Math Starters! 5- to 10-Minute Activities Aligned with the Common Core Math Standards, Grades 6-12

Abstract

CONTENTS About the Authors xxv Acknowledgments xxvii About This Book xxix Standards and Problems Chart xxxi PART 1: MAKING MATH STARTERS PART OF YOUR PROGRAM 1 The Value of Math Starters 3 Starting Class with a Math Starter 3 Purpose and Value of a Math-Starter Notebook 4 The Value of Written Explanations 7 Cooperative Problem Solving Using Math Starters 7 Organizing Groups for Problem Solving 7 The Value of Sharing and Discussion 11 Using Problem-Solving Strategies 11 Evaluation 14 Checklists 14 Point Systems 14 Quizzes That Include Math Starters 15 Review of Math-Starter Notebooks 15 Student Participation 15 Portfolios 15 A FinalWord 15 PART 2: MATH STARTERS 17 Section 1: Whole Numbers and Integers: Theory and Operations 19 1-1 Natural Numbers 19 1-2 Natural Numbers 19 1-3 Whole Numbers 20 1-4 Whole NumbersG 20 1-5 Place Value with Whole Numbers 20 CONTENTS About the Authors xxv Acknowledgments xxvii About This Book xxix Standards and Problems Chart xxxi PART 1: MAKING MATH STARTERS PART OF YOUR PROGRAM 1 The Value of Math Starters 3 Starting Class with a Math Starter 3 Purpose and Value of a Math-Starter Notebook 4 The Value of Written Explanations 7 Cooperative Problem Solving Using Math Starters 7 Organizing Groups for Problem Solving 7 The Value of Sharing and Discussion 11 Using Problem-Solving Strategies 11 Evaluation 14 Checklists 14 Point Systems 14 Quizzes That Include Math Starters 15 Review of Math-Starter Notebooks 15 Student Participation 15 Portfolios 15 A FinalWord 15 PART 2: MATH STARTERS 17 Section 1: Whole Numbers and Integers: Theory and Operations 19 1-1 Natural Numbers 19 1-2 Natural Numbers 19 1-3 Whole Numbers 20 1-4 Whole NumbersG 20 1-5 Place Value with Whole Numbers 20 vi Contents 1-6 Place Value with Whole Numbers 21 1-7 Numerical Operations 21 1-8 Numerical OperationsG 21 1-9 Adding Whole Numbers 22 1-10 Subtracting Whole Numbers 22 1-11 Subtracting Whole Numbers 22 1-12 Multiplying Whole Numbers 23 1-13 Multiplying Whole Numbers 23 1-14 Dividing Whole Numbers (6.NS.2) 23 1-15 Dividing Whole Numbers (6.NS.2) 24 1-16 Dividing Whole Numbers (6.NS.2) 24 1-17 Whole Numbers—Multistep Problem (7.EE.3) 24 1-18 Whole Numbers—Multistep Problem G (7.EE.3) 25 1-19 Estimation with Compatible Numbers (7.EE.3) 25 1-20 Rounding Whole Numbers 25 1-21 Divisibility by 2, 4, and 8 26 1-22 Divisibility by 3, 6, 9, and 12 26 1-23 Divisibility by 5 and 10 27 1-24 Factors 27 1-25 Factors 27 1-26 Greatest Common Factor (6.NS.4) 28 1-27 Greatest Common Factor (6.NS.4) 28 1-28 Multiples 28 1-29 Multiples 29 1-30 Least Common Multiple (6.NS.4) 29 1-31 Least Common Multiple (6.NS.4) 30 1-32 Multiples and the Distributive Property (6.NS.4) 30 1-33 Prime Numbers 31 1-34 Prime Numbers 31 1-35 Composite Numbers 31 1-36 Prime and Composite Numbers 32 1-37 Perfect Squares 32 1-38 Perfect Squares and Prime Numbers 32 1-39 Order of Operations 33 1-40 Order of Operations G 33 1-41 Powers of Numbers (6.EE.1) 33 1-42 Simplifying Expressions with Exponents (6.EE.1) 34 1-43 Simplifying Expressions with Exponents (6.EE.1) 34 1-44 Simplifying Expressions with Exponents (6.EE.1) 34 Contents vii 1-45 Writing Numerical Expressions 35 1-46 Identifying Parts of a Numerical Expression 35 1-47 Integers (6.NS.5) 36 1-48 Integers (6.NS.5) 36 1-49 Opposites (6.NS.5) 36 1-50 The Number Line (6.NS.6) 37 1-51 The Number Line (6.NS.6) 37 1-52 Absolute Value (6.NS.7) 37 1-53 Absolute Value G (6.NS.7) 38 1-54 Comparing Integers 38 1-55 Inequality Symbols 38 1-56 Ordering Integers on a Number Line (6.NS.7) 39 1-57 Understanding Statements of Order (6.NS.7) 39 1-58 The Coordinate Plane (6.NS.6) 40 1-59 Graphing Points in the Coordinate Plane (6.NS.6) 41 1-60 Solving Problems by Graphing Points in the Coordinate Plane (6.NS.8) 41 1-61 Adding Integers (7.NS.1) 41 1-62 Adding Integers (7.NS.1) 42 1-63 Subtracting Integers (7.NS.1) 42 1-64 Subtracting Integers (7.NS.1) 42 1-65 Adding and Subtracting Integers (7.NS.1) 43 1-66 Representing Addition and Subtraction on a NumberLine (7.NS.1) 43 1-67 Multiplying Two Integers (7.NS.2) 44 1-68 Multiplying More Than Two Integers (7.NS.2) 44 1-69 Multiplying More Than Two Integers (7.NS.2) 45 1-70 Dividing Two Integers (7.NS.2) 45 1-71 Dividing Two Integers (7.NS.2) 46 1-72 Multiplying and Dividing Integers (7.NS.2) 46 1-73 Four Operations with Integers (7.NS.3) 46 1-74 Four Operations with Integers (7.NS.3) 47 1-75 Four Operations