addition

➢ A + B = B + A

➢ When you add two values, the order in which you add the values does not matter

• 8 + 5 = 13

• 5 + 8 = 13

• This principle also works with negative numbers

o -8 + 5 = -3

o 5 + (-8) = -3

Commutative Property of Multiplication

➢ A x B = B x A

➢ When you multiply two values, the order in which you multiply the values does not matter

• 8 x 5 = 40

• 5 x 8 = 40


o (-8) x 5 = -40

o 5 x (-8) = -40

Associative Property of Addition

➢ (A + B) + C = A + (B + C)

➢ When you add three values, the order in which you add the values does not matter

• (8 + 5) + 2 = 13 + 2 = 15

• 8 + (5 + 2) = 8 + 7 = 15


o (8 + 5) + (-2) = 13 + (-2) = 11

o 8 + (5 + (-2)) = 8 + 3 = 11

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Section 7.1 Number Properties (continued)

Associative Property of Multiplication

➢ (A x B) x C = A x (B x C)

➢ When you multiply three values, the order in which you multiply the values does not matter

• (8 x 5) x 2 = 40 x 2 = 80

• 8 x (5 x 2) = 8 x 10 = 80


o (8 x 5) x (-2) = 40 x (-2) = -80

o 8 x (5 x (-2)) = 8 x (-10) = -80

Distributive Property

➢ A x (B + C) = (A x B) + (A x C)

➢ When you multiply a value by the sum of two numbers, you can multiply the value by each number

first and then add after you multiply

• 8 x (5 + 2) = 8 x 7 = 56

• (8 x 5) + (8 x 2) = 40 + 16 = 56


o -8 x (5 + 2) = -8 x 7 = -56

o (-8 x 5) + (-8 x 2) = -40 + (-16) = -56

Identity Property of Addition

➢ A + 0 = A

➢ Any number plus zero is the original number

• 8 + 0 = 8

• -8 + 0 = -8

Identity Property of Multiplication

➢ A x 1 = A

➢ Any number times one is the original number

• 8 x 1 = 8

• -8 x 1 = -8

Property of Zero

➢ A x 0 = 0

➢ Any number times zero is always zero

• 8 x 0 = 0

• -8 x 0 = 0

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Number Properties – Guided Practice

_______ 1. Using the Commutative Property of Addition, how can you rewrite 4 + 25?

_______ 2. Using the Commutative Property of Addition, how can you rewrite 23 + 14?

_______ 3. Using the Commutative Property of Multiplication, how can you rewrite 4 x 25?

_______ 4. Using the Commutative Property of Multiplication, how can you rewrite 12 x 9?

_______ 5. Using the Associative Property of Addition, how can you rewrite (8 + 7) + 6?

_______ 6. Using the Associative Property of Addition, how can you rewrite 9 + (7 + 6)?

_______ 7. Using the Associative Property of Multiplication, how can you rewrite (8 x 7) x 6?

_______ 8. Using the Associative Property of Multiplication, how can you rewrite 4 x (5 x 6)?

_______ 9. Using the Distributive Property, how can you rewrite (5 x 6) + (5 x 9)?

_______ 10. Using the Distributive Property, how can you rewrite 9 x (8 + 2)?

_______ 11. Using the Identity Property of Addition, how can you rewrite 45 + 0?

_______ 12. Using the Identity Property of Multiplication, how can you rewrite 45 x 1?

_______ 13. Using the Property of Zero, how can you rewrite 45 x 0?

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Number Properties – Guided Practice (continued)

Name the Property

_________________________________ 1. 6 x (4 + 3) = 6 x 4 + 6 x 3

_________________________________ 2. 8 x 0 = 0

_________________________________ 3. 4 + 5 = 5 + 4

_________________________________ 4. 18 x 1 = 18

_________________________________ 5. (8 x 9) x 4 = 8 x (9 x 4)

_________________________________ 6. (6 + 8) + 9 = 6 + (8 + 9)

_________________________________ 7. 25 x 0 = 0

_________________________________ 8. 6 x 7 = 7 x 6

_________________________________ 9. 37 + 0 = 37

_________________________________ 10. 4 x 5 + 4 x 7 = 4 x (5 + 7)

_________________________________ 11. (4 x 9) x 10 = 4 x (9 x 10)

_________________________________ 12. (4 + 9) + 10 = 4 + (9 + 10)

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Section 7.2 Multiplication – The Foundation to Academic Excellence in Mathematics

The Importance of Mastering Multiplication

➢ Most math principles require you to have solid multiplication skills

➢ In order to achieve “Academic Excellence in Mathematics”, you must be able to perform:

• Level 1 Multiplication (single-digit by single-digit)

• Level 2 Multiplication (single-digit by double-digit and double-digit by single-digit)

• Level 3 Multiplication (double-digit by double-digit)

Level 1 (Single-Digit by Single-Digit Multiplication)

➢ Before you proceed any further in mathematics, memorize these facts

➢ If you do not learn all of these multiplication facts, you are on pace to struggle in mathematics during

middle school, high school, college, and beyond

Level 2 (Single-Digit by Double-Digit and Double-Digit by Single-Digit Multiplication)

➢ Uses the following two math fundamentals

• Level 1 Multiplication

• The Distributive property

➢ Combining these two fundamentals, you can perform Level 2 Multiplication in your head (in a matter

of seconds)

➢ Steps to Level 2 Multiplication

• Convert the double-digit number into a sum, where one value is a tens-value

o 11 → 10 + 1

o 16 → 10 + 6

o 23 → 20 + 3

o 27 → 20 + 7

o 38 → 30 + 8

• Use the Distributive Property to multiply the single-digit number by the sum of the converted

double-digit number

o 12 x 8 → (10 + 2) x 8 → (10 x 8) + (2 x 8) → 80 + 16 = 96

o 15 x 9 → (10 + 5) x 9 → (10 x 9) + (5 x 9) → 90 + 45 = 135

o 23 x 7 → (20 + 3) x 7 → (20 x 7) + (3 x 7) → 140 + 21 = 161

o 38 x 6 → (30 + 8) x 6 → (30 x 6) + (8 x 6) → 180 + 48 = 228

1 2 3 4 5 6 7 8 9

1 1 2 3 4 5 6 7 8 9

2 2 4 6 8 10 12 14 16 18

3 3 6 9 12 15 18 21 24 27

4 4 8 12 16 20 24 28 32 36

5 5 10 15 20 25 30 35 40 45

6 6 12 18 24 30 36 42 48 54

7 7 14 21 28 35 42 49 56 63

8 8 16 24 32 40 48 56 64 72

9 9 18 27 36 45 54 63 72 81

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Section 7.2 Multiplication – The Foundation to Academic Excellence in Mathematics

(continued)

Level 3 (Double-Digit by Double-Digit Multiplication)

➢ Uses the following two math fundamentals

• Level 2 Multiplication

• The Distributive property

➢ Combining these two fundamentals, you can perform Level 3 Multiplication in your head (in a matter

of seconds)

• This is more difficult than Level 2 Multiplication, because the numbers are much larger

• You must regularly practice Level 3 Multiplication to master it

➢ Steps to Level 3 Multiplication

• Convert the smaller double-digit number into a sum, where one value is a tens-value

o 11 → 10 + 1

o 16 → 10 + 6

o 23 → 20 + 3

o 27 → 20 + 7

o 38 → 30 + 8

• Use the Distributive Property to multiply the larger double-digit number by the sum of the

converted double-digit numbers

o 12 x 13 → (10 + 2) x 13 → (10 x 13) + (2 x 13) → 130 + 26 = 156

o 15 x 17 → (10 + 5) x 17 → (10 x 17) + (5 x 17) → 170 + 85 = 255

o 23 x 14 → 23 x (10 + 4) → (10 x 23) + (4 x 23) → 230 + 92 = 322

o 24 x 23 → 24 x (20 + 3)→ (20 x 24) + (3 x 24) → 480 + 72 = 552

➢ The “Steps to Level 3 Multiplication” work every single time, regardless of the values of the double-

digit numbers

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Multiplication – Guided Practice

Solve the following multiplication problems without using a calculator.

__________ 1. What is 13 x 7?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 2. What is 27 x 8?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 3. What is 53 x 9?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 4. What is 86 x 8?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

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Multiplication – Guided Reinforcement (continued)

__________ 5. What is 15 x 17?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 6. What is 24 x 13?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 7. What is 37 x 12?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 8. What is 62 x 14?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

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Section 7.3 Multiplication Facts (1 – 100)

Importance of Multiplication Facts

➢ Mastering Multiplication Facts ( 1 – 100) will help you perform well in the following areas

• Algebra

• Geometry

• Trigonometry

• Statistics and Probability

• Standardized Tests

o CRCT

o ACT

o PSAT

o SAT

o Graduation Exams (Example: Georgia Graduation Exam in Mathematics)

➢ What Multiplication Facts should you master?

• Every product from 1 – 100 (besides 1 and the number)

o 4 = 2 x 2

o 6 = 2 x 3

o 8 = 2 x 4

o 9 = 3 x 3

o 10 = 2 x 5

o 12 = 2 x 6 and 3 x 4

o 14 = 2 x 7

o 15 = 3 x 5

o 16 = 2 x 8 and 4 x 4

o 18 = 2 x 9 and 3 x 6

o 20 = 2 x 10 and 4 x 5

o 21 = 3 x 7

o 22 = 2 x 11

o 24 = 2 x 12, 3 x 8, and 4 x 6

o 25 = 5 x 5

o 26 = 2 x 13

o 27 = 3 x 9

o 28 = 2 x 14 and 4 x 7

o 30 = 2 x 15, 3 x 10, and 5 x 6

o 32 = 2 x 16 and 4 x 8

o 33 = 3 x 11

o 34 = 2 x 17

o 35 = 5 x 7

o 36 = 2 x 18, 3 x 12, 4 x 9, and 6 x 6

o 38 = 2 x 19

o 39 = 3 x 13

o 40 = 2 x 20, 4 x 10, and 5 x 8

o 42 = 2 x 21, 3 x 14, and 6 x 7

o 44 = 2 x 22 and 4 x 11

o 45 = 3 x 15 and 5 x 9

o 46 = 2 x 23

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Section 7.3 Multiplication Facts (1 – 100) – continued

➢ What Multiplication Facts should you master? (continued)

• Examples (continued)

o 48 = 2 x 24, 3 x 16, 4 x 12, and 6 x 8

o 49 = 7 x 7

o 50 = 2 x 25 and 5 x 10

o 52 = 2 x 26 and 4 x 13

o 54 = 2 x 27, 3 x 18, and 6 x 9

o 55 = 5 x 11

o 56 = 2 x 28, 4 x 14, and 7 x 8

o 57 = 3 x 19

o 58 = 2 x 29

o 60 = 2 x 30, 3 x 20, 4 x 15, 5 x 12, and 6 x 10

o 62 = 2 x 31

o 63 = 3 x 21 and 7 x 9

o 64 = 2 x 32, 4 x 16, and 8 x 8

o 65 = 5 x 13

o 66 = 2 x 33, 3 x 22, and 6 x 11

o 68 = 2 x 34 and 4 x 17

o 69 = 3 x 23

o 70 = 2 x 35, 5 x 14, and 7 x 10

o 72 = 2 x 36, 3 x 24, 4 x 18, 6 x 12, and 8 x 9

o 74 = 2 x 37

o 75 = 3 x 25 and 5 x 15

o 76 = 2 x 38 and 4 x 19

o 77 = 7 x 11

o 78 = 2 x 39 and 6 x 13

o 80 = 2 x 40, 4 x 20, 5 x 16, and 8 x 10

o 81 = 3 x 27 and 9 x 9

o 82 = 2 x 41

o 84 = 2 x 42, 3 x 28, 4 x 21, 6 x 14, and 7 x 12

o 85 = 5 x 17

o 86 = 2 x 43

o 87 = 3 x 29

o 88 = 2 x 44, 4 x 22, and 8 x 11

o 90 = 2 x 45, 3 x 30, 5 x 18, 6 x 15, and 9 x 10

o 91 = 7 x 13

o 92 = 2 x 46 and 4 x 23

o 93 = 3 x 31

o 94 = 2 x 47

o 95 = 5 x 19

o 96 = 2 x 48, 3 x 32, 4 x 24, 6 x 16, and 8 x 12

o 98 = 2 x 49, and 7 x 14

o 99 = 3 x 33 and 9 x 11

o 100 = 2 x 50, 4 x 25, 5 x 20, and 10 x 10

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Multiplication Facts – Guided Practice

Write the Multiplication Fact Specified (Without Using the Previous Pages)

* Do not use 1 or the number specified

1) Two multiplication combinations for 12 → _______________________________________




5) One multiplication combinations for 21 → ________________________________________

6) Three multiplication combinations for 24 → ______________________________________

7) Two multiplication combinations for 28 → ______________________________________

8) Four multiplication combinations for 36 → ______________________________________

9) Three multiplication combinations for 40 → _____________________________________

10) Four multiplication combinations for 48 → _____________________________________

11) Three multiplication combinations for 54 → _____________________________________

12) Five multiplication combinations for 60 → _____________________________________



15) Three multiplication combinations for 88 → ____________________________________


17) Two multiplication combinations for 92 → _____________________________________


19) Two multiplication combinations for 99 → _____________________________________

20) Three multiplication combinations for 100 → ____________________________________

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Section 7.4 Multiplying With Multiples of 10

Master Multiplication with Multiples of 10

➢ When performing multiplication with a multiple of 10

• Temporarily ignore the zero on the number that is a multiple of 10

• Multiply the number as if the zero is not present

• After the multiplication is complete, just add the zero to the answer

➢ Examples

• 23 x 20

o Temporarily ignore the 0 on 20 and just treat is as if it is 2

o Multiply 23 x 2 → 46

o Now add the zero to the answer → 460

o So the final answer for 23 x 20 is 460

• 14 x 30





• 16 x 40





• 15 x 80




o So the final answer for 15 x 80 is 1,200

• 22 x 90





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Multiplying With Multiples of 10 – Guided Practice

Solve the Following Multiples of 10 Multiplication Problems

1. 11 x 20 = _____ 2. 12 x 30 = _____ 3. 13 x 40 = _____ 4. 14 x 50 = _____

5. 13 x 60 = _____ 6. 12 x 70 = _____ 7. 11 x 80 = _____ 8. 7 x 90 = _____

9. 21 x 40 = _____ 10. 23 x 50 = _____ 11. 22 x 30 = _____ 12. 31 x 40 = _____

13. 35 x 40 = _____ 14. 45 x 50 = _____ 15. 55 x 60 = _____ 16. 12 x 80 = _____

17. 34 x 20 = _____ 18. 42 x 30 = _____ 19. 81 x 40 = _____ 20. 95 x 40 = _____

21. 63 x 20 = _____ 22. 71 x 30 = _____ 12. 82 x 40 = _____ 24. 91 x 80 = _____

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• Multiply the number as if the zeros are not present

• After the multiplication is complete, just add two zeros to the answer

➢ Examples

• 23 x 200

o Temporarily ignore the two 0’s on 200 and just treat is as if it is 2


o Now add the two zeros to the answer → 4600


• 14 x 300





• 16 x 4000





• 15 x 800





• 22 x 900





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1. 11 x 200 = _____ 2. 12 x 300 = _____ 3. 13 x 400 = _____ 4. 14 x 500 = _____

5. 13 x 600 = _____ 6. 12 x 700 = _____ 7. 11 x 800 = _____ 8. 7 x 900 = _____

9. 21 x 400 = _____ 10. 23 x 500 = _____ 11. 22 x 300 = _____ 12. 31 x 400 = _____

13. 35 x 400 = _____ 14. 45 x 500 = _____ 15. 55 x 600 = _____ 16. 12 x 800 = _____

17. 34 x 200 = _____ 18. 42 x 300 = _____ 19. 81 x 400 = _____ 20. 95 x 400 = _____

21. 63 x 200 = _____ 22. 71 x 300 = _____ 12. 82 x 400 = _____ 24. 91 x 800 = _____

24





• Multiply the number as if the zeros are not present

• After the multiplication is complete, just add three zeros to the answer

➢ Examples

• 23 x 2000

o Temporarily ignore the three 0’s on 2000 and just treat is as if it is 2


o Now add the two zeros to the answer → 46000


• 14 x 3000





• 16 x 4000





• 15 x 8000





• 22 x 900,





25



1. 11 x 2000 = _____ 2. 12 x 3000 = _____ 3. 13 x 4000 = _____ 4. 14 x 5000 = _____

5. 13 x 6000 = _____ 6. 12 x 7000 = _____ 7. 11 x 8000 = _____ 8. 7 x 9000 = _____

9. 21 x 4000 = _____ 10. 23 x 5000 = _____ 11. 22 x 3000 = _____ 12. 31 x 4000 = _____

13. 35 x 4000 = _____ 14. 45 x 5000 = _____ 15. 55 x 6000 = _____ 16. 12 x 8000 = _____

17. 34 x 2000 = _____ 18. 42 x 3000 = _____ 19. 81 x 4000 = _____ 20. 95 x 4000 = _____

21. 63 x 2000 = _____ 22. 71 x 3000 = _____ 23. 82 x 4000 = _____ 24. 91 x 8000 = _____

26

Section 7.7 Division Facts (1 – 100)

Importance of Division Facts

➢ Similar to mastering multiplication facts ( 1 – 100), mastering division facts (1 – 100) will help you

perform well in the following areas

• Algebra

• Geometry

• Statistics and Probability

• Standardized Tests

o CRCT

o ACT

o PSAT

o SAT

o Graduation Exams (Example: Georgia Graduation Exam in Mathematics)

➢ What Division Facts should you master?

• Every quotient from 1 – 100 (besides 1 and the number)

• Examples

o 4 ÷ 2 = 2

o 6 ÷ 2 = 3

o 6 ÷ 3 = 2

o 8 ÷ 2 = 4

o 8 ÷ 4 = 2

o 9 ÷ 3 = 3

o 10 ÷ 2 = 5

o 10 ÷ 5 = 2

o 12 ÷ 2 = 6

o 12 ÷ 3 = 4

o 12 ÷ 4 = 3

o 12 ÷ 6 = 2

o 14 ÷ 2 = 7

o 14 ÷ 7 = 2

o 15 ÷ 3 = 5

o 15 ÷ 5 = 3

o 16 ÷ 2 = 8

o 16 ÷ 4 = 4

o 16 ÷ 8 = 2

o 18 ÷ 2 = 9

o 18 ÷ 3 = 6

o 18 ÷ 6 = 3

o 18 ÷ 9 = 2

o 20 ÷ 2 = 10

o 20 ÷ 4 = 5

o 20 ÷ 5 = 4

o 20 ÷ 10 = 2

27

Section 7.7 Division Facts (1 – 100) – continued

➢ What Division Facts should you master? (continued)


