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

Judith A. Muschla (Jackson, NJ) taught mathematics in South River, New Jersey, at the middle school and high school for over twenty-five years. At South River Middle School she helped revise the mathematics curriculum to reflect the Standards of the National Council of Teachers of Mathematics (NCTM), coordinated interdisciplinary units, and conducted mathematics workshops for teachers and parents.

Gary Robert Muschla (Jackson, NJ) taught at Appleby School in Spotswood, New Jersey, for more than twenty-five years. Judith and Gary have coauthored several very successful math activity books, including The Algebra Teacher's Activities Kit and Hands-On Math Projects with Real Life Applications.

Erin Muschla (Freehold, NJ) teaches middle school math and algebra at Monroe Township Middle School in Monroe, New Jersey. She is also a coauthor with Judith and Gary of The Math Teacher's Survival Guide, and The Elementary Teacher's Book of Lists.

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

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