with Integers (7.NS.3) 47 1-76 Using Positive Exponents with Integers 48 1-77 Using Scientific Notation to Express Large Numbers (8.EE.3) 48 1-78 Computing with Numbers Written in ScientificNotation (8.EE.3) 49 1-79 Changing Numbers in Scientific Notation to StandardForm (8.EE.4) 50 1-80 A Quotation Applicable to Mathematics 50 viii Contents Section 2: Rational Numbers: Fractions, Decimals, and Percents 51 2-1 Equivalent Fractions 51 2-2 Simplifying Fractions 51 2-3 Simplifying Fractions 52 2-4 Writing Improper Fractions as Mixed Numbers 52 2-5 Writing Mixed Numbers as Improper Fractions 53 2-6 Comparing Fractions 53 2-7 Ordering Fractions 54 2-8 Adding Fractions 54 2-9 Adding Fractions 54 2-10 Adding Mixed Numbers 55 2-11 Adding Mixed Numbers 55 2-12 Subtracting Fractions 55 2-13 Subtracting Fractions 56 2-14 Subtracting Mixed Numbers 56 2-15 Subtracting Mixed Numbers 56 2-16 Subtracting Mixed Numbers 57 2-17 Multiplying Fractions 57 2-18 Multiplying Fractions 57 2-19 Multiplying Fractions and Mixed Numbers 58 2-20 Multiplying Mixed Numbers 58 2-21 Estimating and Multiplying Mixed Numbers 58 2-22 Dividing Fractions (6.NS.1) 59 2-23 Dividing Fractions (6.NS.1) 59 2-24 Dividing Fractions G (6.NS.1) 59 2-25 Dividing Fractions and Mixed Numbers (6.NS.1) 60 2-26 Dividing Mixed Numbers (6.NS.1) 60 2-27 Dividing Mixed Numbers (6.NS.1) 60 2-28 Fractions—Multistep Problem (7.EE.3) 61 2-29 Fractions—Multistep Problem (7.EE.3) 61 2-30 Decimals 61 2-31 Ordering Decimals 62 2-32 Ordering Decimals 62 2-33 Place Value with Decimals 62 2-34 Writing Fractions as Decimals 63 2-35 Writing Decimals as Fractions 63 2-36 Writing Decimals as Fractions 64 2-37 Repeating Decimals (7.NS.2) 64 2-38 Repeating Decimals (7.NS.2) 65 Contents ix 2-39 Comparing Fractions and Decimals (7.EE.3) 65 2-40 Estimating with Decimals (7.EE.3) 65 2-41 Rounding Decimals 66 2-42 Rounding Decimals and Unit Pricing (6.RP.2) 66 2-43 Adding Decimals (6.NS.3) 66 2-44 Adding Decimals (6.NS.3) 67 2-45 Subtracting Decimals (6.NS.3) 67 2-46 Subtracting Decimals (6.NS.3) 68 2-47 Multiplying Decimals (6.NS.3) 68 2-48 Multiplying Decimals (6.NS.3) 69 2-49 Dividing a Decimal by a Whole Number (6.NS.3) 69 2-50 Dividing Decimals (6.NS.3) 69 2-51 Dividing Decimals (6.NS.3) 70 2-52 Decimals—Multistep Problem (6.NS.3) 70 2-53 Decimals—Multistep Problem (6.RP.3) 71 2-54 Decimals—Multistep Problem (7.EE.3) 71 2-55 Decimals—Multistep Problem (7.EE.3) 71 2-56 Decimals—Multistep Problem (7.EE.3) 72 2-57 Decimals—Multistep Problem (7.EE.3) 72 2-58 Decimals—Multistep Problem (7.EE.3) 72 2-59 Order of Operations—Decimals (6.NS.3) 73 2-60 Order of Operations—Decimals (6.NS.3) 73 2-61 Order of Operations—Decimals (6.NS.3) 73 2-62 Ratio (6.RP.1) 74 2-63 Ratio (6.RP.1) 74 2-64 Ratio Reasoning (6.RP.3) 75 2-65 Proportional Relationships (7.RP.2) 75 2-66 Proportional Relationships (7.RP.2) 75 2-67 Proportional Relationships and Scale (7.RP.1) 76 2-68 Ratio and Rate Reasoning (6.RP.3) 76 2-69 Equivalent Ratios and the Coordinate Plane (6.RP.3) 77 2-70 Percents 77 2-71 Percents 78 2-72 Equivalencies—Fractions, Decimals, and Percents 78 2-73 Equivalencies—Fractions, Decimals, and Percents 79 2-74 Equivalencies—Fractions, Decimals, and Percents 80 2-75 Equivalencies—Repeating Decimals (7.NS.2) 80 2-76 Finding the Percent of a Number (6.RP.3) 81 2-77 Finding the Percent of a Number (6.RP.3) 81 2-78 Finding the Percent of a Number (6.RP.3) 81 2-79 Finding the Percent of a Number (6.RP.3) 82 2-80 Using Proportional Relationships to Find the Percentof a Number (7.RP.2) 82 2-81 Finding a Number When a Percent of It Is Known (6.RP.3) 82 2-82 Using Proportional Relationships to Find a Number When a Percent of It Is Known (7.RP.2) 83 2-83 Finding What Percent a Number Is of Another Number 83 2-84 Using Proportional Relationships to Find What Percent a Number Is of Another Number (7.RP.2) 84 2-85 The Three Types of Percentage Problems 85 2-86 Percents and Sales Tax—Multistep Problem (7.EE.2) 85 2-87 Percents and Discounts 86 2-88 Percents and Discounts—Multistep Problem (7.RP.3) 86 2-89 Percents and Sales Price—Multistep Problem (7.RP.3) 86 2-90 Percents and Tips—Multistep Problem (7.RP.3) 87 2-91 Percent of Increase (7.RP.3) 87 2-92 Percent of Decrease (7.RP.3) 87 2-93 Adding Positive and Negative Fractions (7.NS.1) 88 2-94 Adding Positive and Negative Fractions (7.NS.1) 88 2-95 Subtracting Positive and Negative Fractions (7.NS.1) 89 2-96 Subtracting Positive and Negative Fractions (7.NS.1) 89 2-97 Multiplying Positive and Negative