o 21 ÷ 3 = 7

o 21 ÷ 7 = 3

o 22 = 2 = 11

o 22 ÷ 11 = 2

o 24 ÷ 2 = 12

o 24 ÷ 3 = 8

o 24 ÷ 4 = 6

o 24 ÷ 6 = 4

o 24 ÷ 8 = 3

o 24 ÷ 12 = 2

o 25 ÷ 5 = 5

o 26 ÷ 2 = 13

o 26 ÷ 13 = 2

o 27 ÷ 3 = 9

o 27 ÷ 9 = 3

o 28 ÷ 2 = 14

o 28 ÷ 4 = 7

o 28 ÷ 7 = 4

o 28 ÷ 14 = 2

o 30 ÷ 2 = 15

o 30 ÷ 3 = 10

o 30 ÷ 5 = 6

o 30 ÷ 6 = 5

o 30 ÷ 10 = 3

o 30 ÷ 15 = 2

o 32 ÷ 2 = 16

o 32 ÷ 4 = 8

o 32 ÷ 8 = 4

o 32 ÷ 16 = 2

o 33 ÷ 3 = 11

o 33 ÷ 11 = 3

o 34 ÷ 2 = 17

o 34 ÷ 17 = 2

o 35 ÷ 5 = 7

o 35 ÷ 7 = 5

o 36 ÷ 2 = 18

o 36 ÷ 3 = 12

o 36 ÷ 4 = 9

o 36 ÷ 6 = 6

o 36 ÷ 9 = 4

o 36 ÷ 12 = 3

o 36 ÷ 18 = 2

28




o 38 ÷ 2 = 19

o 38 ÷ 19 = 2

o 39 ÷ 3 = 13

o 39 ÷ 13 = 3

o 40 ÷ 2 = 20

o 40 ÷ 4 = 10

o 40 ÷ 5 = 8

o 40 ÷ 8 = 5

o 40 ÷ 10 = 4

o 40 ÷ 20 = 2

o 42 ÷ 2 = 21

o 42 ÷ 3 = 14

o 42 ÷ 6 = 7

o 42 ÷ 7 = 6

o 42 ÷ 14 = 3

o 42 ÷ 21 = 2

o 44 ÷ 2 = 22

o 44 ÷ 4 = 11

o 44 ÷ 11 = 4

o 44 ÷ 22 = 2

o 45 ÷ 3 = 15

o 45 ÷ 5 = 9

o 45 ÷ 9 = 5

o 45 ÷ 15 = 3

o 46 ÷ 2 = 23

o 46 ÷ 23 = 2

o 48 ÷ 2 = 24

o 48 ÷ 3 = 16

o 48 ÷ 4 = 12

o 48 ÷ 6 = 8

o 48 ÷ 8 = 6

o 48 ÷ 12 = 4

o 48 ÷ 16 = 3

o 48 ÷ 24 = 2

o 49 ÷ 7 = 7

o 50 ÷ 2 = 25

o 50 ÷ 5 = 10

o 50 ÷ 10 = 5

o 50 ÷ 25 = 2

29




o 52 ÷ 2 = 26

o 52 ÷ 4 = 13

o 52 ÷ 13 = 4

o 52 ÷ 26 = 2

o 54 ÷ 2 = 27

o 54 ÷ 3 = 18

o 54 ÷ 6 = 9

o 54 ÷ 9 = 6

o 54 ÷ 18 = 3

o 54 ÷ 27 = 2

o 55 ÷ 5 = 11

o 55 ÷ 11 = 5

o 56 ÷ 2 = 28

o 56 ÷ 4 = 14

o 56 ÷ 7 = 8

o 56 ÷ 8 = 7

o 56 ÷ 14 = 4

o 56 ÷ 28 = 2

o 57 ÷ 3 = 19

o 57 ÷ 19 = 3

o 58 ÷ 2 = 29

o 58 ÷ 29 = 2

o 60 ÷ 2 = 30

o 60 ÷ 3 = 20

o 60 ÷ 4 = 15

o 60 ÷ 5 = 12

o 60 ÷ 6 = 10

o 60 ÷ 10 = 6

o 60 ÷ 12 = 5

o 60 ÷ 15 = 4

o 60 ÷ 20 = 3

o 60 ÷ 30 = 2

o 62 ÷ 2 = 31

o 62 ÷ 31 = 2

o 63 ÷ 3 = 21

o 63 ÷ 7 = 9

o 63 ÷ 9 = 7

o 63 ÷ 21 = 3

30




o 64 ÷ 2 = 32

o 64 ÷ 4 = 16

o 64 ÷ 8 = 8

o 64 ÷ 16 = 4

o 64 ÷ 32 = 2

o 65 ÷ 5 = 13

o 65 ÷ 13 = 5

o 66 ÷ 2 = 33

o 66 ÷ 3 = 22

o 66 ÷ 6 = 11

o 66 ÷ 11 = 6

o 66 ÷ 22 = 3

o 66 ÷ 33 = 2

o 68 ÷ 2 = 34

o 68 ÷ 4 = 17

o 68 ÷ 17 = 4

o 68 ÷ 34 = 2

o 69 ÷ 3 = 23

o 69 ÷ 23 = 3

o 70 ÷ 2 = 35

o 70 ÷ 5 = 14

o 70 ÷ 7 = 10

o 70 ÷ 10 = 7

o 70 ÷ 14 = 5

o 70 ÷ 35 = 2

o 72 ÷ 2 = 36

o 72 ÷ 3 = 24

o 72 ÷ 4 = 18

o 72 ÷ 6 = 12

o 72 ÷ 8 = 9

o 72 ÷ 9 = 8

o 72 ÷ 12 = 6

o 72 ÷ 18 = 4

o 72 ÷ 24 = 3

o 72 ÷ 36 = 2

o 74 ÷ 2 = 37

o 74 ÷ 37 = 2

o 75 ÷ 3 = 25

o 75 ÷ 5 = 15

o 75 ÷ 15 = 5

o 75 ÷ 25 = 3

31




o 76 ÷ 2 = 38

o 76 ÷ 4 = 19

o 76 ÷19 = 4

o 76 ÷ 38 = 2

o 77 ÷ 7 = 11

o 77 ÷ 11 = 7

o 78 ÷ 2 = 39

o 78 ÷ 6 = 13

o 78 ÷ 13 = 6

o 78 ÷ 39 = 2

o 80 ÷ 2 = 40

o 80 ÷ 4 = 20

o 80 ÷ 5 = 16

o 80 ÷ 8 = 10

o 80 ÷ 16 = 5

o 80 ÷ 20 = 4

o 80 ÷ 40 = 2

o 81 ÷ 3 = 27

o 81 ÷ 9 = 9

o 81 ÷ 27 = 3

o 82 ÷ 2 = 41

o 82 ÷ 41 = 2

o 84 ÷ 2 = 42

o 84 ÷ 3 = 28

o 84 ÷ 4 = 21

o 84 ÷ 6 = 14

o 84 ÷ 7 = 12

o 84 ÷ 12 = 7

o 84 ÷ 14 = 6

o 84 ÷ 21 = 4

o 84 ÷ 28 = 3

o 84 ÷ 42 = 2

o 85 ÷ 5 = 17

o 85 ÷ 17 = 5

o 86 ÷ 2 = 43

o 86 ÷ 43 = 2

o 87 ÷ 3 = 29

o 87 ÷ 29 = 3

32




o 88 ÷ 2 = 44

o 88 ÷ 4 = 22

o 88 ÷ 8 = 11

o 88 ÷ 11 = 8

o 88 ÷ 22 = 4

o 88 ÷ 44 = 2

o 90 ÷ 2 = 45

o 90 ÷ 3 = 30

o 90 ÷ 5 = 18

o 90 ÷ 6 = 15

o 90 ÷ 9 = 10

o 90 ÷ 10 = 9

o 90 ÷ 15 = 6

o 90 ÷ 18 = 5

o 90 ÷ 30 = 3

o 90 ÷ 45 = 2

o 91 ÷ 7 = 13

o 91 ÷ 13 = 7

o 92 ÷ 2 = 46

o 92 ÷ 4 = 23

o 92 ÷ 23 = 4

o 92 ÷ 46 = 2

o 93 ÷ 3 = 31

o 93 ÷ 31 = 3

o 94 ÷ 2 = 47

o 94 ÷ 47 = 2

o 95 ÷ 5 = 19

o 95 ÷ 19 = 5

o 96 ÷ 2 = 48

o 96 ÷ 3 = 32

o 96 ÷ 4 = 24

o 96 ÷ 6 = 16

o 96 ÷ 8 = 12

o 96 ÷ 12 = 8

o 96 ÷ 16 = 6

o 96 ÷ 24 = 4

o 96 ÷ 32 = 3

o 96 ÷ 48 = 2

33




o 98 ÷ 2 = 49

o 98 ÷ 7 = 14

o 98 ÷ 14 = 7

o 98 ÷ 49 = 2

o 99 ÷ 3 = 33

o 99 ÷ 9 = 11

o 99 ÷ 11 = 9

o 99 ÷ 33 = 3

o 100 ÷ 2 = 50

o 100 ÷ 4 = 25

o 100 ÷ 5 = 20

o 100 ÷ 10 = 10

o 100 ÷ 20 = 5

o 100 ÷ 25 = 4

o 100 ÷ 50 = 2

34

Division Facts – Guided Practice

Write the Division Fact Specified (Without Using the Previous Pages)

* Do not use 1 or the number specified (Example: 8 → 8 ÷ 2 = 4 and 8 ÷ 4 = 2)

1) Two division combinations for 12 → _______________________________________




5) One division combinations for 21 → ________________________________________

6) Three division combinations for 24 → ______________________________________

7) Two division combinations for 28 → ______________________________________

8) Four division combinations for 36 → ______________________________________

9) Three division combinations for 40 → _____________________________________

10) Four division combinations for 48 → _____________________________________

11) Three division combinations for 54 → _____________________________________

12) Five division combinations for 60 → _____________________________________



15) Three division combinations for 88 → ____________________________________


17) Two division combinations for 92 → _____________________________________


19) Two division combinations for 99 → _____________________________________

20) Three division combinations for 100 → ____________________________________

35

Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10)

➢ Divisible By 2

• Every even number is divisible by 2

• No odd number is divisible by 2

• 2,754 (an even number) is divisible by 2

• 1,983 (an odd number) is not divisible by 2

➢ Divisible By 3

• Add the individual digits of the number

• If the sum of the digits is divisible by 3, then the original number is divisible by 3

• If the sum of the digits is not divisible by 3, then the original number is not divisible by 3

• 258

o Sum the digits of 258 → 2 + 5 + 8 = 15

o Since the sum of the digits (15) is divisible by 3, the number 258 is divisible by 3

• 718

o Sum the digits of 718→ 7 + 1 + 8 = 16

o Since the sum of the digits (16) is not divisible by 3, the number 718 is not divisible by 3

➢ Divisible By 4

• Look at the last (rightmost) 2 digits of the number

• If the rightmost 2-digit value is divisible by 4, then the original number is divisible by 4

• If the rightmost 2-digit value is not divisible by 4, then the original number is not divisible by 4

• 3,476

o 76 is made from the last (rightmost) 2 digits of 3,476

o Since 76 is divisible by 4 (76 ÷ 4 = 19), then 3,476 is divisible by 4

• 1,986


o Since 86 is not evenly divisible by 4 (86 ÷ 4 = 21½ ), then 1,986 is not divisible by 4

➢ Divisible By 5

• Look at the last (rightmost) digit

• If the rightmost digit is 0 or 5, the number is divisible by 5

• If the rightmost digit is not 0 or 5, the number is not divisible by 5

• 3,475

o The rightmost digit of 3,475 is 5

o The number 3,475 is therefore divisible by 5

• 3,480

o The rightmost digit of 3,480 is 0

o The number 3,480 is therefore divisible by 5

• 1,986

o The rightmost digit of 1,986 is not 0 or 5

o The number 1,986 is therefore not divisible by 5

36

Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10) – continued

➢ Divisible By 6

• Every even number that is divisible by 3 is also divisible by 6

• No odd number that is divisible by 3 is divisible by 6

• If the sum of the digits is divisible by 3 and the number is even, then the original number is divisible

by 6

• If the sum of the digits is not divisible by 3 or the number is odd, then the original number is not

divisible by 3

• 258

o Since the number is even, sum the digits of 258 → 2 + 5 + 8 = 15

o Since the sum of the digits (15) is divisible by 3 and 258 is even, the number 258 is divisible by

3

• 735

o Although the sum of the digits of 735 (7 + 3 + 5 = 15) is evenly divisible by 3, the number 735

is odd and therefore cannot by divisible by 6

➢ Divisible By 7

• Double the rightmost digit

• Subtract this product from the value of the remaining digits

• If the difference is divisible by 7, then the original number is divisible by 7

• If the difference is not divisible by 7, then the original number is not divisible by 7

• 574

o Double the rightmost digit (4) → 4 x 2 = 8

o Subtract 8 from the value of the remaining digits (57) → 57 – 8 = 49

o Since 49 is divisible by 7, the value 574 is divisible by 7

• 734

o Double the rightmost digit (4) → 4 x 2 = 8

o Subtract 8 from the value of the remaining digits (73) → 73 – 8 = 65

o Since 65 is not divisible by 7, the value 734 is not divisible by 7

➢ Divisible By 8

• In order for a number to be divisible by 8, it must by divisible by 4

• Look at the last (rightmost) 2 digits of the number

• If the rightmost 2-digit value is not divisible by 4, then the number cannot by divisible by 8

• If the rightmost 2-digit value is divisible by 4, then perform the following steps

o If the rightmost 2-digit value is divisible by 8 and the 3rd digit from the right is even, then the

number is divisible by 8

o If the rightmost 2-digit value is not divisible by 8, but the 3rd digit from the right is odd, then the

number is divisible by 8

o For any other combination, the number is not divisible by 8

• 4,832


o 32 is divisible by 4

o Since 32 is divisible by 8 and the 3rd digit from the right (8) is even, then 4,832 is divisible by 8

37

Section 7.8 Mastering Divisibility Rules (By 2, 3, 4, 5, 6, 7, 8, 9, 10) – continued

➢ Divisible By 8 (continued)

• 3,476



o Since 76 is not divisible by 8 and the 3rd digit from the right (4) is even, then 3,476 is not

divisible by 8

• 5,728



o Since 28 is not divisible by 8 and the 3rd digit from the right (7) is odd, then 5,728 is divisible by

8

• 6,916



o Since 16 is divisible by 8 but the 3rd digit from the right (9) is odd, then 6,916 is not divisible by

8

➢ Divisible By 9

• Add the individual digits of the number

• If the sum of the digits is divisible by 9, then the original number is divisible by 9

• If the sum of the digits is not divisible by 9, then the original number is not divisible by 9

• 558

o Sum the digits of 558 → 5 + 5 + 8 = 18

o Since the sum of the digits (18) is divisible by 9, the number 558 is divisible by 9

• 258

o Sum the digits of 258→ 2 +5 + 8 = 15

o Since the sum of the digits (15) is not divisible by 9, the number 258 is not divisible by 9

➢ Divisible By 10

• Look at the last (rightmost) digit of the number

• If the rightmost digit is 0, then the original number is divisible by 10

• If the rightmost digit is not 0, then the original number is not divisible by 10

• 1,340

o Since the rightmost digit of 1,340 is 0, the number 1,340 is divisible by 10

• 2,458

o Since the rightmost digit of 2,458 is not 0, the number 2,458 is not divisible by 10

➢ General Divisibility Rule

• An odd number cannot ever be divisible by an even number

• An even number can be divisible by an odd number and/or and even number

38

Divisibility – Guided Practice

Determine which number(s), if any, will divide evenly the specified value. Circle the answer(s).

1. 67 2 3 4 5 6 7 8 9 10 none

2. 84 2 3 4 5 6 7 8 9 10 none

3. 98 2 3 4 5 6 7 8 9 10 none

4. 129 2 3 4 5 6 7 8 9 10 none

5. 240 2 3 4 5 6 7 8 9 10 none

6. 375 2 3 4 5 6 7 8 9 10 none

7. 484 2 3 4 5 6 7 8 9 10 none

8. 720 2 3 4 5 6 7 8 9 10 none

9. 840 2 3 4 5 6 7 8 9 10 none

10. 915 2 3 4 5 6 7 8 9 10 none

39

Number Principles – Guided Reinforcement

Number Properties

__________ 1. Using the Commutative Property of Addition, how can you rewrite 7 + 18?

__________ 2. Using the Commutative Property of Addition, how can you rewrite 35 + –15?

__________ 3. Using the Commutative Property of Addition, how can you rewrite –13 + 9?

__________ 4. Using the Commutative Property of Addition, how can you rewrite –17 + –23?

__________ 5. Using the Associative Property of Addition, how can you rewrite (9 + 2) + 7?

__________ 6. Using the Associative Property of Addition, how can you rewrite ((6 + (–2)) + 8?

__________ 7. Using the Associative Property of Addition, how can you rewrite (5 + 4) + (–11)?

__________ 8. Using the Associative Property of Addition, how can you rewrite –14 + (8 + (–3))?

__________ 9. Using the Commutative Property of Multiplication, how can you rewrite 23 x 4?

__________ 10. Using the Commutative Property of Multiplication, how can you rewrite 18 x (–3)?

__________ 11. Using the Commutative Property of Multiplication, how can you rewrite (–6) x 12?

__________ 12. Using the Commutative Property of Multiplication, how can you rewrite (–15) x (–5)?

40

Numbers Principles – Guided Reinforcement (continued)

Number Properties (continued)

__________ 13. Using the Associative Property of Multiplication, how can you rewrite (4 x 5) x 9?

__________ 14. Using the Associative Property of Multiplication, how can you rewrite ((3 x (–2)) x 7?

__________ 15. Using the Associative Property of Multiplication, how can you rewrite (5 x 4) x (–8)?

__________ 16. Using the Associative Property of Multiplication, how can you rewrite –6 x (2 x 3)?

__________ 17. Using the Distributive Property, how can you rewrite (8 x 3) + (8 x 5)?

__________ 18. Using the Distributive Property, how can you rewrite ((–7) x 5) + ((–7) x 8)?

__________ 19. Using the Distributive Property, how can you rewrite 6 x (4 + 7)?

__________ 20. Using the Distributive Property, how can you rewrite –3 x (9 + (–4))?

__________ 21. Using the Identity Property of Addition, how can you rewrite 18 + 0?

__________ 22. Using the Identity Property of Addition, how can you rewrite –11 + 0?

__________ 23. Using the Identity Property of Multiplication, how can you rewrite 18 x 1?

__________ 24. Using the Identity Property of Multiplication, how can you rewrite (–11) x 1?

41


Multiplication

Solve the following multiplication problems without using a calculator.

__________ 1. What is 16 x 7?

(_____ + _____) x 7

(_____ x 7) + (_____ x 7)

_____ + _____

__________ 2. What is 24 x 9?

(_____ + _____) x 9

(_____ x 9) + (_____ x 9)

_____ + _____

__________ 3. What is 36 x 7?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 4. What is 42 x 8?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

42


Multiplication (continued)

__________ 5. What is 48 x 7?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 6. What is 53 x 6?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 7. What is 66 x 5?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 8. What is 72 x 8?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

43



__________ 9. What is 83 x 6?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 10. What is 94 x 6?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 11. What is 12 x 17?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 12. What is 13 x 18?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

44



__________ 13. What is 24 x 15?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 14. What is 28 x 12?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 15. What is 33 x 15?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 16. What is 38 x 13?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

45



__________ 17. What is 43 x 14?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 18. What is 47 x 14?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 19. What is 56 x 13?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

__________ 20. What is 64 x 16?

(_____ + _____) x _____

(_____ x _____) + (_____ x _____)

_____ + _____

46


Multiplication with Multiples of 10, 100, and 1000

1. 17 x 40 = _____ 2. 18 x 500 = _____ 3. 19 x 6000 = _____ 4. 21 x 4000 = _____

5. 13 x 60 = _____ 6. 17 x 500 = _____ 7. 18 x 300 = _____ 8. 8 x 9000 = _____

9. 21 x 30 = _____ 10. 23 x 40 = _____ 11. 22 x 500 = _____ 12. 24 x 4000 = _____

13. 34 x 40 = _____ 14. 45 x 300 = _____ 15. 55 x 400 = _____ 16. 22 x 4000 = _____

17. 36 x 20 = _____ 18. 38 x 200 = _____ 19. 43 x 2000 = _____ 20. 65 x 2000 = _____

21. 63 x 30 = _____ 22. 71 x 200 = _____ 23. 82 x 3000 = _____ 24. 90 x 4000 = _____

47


Divisibility Rules

__________ 1. Circle the numbers by which 147 is divisible. 2 3 4 5 6 7 8 9 10








__________ 9. Circle the numbers by which 1,200 is divisible. 2 3 4 5 6 7 8 9 10


48


Divisibility Rules (continued)











49

Module 8: Number Theory and Terminology

50


51

Section 8.1 Factors and Multiples

Factor (Definition and Examples)

➢ An Integer (positive or negative) that can be multiplied by another Integer to result in an Integer

product

➢ Factors of 12

• 1 → 1 x 12 = 12

• 2 → 2 x 6 = 12

• 3 → 3 x 4 = 12

• 4 → 4 x 3 = 12

• 6 → 6 x 2 = 12

• 12 → 12 x 1 = 12

• -1 → -1 x 12 = 12

• -2 → -2 x 6 = 12

• -3 → -3 x 4 = 12

• -4 → -4 x 3 = 12

• -6 → -6 x 2 = 12

• -12 → -12 x 1 = 12

Greatest Common Factor (GCF)

➢ The largest factor common to two or more Integers

➢ Greatest Common Factor of 12 and 16

• Factors of 12: 1, 2, 3, 4, 6, 12

• Factors of 16: 1, 2, 4, 8, 16

• The largest factor common to both 12 and 16 (GCF) → 4

➢ Greatest Common Factor of 16, 24, and 40

• Factors of 16: 1, 2, 4, 8, 16

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

• The largest factor common to 16, 24, and 40 (GCF) → 8

Multiple (Definition and Examples)

➢ A number which is the product of two or more of its factors

➢ Every value has an infinite number of multiples

➢ Multiples of 12

• 12 → 1 x 12 = 12

• 24 → 2 x 12 = 24

• 36 → 3 x 12 = 36

• 48 → 4 x 12 = 48

• 60 → 5 x 12 = 60

52

Section 8.1 Factors and Multiples (continued)

Least Common Multiple (LCM)

