The Algebra Teacher's Guide to Reteaching Essential Concepts and Skills: 150 Mini-Lessons for Correcting Common Mistakes

About the Authors v

Acknowledgments vii

SECTION 1: INTEGERS, VARIABLES, AND EXPRESSIONS 1

1.1: Using the Order of Operations  2

1.2: Simplifying Expressions That Have Grouping Symbols  4

1.3: Simplifying Expressions with Nested Grouping Symbols  6

1.4: Using Positive Exponents and Bases Correctly  8

1.5: Simplifying Expressions with Grouping Symbols and Exponents  10

1.6: Evaluating Expressions  12

1.7: Writing Expressions  14

1.8: Writing Expressions Involving Grouping Symbols  16

1.9: Identifying Patterns by Considering All of the Numbers  18

1.10: Writing Prime Factorization  20

1.11: Finding the Greatest Common Factor  22

1.12: Finding the Least CommonMultiple 24

1.13: Classifying Counting Numbers, Whole Numbers, and Integers  26

1.14: Finding Absolute Values and Opposites 28

1.15: Adding Integers with Different Signs  30

1.16: Subtracting Integers 32

1.17: Multiplying Two Integers 34

1.18: MultiplyingMore Than Two Integers  36

1.19: Using Integers as Bases  38

1.20: Dividing Integers  40

1.21: Finding Absolute Values of Expressions 42

1.22: Finding Square Roots of Square Numbers  44

SECTION 2: RATIONAL NUMBERS 47

2.1: Classifying Counting Numbers, Whole Numbers, Integers, and Rational Numbers  48

2.2: Simplifying Fractions 50

2.3: RewritingMixed Numbers as Improper Fractions  52

2.4: Comparing Rational Numbers 54

2.5: Expressing Rational Numbers as Decimals  56

2.6: Expressing Terminating Decimals as Fractions or Mixed Numbers 58

2.7: Expressing Repeating Decimals as Fractions or Mixed Numbers  60

2.8: Adding Rational Numbers 62

2.9: Subtracting Rational Numbers  64

2.10: Multiplying and Dividing Rational Numbers 66

2.11: Expressing Large Numbers in Scientific Notation  68

2.12: Evaluating Rational Expressions  70

2.13: Writing Ratios Correctly  72

2.14: Writing and Solving Proportions 74

2.15: Expressing Fractions as Percents  76

2.16: Expressing Percents as Fractions 78

2.17: Solving Percent Problems  80

2.18: Finding the Percent of Increase or Decrease  82

2.19: Converting from One Unit of Measurement to Another Using theMultiplication Property of One  84

SECTION 3: EQUATIONS AND INEQUALITIES 87

3.1: Writing Equations  88

3.2: Solving Equations by Adding or Subtracting  90

3.3: Solving Equations byMultiplying or Dividing  92

3.4: Solving Two-Step Equations with the Variable on One Side 94

3.5: Solving Equations Using the Distributive Property 96

3.6: Solving Equations with Variables on Both Sides  98

3.7: Solving Equations with Variables on Both Sides, Including Identities and Equations That Have No Solution 100

3.8: Solving Absolute Value Equations 102

3.9: Solving Absolute Value Equations That Have Two Solutions, One Solution, or No Solution  104

3.10: Classifying Inequalities as True or False 106

3.11: Writing Inequalities  108

3.12: Solving Inequalities with Variables on One Side 110

3.13: Rewriting Combined Inequalities as One Inequality 112

3.14: Solving Combined Inequalities—Conjunctions  114

3.15: Solving Combined Inequalities—Disjunctions  116

3.16: Solving Absolute Value Inequalities  118

3.17: Solving Systems of Equations Using the Substitution Method  120

3.18: Solving Systems of Equations Using the Addition-or-SubtractionMethod 122

3.19: Solving Systems of Equations Using Multiplication with the Addition-or-SubtractionMethod  124

3.20: Solving Systems of Equations Using a Variety of Methods 126

3.21: Solving Systems of Equations That Have One Solution, No Solution, or an Infinite Number of Solutions 128