Fractions (7.NS.2) 89 2-98 Multiplying Positive and Negative Fractions G (7.NS.2) 90 2-99 Dividing Positive and Negative Fractions (7.NS.2) 90 2-100 Dividing Positive and Negative Fractions G (7.NS.2) 90 2-101 Four Operations with Positive and NegativeFractions (7.NS.3) 91 2-102 Simplifying Complex Fractions (7.NS.3) 91 2-103 Simplifying Complex Fractions (7.NS.3) 92 2-104 Adding Positive and Negative Decimals (7.NS.1) 92 2-105 Subtracting Positive and Negative Decimals (7.NS.1) 93 2-106 Multiplying Positive and Negative Decimals (7.NS.2) 93 2-107 Dividing Positive and Negative Decimals (7.NS.2) 93 2-108 Four Operations with Positive and NegativeDecimals G (7.NS.3) 94 2-109 Classifying Numbers as Rational or Irrational (8.NS.1) 94 2-110 Changing Repeating Decimals to RationalNumbers G (8.NS.1) 94 2-111 Changing Repeating Decimals to RationalNumbers (8.NS.1) 95 2-112 Rational Approximations of Irrational Numbers (8.NS.2) 95 Contents xi 2-113 Integer Exponents (8.EE.1) 96 2-114 Integer Exponents (8.EE.1) 96 2-115 Square and Cube Roots (8.EE.2) 97 2-116 Using Scientific Notation to Express Small Numbers (8.EE.3) 97 2-117 Using Scientific Notation to Express Large and Small Numbers (8.EE.3) 98 2-118 Performing Operations with Numbers Expressed in Scientific Notation (8.EE.4) 98 2-119 Expressing Large and Small Numbers in Standard Form 99 2-120 A Quotation about Mathematics 99 Section 3: Algebra and Beyond 101 3-1 Simplifying Numerical Expressions with Exponents (6.EE.1) 101 3-2 Simplifying Numerical Expressions with Exponents (6.EE.1) 101 3-3 Simplifying Numerical Expressions with Exponents (6.EE.1) 102 3-4 Words and Phrases as Mathematical Expressions G (6.EE.2) 102 3-5 Writing Phrases as Algebraic Expressions (6.EE.2) 102 3-6 Evaluating Expressions without Exponents (6.EE.2) 103 3-7 Evaluating Expressions with Exponents (6.EE.2) 103 3-8 Evaluating Expressions with Exponents (6.EE.2) 104 3-9 Generating Equivalent Expressions (6.EE.3) 104 3-10 Generating Equivalent Expressions (6.EE.3) 105 3-11 Identifying Equivalent Expressions (6.EE.4) 105 3-12 Identifying Equivalent Expressions (6.EE.4) 105 3-13 Identifying the Solution of an Equation (6.EE.5) 106 3-14 Identifying the Solutions of an Inequality (6.EE.5) 106 3-15 Variables (6.EE.6) 106 3-16 Using Variables to Represent Numbers (6.EE.6) 107 3-17 Solving One-Step Equations—Addition (6.EE.7) 107 3-18 Solving One-Step Equations—Subtraction (6.EE.7) 108 3-19 Solving One-Step Equations—Addition and Subtraction (6.EE.7) 108 3-20 Solving One-Step Equations—Multiplication (6.EE.7) 109 3-21 Solving One-Step Equations—Division (6.EE.7) 109 3-22 Solving One-Step Equations—Multiplication and Division (6.EE.7) 109 3-23 Writing Inequalities (6.EE.8) 110 3-24 Solving Inequalities (7.EE.4) 110 3-25 Representing Relationships between Dependent and Independent Variables G (6.EE.9) 111 3-26 Generating Equivalent Expressions (7.EE.1) 111 3-27 Rewriting Expressions in Different Forms (7.EE.2) 112 xii Contents 3-28 Solving Two-Step Equations (7.EE.3) 112 3-29 Solving Two-Step Equations G (7.EE.4) 112 3-30 Solving Two-Step Equations (7.EE.4) 113 3-31 Square Roots, Cube Roots, and Equations (8.EE.2) 113 3-32 Interpreting the Unit Rate G (8.EE.5) 114 3-33 Using Similar Triangles to Explain Slope (8.EE.6) 114 3-34 Finding the Slope of a Line 115 3-35 Slopes of Horizontal and Vertical Lines 115 3-36 Application of Finding the Slope 116 3-37 Identifying Like Terms 116 3-38 Simplifying Expressions 117 3-39 Simplifying Expressions 117 3-40 Simplifying and Evaluating Expressions 118 3-41 Solving Equations involving Several Steps with Variables on the Same Side (8.EE.7) 118 3-42 Solving Equations involving Several Steps with Variables on the Same Side (8.EE.7) 119 3-43 Solving Equations involving Several Steps with Variables on Both Sides (8.EE.7) 119 3-44 Solving Equations involving Several Steps with Variables on Both Sides (8.EE.7) 119 3-45 Points of Intersection of Linear Equations (8.EE.8) 120 3-46 Estimating Solutions to Systems of Linear Equations by Graphing (8.EE.8) 120 3-47 Using the Graphing Method to Solve Systems of Linear Equations (8.EE.8) 120 3-48 Using the Substitution Method to Solve Systems of Linear Equations (8.EE.8) 121 3-49 Using the Addition-or-Subtraction Method to Solve Systems of Linear Equations (8.EE.8) 122 3-50 Using Multiplication with the Addition-or-Subtraction Method to Solve Systems of Linear Equations (8.EE.8) 123 3-51 Choosing Methods and Solving Systems of Linear Equations (A-REI.6) 123 