➢ The smallest multiple common to two or more Integers

➢ Least Common Multiple of 12 and 16

• Multiples of 12: 12, 24, 36, 48, 60

• Multiples of 16: 16, 32, 48, 64, 80

• The smallest multiple common to both 12 and 16 (LCM) → 48

➢ Least Common Multiple 16, 24, and 40

• Multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240

• Multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240

• Multiples of 40: 40, 80, 120, 160, 200, 240, 280

• The smallest multiple common to 16, 24, and 40 (LCM) → 240

53

Factors and Multiples – Guided Practice

____________________ 1. List 3 Factors of 36.

____________________ 2. What is the Greatest Common Factor of 36 and 48?

____________________ 3. List 3 Multiples of 9.

____________________ 4. What is the Least Common Multiple of 15 and 12?

54

Section 8.2 Prime Numbers and Composite Numbers

Prime Number (Definition and Examples)

➢ A positive integer that can be divided only by itself and by 1

• By definition 0 and 1 are not prime numbers

• 2 is the smallest prime number

• 2 is the only even prime number

• All other prime numbers are odd, but not all odd numbers are prime

➢ Memorize all Prime Numbers less than 100

• 2

• 3

• 5

• 7

• 11

• 13

• 17

• 19

• 23

• 29

• 31

• 37

• 41

• 43

• 47

• 53

• 59

• 61

• 67

• 71

• 73

• 79

• 83

• 97

➢ The following number are often incorrectly thought to be Prime Numbers

• 21 (7 x 3)

• 27 (9 x 3)

• 33 (11 x 3)

• 39 (13 x 3)

• 49 (7 x 7)

• 51 (17 x 3)

• 57 (19 x 3)

• 87 (29 x 3)

• 91 (13 x 7)

55

Section 8.2 Prime Numbers and Composite Numbers (continued)

Composite Number (Definition and Examples)

➢ A positive integer that can be divided not only by itself and by 1, but also by other positive Integers

• A positive Integer that contains factors in addition to 1 and itself

• A positive Integer that is not a Prime Number

• All even numbers (besides 2) are composite

• Many odd numbers are composite

➢ Odd Composite Numbers less than 100

• 9 (3 x 3)

• 15 (5 x 3)

• 21 (7 x 3)

• 25 (5 x 5)

• 27 (9 x 3)

• 33 (11 x 3)

• 35 (7 x 5)

• 39 (13 x 3)

• 45 (15 x 3)

• 49 (7 x 7)

• 51 (17 x 3)

• 55 (11 x 5)

• 57 (19 x 3)

• 63 (21 x 3)

• 65 (13 x 3)

• 69 (23 x 3)

• 75 (25 x 3 & 15 x 5)

• 77 (11 x 7)

• 81 (27 x 3 & 9 x 9)

• 85 (17 x 5)

• 87 (29 x 3)

• 91 (13 x 7)

• 93 (31 x 3)

• 95 (19 x 5)

• 99 (33 x 3)

56


➢ Prime Numbers

• A prime number is a number divisible only by 1 and itself

• Recognize these numbers are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,53, 59, 61,

67, 71, 73, 79, 83, 89, 97

➢ Factors of Composite Numbers

• A composite number has factors besides 1 and itself

• Recognize the numbers below as composite

• Learn the factors below of all composite numbers between 1 and 100

Key Math Factors

4 2

6 2 3

8 2 4

9 3

10 2 5

12 2 3 4 6

14 2 7

15 3 5

18 2 3 6 9

20 2 4 5 10

21 3 7

22 2 11

24 2 3 4 6 8 12

25 5

26 2 13

27 3 9

28 2 4 7 14

30 2 3 5 6 10 15

32 2 4 8 16

33 3 11

34 2 17

35 5 7

36 2 3 4 6 9 12 18

38 2 19

39 3 13

40 2 4 5 8 10 20

42 2 3 6 7 14 21

44 2 4 11 22

45 3 5 9 15

46 2 23

48 2 3 4 6 8 12 16 24

49 7

50 2 5 10 25

57


Key Math Factors (continued)

➢ Factors of Composite Numbers (continued)

51 3 17

52 2 4 13 26

54 2 3 6 9 18 27

55 5 11

56 2 4 7 8 14 28

57 3 19

58 2 29

60 2 3 4 5 6 10 12 15 20 30

62 2 31

63 3 7 9 21

64 2 4 8 16 32

65 5 13

66 2 3 6 11 22 33

68 2 4 17 34

69 3 23

70 2 5 7 10 14 35

72 2 3 4 6 8 9 12 18 24 36

74 2 37

75 3 5 15 25

76 2 4 19 38

77 7 11

78 2 3 6 13 26 39

80 2 4 8 10 20 40

81 3 9 27

82 2 41

84 2 4 6 7 12 14 21 42

85 5 17

86 2 23

87 3 29

88 2 4 8 11 22 44

90 2 3 5 6 9 10 15 18 30 45

91 7 13

92 2 4 23 46

93 3 31

94 2 47

95 5 19

96 2 3 4 6 8 12 16 24 32 48

98 2 7 14 49

99 3 9 11 33

100 2 4 5 10 20 25 50

58


➢ Multiplication Fact of Composite Numbers (1 – 100)

• Learn all of the following multiplication fact for Composite Numbers between 1 and 100

• It will tremendously help you in the following subjects

o Algebra

o Geometry

o Probability

o Statistics

o Trigonometry

o Pre-Calculus

Key Multiplication Fact of Composite Numbers (1 – 100)

4 2 x 2

6 2 x 3

8 2 x 4

9 3 x 3

10 2 x 5

12 2 x 6 3 x 4

14 2 x 7

15 3 x 5

18 2 x 9 3 x 6

20 2 x 10 4 x 5

21 3 x 7

22 2 x 11

24 2 x 12 3 x 8 4 x 6

25 5 x 5

26 2 x 13 13

27 3 x 9

28 2 x 14 4 x 7

30 2 x 15 3 x 10 5 x 6

32 2 x 16 4 x 8

33 3 x 11

34 2 x 17

35 5 x 7

36 2 x 18 3 x 12 4 x 9 6 x 6

38 2 x 19

39 3 x 13

40 2 x 20 4 x 10 5 x 8

42 2 x 21 3 x 14 6 x 7

44 2 x 22 4 x 11

45 3 x 15 5 x 9

46 2 x 23

48 2 x 24 3 x 16 4 x 12 6 x 8

49 7 x 7

50 2 x 25 5 x 10

59


Key Math Factors (continued)

➢ Factors of Composite Numbers (continued)

51 3 x 17

52 2 x 26 4 x 13

54 2 x 27 3 x 18 6 x 9

55 5 x 11

56 2 x 28 4 x 14 7 x 8

57 3 x 19

58 2 x 29

60 2 x 30 3 x 20 4 x 15 5 x 12 6 x 10

62 2 x 31

63 3 x 21 7 x 9

64 2 x 32 4 x 16 8 x 8

65 5 x 13

66 2 x 33 3 x 22 6 x 11

68 2 x 34 4 x 17

69 3 x 23

70 2 x 35 5 x 14 7 x 10

72 2 x 36 3 x 24 4 x 18 6 x 12 8 x 9

74 2 x 37

75 3 x 25 5 x 15

76 2 x 38 4 x 19

77 7 x 11

78 2 x 39 3 x 26 6 x 13

80 2 x 40 4 x 20 8 x 10

81 3 x 27 9 x 9

82 2 x 41

84 2 x 42 4 x 21 6 x 14 7 x 12

85 5 x 17

86 2 x 23

87 3 x 29

88 2 x 44 4 x 22 8 x 11

90 2 x 45 3 x 30 5 x 18 6 x 15 9 x 10

91 7 x 13

92 2 x 46 4 x 23

93 3 x 31

94 2 x 47

95 5 x 19

96 2 x 48 3 x 32 4 x 24 6 x 16 8 x 12

98 2 x 49 7 x 14

99 3 x 33 9 x 11

100 2 x 50 4 x 25 5 x 10 10 x 10

60


Prime Factorization (Definition and Examples)

➢ A process of breaking down a Composite Number into the product of its Prime Factors

➢ Prime Factorization of 12

12 12

/ \ / \

6 x 2 4 x 3

/ \ \ / \ \

3 x 2 x 2 2 x 2 x 3

➢ The bottom line of Prime Factorization contains only Prime Numbers multiplied by one another

• The order of the Prime Numbers does not matter

• The order of the Prime Numbers will depend on the Factors initially chosen

• Regardless of the Factors initially chosen, the same number of Prime Numbers will also result in the

Prime Factorization (bottom) line

o The Prime Factorization of 12 will always result in the product of two 2’s and one 3

61


Equivalent Products

➢ Equivalent Products results from multiplying combinations of factors of a number

➢ Equivalent Products of 72

• 1 x 72

• 2 x 36 (which is the double of 1 and half of 72)

• 3 x 24 (which is the triple of 1 and one third of 72)

• 4 x 18 (which is quadruple of 1 and one fourth of 72)

• 6 x 12 (which is six times 1 and one sixth of 72)

• 8 x 9 (which is eight times 1 and one eighth of 72)

• 9 x 8 (which is nine times 1 and one ninth of 72)

• 12 x 6 (which is twelve times 1 and one twelfth of 72)

• 18 x 6 (which is eighteen times 1 and one eighteenth of 72)

• 24 x 3 (which is twenty-four times 1 and one twenty-fourth of 72)

• 36 x 2 (which is thirty-six times 1 and one thirty-sixth of 72)

➢ Results when you multiply a number by a value and divide the number by the same value

• The multiplication is offset by the division

• The division is offset by the multiplication

• Multiplying by 2 if offset by dividing by 2 (which is the same as taking one half)

• Multiplying by 3 is offset by dividing by 3 (which is the same as taking one third)

• Multiplying by 4 is offset by dividing by 4 (which is the same as taking one fourth)

• Multiplying by 5 is offset by dividing by 5 (which is the same as taking one fifth)

• Multiplying by 6 is offset by dividing by 6 (which is the same as taking one sixth)

• Multiplying by 7 is offset by dividing by 7 (which is the same as taking one seventh)

• Multiplying by 8 is offset by dividing by 8 (which is the same as taking one eighth)

• Multiplying by 9 is offset by dividing by 9 (which is the same as taking one ninth)

• Multiplying by 10 is offset by dividing by 10 (which is the same as taking one tenth)

• This pattern continues forever

62

Prime Numbers and Composite Numbers – Guided Practice

____________________ 1. List 3 Prime Numbers between 30 and 45.

____________________ 2. List 3 Odd Composite Numbers between 30 and 45.

____________________ 3. Perform Prime Factorization on 24 (fill in the blanks where appropriate)

24

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____

____________________ 4. What is an equivalent Product of 12 x 8?

63

Section 8.3 Principles of Numbers

Positive Numbers

➢ Numbers greater than zero

➢ Simple examples : 1 1½ 37.6 ½ 1,000 1,000,000 1,000,000,000

Negative Numbers

➢ Numbers less than zero

➢ Simple examples : -1 -1½ -37.6 -½ -1,000 -1,000,000 -1,000,000,000

Zero

➢ A number which is neither negative nor positive

➢ Zero is “nothing”

Number Line Representation of Numbers

➢ Positive Numbers

• Positioned to the right of zero

• The further to the right the number is, the more positive (larger value) the number is

➢ Negative Numbers

• Positioned to the left of zero

• The further to the left the number is, the more negative (smaller value) the number is

64

Principles of Numbers – Guided Practice

__________ 1. Which of the following numbers are negative? 25, -37.8, 15.3, 45, -80 10½

__________ 2. Which of the following numbers are positive? 25, -37.8, 15.3, 45, -80 10½

__________ 3. Shade in the portion of the number line below (between -4 and 5) that is negative.

65

Section 8.4 Number Terminology

Integer

➢ A positive or negative number (including zero) that has neither a fractional nor decimal part

➢ Simple examples: -1,000,000 -25 -2 -1 0 1 2 25 1,000,000

➢ The following number are not Integers: 8

7 -

3

10 1

2

1 3.1415926 5

Consecutive Integers

➢ Integers listed in increasing order of size without any integers missing in between

• Can include negative and/or positive values

➢ Simple example: -4, -3, -2, -1, 0, 1, 2, 3, 4

➢ The following examples are not Consecutive Integers:

• 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6 → decreasing (not increasing) order of size

• -6, -4, -2, 0, 2, 4, 6 → missing (odd) Integers in between

• -2.0, -1.5, -1.0, -0.5, 0, 0.5, 1.0, 1.5, 2.0 → not Integers

Even Number

➢ An Integer that divides evenly by 2

➢ Must end in 0, 2, 4, 6, or 8

➢ Simple examples : 14 1,000 3,576 1,000,000 1,000,000,000

Odd Number

➢ An Integer that does not divide evenly by 2

➢ Must end in 1, 3, 5, 7, or 9

➢ Simple examples : 15 1,001 3,577 1,000,001 1,000,000,001

Rational Number

➢ Any number that can be written as a ratio of two numbers

• It can be written as a fraction where both the numerator and denominator are Integers

➢ Simple examples : -1,000,000 -1,000 -11

5 0

11

5 1,000 1,000,000

Irrational Number

➢ Any number that cannot be written as a ratio of two numbers

• Any number that is not a Rational Number

➢ Simple examples : - 13 - 5 - 3 3 5

66

Section 8.4 Number Terminology (continued)

Remainder

➢ The “left over” value when one Integer does not evenly divide into another Integer

• Remainder of 17 3 → 2



Digits and Place Value

➢ In the Decimal Numbering System, the digits are

• 0

• 1

• 2

• 3

• 4

• 5

• 6

• 7

• 8

• 9

➢ The value of an Integer is determined by the value and position of the digits of the Integer

➢ The rightmost digit position is referred to as the “units digit”

➢ 2,475 is

• A “four-digit” number

• Comprised of the following digits and values

o 2 (thousands position)

o 4 (hundreds position)

o 7 (tens position)

o 5 (units position)

67

Number Terminology – Guided Practice

__________ 1. Which of the following values are Integers? -50, ½ , -1,000,000, 3.87, 1,000.01

__________ 2. Which of the following values are not Integers? -50, ½ , -1,000,000, 3.87, 1,000.01

__________ 3. List three Consecutive Integers starting at 22.

__________ 4. List three Consecutive Integers starting at – 8.

__________ 5. List the four Even Numbers immediately larger than 18.

__________ 6. List the four Even Numbers immediately smaller than 18.

__________ 7. List the four Even Numbers immediately larger than – 25.

__________ 8. List the four Even Numbers immediately smaller than – 25.

__________ 9. List the four Odd Numbers immediately larger than 18.

__________ 10. List the four Odd Numbers immediately smaller than 18.

__________ 11. List the four Odd Numbers immediately larger than – 25.

__________ 12. List the four Odd Numbers immediately smaller than – 25.

68

Number Terminology – Guided Practice (continued)

13. Categorize the following numbers either as Rational or Irrational

__________ 7

1 __________ 7 __________ 33.3333

__________ 14. What is the remainder of 38 4?

__________ 15. In the number 12,874,935, what digit is in the millions position?

69

Section 8.5 Arithmetic Operations on Numbers

Common Arithmetic Operations on Numbers

➢ Addition → +

➢ Subtraction → –

➢ Multiplication → x

➢ Division → ÷

Standard Arithmetic Symbols

➢ = : is equal to

➢ : is not equal to

➢ < : is less than

➢ > : is greater than

➢ : is less than or equal to

➢ : is greater than or equal to

Addition (Sum)

➢ When a number is added to another number, the answer is call the Sum

➢ Adding a positive number to a positive number → 8 + 5

• Start at the first number (8)

• Move to the right the number of positions of the second number (5)

• 8 + 5 = 13

➢ Adding a positive number to a negative number → -8 + 5

• Start at the first number (-8)


• -8 + 5 = -3

• Notice that the sign of the answer (negative) is the same as the sign of the larger number (-8)

➢ Adding a negative number to a positive number → 8 + (-5)


• Move to the left the number of positions of the second number (5)

o This is the same as subtracting the second number from the first number

o 8 + (-5) → 8 – 5

• 8 + (-5) = 3

• Notice that the sign of the answer (positive) is the same as the sign of the larger number (8)

➢ Adding a negative number to a negative number → -8 + (-5)



o This is the same as subtracting the second number from the first number

o -8 + (-5) → -8 – 5

o Because both numbers are negative, you actually add the values and keep the negative sign

• -8 + (-5) = -13

• Notice that the sign of the answer (negative) is the same as the sign of both numbers

70

Section 8.5 Arithmetic Operations on Numbers (continued)

Subtraction (Difference)

➢ When a number is subtracted from another number, the answer is call the Difference

➢ Subtracting a positive number from a positive number → 8 – 5



o The is the same as adding a negative number to a positive number

o 8 – 5 → 8 + (-5)

• 8 – 5 = 3

• Notice that the sign of the answer (positive) is the same as the sign of the larger (8) value

➢ Subtracting a positive number from a negative number → -8 – 5



o This is the same as adding a negative number to another negative number

o -8 – 5 → -8 + (-5)

o Because both numbers are negative, you actually add the values and keep the negative sign

• -8 – 5 = -13

➢ Subtracting a negative number from a positive number → 8 – (-5)



o The is the same as adding the second number to the first number

o 8 – (-5) → 8 + 5

o Because both numbers are positive, you actually add the values and keep the positive sign

• 8 – (-5) → 8 + 5 → 13

➢ Subtracting a negative number from a negative number → -8 – (-5)



o This is the same as adding the second number to the first number

o -8 – (-5) → -8 + 5

• -8 – (-5) → -8 + 5 → -3

• Notice that the sign of the answer (negative) is the same as the sign of the larger (-8) value

71


Multiplication (Product)

➢ When a number is multiplied by another number, the answer is call the Product

➢ Multiplication is “repeated addition”

• 8 x 5 means “repeatedly add the value 8 (5 sets of times)” → 8 + 8 + 8 + 8 + 8

• 5 x 8 means “repeatedly add the value 5 (8 sets of times)” → 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

• Multiplication is a “short hand” way is saying “add this value over and over this many times”

➢ Multiplying a positive number with a positive number → 8 x 5

• Results in a positive answer

• 8 x 5 = 40

➢ Multiplying a positive number with a negative number → 8 x (-5)

• Results in a negative answer

• 8 x (-5) = -40

➢ Multiplying a negative number with a positive number → (-8) x 5


• (-8) x 5 = -40

➢ Multiplying a negative number with a negative number → (-8) x (-5)


• (-8) x (-5) = 40

➢ General Multiplication Principles

• When multiplying two values:

o When the signs are the same, the answer is positive

o When the signs are different, the answer is negative

• When multiplying more than two values:

o When there is an even number of negative values, the answer is positive

o When there is an odd number of negative values, the answer is negative

72


Division (Quotient)

➢ When a number is divided into another number, the answer is call the Quotient

➢ Division is “repeated subtraction”

• 40 ÷ 5 means “repeatedly divide 40 in to groups containing 5 elements”

• The final answer specifies “how many groups” there are

• Repeatedly “dividing 40 into groups containing 5 elements” will result in 8 groups

o Therefore, 40 ÷ 5 = 8

➢ Dividing a positive number by a positive number → 40 ÷ 5


• 40 ÷ 5 = 8

➢ Dividing a positive number by a negative number → 40 ÷ (-5)


• 40 ÷ (-5) = -8

➢ Dividing a negative number by a positive number → (-40) ÷ 5


• (-40) ÷ 5 = -8

➢ Dividing a negative number with a negative number → (-40) ÷ (-5)


• (-40) ÷ (-5) = 8

➢ General Division Principles

• When dividing two values:

o When the signs are the same, the answer is positive

o When the signs are different, the answer is negative

• When dividing more than two values:

o When there is an even number of negative values, the answer is positive

o When there is an odd number of negative values, the answer is negative

73

Module 8 (Number Theory and Terminology) – Review Exercises

Factors and Multiples

For Factor questions, use factors other than 1 and the number.