3.22: Using Matrices—Addition, Subtraction, and Scalar Multiplication 130

3.23: Identifying Conditions forMultiplying TwoMatrices  132

3.24: Multiplying TwoMatrices 134

SECTION 4: GRAPHS OF POINTS AND LINES 137

4.1: Graphing on a Number Line  138

4.2: Graphing Conjunctions  140

4.3: Graphing Disjunctions  142

4.4: Graphing Ordered Pairs on the Coordinate Plane  144

4.5: Completing T-Tables  146

4.6: Finding the Slope of a Line, Given Two Points on the Line 148

4.7: Identifying the Slope and Y-Intercept from an Equation . . 150

4.8: Using Equations to Find the Slopes of Lines  152

4.9: Identifying Parallel and Perpendicular Lines, Given an Equation  154

4.10: Using the X-Intercept and the Y-Intercept to Graph a Linear Equation  156

4.11: Using Slope-Intercept Form to Graph the Equation of a Line  158

4.12: Graphing Linear Inequalities in the Coordinate Plane 160

4.13: Writing a Linear Equation, Given Two Points 162

4.14: Finding the Equation of the Line of Best Fit  164

4.15: Using theMidpoint Formula  166

4.16: Using the Distance Formula to Find the Distance Between Two Points  168

4.17: Graphing Systems of Linear Equations When Lines Intersect 170

4.18: Graphing Systems of Linear Equations if Lines Intersect, Are Parallel, or Coincide 172

SECTION 5: MONOMIALS AND POLYNOMIALS 175

5.1: ApplyingMonomial Vocabulary Accurately  176

5.2: Identifying Similar Terms  178

5.3: Adding Polynomials  180

5.4: Subtracting Polynomials  182

5.5: MultiplyingMonomials  184

5.6: Using Powers ofMonomials  186

5.7: Multiplying a Polynomial by aMonomial  188

5.8: Multiplying Two Binomials  190

5.9: Multiplying Two Polynomials  192

5.10: DividingMonomials  194

5.11: Dividing Polynomials  196

5.12: Finding the Greatest Common Factor of Two or More Monomials 198

5.13: Factoring Polynomials by Finding the Greatest Monomial Factor 200

5.14: Factoring the Difference of Squares  202

5.15: Factoring Trinomials if the Last TermIs Positive 204

5.16: Factoring Trinomials if the Last TermIs Negative  206

5.17: Factoring by Grouping 208

5.18: Factoring Trinomials if the Leading Coefficient Is an Integer Greater Than 1 210

5.19: Factoring the Sums and Differences of Cubes  212

5.20: Solving Quadratic Equations by Factoring 214

5.21: Solving Quadratic Equations by Finding Square Roots  216

5.22: Solving Quadratic Equations Using the Quadratic Formula  218

5.23: Using the Discriminant  220

SECTION 6: RATIONAL EXPRESSIONS 223

6.1: Using Zero and Negative Numbers as Exponents 224

6.2: Using the Properties of Exponents That Apply to Division 226

6.3: Using the Properties of Exponents That Apply to Multiplication and Division 228

6.4: Identifying Restrictions on the Variable 230

6.5: Simplifying Algebraic Fractions  232

6.6: Adding and Subtracting Algebraic Fractions with Like Denominators 234

6.7: Finding the Least CommonMultiple of Polynomials 236

6.8: Writing Equivalent Algebraic Fractions  238

6.9: Adding and Subtracting Algebraic Fractions with Unlike Denominators  240

6.10: Multiplying and Dividing Algebraic Fractions  242

6.11: Solving Proportions  244

6.12: Solving Equations That Have Fractional Coefficients  246

6.13: Solving Fractional Equations 248

SECTION 7: IRRATIONAL AND COMPLEX NUMBERS 251

7.1: Simplifying Radicals  252

7.2: Multiplying Radicals  254

7.3: Rationalizing the Denominator  256

7.4: Dividing Radicals  258

7.5: Adding and Subtracting Radicals  260

7.6: Multiplying Two Binomials Containing Radicals  262

7.7: Using Conjugates to Simplify Radical Expressions  264

7.8: Simplifying Square Roots of Negative Numbers  266

7.9: Multiplying Imaginary Numbers 268

7.10: Simplifying ComplexNumbers  270

SECTION 8: FUNCTIONS 273

8.1: Determining if a Relation Is a Function  274

8.2: Finding the Domain of a Function 276

8.3: Finding the Range of a Function  278

8.4: Using the Vertical Line Test  280

8.5: Describing Reflections of the Graph of a Function  282

8.6: Describing Vertical Shifts of the Graph of a Function  284

8.7: Describing Horizontal and Vertical Shifts of the Graph of a Function  286

8.8: Describing Dilations of the Graph of a Function  288

8.9: Finding the Composite of Two Functions  290

8.10: Finding the Inverse of a Function 292

8.11: Evaluating the Greatest Integer Function 294

8.12: Identifying Direct and Indirect Variation 296

8.13: Describing the Graph of the Quadratic Function 298

8.14: Using Rational Numbers as Exponents  300

8.15: Using Irrational Numbers as Exponents  302

8.16: Solving Exponential Equations 304

8.17: Using the Compound Interest Formula  306

8.18: Solving Radical Equations  308

8.19: Writing Logarithmic Equations as Exponential Equations  310

8.20: Solving Logarithmic Equations  312

8.21: Using the Properties of Logarithms 314

Common Core State Standards for Mathematics 316