3-52 Multiplying Monomials 124 3-53 Multiplying Monomials G 124 3-54 Powers of Monomials 125 3-55 Powers of Monomials 125 3-56 Rewriting Monomials (A-SSE.2) 125 3-57 Dividing Monomials 126 3-58 Dividing Monomials 126 3-59 Dividing Monomials 126 3-60 Interpreting Algebraic Expressions (A-SSE.1) 127 Contents xiii 3-61 Finding the Greatest Common Factor (GCF) of Monomials 127 3-62 Polynomials G (A-APR.1) 128 3-63 Adding and Subtracting Polynomials (A-APR.1) 128 3-64 Multiplying a Monomial by a Binomial (A-APR.1) 128 3-65 Dividing a Polynomial by a Monomial (A-APR.1) 129 3-66 Multiplying Binomials (A-APR.1) 129 3-67 Multiplying Binomials (A-APR.1) 130 3-68 Cubes of Binomials (A-APR.4) 130 3-69 Rewriting Differences of Squares (A-SSE.2) 131 3-70 Factoring Squares of Binomials (A-SSE.3) 131 3-71 Factoring Trinomials of the Form x2 + bx + c Where c > 0 (A-SSE.3) 132 3-72 Factoring Trinomials of the Form x2 + bx + c Where c < 0 (A-SSE.3) 132 3-73 Factoring Polynomials of the Form ax2 + bx + c Where a Is an Integer > 1 133 3-74 Factoring by Grouping (A-SSE.3) 133 3-75 Sums and Differences of Cubes 134 3-76 Completing the Square (A-SSE.3) 134 3-77 Arithmetic and Geometric Sequences 135 3-78 Finding the Partial Sums of Infinite Series 135 3-79 Deriving the Formula for Finding the Sums of a Geometric Series G (A-SSE.4) 136 3-80 Using the Remainder Theorem and the Factor Theorem (A-APR.2) 137 3-81 Identifying Zeros of Polynomials (A-APR.3) 137 3-82 Using Zeros to Sketch Graphs of Functions Defined by Polynomials (A-APR.3) 137 3-83 Generating Pythagorean Triples (A-APR.4) 138 3-84 Pascal’s Triangle and the Binomial Theorem G (A-APR.5) 138 3-85 Rewriting Rational Expressions (A-APR.6) 139 3-86 Rewriting Rational Expressions (A-APR.6) 139 3-87 Rewriting Rational Expressions (A-APR.6) 139 3-88 Simplifying Rational Expressions (A-APR.6) 140 3-89 Multiplying and Dividing Rational Expressions (A-APR.7) 140 3-90 Multiplying and Dividing Rational Expressions (A-APR.7) 141 3-91 Adding and Subtracting Rational Expressions with the Same Denominator (A-APR.7) 141 3-92 Finding the Least Common Denominator of Rational Expressions (A-APR.7) 142 3-93 Adding and Subtracting Rational Expressions with Different Denominators (A-APR.7) 142 xiv Contents 3-94 Adding, Subtracting, Multiplying, and Dividing Rational Expressions G (A-APR.7) 143 3-95 Explaining the Steps in Solving an Equation (A-REI.1) 143 3-96 Explaining the Steps in Solving an Equation (A-REI.1) 144 3-97 Solving Rational Equations (A-REI.2) 144 3-98 Solving Rational Equations—Extraneous Solutions (A-REI.2) 145 3-99 Expressing Square Roots in Radical Form (A-REI.2) 145 3-100 Finding Square Roots 146 3-101 Expressing Radical Expressions in Simplest Form (A-REI.2) 146 3-102 Adding and Subtracting Radicals (A-REI.2) 147 3-103 Multiplying Binomials Containing Radicals (A-REI.2) 147 3-104 Rationalizing the Denominator That Contains Radicals (A-REI.2) 148 3-105 Solving Radical Equations (A-REI.2) 148 3-106 Solving Simple Radical Equations—Extraneous Solutions (A-REI.2) 149 3-107 Transforming Equations G (A-CED.4) 149 3-108 Solving Linear Equations in One Variable—Coefficients Are Letters (A-REI.3) 149 3-109 Solving Two-Step Inequalities (A-REI.3) 150 3-110 Using Squares of Binomials and Perfect Squares (A-REI.4) 150 3-111 Using the ± Symbol 151 3-112 Solving Quadratic Equations by Completing the Square (A-REI.4) 151 3-113 Deriving the Quadratic Formula (A-REI.4) 152 3-114 Transforming Equations into the Form ax2 + bx + c = 0, a = 0 (A-CED.4) 152 3-115 Solving Quadratic Equations Using the Quadratic Formula If b2 − 4ac ≥ 0 (A-REI.4) 153 3-116 Using the Zero-Product Property (A-REI.4) 153 3-117 Solving Quadratic Equations by Factoring (A-REI.4) 154 3-118 Solving Quadratic Equations of the Form ax2 = c, a = 0 (A-REI.4) 155 3-119 Sums and Products of Roots (A-REI.4) 155 3-120 Using the Discriminate G (A-REI.4) 156 3-121 Producing Systems of Equations with the Same Solution (A-REI.5) 156 3-122 Solving a System Consisting of a Linear Equation and a Quadratic Equation (A-REI.7) 157 3-123 Representing a System of Linear Equations as a Matrix Equation (A-REI.8) 157 3-124 Using the Inverse of a Matrix to Solve a Matrix Equation (A-REI.9) 158 Contents xv 3-125 Using the Inverse of a 3 × 3 Matrix to Solve a Matrix Equation (A-REI.9) 158 3-126 Graphs and Solutions of Equations (A-REI.10) 159 3-127 Finding the Point Where Two Graphs Intersect (A-REI.11) 159 3-128 Graphing Solutions to a System of Linear Inequalities (A-REI.12) 160 3-129 