__________ 1. List 3 Factors of 24.

__________ 2. List 4 Factors of 36.

__________ 3. List 4 Factors of 48.

__________ 4. List 4 Factors of 60.

__________ 5. List 4 Factors of 75.

__________ 6. What is the Greatest Common Factor of 12 and 16?





__________ 11. List 3 Multiples of 8.



74

Module 8 (Number Theory and Terminology) – Review Exercises (continued)

Factors and Multiples (continued)



__________ 16. What is the Least Common Multiple of 6 and 15?





75


Prime Numbers and Composite Numbers

__________ 1. List 3 Prime Numbers between 5 and 15.





__________ 6. List 3 Odd Composite Numbers between 4 and 16.



__________ 9. List 3 Odd Composite Numbers between 50 and 60

__________ 10. List 3 Odd Composite Numbers between 16 and 34

76


Prime Numbers and Composite Numbers (continued)

__________ 11. Perform Prime Factorization on 30 (fill in the blanks where appropriate)

30

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____


36

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____


40

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

77


Prime Numbers and Composite Numbers (continued)


48

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____


60

/ \

_____ x _____

/ \ / \

_____ x _____ _____ x _____

/ \ / \ / \ / \

_____ x _____ _____ x _____ _____ x _____ _____ x _____

__________ 16. What is an equivalent Product of 9 x 4?





78


Positive & Negative Number Arithmetic

__________ 1. What arithmetic operation does the symbol – perform?

__________ 2. What arithmetic operation does the symbol x perform?

3. What is the value of the following arithmetic operations?

__________ 8 + 13

__________ 8 + (– 13)

__________ –8 + 13

__________ –8 + (–13)

__________ 5 – 11

__________ 5 – (–11)

__________ –5 – 11

__________ –5 – (–11)

__________ 7 x 8

__________ 7 x (– 8)

__________ (–7) x 8

__________ (–7) x (– 8)

__________ 72 ÷ 8

__________ 72 ÷ (–8)

__________ (– 72) ÷ 8

__________ (– 72) ÷ (–8)

79

Module 9: Principles of Fractions

80


81

Section 9.1 Introduction to Fractions

Introduction to Fractions

➢ A comparison of the “part” (numerator) to the “whole” (denominator)

• Three pieces out of eight equally cut pieces of pie → 8

3

• Five free throws made out of six free throws attempted → 6

5

• Four complete pizzas and seven pieces out of ten equally cut pieces of pizza → 410

7

Types of Fractions

➢ Proper

• The numerator is smaller than the denominator

• 8

7 is a Proper Fraction

➢ Improper

• The numerator is larger than the denominator

• 7

8 is a Proper Fraction

➢ Mixed

• Contains both a whole number part and a fractional part

• 18

7

is a Mixed Fraction

Simplest-Form Fraction

➢ A fraction where the greatest common factor of the numerator and the denominator is 1

• 8

3 → the greatest common factor of 3 and 8 is 1 (this is a simplest-form fraction)

• 8

6 → the greatest common factor of 6 and 8 is 2 (this is not a simplest-form fraction)

82

Section 9.1 Introduction to Fractions (continued)

Equivalent Fractions

➢ Fractions with different numerators and denominators that represent the same value

• The value 2

1 can be represented by the fractions

4

2 and

6

3

• Therefore, the fractions 2

1,

4

2, and

6

3 are Equivalent Fractions

➢ Equivalent fractions will eventually reduce to the same Simplest-Form Fraction

• 72

60 is an equivalent fraction to

42

35

• Both fractions eventually reduce to 6

5

Reciprocal

➢ Switching the numerator and denominator of a fraction

➢ Reciprocal of 9

4 →

4

9

➢ Reciprocal of 8 (or 1

8) →

8

1

➢ Reciprocal of 27

4 (or

7

18) →

18

7

Least Common Denominator (LCD)

➢ This is an extremely important concept when working with multiple fractions

➢ The LCD is the least common multiple (LCM) of the denominators of all fractions involved

➢ Example 1: what is the LCD of 3

2,

4

1, and

6

5

• You must determine the LCM of denominators 3, 4, and 6

• The smallest number that is a multiple of 3, 4, and 6 is 12

➢ Example 2: what is the LCD of 4

3,

5

4,

8

5 and

10

7

• You must determine the LCM of denominators 4, 5, 8, and 10

• The smallest number that is a multiple of 4, 5, 8, and 10 is 40

83


Reducing (Simplifying) Fractions

➢ Determine the greatest common (GCF) factor of the numerator and denominator

➢ Divide the numerator and denominator that GCF (making the resulting GCF 1)

➢ Example: Reduce 60

45

• Determine GCF of 45 and 60 → 15

• Divide numerator (45) and denominator (60) by the GCF (15)

• 1560

1545

→

4

3

• The GCF of 3 and 4 is 1 (therefore, the fraction has been fully reduced)

Unreducing (Unsimplifying) Fractions

➢ Multiply both the numerator and denominator by the a common value

➢ Example: Unreduce 4

3

• Determine a common value by which you multiply the numerator and denominator (i.e. 15)

• 154

153

x

x→

60

45

Converting Improper Fractions to Mixed Fractions

➢ Divide the denominator into the numerator → becomes the whole number part

➢ Place the remainder over the denominator → becomes the fractional part

➢ Put together the whole part and the fractional part

➢ Example: Convert 4

11 to a mixed fraction

• Divide the denominator into the numerator → 11 ÷ 4 = 2 (whole part with remainder 3)

• Place the remainder over the denominator → 4

3 (fractional part)

• Put together the whole part and the fractional part → 24

3

Converting Mixed Fractions to Improper Fractions

➢ Multiply the integer by the denominator of the fraction

➢ Add this product to the numerator of the fraction

➢ Place this sum over the original denominator

➢ Example: Convert 2 4

3 to an improper fraction

• Multiply the integer by the denominator of the fraction → 2 x 4 = 8

• Add this product to the numerator of the fraction → 8 + 3 → 11

• Place this sum over the original denominator → 4

11

84


Comparing Fractions

➢ Multiply the numerator of the first fraction by the denominator of the second fraction and the numerator

of the second fraction by the denominator of the first fraction

➢ Compare the two multiplied values

• The first multiplied value corresponds to the first fraction

• The second multiplied value corresponds to the second fraction

➢ Example: Compare 8

7 and

15

13

• Multiply 7 and 15 (corresponds to first fraction) → 105

• Multiply 13 and 8 (corresponds to second fraction) → 104

• Since the first product (7 x 15) is larger than the second product (13 x 8), the first fraction (8

7) is

larger than the second fraction (15

13)

Converting Fractions to have a Common Denominator

➢ This is an extremely important concept when preparing to add or subtract fractions with different

denominators

➢ Steps to converting fractions to have a common denominator

• First, determine the least common denominator (LCD)

• Then determine for each fraction what number to multiply the current denominator to reach the

LCD

• For each fraction, multiply both the numerator and denominator by the value that will make the

fraction contain the LCD, making an equivalent fraction

➢ Example: Convert the following fractions to have a common denominator 3

2,

4

1, and

6

5

• Determine the LCD for the fractions → 12

• Determine the number to multiply each fraction’s denominator’s to reach the LCD

o 3

2 → multiply the denominator (3) by 4 to become 12

o 4

1 → multiply the denominator (4) by 3 to become 12

o 6

5→ multiply the denominator (6) by 2 to become 12

• For each fraction, multiply both the numerator and denominator by the value that will make the

fraction contain the LCD

o 3

2 → multiply the numerator and denominator by 4:

3

2 x

4

4 =

12

8

o 4


4

1 x

3

3 =

12

3

o 6


6

5 x

2

2 =

12

10

85

Introduction to Fractions – Guided Practice

Write the following as a fraction (questions 1 – 3)

__________ 1. 7 points received out of 10 possible points

__________ 2. 11 pieces of pie eaten out of 15 pieces of pie cut

__________ 3. 17 dogs out of 23 total animals

Write yes or no to indicate if the specified is a Simplest-Form Fraction (question 4 – 7)

__________ 4. 5

4 ________ 5.

21

12 ________ 6.

26

15 ________ 7.

51

17

Reduce the following fractions to Simplest-Form (question 8 – 11)

__________ 8. 20

12 ________ 9.

24

16 ________ 10.

35

20 ________ 11.

72

60

Create an Equivalent Fraction with the specified numerator or denominator (question 12 – 17)

_____ 12. 7

4 (numerator 20) _____ 13.

4

3 (denominator 16) _____ 14.

5

2 (numerator 12)

_____ 15. 8

7 (denominator 64) _____ 16.

6

5 (numerator 35) _____ 17.

9

7 (denominator 72)

Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 – 17)

______ 18. 6

35 ________ 19. 4

5

3 ________ 20.

8

53 ________ 21. 8

7

5

86

Introduction to Fractions – Guided Practice (continued)

What is the reciprocal of each fraction, specified as proper or improper? (question 22 – 25)

__________ 22. 7

18 ________ 23. 1

4

3 ________ 24.

23

12 ________ 25. 5

8

7

What is the least common denominator (LCD) of each set of fractions? (question 26 – 28)

__________ 26. 5

2,

6

5,

15

7 __________ 27.

8

3,

6

1,

12

7 __________ 28.

15

7,

9

4,

5

3

Convert the following fractions to have a least common denominator (question 29 – 31)

__________ 29. 5

2,

6

5,

15

7 __________ 30.

8

3,

6

1,

12

7 __________ 31.

15

7,

9

4,

5

3

87

Section 9.2 Fraction Arithmetic

Adding and Subtracting Fractions with Like Denominators

➢ Just add or subtract the numerators

➢ Keep the same denominator

➢ Reduce the final answer if possible

➢ Example: 9

5 +

9

1 =

9

15 + =

9

6 =

3

2

➢ Example: 9

5 –

9

1 =

9

15 − =

9

4

Adding and Subtracting Fractions with Unlike Denominators

➢ Determine the “Least Common Denominator” (LCD) for the two fractions

➢ Now “unreduced” each fraction to have that LCD

➢ Since each fraction now has the same denominator, just added the numerators

➢ Keep the same denominator


➢ Example 9

4 +

6

1

• Least Common Denominator (LCD) = 18

• Unreduce 9

4 to have a denominator of 18 (LCD) → fraction must be multiplied by

2

2

• Unreduce 6


3

3

• 9

4 x

2

2 =

18

8 &

6

1 x

3

3 =

18

3 →

18

8 +

18

3 →

18

11 (already reduced)

➢ Example 9

4 –

6

1

• Least Common Denominator (LCD) = 18

• Unreduce 9


2

2

• Unreduce 6


3

3

• 9

4 x

2

2 =

18

8 &

6

1 x

3

3 =

18

3 →

18

8 –

18

3 →

18

5 (already reduced)

88

Section 9.2 Fraction Arithmetic (continued)

Adding Mixed Fractions

➢ Add the whole parts to one another and the fractional parts to one another

• If the fractions have the same denominators, use the steps for “Adding Fractions with Like

Denominators”

• If the fractions have different denominators, use the steps for “Adding Fractions with Unlike

Denominators”

➢ If the sum of the fractional parts exceeds 1, initially specify that sum as an improper fraction

• Convert the sum’s improper fraction to the corresponding mixed fraction

• Add this mixed fraction to the sum of the whole parts


➢ Example 1: 49

5 + 3

9

1 = 4 + 3 +

9

5 +

9

1 = 7 +

9

15 + = 7 +

9

6 = 7 +

3

2 = 7

3

2

➢ Example 2: 67

4 + 2

7

5 = 6 + 2 +

7

4 +

7

5 = 8 +

7

54 + = 8 +

7

9 = 8 + 1

7

2 = 9

7

2

➢ Example 3: 55

3 + 6

5

4 + 2

5

4 = 5 + 6 + 2 +

5

3 +

5

4 +

5

4= 13 +

5

443 ++ = 13 +

5

11 = 13 + 2

5

1 = 15

5

1

➢ Example 4: 38

3 + 2

6

1 = 3 + 2 +

8

3 +

6

1 = 5 +

24

9 +

24

4 = 5 +

24

49 + = 5

24

13

➢ Example 5: 48

7 + 5

6

5 = 4 + 5 +

8

7 +

6

5 = 9 +

24

21 +

24

20 = 9 +

24

2021+ = 5

24

41 = 5 + 1

24

17= 6

24

17

➢ Example 6: 25

4 + 3

8

5 + 6

10

7 = 2 + 3 + 6 +

5

4 +

8

5 +

10

7= 11 +

40

32 +

40

25 +

40

28 = 11 +

40

85 = 11

+ 240

5 = 13

40

5 = 13

8

1

89


Subtracting Mixed Fractions

➢ Subtract the whole parts from one another and the fractional parts from one another

• If the fractions have the same denominators, use the steps for “Subtracting Fractions with Like

Denominators”

• If the fractions have different denominators, use the steps for “Subtracting Fractions with Unlike

Denominators”

➢ If the fraction to the right of the minus sign is great than the fraction to the left of the minus sign, you

will have to “borrow” from the whole number to the left of the minus sign

• Subtract 1 from the fraction to the left of the minus sign

• Add the denominator value on the fraction to the left of the minus sign to the numerator value to the

fraction to the left of the minus sign (creating an improper fraction)

• Subtract the whole number to the right of the minus sign from the whole number to the left of the

minus sign

• Subtract the fraction part of the number to the right of the minus sign from the improper fraction to

the left of the minus sign


➢ Example 1: 66

5 – 2

6

1 = 6 – 2 + (

6

5–

6

1) = 4 +

6

15 − = 4 +

6

4 = 4 +

3

2 = 4

3

2

➢ Example 2: 66

1 – 2

6

5 = 5

6

7 – 2

6

5 = 5 – 2 + (

6

7–

6

5) = 3 +

6

57 − = 3 +

6

2 = 4 +

3

1 = 3

3

1

➢ Example 3: 88

3 – 5

6

1 = 8

24

9 – 5

24

4 = 8 – 5 + (

24

9–

24

4) = 3 +

24

49 − = 3 +

24

5 = 3

24

5

➢ Example 4: 86

1 – 5

8

3 = 8

24

4 – 5

24

9 = 7

24

28 – 5

24

9 = 7 – 5 + (

24

28–

24

9) = 2 +

24

928 − = 2 +

24

19 =

224

19

90


Multiplying Fractions

➢ Reduce fractions if possible

• You can combine the numerator of one fraction with the denominator of another

• Continue the process until no further reduction is possible

➢ Multiply the numerators

➢ Multiply the denominators

➢ If you fully reduce all fractions before multiplying, the final answer will already be reduced

➢ Example: 9

4 x

8

3 x

6

5 x

5

3 → (

9

4 x

8

3) x (

6

5 x

5

3) → (

3

1 x

2

1) x (

2

1 x

1

1) →

3

1 x

2

1 x

2

1 x

1

1) →

12

1

• If you multiply first: 9

4 x

8

3 x

6

5 x

5

3 =

2160

180→

12

1 (reducing takes much more effort)

• For multiplying fractions, remember this phrase → “Simplify before you multiply”

Multiplying Mixed Fractions

➢ Convert each mixed fraction to the corresponding improper fraction

➢ Reduce the fractions if possible by combining numerators and denominators (whether from the same

fraction or from different fractions)

➢ Multiply the numerators

➢ Multiply the denominators

➢ If the answer results in an improper fraction, convert the final answer back to a mixed fraction

➢ Example: 29

4 x 1

11

3 →

9

22 x

11

14 →

9

2 x

1

14 →

9

28 → 3

9

1

Dividing Fractions by Fractions

➢ Multiply first fraction by the reciprocal of the second fraction

➢ Often phrased by teachers as “keep it, change it, flip it”

• “Keep” the first fraction as is

• “Change” the operation from division to multiplication

• “Flip” the numerator and denominator of the second fraction

➢ Do not attempt to divide fractions, but always multiply them

➢ Example: 4

3 ÷

5

2 →

4

3 x

2

5 →

8

15 (or 1

8

7)

• “Keep” 4

3as is


• “Flip” 5

2to

2

5

91


Dividing Fractions by Whole Numbers

➢ Use the “keep it, change it, flip it” concept

• “Keep” the first fraction as is


• “Flip” the whole number to become a fraction (the reciprocal of the whole number)

➢ Now you multiply the first fraction by the second fraction

• See if you can reduce the fractions’ numerator and/or denominator prior to multiplying

• One all numerators and denominators have been reduced, multiply the numerators

• Then multiply the denominators

• At this point, the final answer should already be reduced

➢ Example 1: 4

3 ÷ 5 →

4

3 x

5

1 →

20

3

➢ Example 2: 5

4 ÷ 8 →

5

4 x

8

1 →

5

1 x

2

1 →

10

1

Dividing Whole Numbers by Fractions


• “Keep” the first value (whole number) as is


• “Flip” the numerator and denominator of the second value (the fraction)

➢ Now you multiply the whole number by the “flipped” second value

• See if you can reduce the fractions’ numerator and/or denominator prior to multiplying

• One all numerators and denominators have been reduced, multiply the numerators

• Then multiply the denominators

• At this point, the final answer should already be reduced

• If the answer is an improper fraction, you can convert it to a mixed fraction

➢ Example 1: 5 ÷ 4

3 → 5 x

3

4 →

3

20 → 6

3

2

➢ Example 2: 6 ÷ 11

9 → 6 x

9

11 → 2 x

3

11 →

3

22 → 7

3

1

92


Dividing Mixed Fractions

➢ Convert each mixed fraction to the corresponding improper fraction


• “Keep” the first value (the original fraction) as is


• “Flip” the numerator and denominator of the second value (the improper fraction)

➢ Reduce the fractions if possible by combining numerators and denominators (whether from the same

fraction or from different fractions)

➢ Multiply the first value by the second

➢ At this point, the final answer should already be reduced

➢ If the answer is an improper fraction, you can convert it to a mixed fraction

➢ Example 1: 25

4 ÷ 5

5

3 →

5

14 ÷

3

28 →

5

14 x

28

3 →

5

1 x

2

3 →

10

3

➢ Example 2: 55

3 ÷ 2

5

4 →

5

28 ÷

5

14 →

5

28 x

14

5 →

5

2 x

1

5 →

1

2 → 2

93

Fraction Arithmetic – Guided Practice

__________ 1. What is 13

5 +

13

7?

__________ 2. What is 8

3 +

6

1?

__________ 3. What is 45

4 + 6

8

3?

__________ 4. What is 13

7 –

13

5?

__________ 5. What is 8

3 –

6

1?.

__________ 6. What is 68

3 – 4

5

4?.

__________ 7. What is 4

3 x

3

2?

__________ 8. What is 34

3 x 6

3

2?

__________ 9. What is 34

3 ÷ 6

3

2 ?

__________ 10. Which fraction has the largest value: 5

4 ,

8

7,

6

5,

4

3?

94

Section 9.3 Advanced Fraction Principles

Fractions (Tenths, Hundredths, Thousandths, and Beyond)

➢ Similar to integers with place value, fractions can have place value

➢ Special fraction place value occurs when the numerator is 1 and the denominator contains 1 followed

by one or more zeros

• The fractions turn out to be the reciprocals of ten, hundred, thousand, ten thousand, etc.

• They have the corresponding names of tenth, hundredth, thousandth, ten thousandth, etc.

➢ Here are examples of these special fractions

• 10

1 → “one tenth”

• 100

1 → “one hundredth”

• 1000

1 → “one thousandth”

• 10000

1 → “one the thousandth”

➢ To add or subtract these types of fractions, you must convert one or more of the fractions to have a

common denominator

• 10

1 +

100

1→

100

10 +

100

1 →

100

11 (“eleven hundredths”)

• 10

1 –

100

1→

100

10 –

100

1 →

100

9 (“nine hundredths”)

• 100

1 +

10000

1→

10000

100 +

10000

1 →

10000

101 (“one hundred one ten thousandths”)

• 100

1 –

10000

1→

10000

100 –

10000

1 →

10000

99 (“ninety nine ten thousandths”)

➢ To multiply these types of fractions, the final answer will have the sum of the number of zeros that the

fractions involved in the multiplication have

• 10

1 x

100

1→

1000

1 (“one thousandth”)

• 100

1 x

10000

1→

100000

1 (“one millionth”)

➢ To multiply these types of fractions, multiply the first fraction by the reciprocal of the second fraction

• 10

1 ÷

1000

1→

10

1 x

1

1000 →

10

1000 → 100

• 100000

1 ÷

100

1→

100000

1 x

1

100 →

1000

1 (“one thousandth”)

95

Section 9.3 Advanced Fraction Principles (continued)

Introduction to Complex Fractions

➢ A complex fraction contains a fraction within a fraction

• A fraction in the numerator.

• A Fraction in the denominator

• A fraction in both the numerator and the denominator.

➢ Examples of complex fractions

• 8

31

•

52

5

•

54

73

Solving Complex Fractions

➢ Complex fractions must be converted to “simple fractions” where there is neither a fraction in the

numerator nor a fraction in the denominator

➢ Treat the “division bar” of the main fraction as a “division symbol” between the main fractions

numerator and denominator

• 8

31

→ 3

1 ÷ 8 (“Dividing Fractions by Whole Numbers”) →

3

1 x

8

1 →

24

1

•

52

5→ 5 ÷

5

2 (“Dividing Whole Numbers by Fractions”) → 5 x

2

5→

2

25→12

2

1

•

54

73

→ 7

3 ÷

5

4 (“Dividing Fractions by Fractions”) →

7

3 x

4

5→

28

15

Introduction to Word Problems with Fractions

➢ Word problems that cause any of the following fraction operations

• Specifying fractions

• Reducing (simplifying) fractions

• Adding fractions

• Subtracting fractions

• Multiplying fractions

• Dividing fractions

96

Section 9.3 Advanced Fraction Principles (continued)

Solving Word Problems with Fractions

➢ Understand what the word problem is telling you

➢ Understand what the word problem is asking you

➢ Example 1: John had 10 marbles in his pocket. He lost 3 marbles because of a hole in his pocket.

