The Algebra Teacher's Guide to Reteaching Essential Concepts and Skills

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About This Book iii 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

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 .

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