Writing and Solving Equations and Inequalities (A-CED.1) 160 3-130 Creating and Graphing Equations G (A-CED.2) 161 3-131 Interpreting Solutions as Viable Options (A-CED.3) 161 3-132 A Quotation about Algebra 161 Section 4: Functions 163 4-1 Domain and Range (8.F.1) 163 4-2 Describing Graphs of Linear Functions (8.F.1) 163 4-3 Finding and Comparing Rates of Change (8.F.2) 164 4-4 Identifying Linear Functions (8.F.3) 164 4-5 Identifying Linear Functions (8.F.3) 165 4-6 Interpreting the Initial Value of a Function (8.F.4) 165 4-7 Functions and Graphs (8.F.5) 166 4-8 Increasing and Decreasing Functions G (8.F.5) 167 4-9 Understanding the Concept of a Function (F-IF.1) 167 4-10 Evaluating Functions (F-IF.2) 168 4-11 Using Sequences Defined Recursively (F-IF.3) 168 4-12 Identifying Key Features of a Graph (F-IF.4) 168 4-13 Relating the Domain to the Relationships It Describes (F-IF.5) 169 4-14 Finding the Average Rate of Change (F-IF.6) 169 4-15 Graphs of Functions (F-IF.7) 170 4-16 Step Graphs G (F-IF.7) 171 4-17 Using Factoring and Completing the Square in Quadratic Functions (F-IF.8) 172 4-18 Classifying Functions as Exponential Growth or Exponential Decay (F-IF.8) 172 4-19 Comparing Properties of Functions (F-IF.9) 173 4-20 Writing Functions (F-BF.1) 173 4-21 Writing Sequences (F-BF.2) 174 4-22 Translations and Dilations of Graphs (F-BF.3) 174 4-23 Odd and Even Functions (F-BF.3) 175 4-24 Finding the Inverse of a Function (F-BF.4) 175 4-25 Determining If Two Functions Are Inverses (F-BF.4) 176 4-26 Using Exponential and Logarithmic Functions (F-BF.5) 176 4-27 Linear and Exponential Models (F-LE.1) 177 xvi Contents 4-28 Constructing Linear and Exponential Functions (F-LE.2) 177 4-29 Comparing Exponential, Linear, and Polynomial Functions (F-LE.3) 178 4-30 Solving Exponential Equations (F-LE.4) 178 4-31 Interpreting Parameters in the Compound Interest Formula (F-LE.5) 179 4-32 Understanding Radian Measures (F-TF.1) 179 4-33 Extending Trigonometric Functions to All Real Numbers (F-TF.2) 180 4-34 Using Special Right Triangles (F-TF.3) 181 4-35 Using the Unit Circle to Explain the Symmetry and Periodicity of the Trigonometric Functions (F-TF.4) 182 4-36 Choosing Trigonometric Functions to Model Periodic Phenomena (F-TF.5) 182 4-37 Restricting the Domain of a Trigonometric Function to Find Its Inverse G (F-TF.6) 183 4-38 Using Inverse Trigonometric Functions (F-TF.7) 183 4-39 Proving a Pythagorean Identity (F-TF.8) 183 4-40 Proving the Addition Formula for the Sine Function (F-TF.9) 184 4-41 A Quotation Applicable to Functions 184 Section 5: Geometry 185 5-1 Naming Lines, Rays, and Segments 185 5-2 Intersection of Lines, Segments, and Rays 186 5-3 Unions of Lines, Segments, and Rays 187 5-4 Naming Angles 188 5-5 Types of Angles 189 5-6 Complementary and Supplementary Angles 190 5-7 Pairs of Angles—Adjacent, Vertical, Complementary, and Supplementary Angles 191 5-8 Angles Formed by a Transversal 192 5-9 Parallel Lines and Transversals (8.G.5) 193 5-10 Perpendicular Lines and Unknown Angle Measurements (7.G.5) 194 5-11 Identifying and Sketching Common Polygons 195 5-12 Drawing Polygons in the Coordinate Plane (6.G.3) 195 5-13 Diagonals of Polygons 196 5-14 Sum of the Angles of a Polygon 197 5-15 The Measure of Each Interior Angle of a Regular Polygon 197 5-16 The Measure of Each Exterior Angle of a Regular Polygon 198 5-17 The Sum of the Measures of Each Exterior Angle of a Polygon 199 5-18 Classifying Triangles by the Lengths of Their Sides 200 Contents xvii 5-19 Classifying Triangles by the Measures of Their Angles 201 5-20 Included Sides and Angles of a Triangle 202 5-21 Opposite Sides and Angles of a Triangle 203 5-22 Finding the Measures of the Angles in a Triangle G 204 5-23 Using the Triangle Inequality Theorem 205 5-24 Drawing Triangles with Given Conditions G (7.G.2) 206 5-25 Using the Pythagorean Theorem to Find the Length of the Hypotenuse (8.G.7) 207 5-26 Explaining a Proof of the Pythagorean Theorem (8.G.6) 207 5-27 Using the Pythagorean Theorem to Find the Length of a Leg (8.G.7) 208 5-28 Applying the Pythagorean Theorem to Find the Distance between Two Points (8.G.8) 209 5-29 Testing for Acute and Obtuse Triangles 210 5-30 Finding the Length of the Hypotenuse in a 45◦-45◦-90◦ Triangle 210 5-31 Finding the Length of a Leg in a 45◦-45◦-90◦ Triangle 211 5-32 Finding the Length of the Hypotenuse in a 30◦-60◦-90◦ Triangle 211 5-33 Finding the Lengths of the Legs in a 30◦-60◦-90◦ Triangle 212 5-34 Finding the Missing Lengths of the Sides of a 