Specify the fractions of marbles lost.

• Total marbles that John had → 10

• Total marbles lost by John → 3

• Fraction of marbles lost → 10

3

➢ Example 2: John had 10 marbles in his pocket. He lost 3 marbles because of a hole in his pocket.

Specify the fractions of marbles that John still has.

• Total marbles that John had → 10

• Total marbles lost by John → 3

• Total marbles that John still has → 10 – 3 = 7

• Fraction of marbles John still has → 10

7

97

Advanced Fraction Principles – Guided Practice

__________ 1. What is 1000

1 +

10000

1?

__________ 2. What is 10000

1 +

1000000

1?

__________ 3. What is 10

1 +

1000

1?

__________ 4. What is 100000

1 +

1000000

1?

__________ 5. What is 10

1 +

1000000

1?

__________ 6. Simplify 6

41

__________ 7. Simplify 9

21

__________ 8. Simplify 7

31

__________ 9. Simplify 3

43

__________ 10. Simplify 8

54

98

Advanced Fraction Principles – Guided Practice (continued)

__________ 11. Simplify 43

8

__________ 12. Simplify 52

10

__________ 13. Simplify 32

13

__________ 14. Simplify 51

16

__________ 15. Simplify 109

9

__________ 16. Simplify 43

52

__________ 17. Simplify 52

43

__________ 18. Simplify 21

85

__________ 19. Simplify 21

85

__________ 20. Simplify 52

75

99

Advanced Fraction Principles – Guided Practice (continued)

__________ 21. Mary was planning to bake 12 pies for the family reunion. She ran out of time and only

baked 10 pies. Specify the pies baked as a simplified fraction of the pies planned.

__________ 22. In the state basketball tournament, Bill shot 15 free throws. He made 12 and missed 3.

Specify the free throws made as a simplified fraction of the free throws attempted.

__________ 23. In the state basketball tournament, Robert shot 18 free throws. He made 14 and missed 4.

Specify the free throws missed as a simplified fraction of the free throws attempted.

__________ 24. In the state basketball tournament, Henry shot 14 free throws. He made 9 and missed 5.

Specify the fraction of free throws missed to free throw made.

100


101

Module 9 (Principle of Fractions) – Review Exercises

Fraction Fundamentals

Write the following as a fraction (questions 1 – 3)

__________ 1. 20 cars sold out of 27 cars available

__________ 2. 34 book read out of 35 books assigned

__________ 3. 21 green marbles out of 25 total marbles

Write yes or no to indicate if the specified is a Simplest-Form Fraction (question 4 – 7)

__________ 4. 45

24 ________ 5.

53

31 ________ 6.

57

39 ________ 7.

84

49

Reduce the following fractions to Simplest-Form (question 8 – 11)

__________ 8. 27

15 ________ 9.

45

18 ________ 10.

63

35 ________ 11.

65

39

Create an Equivalent Fraction with the specified numerator or denominator (question 12 – 17)

_____ 12. 7

4 (numerator 28) _____ 13.

4

3 (denominator 24) _____ 14.

5

2 (numerator 16)

_____ 15. 8

7 (denominator 72) _____ 16.

6

5 (numerator 55) _____ 17.

9

7 (denominator 108)

102

Module 9 (Principles of Fractions) – Review Exercises (continued)

Fractional Fundamentals (continued)

Convert the following fractions (Improper to Mixed or Mixed to Improper) (question 18 – 17)

______ 18. 8

35 ________ 19. 7

5

3 ________ 20.

9

53 ________ 21. 9

7

4

What is the reciprocal of each fraction, specified as proper or improper? (question 22 – 25)

__________ 22. 7

26 ________ 23. 4

4

3 ________ 24.

27

16 ________ 25. 8

9

7

What is the least common denominator (LCD) of each set of fractions? (question 26 – 28)

__________ 26. 5

2,

6

5,

4

3 __________ 27.

8

3,

6

1,

16

3 __________ 28.

8

3,

6

1,

16

3

Convert the following fractions to have a least common denominator (question 29 – 31)

__________ 29. 5

2,

6

5,

4

3 __________ 30.

8

3,

6

1,

16

3 __________ 31.

8

3,

6

1,

16

3

103


Fraction Arithmetic

Fractional answers should be fully reduced unless otherwise specified.

__________ 1. Fully reduce the fraction 20

12.


36.


36.


48.


80.

__________ 6. Unreduce the fraction 6

5 to create an Equivalent Fraction (numerator 40).


3 to create an Equivalent Fraction (denominator 45).


7 to create an Equivalent Fraction (numerator 48).





104


Fractional Arithmetic (continued)

__________ 11. Convert the fraction 5

33 to Mixed.


43 to Mixed.


55 to Mixed.


55 to Mixed.


67 to Mixed.


5 to Improper.


3 to Improper.


7 to Improper.


3 to Improper.


1 to Improper.

105



__________ 21. What is the reciprocal of 9

5?.


13?.


4?.


5?.


2?.

__________ 26. What is 7

3 +

7

2?

__________ 27. What is 8

1 +

8

5?

__________ 28. What is 57

1 + 4

7

4?

__________ 29. What is 55

4 + 7

5

3?

__________ 30. What is 129

7 + 9

9

8?

106



__________ 31. What is 5

1 +

8

5?

__________ 32. What is 8

3 +

6

1?

__________ 33. What is 5

4 +

8

7?

__________ 34. What is 43

2 + 7

8

5?

__________ 35. What is 95

1 + 4

8

7?

__________ 36. What is 7

3 –

7

2?

__________ 37. What is 8

1 –

8

5?

__________ 38. What is 57

1 – 4

7

4?

__________ 39. What is 55

4 – 7

5

3?

__________ 40. What is 129

7 – 9

9

8?

107



__________ 41. What is 5

1 –

8

5?

__________ 42. What is 8

3 –

6

1?

__________ 43. What is 5

4 –

8

7?

__________ 44. What is 43

2 – 7

8

5?

__________ 45. What is 95

1 – 4

8

7?

__________ 46. What is 8

5 x

5

3?

__________ 47. What is 5

8 x

16

15?

__________ 48. What is 23

1 x 5

4

1?

__________ 49. What is 48

5 x 2

3

2?

__________ 50. What is 68

5 x 1

3

1?

108



__________ 51. What is 4

3 ÷

4

1?

__________ 52. What is 4

3 ÷ 1

3

1?

__________ 53. What is 25

3 ÷ 1

5

2?

__________ 54. What is 56

5 ÷ 3

2

1?

__________ 55. What is 88

3 ÷ 1

8

7?


3 ,

3

1,

10

3,

5

2?


3 ,

10

4,

3

1,

6

1?


1 ,

8

5,

5

3,

3

2?


3 ,

3

2,

8

5,

5

3?


7 ,

5

4,

6

5,

4

3?

109


Advanced Fraction Principles

__________ 61. What is 10

1 +

100

1?

__________ 62. What is 10

1 +

1000

1?

__________ 63. What is 10

1 +

10000

1?

__________ 64. What is 10

1 +

100000

1?

__________ 65. What is 10

1 +

1000000

1?

__________ 66. What is 100

1 +

1000

1?

__________ 67. What is 100

1 +

10000

1?

__________ 68. What is 100

1 +

100000

1?

__________ 69. What is 1000

1 +

100000

1?

__________ 70. What is 1000

1 +

1000000

1?

110


Advanced Fraction Principles (continued)

__________ 71. Simplify 12

51

__________ 72. Simplify 12

61

__________ 73. Simplify 12

81

__________ 74. Simplify 12

54

__________ 75. Simplify 12

65

__________ 76. Simplify 12

83

__________ 77. Simplify 51

12

__________ 78. Simplify 54

12

__________ 79. Simplify 83

12

__________ 80. Simplify 38

12

111


Advanced Fraction Principles (continued)

__________ 81. Simplify 43

87

__________ 82. Simplify 32

65

__________ 83. Simplify 23

65

__________ 84. Simplify 109

41

__________ 85. Simplify 53

56

The following scenario applies to questions 85 – 90.

During target practice with a bow and arrow, John had the following results:

1) Hit the bulls eye 3 times

2) Hit the target in an area other than the bulls eye 7 times

3) Missed the target completely 5 times

__________ 86. Write the fraction of bulls eyes made to total shots attempted.

__________ 87. Write the fraction of bulls eyes made to total missed target.

__________ 88. Write the fraction of target hit (not in the bulls eye area) to total shots attempted.

__________ 89. Write the fraction of target hit (in any area) to total shots attempted.

__________ 90. Write the fraction of target hit (in any area) to target missed completely.

112


113

Module 10: Principles of Decimals

114


115

Section 10.1 Introduction to Decimals

Decimal

➢ Another way of expressing a fraction (divide denominator into numerator)

➢ The fraction 8

7 is the decimal of 7 ÷ 8 → 0.875

Decimal Place Value

➢ Decimals have “place value” to the right of the decimal point just as whole numbers have place value to

the left of the decimal point

➢ Common decimal place values are

• Tenths : 0.1 (one tenth)

• Hundredths : 0.01 (one hundredth)

• Thousandths : 0.001 (one thousandth)

• Ten Thousandths : 0.0001 (one ten thousandth)

• Hundred Thousandths : 0.00001 (one hundred thousandth)

• Millionths : 0.000001 (one millionth)

➢ In stating the value of a decimal, use the rightmost non-zero digit to determine the place value name

• 0.3 → three tenths

• 0.45 → forty-five hundredths

• 0.317 → three hundred seventeen thousandths

• 0.4208 → four thousand two hundred eight ten thousandths

• 0.13853 → thirteen thousand eight hundred fifty three hundred thousandths

• 0.426712 → four hundred twenty six thousand seven hundred twelve millionths

116

Section 10.1 Introduction to Decimals (continued)

Comparing Decimals

➢ Starting from the left most position, the decimal containing the largest corresponding digit has the

largest value

➢ Starting from the left most position, the decimal containing the smallest corresponding digit has the

smallest value

➢ Example: What are the largest and smallest numbers: 1.254, 1.387, 2.038, and 1.099, 1.987

• For the largest value, first look at the digits to the left of the decimal point

o The values of the corresponding numbers are : first (1), second (1), third (2), fourth(1), fifth (1)

o Since the third number has a 2, it is the largest number, regardless of the digits to the right of the

decimal point

• For the smallest value, first look at the digits to the left of the decimal point

o The values of the corresponding numbers are : first (1), second (1), third (2), fourth(1), fifth (1)

o Since the first, second, fourth, and fifth numbers contain 1, you must now look at their digits to

the right of the decimal point

o The values digits immediately to the right of the decimal point of the corresponding number are:

first (2), second (3), fourth (0), and fifth (9)

o Since the fourth number contains the smallest value (0) in that position, it is the smallest

number, regardless of the digits to the right of that place value

Ordering Decimals

➢ To order decimal values (such as smallest to largest or largest to smallest), perform the steps above for

comparing decimals

• Based upon the results of comparing the decimals, place the decimals in the proper order

o If ordering the decimals from smallest to largest, list the smallest value first, then the next

smallest, and proceed with this pattern to include all the values

o If ordering the decimals from largest to smallest, list the largest value first, then the next largest,

and proceed with this pattern to include all the values

➢ Example: Order from smallest to largest the following numbers: 1.254, 1.387, 2.038, and 1.099, 1.987

• Determine the smallest value

o Starting with the leftmost (ones) digit of each number, comparing the corresponding number,

and proceeding with the corresponding digits to the right, the smallest value is 1.099

o Following the same process for the remaining numbers, the second smallest value is 1.254

o The third smallest value is 1.387

o The fourth smallest vale is 1.987

o The fifth smallest (largest) value is 2.038

o Order the values → 1.099, 1.254, 1.387, 1.987, 2.038

117

Introduction to Decimals – Guided Reinforcement

Name the following decimals [Example: 0.37 → thirty-seven hundredths] (questions 1 – 6)

1. 0.7 → __________________________________________________________________

2. 0.14 → _________________________________________________________________

3. 0.212 → ________________________________________________________________

4. 0.4738 → _______________________________________________________________

5. 0.51617 → ______________________________________________________________

6. 0.000045 → _____________________________________________________________

Answer the following decimal comparisons (questions 7 – 8)

__________ 7. Which number is the smallest: 3.407, 3.047, 3.704, 3.074, 3.470, 3.740?

__________ 8. Which number is the largest: 3.407, 3.047, 3.704, 3.074, 3.470, 3.740?

Order the following numbers as specified (questions 9 – 10)

9. 3.407, 3.047, 3.704, 3.074, 3.470, 3.740 (smallest to largest)?

____________________________________________________________________

10. 3.407, 3.047, 3.704, 3.074, 3.470, 3.740 (largest to smallest)?

____________________________________________________________________

118

Section 10.2 Rounding Decimal Numbers

What is Decimal Rounding?

➢ As numbers get smaller (tenths, hundredths, thousandths), you may often not be concerned with the

precise value

➢ You may want a “ballpark” value

➢ Rounding decimal numbers allows you to eliminate some of the precision of the number and just deal

with a “ballpark” value of the decimal number instead

Principles of Rounding Decimal Numbers

➢ You are normally required to round a number to a certain place value, such as

• Round to the nearest tenths

• Round to the nearest hundredths

• Round to the nearest hundred thousandths

➢ Notice the phrasing “round to the nearest . . .”

• Your final (rounded) answer will contain no more precision than the place value stated

• Your final (rounded) answer will contain zeros for all digits to the right of the place value stated

o When those zeros appear after the decimal point, the do not have to be written

o Round to the nearest tenths → zeros in the hundredths place value and beyond do not have to be

written

o Round to the nearest hundredths → zeros in the thousandths place value and beyond do not have

to be written

o Round to the nearest thousandths → zeros in the ten thousandths place value and beyond do not

have to be written

• Rounding to a certain place value tells you what value your original value is closer to using the

stated place value

➢ Steps in Decimal Rounding

• Look at the place value which was stated in the rounding requirement

• Now look at the digit immediately to the right place of the place value which was stated in the

rounding requirement

o If that digit is less than five (0, 1, 2, 3, or 4), keep the digit stated in the rounding requirement

the same and make all digits to the right of it become zeros

o If that digit is five or more (5, 6, 7, 8, or 9), increase by one the digit stated in the rounding

requirement and make all digits to the right of it become zeros

o Zeros after the required decimal place value do not have to be written

119

Section 10.2 Rounding Decimal Numbers (continued)

Examples of Rounding Decimal Numbers

➢ Round 24.387 to the nearest tenths

• Look at the place value of the “tenths” position → 3

• Look at the digit immediately to the right of the “tenths” position → 8 is in the hundredths position

• Since the value in the hundredths position (8) is five or larger, increase the tenths position value by

1 → from 3 to 4

• All digits to the right of the tenths position will now be ignored (and not written)

• 24.387 rounded to the nearest tenths → 24.4

• The number 24.387 is closer to 24.4 than it is to 24.3

➢ Round 13.56268 to the nearest hundredths

• Look at the place value of the “hundredths” position → 6

• Look at the digit immediately to the right of the “hundredths” position → 2 in the thousandths

position

• Since the value of the thousandths position (2) is less than five, keep the digit in the hundredths

position (6) the same

• All digits to the right of the tenths position will now be ignored (and not written)

• 13.56268 rounded to the nearest hundredths → 13.56


➢ Round 13.56268 to the nearest thousandths

• Look at the place value of the “thousandths” position → 2

• Look at the digit immediately to the right of the “thousandths” position → 6 is in the ten

thousandths position

• Since the value in the ten thousandths position (6) is five or larger, increase the thousandths position

value by 1 → from 2 to 3

• All digits to the right of the thousandths position will now be ignored (and not written)

• 13.56268 rounded to the nearest thousandths → 13.563


120

Rounding Decimal Numbers – Guided Practice

Write the correct value from the specified rounding

1) 18.47584 Rounded to the nearest tenth → _________________________________

2) 18.47584 Rounded to the nearest hundredth → _____________________________

3) 18.47584 Rounded to the nearest hundred thousandth → _____________________

4) 18.42649 Rounded to the nearest tenth → _________________________________

5) 18.42649 Rounded to the nearest hundredth → _____________________________

6) 18.42649 Rounded to the nearest thousandth → ____________________________

7) 18.42649 Rounded to the nearest ten thousandth → _________________________

8) 278.42649 Rounded to the nearest unit (ones) → ___________________________

9) 278.72649 Rounded to the unit nearest (ones)→ ___________________________

10) 278.72649 Rounded to the nearest ten → ________________________________

121

Section 10.3 Decimal Arithmetic

Common Arithmetic Operations Involving Decimals

➢ Addition

➢ Subtraction

➢ Multiplication

• Multiplying a number with a decimal by a number without a decimal

• Multiplying a number with a decimal by another number with a decimal

➢ Division

• Dividing Decimals by Whole Numbers

• Dividing Whole Numbers by Decimals

• Dividing Decimals by Decimals

• Dividing Decimals with Zeros in the Quotient

Decimal Addition

➢ To add one or more numbers with a decimal point, perform the following steps

• Line up the decimal points of all values to be added

o Remember that if a number does not have a decimal point, you can place a decimal point just to

the right of the rightmost digit

o You can also add zeros to the right of the rightmost digit after the decimal point

o Examples: 257 → 257. 385 → 385.000 415 → 415.00000

• Add the digits of the numbers, starting from the right and proceeding to the left

o The same rules apply for carrying and regrouping with a decimal point as apply for carrying and

regrouping without a decimal point

• The number of digits after the decimal in the final answer will be the same as the number of digits

in whichever original number has the most digits after the decimal

➢ Example: 5.53 + 3.475

• Rewrite the numbers vertically, and line up the decimal points of the values

• You can also add zeros to the right of the rightmost digit after the decimal point to make all values

have the same number of digits after the decimal point

o 5.53 can be written as 5.530 to have three digits after the decimal point (like 3.475 has)

• Add the digits of the numbers starting from the right and proceeding to the left

• Since 3.475 has three digits to the right of the decimal point, the final answer will also have three

digits to the right of the decimal point

5.530

+ 3.475

-------------

9.005

122

Section 10.3 Decimal Arithmetic (continued)

Decimal Subtraction

➢ To subtract one or more numbers with a decimal point, perform the following steps

• Line up the decimal points of all values to be subtracted

o Just like with decimal addition, if a number does not have a decimal point, you can place a

decimal point just to the right of the rightmost digit and add zeros if necessary

• Subtract the digits of the numbers, starting from the right and proceeding to the left

o The same rules apply for borrowing and regrouping with a decimal point as apply for borrowing

and regrouping without a decimal point


in whichever original number has the most digits after the decimal

➢ Example: 5.53 – 3.475

• Rewrite the numbers vertically, and line up the decimal points of the values

• You can also add zeros to the right of the rightmost digit after the decimal point to make all values

have the same number of digits after the decimal point

o 5.53 can be written as 5.530 to have three digits after the decimal point (like 3.475 has)

• Subtract the digits of the numbers starting from the right and proceeding to the left

• Since 3.475 has three digits to the right of the decimal point, the final answer will also have three

digits to the right of the decimal point

5.530

– 3.475

-------------

1.055

123

Decimal Addition and Subtraction – Guided Practice

__________ 1. What is 6.357 + 7.3?

__________ 2. What is 6.357 + 7.38?

__________ 3. What is 6.357 + 7.386?

__________ 4. What is 18.865 – 12.9?

__________ 5. What is 18.865 – 12.98?

__________ 6. What is 18.865 – 12.987?

124


Introduction to Decimal Multiplication

➢ You will typically perform one of the following for decimal multiplication

• Multiplying a number with a decimal by a number without a decimal

• Multiplying a number with a decimal by another number with a decimal

➢ In multiplying numbers with a decimal, you will use the same rules that apply for multiplying numbers

without a decimal

Multiplying a Number with a Decimal by a Number without a Decimal

➢ To multiply a number with a decimal by a number without a decimal, perform the following steps

• Write the numbers vertically

o Place the number with the greater number of digits on top

o Place the number with the fewer number of digits on the bottom

• Multiply the digits of the bottom number by each of the digits of the top number




after the decimal point of the original number that contained a decimal point

➢ Example: 2.37 x 21

• Rewrite the numbers vertically

o 2.37 has three digits → place this number on top

o 21 has two digits → place this number on the bottom


• Since the number with a decimal point (2.37) contains two digits to the right of the decimal point,

the final answer will contain two digits to the right of the decimal point

2.37

x 21

-----------

237

4740

-----------

49.77

125


Multiplying a Number with a Decimal by another Number with a Decimal

➢ To multiply a number with a decimal by another number with a decimal, perform the following steps

• Write the numbers vertically

o Place the number with the greater number of digits on top

o Place the number with the fewer number of digits on the bottom




• Add the number of digits after the decimal point of the first number to the number f digits after the

decimal point to the second number

o The number of digits after the decimal point in the final answer will be this sum

➢ Example: 2.37 x 2.1

• Rewrite the numbers vertically

o 2.37 has three digits → place this number on top

o 2.1 has two digits → place this number on the bottom


• Since the first number (2.37) contains two digits to the right of the decimal point and the second

number (2.1) contains one digit to the right of the decimal point, the final answer will contain three

(2 + 1) digits to the right of the decimal point

2.37

x 2.1

-----------

237

4740

-----------

4.977

➢ The number of digits to the right of the decimal in the final answer is always the sum of the number of

digits to the right of the decimal of each numbers being multiplied

• Even when multiplying a number with a decimal by a number without a decimal, the number of

digits to the right of the decimal in the final answer will be the sum of the number digits to the right

of the decimal of each number

• In the example on the previous page (2.37 x 21), the number of digits to the right of the decimal in

the final answer was the sum of the number of digits to the right of the decimal of the first number

2.37 (2 digit) and the number of digits to the right of the decimal of the second number 21 (0 digits)

o 49.77 contains two digits to the right of the decimal

126


Introduction to Decimal Division

➢ You will typically perform one of the following for decimal division

• Dividing numbers without a decimal, but the answer contains a decimal

• Dividing a number without a decimal point into a number with a decimal point

• Dividing a number with a decimal point into a number without a decimal point

• Dividing a number with a decimal point into another number with a decimal point

➢ In dividing numbers with a decimal

• Place the decimal point in the answer above the decimal point in the original problem

• Write the answer in the proper place value position

Dividing Numbers without a Decimal Point (Resulting in an Answer with a Decimal Point)

➢ Write the division fact out the normal way

➢ Place the decimal point in the original number after the last digit, and add one or more 0’s

➢ Place the decimal point for the answer immediately above the decimal point in the original number

➢ Example 14 ÷ 40

• Write 14 ÷ 40 the normal way and place the decimal point after

.