45◦-45◦-90◦ and a 30◦-60◦-90◦ Triangle 213 5-35 Properties of Rotations, Reflections, and Translations (8.G.1) 214 5-36 Translations, Rotations, and Reflections (8.G.2) 214 5-37 Identifying Congruent Triangles (8.G.2) 215 5-38 Translations, Rotations, and Reflections of a Right Triangle (8.G.2) 216 5-39 Using Undefined Terms (G-CO.1) 217 5-40 Describing Transformations as Functions (G-CO.2) 217 5-41 Describing Dilations (G-CO.2) 218 5-42 Rotations and Reflections of Regular Polygons (G-CO.3) 218 5-43 Defining Rotations, Reflections, and Translations in Terms of Line Segments G (G-CO.4) 219 5-44 Specifying a Sequence of Transformations That Will Carry a Given Figure onto Another (G-CO.5) 220 5-45 Predicting the Effects of Rigid Motions (G-CO.6) 220 5-46 Writing a Statement of Congruence and Identifying Corresponding Parts (G-CO.7) 221 5-47 Identifying Corresponding Parts in Overlapping Triangles 222 5-48 Using SSS, SAS, and ASA to Verify Congruent Triangles 223 5-49 Explaining the Criteria for Triangle Congruence (G-CO.8) 223 5-50 Proving Vertical Angles Are Congruent (G-CO.9) 224 5-51 Proving the Isosceles Triangle Theorem (G-CO.10) 225 xviii Contents 5-52 Proving the Diagonals of a Parallelogram Bisect Each Other (G-CO.11) 225 5-53 Constructing the Perpendicular Bisector of a Segment (G-CO.12) 226 5-54 Constructing a Regular Hexagon Inscribed in a Circle (G-CO.13) 226 5-55 Describing the Effects of Dilations, Translations, Rotations, and Reflections (8.G.3) 227 5-56 Describing a Sequence That Exhibits Similarity between Two Figures (8.G.4) 228 5-57 Verifying the Properties of Dilations (G-SRT.1) 229 5-58 Using the Definition of Similarity to Decide If Two Figures Are Similar (G-SRT.2) 230 5-59 Establishing the AA Criterion for Similar Triangles (G-SRT.3) 230 5-60 Using AA, SSS, and SAS to Prove That Triangles Are Similar 231 5-61 Writing a Similarity Statement and Finding the Scale Factor 232 5-62 Proving the Triangle Proportionality Theorem (G-SRT.4) 233 5-63 Finding the Lengths of the Sides of Similar Triangles (G-SRT.5) 234 5-64 Working with Scale Drawings of Geometric Figures (7.G.1) 235 5-65 Finding the Area of a Triangle by Using a Rectangle (6.G.1) 236 5-66 Finding the Area of a Triangle 237 5-67 Finding the Area of a Triangle 238 5-68 Definitions of Trigonometric Ratios for Acute Angles of a Right Triangle (G-SRT.6) 238 5-69 Using the Sine and Cosine of Complementary Angles (G-SRT.7) 239 5-70 Using Trigonometric Ratios and the Pythagorean Theorem to Solve Problems G (G-SRT.8) 239 5-71 Deriving the Formula A = 12 ab sin C to Find the Area of a Triangle (G-SRT.9) 240 5-72 Proving the Law of Sines (G-SRT.10) 241 5-73 Using the Law of Sines (G-SRT.11) 242 5-74 Applying the Law of Cosines (G-SRT.11) 243 5-75 Identifying Types of Quadrilaterals 244 5-76 Classifying Quadrilaterals 245 5-77 Classifying Quadrilaterals in the Coordinate Plane (G-GPE.4) 245 5-78 Parallelograms and Kites 246 5-79 Properties of Quadrilaterals 247 5-80 Properties of Diagonals of Quadrilaterals 247 5-81 Finding the Equations of Parallel and Perpendicular Lines (G-GPE.5) 248 5-82 Partitioning Line Segments (G-GPE.6) 249 5-83 Finding the Area of a Square 249 Contents xix 5-84 Area and Perimeter of Squares G 250 5-85 Finding the Area of a Rectangle 250 5-86 Finding the Area of a Rectangle 250 5-87 Finding the Area of a Rectangle by Using Other Figures (6.G.1) 251 5-88 Finding the Area and Perimeter of a Rectangle G 251 5-89 Finding the Area of an Irregular Figure (7.G.6) 252 5-90 Finding the Area of an Irregular Figure G (7.G.6) 252 5-91 Area of a Parallelogram (7.G.6) 253 5-92 Finding the Area of a Trapezoid (7.G.6) 253 5-93 Using Coordinates to Compute Perimeters and Areas of Figures (G-GPE.7) 254 5-94 Circles G 254 5-95 Finding the Diameter and Radius of a Circle 255 5-96 Finding the Circumference of a Circle (7.G.4) 255 5-97 Diameter and Circumference (7.G.4) 256 5-98 Finding the Area of a Circle (7.G.4) 256 5-99 Finding the Area of a Circle (7.G.4) 256 5-100 Comparing the Areas of a Square and a Circle (7.G.4) 257 5-101 Proving All Circles Are Similar (G-C.1) 257 5-102 Types of Arcs G 258 5-103 Central and Inscribed Angles (G-C.2) 259 5-104 Arcs and Angles of Circles (G-C.2) 260 5-105 Secants and Tangents 261 5-106 Measures of Angles—Chord-Tangent Angle Theorem and Chord-Chord Angle Theorem (G-C.2) 262 5-107 Measures of Angles Formed by Secants and Tangents Drawn from a Point outside the Circle 263 5-108 Lengths of Segments—Chords Intersecting in the Interior of a Circle (G-C.2) 264 5-109 Length of Segments—Secant and Tangent Segments 264 5-110 Proving Opposite Angles of a Quadrilateral Inscribed in a Circle Are Supplementary (G-C.3) 265 5-111 Constructing Tangent Lines to a Circle (G-C.4) 266 5-112 Arc Lengths 266 5-113 Area of a Sector (G-C.5) 267 5-114 Deriving the Formula for Finding the Area of a Sector (G-C.5) 267 5-115 Deriving the Equation of a Circle (G-GPE.1) 268 5-116 Deriving the Equation of a Parabola (G-GPE.2) 269 5-117 Deriving the Equation of an Ellipse (G-GPE.3) 270 5-118 Slicing Three-Dimensional Figures (7.G.3) 271 xx Contents 5-119 Identifying Three-Dimensional Objects Generated by Rotations of Two-Dimensional Objects G (G-GMD.4) 272 5-120 Using Geometric Shapes to Describe Objects G (G-MG.1) 272 5-121 Finding the Volume of a Rectangular Prism (6.G.2) 273 5-122 Finding the Volume of a Rectangle Prism G (6.G.2) 273 5-123 Using Nets to Find the Surface Area of a Three-Dimensional Figure (6.G.4) 274 5-124 Finding the Surface Area of a Rectangular Prism (7.G.6) 274 5-125 Finding the Surface Area of a Rectangular Prism (7.G.6) 275 5-126 Finding the Volume and Surface Area of Pyramids (7.G.6) 275 5-127 Finding the Volume of Cones (8.G.9) 276 5-128 Finding the Volume of Spheres (8.G.9) 276 5-129 Using Various Volume Formulas (G-GMD.3) 277 5-130 Providing an Informal Argument for the Area of a Circle (G-GMD.1) 277 5-131 Cavalieri’s Principle and the Volume of a Sphere (G-GMD.2) 278 5-132 Density in Modeling (G-MG.2) 278 5-133 Solving Design Problems (G-MG.3) 279 5-134 A Quotation about Geometry 279 Section 6: Statistics, Probability, and Data Analysis 281 6-1 Statistical and Nonstatistical Questions G (6.SP.1) 281 6-2 Distribution of Data (6.SP.2) 281 6-3 Finding the Mean of a Set of Numbers 282 6-4 Finding the Weighted Mean 282 6-5 Finding the Mode 283 6-6 The Median 283 6-7 Finding the Median and the Mode 283 6-8 Finding the Mean, Median, and Mode (6.SP.3) 284 6-9 Using the Measures of Center and Measure of Variation (6.SP.3) 284 6-10 Making a Histogram (6.SP.4) 285 6-11 Making a Bar Graph 285 6-12 Using Stem-and-Leaf Plots 286 6-13 Completing a Circle Graph 287 6-14 Summarizing Numerical Data (6.SP.5) 288 6-15 Obtaining Information about a Population (7.SP.1) 289 6-16 Using Data from Random Samples to Draw Inferences about a Population (7.SP.2) 289 6-17 Assessing Numerical Data Distributions G (7.SP.3) 290 Contents xxi 6-18 Drawing Comparative Inferences about Two Populations (7.SP.4) 290 6-19 The Probability of Impossible and Certain Events G (7.SP.5) 291 6-20 Finding Simple Probability G 291 6-21 Approximating the Probability of a Chance Event (7.SP.6) 292 6-22 Predicting Relative Frequency (7.SP.6) 292 6-23 Developing a Probability Model G (7.SP.7) 292 6-24 Spinners as Probability Models (7.SP.7) 293 6-25 Representing the Sample Spaces of Compound Events (7.SP.8) 293 6-26 Constructing a Scatter Plot G (8.SP.1) 294 6-27 Using Scatter Plots (8.SP.2) 295 6-28 Positive, Negative, and No Relationship 296 6-29 Representing the Slope and Y-Intercept (8.SP.3) 296 6-30 Using Two-Way Tables (8.SP.4) 297 6-31 Representing Data with a Box-and-Whisker Plot (S-ID.1) 297 6-32 Comparing the Center and Spread of Two Sets of Data (S-ID.2) 298 6-33 Accounting for Outliers (S-ID.3) 298 6-34 Using the Mean and Standard Deviation of a Data Set (S-ID.4) 299 6-35 Summarizing Categorical Data (S-ID.5) 299 6-36 Describing How Variables Are Related (S-ID.6) 300 6-37 Interpreting the Slope and Y-Intercept (S-ID.7) 300 6-38 Interpreting the Correlation Coefficient of a Linear Fit (S-ID.8) 301 6-39 Distinguishing between Correlation and Causation (S-ID.9) 301 6-40 Understanding the Value of Statistics G (S-IC.1) 301 6-41 Deciding If Results Are Consistent (S-IC.2) 302 6-42 Recognizing Sample Surveys, Observational Studies, and Experiments (S-IC.3) 302 6-43 Using Data from a Sample Survey (S-IC.4) 303 6-44 Data and Simulations (S-IC.5) 304 6-45 Evaluating Reports Based on Data (S-IC.6) 305 6-46 Describing Events as Subsets of a Sample Space (S-CP.1) 305 6-47 The Probability of Independent Events (S-CP.2) 306 6-48 The Probability of Independent Events G (S-CP.2) 307 6-49 Understanding Conditional Probability (S-CP.3) 308 6-50 Interpreting a Two-Way Frequency Table (S-CP.4) 308 6-51 Conditional Probability G (S-CP.5) 309 6-52 Finding Conditional Probability (S-CP.6) 309 6-53 Applying the Addition Rules for Finding Probability (S-CP.7) 310 6-54 Applying the General Multiplication Rule (S-CP.8) 310 xxii Contents 6-55 Using