_________

40 ) 14.000

• Now begin the normal steps of division, and place each answer in the proper place value position

• How many times will 40 divide into 14 → 0

o You do not have to write the 0


o Place the 3 above the 0 (from 140), just to the right of the decimal point

.3

_________

40 ) 14.000

- 12 0

-------

200


o Place the 5 to the right of the 3

.35

_________

40 ) 14.000

- 12 0

-------

200

- 200

-------

0

• With remainder 9, you are done: 40 ÷ 140 = .35

127


Dividing a Number without a Decimal Point into a Number with a Decimal Point


➢ Place the decimal point for the answer immediately above the decimal point in the original number

➢ Example 48.6 ÷ 27

• Write 48.6 ÷ 27 the normal way and place the decimal point for the answer immediately above the

decimal point of the original number

.

_____

27 ) 48.6

• Now begin the normal steps of division, and place each answer in the proper place value position


o Place the 1 above the 8 from 48 (just to the left of the decimal point)

1.

_________

27 ) 48.6

- 27

-------

21 6


o Place the 8 above the 6 from 48.6 (just to the right of the decimal point)

1.8

_____

27 ) 48.6

- 27

-------

216

- 216

------

0

• With remainder 0, you are done: 48.6 ÷ 27 = 1.8

128


Dividing a Number with a Decimal Point into a Number without a Decimal Point


➢ Move the decimal point of the “divided by” number to the right of the last digit

➢ Now move the decimal point of the “divided into” number the same number of places, adding 0’s if

necessary

➢ Divide these modified numbers the normal way

➢ Example 256 - .16

• Write 256 ÷ .16 the normal way

_____

.16 ) 256

• Since .16 contains two digits after the decimal point, move the decimal point two places to the right

(becoming 16)

o Because you moved the decimal point two places to the right in .16, you must now move the

decimal point two places to the right in 256 (becoming 25600)

_____ _______

.16 ) 256 → 16 ) 25600


o Place the 1 above the 5 from 25

1

_________

16 ) 25600

- 16

-------

96


o Place the 6 above the 6 from 256 (just to the right of the decimal point)

16

_________

16 ) 25600

- 16

-------

96

- 96

------

0

• As you bring down the remaining two 0’s, 16 will divide into them 0 times

1600

_________

16 ) 25600

• Your final answer is 1600

o .16 divided into 256 gives you the same answer as 16 divided into 25600

129


Dividing a Number with a Decimal Point into another Number with a Decimal Point


➢ Move the decimal point of the “divided by” number to the right of the last digit

➢ Now move the decimal point of the “divided into” number the same number of places, adding 0’s if

necessary

➢ Divide these modified numbers the normal way

➢ Example 5.78 – 1.7

• Write 5.78 ÷ 1.7 the normal way

_____

1.7 ) 5.78

• Since 1.7 contains one digit after the decimal point, move the decimal point one place to the right

(becoming 17)

o Because you moved the decimal point one places to the right in 1.7, you must now move the

decimal point one place to the right in 5.78 (becoming 57.8)

_____ _______

1.7 ) 5.78 → 17 ) 57.8


o Place the 3 above the 7 from 57 (just to the left of the decimal point)

3.

_________

17 ) 57.8

- 51

-------

68


o Place the 4 above the 4 from 57.8 (just to the right of the decimal point)

3.4

_________

17 ) 57.8

- 51

-------

68

- 68

------

0

• With remainder 0, you are done

o Your final answer is 3.4

o 1.7 divided into 5.78 gives you the same answer as 17 divided into 57.8

130


131

Module 10 (Principle of Decimals) – Review Exercises

Decimal Fundamentals

Name the following decimals [Example: 0.37 → thirty-seven hundredths] (questions 1 – 14)

1. 0.9 → _____________________________________________________________________

2. 0.46 → ___________________________________________________________________

3. 0.046 → __________________________________________________________________

4. 0.0219 → _________________________________________________________________

5. 0.00219 → ________________________________________________________________

6. 0.000542 → _______________________________________________________________

7. 0.2 → ____________________________________________________________________

8. 0.39 → ___________________________________________________________________

9. 0.583 → __________________________________________________________________

10. 0.0583 → ________________________________________________________________

11. 0.00039 → _______________________________________________________________

12. 0.000583 → ______________________________________________________________

13. 0.0328 → ________________________________________________________________

14. 0.0637 → ________________________________________________________________

132

Module 10 (Principles of Decimals) – Review Exercises (continued)

Decimal Fundamentals (continued)

Answer the following decimal comparisons (questions 14 – 15)

__________ 14. Which number is the smallest: 2.3748 2.3478 2.3847 2.3784 2.3874 2.3487?

__________ 15. Which number is the largest: 2.3748 2.3478 2.3847 2.3784 2.3874 2.3487?

Order the following numbers as specified (questions 16 – 17)

16. 2.3748 2.3478 2.3847 2.3784 2.3874 2.3487 (smallest to largest)?

_______________________________________________________________

17. 2.3748 2.3478 2.3847 2.3784 2.3874 2.3487 (largest to smallest)?

_______________________________________________________________

Round the following decimals to the specified place value (questions 18 – 25)

18. 23.84925 Rounded to the nearest tenth → ____________________________________

19. 23.84925 Rounded to the nearest hundredth → ________________________________

20. 73.96304 Rounded to the nearest tenth → ____________________________________

21. 73.96304 Rounded to the nearest hundredth → ________________________________

22. 14.849560 Rounded to the nearest thousandth → ______________________________

23. 14.849560 Rounded to the nearest ten thousandth → ___________________________

133


Decimal Fundamentals (continued)

24. 22.98765 Rounded to the nearest hundredth → __________________________________

25. 22.299504 Rounded to the nearest thousandth → ________________________________

Decimal Arithmetic

Complete the following Decimal Arithmetic problems (questions 26 – 43)

____________ 26. What is 7.23 x 4?

____________ 27. What is 7.23 x 43?

____________ 28. What is 7.23 x 4.3?

____________ 29. What is 40 ÷ 16?

____________ 30. What is 40 ÷ 25?

____________ 31. What is 45 ÷ 60?

____________ 32. What is 45 ÷ 120?

134


Decimal Arithmetic (continued)

____________ 33. What is 1.43 ÷ 13?

____________ 34. What is 0.168 ÷ 7?

____________ 35. What is .0192 ÷ 16?

____________ 36. What is 2.16 ÷ 16?

____________ 37. What is 18.9 ÷ 9?

____________ 38. What is 0.189 ÷ 9?

____________ 39. What is 21.6 ÷ 1.6?

____________ 40. What is 1.92 ÷ .16?

____________ 41. What is 16.8 ÷ .07?

____________ 42. What is 0.156 ÷ .0013?

____________ 43. What is 0.156 ÷ .0013?

135


Supplemental Decimal Arithmetic

1. 1196 ÷16 = _____ 2. 576 ÷ 320 = _____ 3. 2198 ÷140 = _____ 4. 1020 ÷ 48 = _____

5. 187.2 ÷12 = _____ 6. 454.8 ÷ 16 = _____ 7. 692.55 ÷19 = _____ 8. 323.95 ÷ 22 = _____

9. 240 ÷.016 = _____ 10. 342 ÷ 1.9 = _____ 11. 621 ÷.27 = _____ 12. 980 ÷ .028 = _____

13. 67.2 ÷.024 = _____ 14. 1.044 ÷ 2.9 = _____ 15. 15.75 ÷ 4.5 = _____ 16. 1.378 ÷ .053 = _____

136


Supplemental Decimal Arithmetic (continued)

__________ 1. What is 12.347 + 11.543?

__________ 2. What is 12.347 + 11.765?

__________ 3. What is 18.865 – 12.623?

__________ 4. What is 12.623 – 18.865?

__________ 5. What is 11.75 x 3.2?

__________ 6. What is 73.8 ÷ 12.3?

__________ 7. What is .0028 ÷ .0004?

__________ 7. What is .00036 ÷ 3?

137



__________ 1. What is 11.123 + 12.246?

__________ 2. What is 13.348 + 21.478?

__________ 3. What is 15.736 + 14.264

__________ 4. What is 17.864 + 12.857?

__________ 5. What is 19.978 + 17.879?

__________ 6. What is 16.967 – 11.635?

__________ 7. What is 21.855 – 11.634?

__________ 8. What is 19.326 – 12.818?

__________ 9. What is 13.316 – 17.649?

__________ 10. What is 11.342 – 16.231?

138



__________ 11. What is 6.4 x 1.5?

__________ 12. What is 22.3 x 2.4?

__________ 13. What is 24.21 x 3.2?

__________ 14. What is 33.715 x (–3.25)?

__________ 15. What is –48.603 x (–4.7)?

__________ 16. What is 79.2 ÷ 13.2?

__________ 17. What is .0048 ÷ .0012?

__________ 18. What is .00065 ÷ .00013?

__________ 19. What is .00065 ÷ .000013?

__________ 20. What is 25 ÷ .004?

139

Module 11: Principles of Percentages

140


141

Section 11.1 Introduction to Percentages

Percentage (Definition & Examples)

➢ A way of expressing a fraction whose denominator is 100

whole

part =

100

x

➢ Percent means “per 100” or “out of 100”

Converting from Percentage to Fraction

➢ Put the percent value over 100

➢ Reduce (if possible)

➢ Example: 75% = 100

75 =

4

3

Converting from Fraction to Percentage

➢ Divide the denominator into the numerator

➢ Multiply by 100

➢ Example: 4

3 = .75 x 100 = 75%

Converting from Decimal to Percentage

➢ Move the decimal point 2 places to the right

➢ Add the percent symbol (%)

➢ Example: .45 = 45% .625 = 62.5%

Converting from Percentage to Decimal

➢ Move the decimal point 2 places to the left

➢ Remove the percent symbol (%)

➢ Example: 87.5% = .875 12.5% = .125

142

Section 11.2 Calculating Percentages

What is a Certain Percent of a Certain Value?

➢ Convert the given percentage to decimal

➢ Multiply (of) the value by this decimal equivalent of the percent

➢ Example: What is 40% of 650?

• Convert 40% to .4

• Multiply 650 by .4 → 260

• 40% of 650 is 260

What Percent is a Value of another Certain Value?

➢ Divide the first value by the second value (or rephrased : divide the second value into the first value)

➢ Multiply the quotient obtained in the previous step by 100

➢ Add the percentage symbol to this product

➢ Example: 85 is what percent of 400?

• Divide 85 by 400 (or rephrased : divide 400 into 85) → .2125

• Multiply .2125 by 100 → 21.25

• Add the percentage symbol to the product → 21.25%

• 85 is 21.25% of 400

What Value is a Certain Percent of a Certain Value?


➢ Divide the value by this decimal equivalent of the percent

➢ If the percent is less than 100%, the resulting value will be larger than the original number

➢ If the percent is greater than 100%, the resulting value will be smaller than the original number

➢ Example 1: 25 is 5% of what number


• Divide 25 by .05 (do not multiply) → 25 ÷ .05 = 500

• 25 is 5% of 500

➢ Example 2: 15 is 300% of what number

• Convert 300% to 3.0

• Divide 15 by 3 (do not multiply) → 15 ÷ .3 = 5

• 15 is 300% of 5

143

Section 11.2 Calculating Percentages (continued)

What is a Certain Percent Increase of a Value?


➢ Add 1.00 to that decimal

➢ Multiply the original value by this decimal

➢ Example 1: What is a 40% increase of 650?


• Add 1.00 to .40 → 1.00 + .40 = 1.40

• Multiply 650 and 1.4 → 650 x 1.40 = 910

• 40% increase of 640 is 910

• “40% increase of 650” is the same as “140% of 640” → both give you 910

➢ Example 2: What is a 130% increase of 650?


• Add 1.00 to 1.30 → 1.00 + 1.30 = 2.30

• Multiply 650 and 2.3 → 650 x 2.30 = 1495

• 130% increase of 650 is 1495

• “130% increase of 650” is the same as “230% of 650” → both give you 1495

What is a Certain Percent Decrease of a Value?


➢ Subtract the decimal from 1.00


➢ Example: What is a 40% decrease of 650?


• Subtract .40 from 1.00 → 1.00 – .40 = .60

• Multiply 650 and .60 → 650 x .60 = 390

• 40% decrease of 650 is 390

• “40% decrease of 650” is the same as “60% of 650” → both give you 390

144

Section 11.2 Calculating Percentages (continued)

What is a Certain Percent More than a Value?

➢ Works just like “a certain percent increase of a value”


➢ Add 1.00 to that decimal


➢ Example 1: What is 40% more than 650?


• Add 1.00 to .40 → 1.00 + .40 = 1.40

• Multiply 650 and 1.4 → 650 x 1.40 = 910

• 40% more than 650 is 910

• “40%” more than 650” is the same as “a 40% increase of 650” → both give you 910

➢ Example 2: What is a 130% more than 650?


• Add 1.00 to 1.30 → 1.00 + 1.30 = 2.30

• Multiply 650 and 2.3 → 650 x 2.30 = 1495

• 130% more than 650 is 1495

• “130% more than 650” is the same as “a 130% increase of 650” → both give you 1495

What is a Certain Percent Less than a Value?

➢ Works just like “a certain percent decrease of a value”


➢ Subtract the decimal from 1.00


➢ Example: What is a 40% less than 650?


• Subtract .40 from 1.00 → 1.00 – .40 = .60

• Multiply 650 and .60 → 650 x .60 = 390

• 40% decrease of 650 is 390

• “40% decrease of 650” is the same as “60% of 650” → both give you 390

145

Section 11.3 Percentage Equivalents

Important Fraction, Decimal, Percentage Equivalents

2

1 = .5 = 50%

3

1 = .333 = 33

3

1%

3

2 = .666 = 66

3

2%

4

1 = .25 = 25%

4

3 = .75 = 75%

5

1 = .2 = 20%

5

2 = .4 = 40%

5

3 = .6 = 60%

5

4 = .8 = 80%

6

1 = .166 = 16

3

2%

6

5 = .833 = 83

3

1%

8

1 = .125 = 12.5%

8

3 = .375 = 37.5%

8

5 = .625 = 62.5%

8

7 = .875 = 87.5%

2

1=

4

2=

6

3=

8

4=

10

5 = .5 = 50%

8

2 =

4

1 = .25 = 25%

9

3 =

6

2 =

3

1 = .333 = 33

3

1%

9

6 =

6

4 =

3

2 = .666 = 66

3

2%

8

6 =

4

3 = .75 = 75%

146

Section 11.4 Word Problems with a Percent

Solving Word Problems with Containing Percentages

➢ Understand what the word problem is telling you

➢ Understand what the word problem is asking you

➢ Example 1: John had 70 marbles in his bag. Bill has 20% more marbles in his bag than John has. How

many marbles does Bill have?

• This requires the concept of “what is a certain percent more than a value?”

• What is 20% more than 70

o Convert 20% to .20

o Add 1.00 to .20 → 1.00 + .20 = 1.20

o Multiply 70 and 1.2 → 70 x 1.2 = 84

o Bill has 84 marbles in his bag (20% more marbles than John has in his bag)

➢ Example 2: Martha currently has $60 in her purse. She wants to increase the amount of money in her

purse by 250%. How much money does Martha want to eventually have in her purse?

• This requires the concept of “what is a certain percent increase of a value?”

• What is a 250% increase of $60

o Convert 250% to 2.50

o Add 1.00 to 2.50 → 1.00 + 2.50 = 3.50

o Multiply $60 and 3.5 → $60 x 3.5 = $210

o Martha needs to eventually have $210 in her purse in order for it to be a 250% increase of $60

➢ Example 3: Henry currently spends $500 per month on travel expenses. His manager has asked him to

reduce is travel expenses by 30% each month. How much money will Henry be able to spend per

month on travel expenses?

• This requires the concept of “what is a certain percent decrease of a value?”

• What is a 30% decrease of $500

o Convert 30% to .30

o Subtract .30 from 1.00 → 1.00 – .30 = .70

o Multiply .70 and $500 → $500 x .70 = $350

o Henry’s will be able to spend $350 per month on travel expenses (a 30% decrease of $500 or

70% of $500)

147

Percentage – Guided Practice

__________ 1. Convert 12.5% to a fraction.

__________ 2. Convert 8

3 to a percentage.

__________ 3. Convert .875 to a percentage.

__________ 4. Convert 62.5% to a decimal.

__________ 5. What is 20% of 65?

__________ 6. 45 is what percent of 80?

__________ 7. 150 is 30% of what value?

__________ 8. What is 25% increase of 80?

__________ 9. What is 25% decrease of 80?

148


149

Module 11 (Principle of Percentages) – Review Exercises

Percentage Fundamentals


__________ 2. Convert 333

1% to a fraction.


__________ 4. Convert 833

1% to a fraction.

__________ 5. Convert 872

1% to a fraction.

__________ 6. Convert 4

1 to a percentage.

__________ 7. Convert 5

4 to a percentage.

__________ 8. Convert 3

2 to a percentage.

__________ 9. Convert 3

2 to a percentage.

__________ 10. Convert 3

2 to a percentage.

150

Module 11 (Principles of Percentages) – Review Exercises (continued)

Percentage Fundamentals (continued)






__________ 16. Convert 20% to a decimal.

__________ 17. Convert 50% to a decimal.

__________ 18. Convert 622

1% to a decimal.

__________ 19. Convert 833

1% to a decimal.

__________ 20. Convert 872

1% to a percentage.

151


Percentage Calculations

__________ 21. What is 30% of 80?

__________ 22. What is 40% of 120?

__________ 23. What is 55% of 240?

__________ 24. What is 85% of 300?

__________ 25. What is 96% of 500?






152


Percentage Calculations (continued)

__________ 31. 40 is 80% of what number?

__________ 32. 25 is 622

1% of what number?

__________ 33. 65 is 20% of what number?

__________ 34. 125 is 333

1% of what number?

__________ 35. 17 is 5% of what number?

__________ 36. What is a 30% increase of 80?





153


Percentage Calculations (continued)

__________ 41. What is a 30% decrease of 80?





154


155

Module 12: Advanced Arithmetic Operations

156


157

Section 12.1 Ratios

Ratio (Definition)

➢ A comparison (slightly different from a fraction)

➢ Fractions give you a part (the numerator) over a whole (the denominator)

➢ Ratios give you the two parts (where you have to come up with the whole)

Ratio (Expressions)

➢ Typically expressed in one of the following ways

• The ratio of x to y

• x:y (the more common expression of a ratio)

Ratio Examples

➢ Example 1: The ratio of girls to boys is 3:2. In a class of 75 students, now many are girls?