the Factorial Counting Rule 310 6-56 Using the Permutations Rule (S-CP.9) 311 6-57 Using the Combinations Rule (S-CP.9) 311 6-58 Defining a Random Variable (S-MD.1) 312 6-59 Calculating the Expected Value of a Random Variable (S-MD.2) 313 6-60 Developing a Probability Distribution (S-MD.3) 314 6-61 A Probability Distribution Using Empirical Data (S-MD.4) 314 6-62 Evaluating a Flood Insurance Plan G (S-MD.5) 315 6-63 Using Probabilities to Make Fair Decisions G (S-MD.6) 316 6-64 Basing Decisions on Probability G (S-MD.7) 317 6-65 A Quotation about Statistics 317 Section 7: Number and Quantity 319 7-1 Using Rational Exponents (N-RN.1) 319 7-2 Rewriting Expressions Using Radicals and Rational Exponents (N-RN.2) 319 7-3 Explaining Products and Sums (N-RN.3) 320 7-4 Using Units as a Way to Understand Problems (N-Q.1) 320 7-5 Defining Appropriate Quantities (N-Q.2) 320 7-6 Levels of Accuracy (N-Q.3) 321 7-7 Using Imaginary Numbers (N-CN.1) 321 7-8 Using Powers of i 322 7-9 Adding, Subtracting, and Multiplying Complex Numbers (N-CN.2) 322 7-10 Finding Quotients of Complex Numbers (N-CN.3) 323 7-11 Using an Argand Diagram (N-CN.4) 323 7-12 Representing Operations with Complex Numbers in the Complex Plane (N-CN.5) 324 7-13 Finding the Distance between Numbers in the Complex Plane (N-CN.6) 325 7-14 Solving Quadratic Equations by Using the Quadratic Formula—Complex Solutions (N-CN.7) 326 7-15 Extending Polynomial Identities to the Complex Numbers (N-CN.8) 326 7-16 The Fundamental Theorem of Algebra (N-CN.9) 326 7-17 Vector Quantities (N-VM.1) 327 7-18 The Components of Vectors (N-VM.2) 328 7-19 Using Vectors to Find a Plane’s Ground Speed and True Course (N-VM.3) 328 7-20 Using the Parallelogram Rule (N-VM.4) 329 7-21 Multiplying a Vector by a Scalar (N-VM.5) 329 7-22 Using Matrices to Represent Data (N-VM.6) 329 Contents xxiii 7-23 Multiplying Matrices by a Scalar (N-VM.7) 330 7-24 Adding and Subtracting Matrices (N-VM.8) 330 7-25 Multiplying Matrices (N-VM.8) 331 7-26 Proving the Properties of Matrix Multiplication for Square Matrices (N-VM.9) 331 7-27 Using the Zero Matrix and the Identity Matrix (N-VM.10) 332 7-28 Translations and Vectors 332 7-29 Using Transformation Matrices: Reflection (N-VM.11) 333 7-30 Using Transformation Matrices: Enlargement (N-VM.11) 333 7-31 Using the Determinate to Find Area (N-VM.12) 334 7-32 A Quotation about the Boundaries of Mathematics 334 Section 8: Potpourri 335 8-1 Emirps 335 8-2 Deficient Numbers 335 8-3 Perfect Numbers 335 8-4 Abundant Numbers 336 8-5 Deficient, Abundant, and Perfect Numbers G 336 8-6 Linear Measurement—The Customary System 336 8-7 Linear Measurement—The Customary System 337 8-8 Linear Measurement—The Metric System 337 8-9 Linear Measurement—The Customary and Metric Systems 338 8-10 Linear Measurement—Obsolete Units 338 8-11 Linear Measurement—Obsolete Units 338 8-12 Measurement—Quotation 339 8-13 Weight—The Customary System 339 8-14 Weight—The Metric System 339 8-15 Weight—Using Balances 340 8-16 Capacity—The Customary System 340 8-17 Capacity—The Metric System 340 8-18 Time and the Calculation of Pi 341 8-19 Interpreting Time 341 8-20 Time—A Tricky Problem 341 8-21 Temperature 342 8-22 Converting Temperatures—Fahrenheit and Celsius 342 8-23 Measurement—Light-Years G 343 8-24 Babylonians and Angles in a Circle 343 8-25 Platonic Solids and Euler’s Formula 344 8-26 Squares on a Checkerboard G 344 8-27 Rectangles on a Checkerboard G 345 xxiv Contents 8-28 Finding the Area of a Rectangular Chicken Coop 345 8-29 Edward I and the Area of a Rectangle 345 8-30 Finding the Area of a Triangle Using Hero’s Formula 346 8-31 An Ancient Palestinian Formula for Finding of a Circle 346 8-32 Palindromes 347 8-33 Palindromes G 347 8-34 Using Cryptarithms 348 8-35 Number Ciphers G 348 8-36 Number Ciphers 349 8-37 Using Number-Box Ciphers 349 8-38 Roman Numerals G 350 8-39 Symbols (Infinity) 350 8-40 Figural Analogies 351 8-41 Fractals 352 8-42 Fractals—Using the Sierpinski Triangle 353 8-43 Figurate Numbers—Square Numbers 354 8-44 Figurate Numbers—Square Numbers 354 8-45 Figurate Numbers—Rectangular Numbers 355 8-46 Figurate Numbers—Rectangular Numbers 355 8-47 Numerical Patterns 356 8-48 Numerical Patterns G 356 8-49 Line Symmetry 357 8-50 Lines of Symmetry 357 8-51 Networks 358 8-52 Traceable Networks G 359 8-53 Using Digraphs 359 8-54 Using Digraphs 360 8-55 Using Digraphs G 361 8-56 Symbols and Letters in Math 361 8-57 Quotation about Mathematics 362 8-58 A Personal Quotation about Mathematics 362 Answer Key 363