• Given the “parts” of the ratio (3 girls and 2 boys), find the “whole”: 3 + 2 → 5 students

• Think of the “whole” as a group (i.e. one group contains 3 girls and 2 boys)

• Determine how many groups exist with 75 students: 75 students (entire class) ÷ 5 students (per

group) → 15 groups

• Determine how many total girls: 3 girls (per group) x 15 groups → 45 girls

• If the questions had asked how many boys, you could easily determine the total boys: 2 boys (per

group) x 15 groups → 30 boys

• Both categories in this example (45 girls and 30 boys) must equal total students (75)

➢ Example 2: Bill’s ratio of red, green, blue, and yellow marbles is 4:5:6:10. If Bill has 30 green

marbles, how many total marbles does he have?

• Given the “parts” of the ratio (4 red, 5 green, 6 blue, and 10 yellow), find the “whole”: 4 + 5 + 6 +

10 → 25 marbles

• Think of the “whole” as a group (i.e. one group contains 4 red, 5 green, 6 blue, and 10 yellow

marbles)

• If Bill has 30 green marbles, determine how many groups of marbles are required → 30 green

marbles ÷ 5 green marbles per group → 30 ÷ 5 → 6 groups of green marbles

• This would also result in 6 groups of red, blue, and yellow marbles.

• The total number of marbles that Bill has is 6 groups of 25 marbles (per group) → 6 x 25 → 150

total marbles

• Using 6 groups, you can determine the total number of each color:

o Red: 4 red (per group) x 6 (groups) → 4 x 6 → 24

o Green: 5 green (per group) x 6 (groups) → 5 x 6 → 30 (given in the question)

o Blue: 6 blue (per group) x 6 (groups) → 6 x 6 → 36

o Yellow: 10 yellow (per group) x 6 (groups) → 10 x 6 → 60

o Total marbles (24 red + 30 green + 36 blue + 60 yellow) → 150 (matches answer calculated

above)

158

Ratio – Guided Practice

__________ 1. If Academic Excellence Section 101 has a ratio of girls to boys of 4:3 and there

are 28 students in the class, how many students are girls?

__________ 2. If the ratio of red, green, blue, and yellow marbles is 4:5:3:6 for a total of 90

marbles, how many marbles are red?


marbles, how many marbles are green?


marbles, how many marbles are blue?


marbles, how many marbles are yellow?

__________ 6. If the ratio of red, green, blue, and yellow marbles is 3:4:5:8 and there are 24

green marbles, how many are marbles are red?


green marbles, how many are marbles are blue?


green marbles, how many are marbles are yellow?


green marbles, how many are total marbles are there?

159

Section 12.2 Proportion

Proportion (Definition & Example)

➢ Equal ratios of fractions (normally with one piece missing)

➢ Steps to finding the missing value in a proportion

• Set up the proportions as equivalent fractions (using a variable or question mark for the missing

number)

• The missing value must make the two fractions equivalent

o One fraction will be a reduced version of the other fraction, or

o Both fractions would be able to reduce to the same fraction

➢ Example: If every 3 boxes contain 8 calculators, how many calculators are in 12 boxes?

3 (boxes) 12 (boxes)

--- -------------- = -----------------

8 (calculators) ? (calculators)

Using 8

3 =

?

12 shows that 12 is a multiple of 3 (multiplied by 4). Now multiply 8 by 4 to get the

missing value: 32. 8

3 =

32

12 (equivalent fractions) → 12 boxes contain 32 calculators.

160

Proportion – Guided Practice

__________ 1. If each box contains 12 ink pens, how many ink pens are in 7 boxes?

__________ 2. If every 2 note books contain 50 sheets of paper, now many sheets of paper are

in 13 notebooks?

__________ 3. If every 5 free throws attempted results in 3 points, how many points will results

from 45 free throws attempted?

161

Section 12.3 Other Advanced Arithmetic Operations

Principles of Exponents

➢ Just as multiplication is repeated addition,

➢ Base : the number being raised to a power

➢ Exponents : the power to which the base is raised

➢ 25

• 2 is the base

• 5 is the exponent

• 25 = 2x2x2x2x2=32

➢ Negative Exponents

• Means raise the reciprocal of the base to the positive power of the exponent

o If the negative exponent is in the numerator, the base will be moved to the denominator and the

negative exponent will become positive

o If the negative exponent is in the denominator, the base will be moved to the numerator and the

negative exponent will become positive

• Example: 3-2

o Since the negative exponent is in the numerator(because 3 = 1

3), move the base (3) to the

denominator and switch the negative exponent to positive

1

o 3-2 becomes ---- → 9

1

32

1

• Example: ----

4-2

o Since the negative exponent is in the denominator, move the base (4) to the numerator and

switch the negative exponent to positive

1

o ---- becomes 42 → 16

4-2

3

• Example: ----

5-2

o Since the negative exponent is in the denominator, move the base (5) to the numerator and

switch the negative exponent to positive

o Multiply this value by any existing value in the numerator

3

o ---- becomes 3 x 52 → 3 x 25 = 75

5-2 d

162

Section 12.3 Other Advanced Arithmetic Operations (continued)

Raising a Number to a Power (continued)

➢ Multiplying numbers with exponents (and the same base)

• Add the exponents

• Keep the base the same

• 34 x 35 = 34+5 = 39

➢ Dividing numbers with exponents (and the same base)

• Subtract the exponents

• Keep the base

38

--- = 38-6 = 32

36

36 1 1

--- = 36-8 = 3-2 = ---- = ----

38 32 9

➢ Raising a power to a power

• Multiply the exponents

• Keep the base the same

(32)4 = 32x4 = 38

Square Root

➢ Denoted by symbol

➢ Multiplying square roots

• Combine the multiplication of the individual numbers under one square root symbol

• Multiply the individual numbers

• Find the square root of the product of the two numbers

• Simplify if possible

x y = xy → 2 8 = 82x = 16 = 4

➢ Dividing square roots

• Combine the division of the individual numbers under one square root symbol

• Divide the individual numbers

• Find the square root of the quotient of the two numbers

• Simplify if possible

x / y = yx / → 45 / 5 = 5/45 = 9 = 3

163


➢ Squares and Square Roots : 1 Through 30

12 = 1 22 = 4 32 = 9 42 = 16 52 = 25

62 = 36 72 = 49 82 = 64 92 = 81 102 = 100

112 = 121 122 = 144 132 = 169 142 = 196 152 = 225

162 = 256 172 = 289 182 = 324 192 = 361 202 = 400

212 = 441 222 = 484 232 = 529 242 = 576 252 = 625

262 = 676 272 = 729 282 = 784 292 = 841 302 = 900

1 = 1 4 = 2 9 = 3 16 = 4 25 = 5

36 = 6 49 = 7 64 = 8 81 = 9 100 =10

121 = 11 144 = 12 169 = 13 196 = 14 225 =15

256 = 16 289 = 17 324 = 18 361 = 19 400 =20

441 = 21 484 = 22 529 = 23 576 = 24 625 =25

676 = 26 729 = 27 784 = 28 841 = 29 900 =30

➢ Fundamental Roots and Exponents

23 = 8 24 = 16 25 = 32 26 = 64 27 = 128

28 = 256 33 = 27 34 = 81 35 = 243 36 = 729

43 = 64 44 = 256 53 = 125 54 = 625 63 = 216

73 = 343 83 = 512 93 = 729 103 = 1,000

3 8 = 2 4 16 = 2 3 27 = 3 5 32 = 2 3 64 = 4

6 64 = 2 4 81 = 3 3 125 = 5 7 128 = 2 3 216 = 6

5 243 = 3 8 256 = 2 4 256 = 4 3 343 = 7 3 512 = 8

4 625 = 5 3 729 = 9 6 729 = 3 3 1000 = 10

164


Order of Operations

➢ Teacher often use the phrase “Please Excuse My Dear Aunt Sally” to indicate the following

• “Please” → P → Parentheses

• “Excuse” → E → Exponents

• “My” → M → Multiplication

• “Dear” → D → Division

• “Aunt” → A → Addition

• “Sally” → S → Subtraction

➢ Realize the “Please Excuse My Dear Aunt Sally” phrase is not totally accurate, because it implies that

Multiplication occurs before Division and that Addition occurs before Subtraction

• Multiplication and Division are peers (always going from left to right)

• Addition and Subtraction are peers (always going from left to right)

➢ Here is the accurate description of the Order of Operations

• Groupings [Parentheses or Brackets, Absolute Value Bars, or Fraction Bars] (left to right)

• Exponents (left to right)

• Multiplication & Division (whichever occurs from left to right)

• Addition & Subtraction (whichever occurs from left to right)

➢ Example: 42 x (2 + 4) – (10 + 12) ÷ 8 x 30 ➔ 16 x 6 – 24 ÷ 8 x 30 ➔ 96 – 3 x 30 ➔ 96 – 90 ➔ 6

Arithmetic Word Problem Fundamentals

➢ Arithmetic word problems require you to apply principles of arithmetic (typically addition, subtraction,

multiplication, division, ratios, and/or proportions) to solve the problem

➢ Example: Billy is placing marbles in plastic bags. If he puts 6 marbles in each plastic bag, he will have

16 bags of marbles and four marbles left over. How many plastic bags are required for Billy to put 5

marbles in each bag?

• 6 marbles in 16 bags make (6 x 16) 96 marbles (in bags)

• Adding the 4 left over marbles makes (96 + 4) 100 total marbles

• With 5 marbles per plastic bag, 100 marbles will require (100 ÷ 5 ) 20 plastic bags

• The answer to the question asked is 20

165

Section 12.3 Advanced Arithmetic Operations (continued)

Sequence

➢ A sequence is an ordered list of numbers, following a specific pattern

• Arithmetic Sequence – adding a common value to the previous one

• Geometric Sequence – multiplying a common value by the previous one

➢ Formula for Sequences

• Arithmetic Sequence: an = a0 + d(n – 1)

o an → Value of desired term

o a0 → Value of initial term

o d → Value being added to each subsequent term of the sequence

o n → Desired Term (position)

• Geometric Sequence: an = (a0)dn-1

o an → Value of desired term

o a0 → Value of initial term

o d → Value being added to each subsequent term of the sequence

o n → Desired Term (position)

➢ Example 1: What is the 8th term of the sequence: 3, 7, 11, 15, …

• Identify each component of the sequence

o a8 → Value of desired (8th) term ➔ to be calculated

o a0 → Value of initial term ➔ 3

o d → Value being added to each subsequent term of the sequence ➔ 4

o n → Desired Term (position) ➔ 8

• Use the appropriate Sequence Formula to calculate the value of the desired term

o an = a0 + d(n – 1)

o a8 = 3 + 4(8 – 1) → a8 = 3 + 4(7) → a8 = 3 + 28 → a8 = 31

➢ Example 2: What is the 6th term of the sequence: 7, 21, 63, 189, …

• Identify each component of the sequence

o a6 → Value of desired (6th) term ➔ to be calculated

o a0 → Value of initial term ➔ 7

o d → Value being multiplied by each subsequent term of the sequence ➔ 3

o n → Desired Term (position) ➔ 6

• Use the appropriate Sequence Formula to calculate the value of the desired term

o an = (a0)dn-1

o a6 = (7)36-1 → a6 = (7)35 → a6 =7 x 243 → a6 = 1701

166

Section 12.3 Advanced Arithmetic Operations (continued)

Set

➢ A set is a collection of things

➢ The things are called elements

➢ The elements of a set are typically denoted in brackets : {1,3,5,7,9}

➢ Union of Two Sets

• The elements that are in either or both sets

• If an element occurs more than once, you only list it once

➢ Intersection of Two Sets

• The elements that are simultaneously in both sets

• Elements in the intersection are common to both sets

➢ Example set A = {2,4,6,8,10} and

set B = {8,10,12,14}

What is the Union of sets A and B?

What is the Intersection of sets A and B?

• Union lists elements in either or both sets : {2,4,6,8,10,12,14}

• Intersection lists elements common to both sets : {8,10}

167

Advanced Arithmetic Operations – Guided Practice

__________ 1. What is 33 ?

__________ 2. What is 4-3 ?

3-2

__________ 3. What is ---- ?

4

3

__________ 4. What is ---- ?

2-3

__________ 5. What is 46 x 45 ?

__________ 6. What is 46 ÷ 45 ?

__________ 7. What is 46 x 42 ?

__________ 8. What is 42 x 45 ?

__________ 9. What is (47)3 ?

__________ 10. What is 2 18 ?

__________ 11. What is 96 / 6 ?

__________ 12. What is (3 + 4)2 - 6 x 8 + 15 ÷ 3.

168

Advanced Arithmetic Operations – Guided Reinforcement (continued)

__________ 13. Billy is placing books on small book cases. If he puts 5 books on the each book

case, he will need 11 box cases and have 2 books left over. If he decides to put

3 books on each box case instead, how many books cases will he need?

__________ 14. What is the 36th number in the sequence 3, 7, 11, 15 . . .?

__________ 15. What is 8th number in the sequence, 2, 6, 18, 54 . . .?.

__________ 16. Set A = {1,2,3,8,} and set B = {2,4,5,8}. What is the union of sets A and B?

__________ 17. Set A = {1,2,3,8,} and set B = {2,4,5,8}. What is the intersection of sets A and

B?

169

Section 12.4 Advanced Counting Principles

Multiplication Principle of Counting

➢ Problems where you have to determine how many different ways you can select or arrange a group of

items

➢ Counting problems involve multiplication

➢ Example: Mrs. Wilson is considering purchasing a computer from five possible brands and a printer

from four possible brands. How many ways can Mrs. Wilson purchase one computer and one printer?

• Two separate events must occur

• To determine the totally possibilities, you must multiply the number of possibilities of each event

• Event 1 (5 possibilities) x Event 2 (4 possibilities) = 5 x 4 = 20

• The answer to the question asked is 20

• If Mrs. Wilson decides to acquire an Internet Service Provider (ISP) from three possible companies,

how many ways can Mrs. Wilson choose a computer, printer, and ISP?

o Now multiply all three categories: 5 computers x 4 printers x 3 ISP’s → 60

Advanced Counting Principles

➢ The total possible ways of counting how many ways and occurrence can take place

➢ You are selecting items one at a time

➢ The order of the items does matter

• A B C is different from C B A

• X Y Z is different from Z Y X

➢ Each time you select an item, the number of possible choices decreases by 1

➢ Example: Mark needs to select a three-character security code for his computer. He can use any

alphabet except the vowels ‘a’, ‘e’, ‘i’, ‘o’ or ‘u’. He cannot repeat a letter once it is used. How many

possible security codes are available for Mark?

• For the first letter of the security code, there are 21 possible choices

• For the second letter, there are now 20 possible choices

• For the third letter, there are now 19 possible choices

• You now multiply the possibilities of each event : 21 x 20 x 19 = 7,980

170

Section 12.4 Fundamental Counting Principles (continued)

Combination

➢ The total possible ways of grouping a certain number of items

➢ The order of the items does not matter

• A B C is the same combination as C B A (each contains A, B, and C)

• X Y Z is the same combination as Z Y X (each contains X, Y, and Z)

➢ Use the following Combinations formula : nCr = )!(!

!

rnr

n

−

• n is the total number of items

• r is the number of items being combined at a time

• ! (Factorial) means multiply starting with that number, going down to 1 (4! = 4 x 3 x 2 x 1 = 24)

➢ Example 1 : ABC High School has 6 students trying out for the doubles tennis team. How many

combinations of two-player teams can be formed from the six players?

6C2 = )!26(!2

!6

− =

!4!2

!6 =

!4!2

!456 xx =

12

56

x

x =

2

30= 15

➢ Example 2: How many combinations of 3-player teams can be drawn from a pool of 5 players?

5C3 = )!35(!3

!5

− =

!2!3

!5 =

!2!3

!345 xx =

12

45

x

x =

2

20= 10

171

Section 12.4 Fundamental Counting Principles (continued)

Permutation

➢ The total possible ways of grouping a certain number of items

➢ You are selecting items one at a time

➢ The order of the items does matter

• A B C is different from C B A

• X Y Z is different from Z Y X

➢ Use the following Permutation formula : nPr = )!(

!

rn

n

−

• n is the total number of items

• r is the number of items being selected at a time

• ! (Factorial) mean multiply starting with that number, going down to 1 (4! = 4 x 3 x 2 x 1 = 24)

Example 1 : John has 8 possible photographs to arrange on his living room table. If he has space to

place only 3 of the photographs, how many ways can the 8 photographs be arranged in

the 3 available slots?

8P3 = )!38(

!8

− =

)!5(

!5678 xxx = 8x7x6 = 336

Example 2 : Bill has 7 marbles, each a different color. If he wants to place 4 of the marbles on his

bedroom dresser in a straight line, how many possible ways can he arrange 4 of the 7

marbles?

7P4 = )!47(

!7

− =

)!3(

!34567 xxxx = 7x6x5x4 = 840

172

Advanced Counting Principles – Guided Practice

__________ 1. John is considering reading 1 fiction book, 1 short story, and 1 biography during

the summer. If john has 2 fiction books, 3 short stories, and 4 biographies, how

many ways can he accomplish his reading plans for the summer?

__________ 2. John needs to select a three-character security code from the letters ‘a’, ‘b’, ‘c’,

‘d’, ‘e’, ‘f’, and ‘g’. He cannot repeat a letter once it is used. How many

Possible security codes are available for John?

__________ 3. Mr. Wilson needs three students from his Science to rake leaves on Saturday. If

Mr. Wilson has 7 students in his class, how many combinations of students can

he select?

173

Module 12 (Advanced Arithmetic Operations) – Review Exercises

Ratio

__________ 1. If Academic Excellence Section 102 has a ratio of girls to boys of 5:3 and there are

32 students in the class, how many students are girls?


32 students in the class, how many students are boys?


12 girls in the class, how many total students are in the class?


21 boys in the class, how many total students are in the class?

__________ 5. The ratio of green, blue, yellow, and red marbles is 4:6:5:3. If there are 126 total

marbles, how many are green?


marbles, how many are blue?


marbles, how many are yellow?


marbles, how many are red?

__________ 9. The ratio of green, blue, yellow, and red marbles is 5:3:7:6. If there are 35 green

marbles, how many are blue?


marbles, how many are yellow?


marbles, how many are red?


marbles, how many total marbles are there?

174

Module 12 (Advanced Arithmetic Operations) – Review Exercises (continued)

Proportion

__________ 1. If each phone contains 23 buttons, how many buttons are on 9 phones?

__________ 2. If every three books contain 11 stickers, how many stickers are on 36 books?

__________ 3. If every five stores contain 8 flashing signs, how many flashing signs are at 70 stores?

__________ 4. If every four folders contain 15 sheets, how many sheets are in 48 folders?

__________ 5. If every seven boxes contain 15 markers, how many markers are in 91 books?

__________ 6. If every nine drawers contain 20 cartridges, how many cartridges are in 108 drawers?

__________ 7. If three boxes contain 25 cards, how many cards are on 33 boxes?

__________ 8. If every six rooms contain 7 pictures, how many pictures are in 54 books?

__________ 9. If every four houses contain 50 windows, how many windows are in 60 houses?

__________ 10. If every seven bags contain 13 badges, how many badges are in 112 bags?

175


Advanced Arithmetic Operations

__________ 1. What is 72 ?

1

__________ 2. What is ----?

82

__________ 3. What is 9-2 ?

__________ 4. What is 6-2 ?

1

__________ 5. What is ----?

6-2

1

__________ 6. What is ----?

8-2

3-2

__________ 7. What is ----?

2

6-2

__________ 8. What is ----?

2

3

__________ 9. What is ----?

4-2

2

__________ 10. What is ----?

7-2

176


Advanced Arithmetic Operations (continued)

__________ 11. What is 65 x 67?

__________ 12. What is 73 x 78?

__________ 13. What is 86 x 82?

__________ 14. What is 513 x 515?

__________ 15. What is 321 x 318?

__________ 16. What is 68 ÷ 62?

__________ 17. What is 57 ÷ 56?

__________ 18. What is 43 ÷ 48?

__________ 19. What is 79 ÷ 715?

__________ 20. What is 811 ÷ 812?

177



__________ 21. What is (65 )2?

__________ 22. What is (56 )3?

__________ 23. What is (5-4 )2?

__________ 24. What is (65 )-3?

__________ 25. What is (84 )-5?

__________ 26. What is (3-6 )5?

__________ 27. What is (5-4 )-2?

__________ 28. What is (5-3 )-5?

__________ 29. What is (5-7 )-6?

__________ 30. What is (5-8 )-5?

178



__________ 31. What is 3 x 5 ?

__________ 32. What is 3 x 3 ?

__________ 33. What is 12 x 3 ?

__________ 34. What is 7 x 5 ?

__________ 35. What is 48 x 3 ?

__________ 36. What is 48 ÷ 3 ?

__________ 37. What is 72 ÷ 8 ?

__________ 38. What is 192 ÷ 12 ?

__________ 39. What is 240 ÷ 15 ?

__________ 40. What is 275 ÷ 11 ?

179



__________ 41. What is 6 + (7 – 2)2 – 12 ÷ 2 x 5?

__________ 42. What is 62 – 14 ÷ 2 – 7 x 3?

__________ 43. What is 5 x 6 – 3 x 8 + (7 – 5) 2?

__________ 44. What is (8 – 5) 2 + (5 – 8) 2 – 6 x 3?

__________ 45. What is (15 ÷ 3) 2 + (2 – 5) 2 – (2 x 3) 2 – 4?

__________ 46. John bought 4 video games for the following prices: $35, $37, $40, $50. He asked

his dad to pay one-fourth the price and his mom to pay half of the remaining price.

If John pays the rest equally divided over the next four weeks, how much will he pay

per week?

__________ 47. Susan chose 3 dresses for the following prices: $52, $63, $65. She paid one-third the

total price to put them on hold. .If she pays $20 per week on the remaining balance,

how many weeks will it take Susan to pay for the three dresses?

__________ 48. Bob borrowed $250 from his brother Bill. If Bob back pays Bill one-fifth of what he

borrowed two days later and then splits the remaining balance into 8 daily payments,

how much will Bob pay Bill each day?

__________ 49. Sarah bought 8 packets of loose leaf paper. If each packet contains 200 sheets of

paper, and Sarah uses 15 sheets of paper each day, how many days will the paper she

purchased last?

__________ 50. Henry received 8 full boxes and 1 half-full box of calculators. He gave 10 calculators

per class to 17 classes and had no calculators left. How many calculators came in

each box full box?

180



__________ 51. What is the 13th term of the sequence: 2, 8, 14, 22, . . .?

__________ 52. What is the 21st term of the sequence: 2, 8, 14, 22, . . .?









181



__________ 61. Set X = {2, 3, 8, 11, 15} and Set Y = {2, 3, 8, 11, 18}. What is the intersection of

Set X and Set Y?

__________ 62. Set X = {2, 3, 8, 11, 15} and Set Y = {2, 3, 8, 11, 18}. What is the union of Set X

and Set Y?

__________ 63. Set A = {1, 3, 5}, Set B = {1, 2, 3}, and Set C = {3, 5, 6}. What is the intersection

of Set A and Set B?


of Set A and Set C?


of Set B and Set C?

__________ 66. Set A = {1, 3, 5}, Set B = {1, 2, 3}, and Set C = {3, 5, 6}. What is the union of Set

A and Set B?


A and Set C?


B and Set C?

__________ 69. Set A = {1, 3, 5}, Set B = {1, 2, 3}, and Set C = {3, 5, 6}. What is the intersection of

Set A, Set B, and Set C?


A, Set B, and Set C

182


Advanced Counting Principles

__________ 1. John wants to purchase a cell phone (available in 2 models), pager (available in 4

models) , and walkie-talkie (available in 3 models). How many ways can John

purchase one cell phone and one pager?

__________ 2. From question 1, how many ways can John purchase one cell phone and one

walkie-talkie?

__________ 3. From question 1, how many ways can John purchase one pager and one

walkie-talkie?

__________ 4. From question 1, how many ways can John purchase one cell phone, one pager, and

walkie-talkie?

__________ 5. Bill plans to put a 3-digit password on his computer. If the digit available must be

odd, and if Bill cannot use a digit more than once, how many passwords are

available on Bill’s computer?

__________ 6. Bill plans to put a 3-character password on his computer from the letters ‘a’, ‘b’, ‘c’,

‘d’, ‘e’, ‘v’, ‘w’, ‘x’, ‘y’, and ‘z’. If Bill cannot use a letter more than once, how

many passwords are available on Bill’s computer?













183


Advanced Counting Principles (continued)

__________ 11. How many combinations are available from a group of 4 students taking 2 at a time?










184


185

Module 13: Principles of Roots and Radicals

186


187

Section 13.1 Introduction to Radicals

What is a Radical?

➢ Radical Definition

• A radical is denoted by √ symbol

➢ Radical Examples

• Square Root

• Cube Root

• Fourth Root

• Eight Root

Radical Terminology

➢ The Radical consists √𝑎𝑛

of two components

• Radicand → the value under the radical (a in the notation above)

• Index → the value on the outside of the radical, which specifies the root (n in the notation above)

➢ In the radical √135

• Radicand → 13

• Index → 5

• Read as “the 5th root of 13”

➢ When the index is 2 (square root), you leave off the index in the radical notation √72

is written

• √72

is written √7

• Read as “the square root of 7”

➢ A Radical may also have a coefficient in front of it

• The coefficient denotes multiplication on the radical

• 10√173

• Has coefficient 10 and is read as “ten times the cube root of 17”

• 1√54

is written as √54

(a coefficient of 1 should not be written)

Learn the Following to Master Radicals

➢ Square Roots (of the Perfect Squares 1 Through 30)

1 = 1 4 = 2 9 = 3 16 = 4 25 = 5

36 = 6 49 = 7 64 = 8 81 = 9 100 =10

121 = 11 144 = 12 169 = 13 196 = 14 225 =15

256 = 16 289 = 17 324 = 18 361 = 19 400 =20

441 = 21 484 = 22 529 = 23 576 = 24 625 =25

676 = 26 729 = 27 784 = 28 841 = 29 900 =30

➢ Fundamental Roots

3 8 = 2 4 16 = 2 3 27 = 3 5 32 = 2 3 64 = 4

6 64 = 2 4 81 = 3

3 125 = 5 7 128 = 2

3 216 = 6

5 243 = 3

8 256 = 2 4 256 = 4 3 343 = 7

3 512 = 8

4 625 = 5 3 729 = 9

6 729 = 3 3 1000 = 10

188

Section 13.2 Rules of Radicals

Product Rule of Radicals

➢ The product of nth roots is equal to the nth root of the products, and the nth root of products is equal to

the product of the nth roots

• √𝑎𝑏𝑛

= √𝑎𝑛

√𝑏𝑛

and that √𝑎𝑛

√𝑏𝑛

= √𝑎𝑏𝑛

• The index of the resulting radicals will be the same when separating out the radicands

• The index of the radicals must be the same in order to combine the radicands under the radical

➢ √35 = √7 √5

➢ √11 √5 = √55

➢ √393

= √33

√133

➢ √54

√94

= √454

➢ √134

√205

→ Cannot be combined because the indexes of the radicals (4 for the first one and 5 for the

second one) are not the same

Quotient Rule of Radicals

➢ The quotient of nth roots is equal to the nth root of the quotients, and the nth root of quotients is equal

to the quotient of the nth roots

• √𝑎𝑏

𝑛 = √𝑎

𝑛

√𝑏𝑛 and that

√𝑎𝑛

√𝑏𝑛 = √𝑎

𝑏𝑛

• The index of the resulting radicals will be the same when separating out the radicands

• The index of the radicals must be the same in order to combine the radicands under the radical

➢ √25

9 =

√25

√9 =

5

3

➢ √45

√5 = √

45

5 = √9 = 3

➢ √216

125

3 =

√2163

√1253 =

6

5

➢ √108

3

√43 = √

108

4

3 = √27

3 = 3

➢ √35

3

√175 → Cannot be combined because the indexes of the radicals (3 for the numerator and 5 for the

denominator) are not the same

189

Section 13.3 Radical Manipulation

Simplifying Radicals

➢ To “simplify” a radical, you must remove all perfect roots from under the radical and place them as

coefficients outside the radical

➢ You will frequently use the Product Rule of Radicals to simplify them

➢ Simplify √32

• Look for the largest “Perfect Square” factor of 32 → 16

• Rewrite √32 as √16 ∙ 2

• Using the Product Rule of Radicals, rewrite √16 ∙ 2 as √16 √2

• Convert √16 to 4 and use as the coefficient → 4√2

➢ Simplify √813

• Look for the largest “Perfect Cube” factor of 81→ 27

• Rewrite √813

as √27 ∙ 33

• Using the Product Rule of Radicals, rewrite √27 ∙ 33

as √273

√33

• Convert √273

to 3 and use as the coefficient → 3√33

Radicals and Rational Exponents

➢ Radicals are frequently denoted by expressions containing rational (fractional) exponents

• 163

2

• 2433

5

➢ For the expression 𝑎𝑛

𝑑 (where n is the numerator of the exponent and d is the denominator of the

exponent), you can convert it to the radical ( √𝑎𝑑

)n

• 163

2 → (√162

)3 → (√16)3

• 2433

5 → (√2435

)3

➢ Now by knowing the fundamental Roots, you can easily solve (simplify) these expressions

• 163

2 → (√162

)3 → (√16)3 → 43 → 64

• 2433

5 → (√2435

)3 → 33 → 27

Positive versus Negative Rational Exponents

➢ The standard form to denote a value with a rational exponent is to specify the value with a positive

rational exponent

• It is improper to specify a value with a negative exponent

• In order to convert a value with a negative rational exponent to an equivalent value with a positive

rational exponent, simple take the reciprocal of the value and change the negative rational exponent

to positive

➢ 𝑎

𝑏

−𝑚

𝑛 = 𝑏

𝑎

𝑚

𝑛

• (8)−2

5 = (1

8)

2

5

• (1

3)−

45 = (3)

45

• (7

8)−2

3 =( 8

7)

23

190

Radical Manipulation – Guided Practice

Simplifying Radicals (If necessary, Use Product Rule and Quotient Rule of Radicals)

___________ 1. Simplify √64

___________ 2. Simplify √256

___________ 3. Simplify √625

___________ 4. Simplify √784

___________ 5. Simplify √900

___________ 6. Simplify √643

___________ 7. Simplify √2163

___________ 8. Simplify √2435

___________ 9. Simplify √5123

___________ 10. Simplify √7296

__________ 11. Simplify √50

__________ 12. Simplify √108

__________ 13. Simplify √4323

191

Radical Manipulation – Guided Practice (continued)

Simplifying Radicals (continued)

__________ 14. Simplify √324

__________ 15. Simplify √1285

__________ 16. Simplify √75

√3

__________ 17. Simplify √169

225

__________ 18. Simplify √128

3

√163

__________ 19. Simplify √81

625

4

__________ 20. Simplify (64)43

__________ 21. Simplify (16)74

__________ 22. Simplify (81)−54

__________ 23. Simplify (1

36)−3

2

__________ 24. Simplify (16

25)−3

2

192

Section 13.4 Radical Arithmetic

Adding Radicals

➢ To add radicals, they must have the same radicand and the same index

➢ Adding radicals involves using the Distributive Property

• Add the coefficients of the radicals

• Keep the radicand and index

➢ 3√5 + 4√5

• The radicands of the two radicals are the same→ 5

• The index of the two radicals are the same → implied 2

• Add the coefficients (3 from the first radical and 4 from the second radical) → 3 + 4 = 7

• Place the sum of the coefficients in front of the radical (keep the radicand and index) → 7√5

➢ 5√73

+ 6√73


• The index of the two radicals are the same → 3

• Add the coefficients (5 from the first radical and 6 from the second radical) → 5 + 6 = 11

• Place the sum of the coefficients in front of the radical (keep the radicand and index) → 11√73

➢ 8√53

+ 9√113

• Cannot be combined because the radicands of the radicals (5 for the first radical and 11 for the

second radical) are not the same

➢ 8√54

+ 9√53

• Cannot be combined because the indexes of the radicals (4 for the first radical and 3 for the second

radical) are not the same

Subtracting Radicals

➢ To subtract radicals, they must have the same radicand and the same index

➢ Subtracting radicals involves using the Distributive Property

• Subtract the coefficient on the right hand side of the minus sign from the coefficient on the left hand

side of the minus sign

• Keep the radicand and index

➢ 8√5 – 2√5


• The index of the two radicals are the same → implied 2

• Subtract the coefficient on the right side of the minus sign (2) from the coefficient on the left hand

side of the minus sign (8) → 8 – 2 = 6

• Place the difference of the coefficients in front of the radical (keep the radicand and index) → 6√5

➢ 13√73

– 5√73


• The index of the two radicals are the same → 3

• Subtract the coefficient on the right side of the minus sign (5) from the coefficient on the left hand

side of the minus sign (13) → 13 –5 = 8

• Place the difference of the coefficients in front of the radical (keep the radicand and index) → 8√73

193

Section 13.4 Radical Arithmetic (continued)

Subtracting Radicals (continued)

➢ 8√53

– 9√113

• Cannot be combined because the radicands of the radicals (5 for the first radical and 11 for the

second radical) are not the same

➢ 8√54

– 9√53

• Cannot be combined because the indexes of the radicals (4 for the first radical and 3 for the second

radical) are not the same

Multiplying Radicals

➢ To multiply radicals, they must have the same index but are not required to have the same radicand

➢ Multiplying radicals involves the following steps

• Multiply the coefficients

• Using the Product Rule of Radicals, multiply the numbers under the radical

• Simplify the radical by finding the largest “perfect root” factor of the product under the radical

• Take the root of the “perfect root” factor and multiply it by the existing coefficient

• Keep the index

➢ (3√5)(2√7)

• Multiply the coefficients → 3 ∙ 2 = 6

• Multiply the numbers under the radical → (√5)(√7) = √35

• Simplify the radical if possible → √35 is already simplified (no “perfect square” factors of 35)

• Final answer → 6√7

➢ (4√3)(5√15)


• Multiply the numbers under the radical → (√3)(√15) = √45

• Simplify the radical if possible → √45 = √9 ∙ 5 = √9 ∙ √5 = 3√5

• Take the root of the “perfect root” factor and multiply it by the existing coefficient → 20 ∙ 3 = 60

• Keep the index in the final answer → 60√5

➢ (3√723

)(4√63

)


• Multiply the numbers under the radical → (√723

)(√63

) = √4323

• Simplify the radical if possible → 216 is the largest perfect cube root of 432 → √4323

= √216 ∙ 22

=

√2162

∙ √23

= 6√23

• Take the root of the “perfect root” factor and multiply it by the existing coefficient → 12 ∙ 6 = 72

• Keep the index in the final answer → 72√23

194


Special Rule for Multiplying Radicals

➢ Any radicand raised to the power of the radical’s index is the radicand

• (√𝑎)2 = a

• ( √𝑎𝑛

)n = a

➢ The index and exponent nullify each other and leave the radicand as the final answer

• (√3)2 = 3

• (√17)2 = 17

• (√23)2 = 23

• (√113

)3 = 11

• (√245

)5 = 24

• (√357

)7 = 35

➢ The square root of any value times the square root itself just becomes that value

• (√𝑎)(√𝑎) = a

• (√3)(√3) = √9 = 3

• (√7)(√7) = √49 = 7

• (√13)(√13) = √169 = 13

• (√29)(√29) = √841 = 29

Multiplying Radicals with Additional or Subtraction

➢ Uses the Distributive Property to distribute the multiplication throughout the parentheses

➢ (3√5)(2 + 4√7) → (3√5)(2) + (3√5)(4√7) = 6√5) + 12√35

➢ (2 – 3√7)(4 + 5√2) → (2)(4) – 3√7)(4) + (2)(5√2) – (3√7)(5√2) = 8 – 12√7 +10√2 – 15√14

195

Radical Arithmetic – Guided Practice

Simplifying Radicals using the Rules of Radical Arithmetic

(Indicate if the radicals are already simplified and cannot be combined)

___________ 1. Simplify 3√2 + 5√2

___________ 2. Simplify 11√5 + 6√5

___________ 3. Simplify 4√5 + 3√6

___________ 4. Simplify 7√2 + 8√23

___________ 5. Simplify 3√2 – 5√2

___________ 6. Simplify 11√5 – 6√5

___________ 7. Simplify 4√5 – 3√6

___________ 8. Simplify 7√2 – 8√23

___________ 9. Simplify (2√3) (4√5)

___________ 10. Simplify (3√2) (5√6)

___________ 11. Simplify (4√3) (5√3)

___________ 12. Simplify (4√2) (3√12)

___________ 13. Simplify (3√53

) (4√63

)

___________ 14. Simplify (4√23

) (5√123

)

___________ 15. Simplify (5√33

) (7√44

)

196

Radical Arithmetic – Guided Practice (continued)

Simplifying Radicals Using the Rules of Radical Arithmetic (continued)

___________ 16. Simplify (√11)(√11)

___________ 17. Simplify (√19)(√19)

___________ 18. Simplify (√30)(√30)

___________ 19. Simplify (√1250)(√1250)

___________ 20. Simplify 5(2 + √7)

___________ 21. Simplify √3(2 + √7)

___________ 22. Simplify (2 + √3)(4 + √7)

___________ 23. Simplify (3 + √2)(3 + √2)

___________ 24. Simplify (4 + √5)(4 – √5)

___________ 25. Simplify (√2 + 3)(√2 – 3)

197


Dividing Radicals

➢ To divide radicals, they must have the same index but are not required to have the same radicand

➢ When dividing radicals, the final answer typically ends up in the form of a fraction

• However, it is improper to specify the final answer with a radical in the denominator

• The process of removing the radical from the denominator is called Rationalizing the Denominator

➢ Examples of Dividing Radicals

• √5 ÷ 4 → √5

4 (answer is in proper form)

• √24 ÷ √2 → √24

√2 = √

24

2 = √12 = √4 ∙ 3 = √4 ∙ √3 = 2√3 (answer is in proper form)

• √27 ÷ √5 → √27

√5 =

√9 ∙3

√5 =

√9∙√3

√5 =

3√3

5 (this is not proper form; requires Rationalizing the

Denominator)

• (2 + √5) ÷ (8 – √7) = 2+√5

8−√7 (this is not proper form; requires Rationalizing the Denominator)

Conjugate Pair

➢ A pair of expressions (in the form a + √𝑏 and a – √𝑏 or in the form √𝑏 + a and √𝑏 – a)

• They contain the same rational parts but opposite radical parts, or

• They contain the same radical parts but opposite rational parts

➢ Conjugate Examples

• 3 + √5 → Conjugate is 3 – √5

• 4 – √3 → Conjugate is 4 + √3

• √7 + 6 → Conjugate is √7 – 6

• √3 + 7 → Conjugate is √3 – 7

Multiplying Conjugates

➢ When conjugates are multiplied, the radical portion disappears completely

• (3 + √5)(3 – √5) → (3)(3) + (√5)(3) – (3)(√5) – (√5)(√5) = 9 +3√5 – 3√5 – 5 = 9 – 5 = 4

• (4 – √3)(4 + √3) → (4)(4) – (√3)(4) + (4)(√3) – (√3)(√3) = 16 – 4√3 + 4√3 – 3 = 16 – 3 = 13

• (√7 + 6)(√7 – 6) → (√7)(√7) + (6)(√7) – (√7)(6) – (6)(6) = 7 +6√7 – 6√7 – 36 = 7 – 36 = –29

• (√3 – 7)(√3 + 7) → (√3)(√3) – (7)(√3) + (√3)(7) – 49 = 3 +7√3 – 7√3 – 49 = 3 – 49 = –46

Rationalizing the Denominator

➢ If the denominator contains a single radical (without addition or subtraction), simply multiply the

fraction by the denominator’s radical over itself

• 4

√5 →

4

√5 ∙

√5

√5 →

4√5

√5∙√5 →

4√5

5 (this answer is now in proper form)

➢ If the denominator contains a radical with addition or subtraction, multiply the fraction by the

denominator’s conjugate over itself

• 4

3+√2 →

4

3+√2 3−√2

3−√2 =

4(3−√2)

(3+√2)(3−√2) =

12−4√2

9−2 =

12−4√2

7 (this answer is now in proper form)

198

Dividing Radicals – Guided Practice

Simplifying Radicals Using the Rules of Radical Arithmetic

___________ 1. Simplify √11 ÷ 7

___________ 2. Simplify √17 ÷ 5

___________ 3. Simplify √68 ÷ 17

___________ 4. Simplify √68 ÷ √17

___________ 5. Simplify (√13 + 4) ÷ 7

___________ 6. Simplify (√19 – 5) ÷ 11

___________ 7. Simplify 6 ÷ √5

___________ 8. Simplify √20 ÷ √5

___________ 9. Simplify 12 ÷ √3

___________ 10. Simplify 14 ÷ √7

___________ 11. Simplify (2 + √11) ÷ √5

___________ 12. Simplify (6 – √3) ÷ √15

199

Dividing Radicals – Guided Practice (continued)

Simplifying Radicals Using the Rules of Radical Arithmetic (continued)

___________ 13. Simplify 5 ÷ (6 – √3)

___________ 14. Simplify 5 ÷ (6 + √3)

___________ 15 Simplify (3 + √5) ÷ (6 + √3)

___________ 16 Simplify (7 – √3) ÷ (7 + √3)

___________ 17 Simplify (7 + √3) ÷ (7 – √3)

___________ 18 Simplify (4 + 2√6) ÷ (5 + √7)

___________ 19 Simplify (7 + 3√5) ÷ (8 + 4√2)

___________ 20 Simplify (7 + 3√5) ÷ (8 – 4√2)

200


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