Contents Getting Started

1

About This Book Eight Standards for Mathematical Practice Test-Taking Strategies

Chapter 1: Review: The Elements of Algebra 1.1 Writing and Translating Algebraic Expressions Writing Algebraic Expressions Translating English into the Language of Algebra

1.2 Translating and Writing Formulas

1 1 3

4 5 6 7 9

1.3 Simple Algebraic Inequalities

11

1.4 Evaluating Algebraic Expressions and Formulas

13

1.5 Algebraic Properties

17 17 19

Algebraic Operations Closure

1.6 Exponents Exponents and Operations

1.7 Roots and Radicals Square Roots Cube Roots and Other Roots Simplifying Square Root Radicals Operations with Radicals

1.8 Scientific Notation, Significant Digits, Precision, and Accuracy Scientific Notation Significant Digits Units of Measure Accuracy and Precision

Chapter 1 Review

21 23 25 25 26 29 30 32 32 35 37 37 40

Key to the icons: The computer icon The globe icon

indicates Digital Activities that can be found at www.amscomath.com. indicates where Real-World Model Problems are found in the text.

Contents

iii

Chapter 2: W  riting and Solving Linear Equations and Inequalities 2.1 Solving Linear Equations A-REI.1; A-REI.3 Solving Equations by Adding and Subtracting Solving Equations by Multiplying and Dividing Solving Multi-Step Equations Solving Equations with Variables on Both Sides Solving Equations Using the Least Common Denominator

44 45 46 48 51 53 56

2.2 Solving for a Variable in Literal Equations A-CED.4; A-REI.3

59

2.3 Ratios, Rates, and Proportions

N-Q.1

62 62 65

A-CED.1; A-REI.3

69

Ratios and Rates Proportions

2.4 Modeling with Linear Equations 2.5 Solving Inequalities

72

A-REI.3

2.6 Modeling with Inequalities A-CED.1; A-REI.3

78

Chapter 2 Review

81

Cumulative Review for Chapters 1–2

85

Chapter 3: Graphing Linear Equations and Functions 3.1 Graphing Linear Equations

89 93

Review

Graphing a Line Parallel to an Axis

3.2 Direct Variation Review

95 98

Graphing Direct Variation

3.3 The Slope of a Line

88

101 106

Review

The Slope of Parallel and Perpendicular Lines

3.4 Graphing and Writing Linear Equations Using the Slope-Intercept and Point-Slope Forms Review Slope-Intercept Form Point-Slope Form Finding the Slope-Intercept Form Given Two Points

3.5 Functions A-REI.10; F-IF.1; F-IF.2

109 111 113 117 121 121 121

Graphing Functions Domain Restrictions

3.6 The Algebra of Functions F-BF.1b

124

3.7 Inverse Functions F-BF.4a

126

3.8 Modeling with Linear Functions F-IF.6; F-IF.7a; F-LE.1a; F-LE.1b; F-LE.2; F-LE.5

N-Q.2; A-CED.2; F-BF.1a; F-IF.4; F-IF.5;

129

Chapter 3 Review

136

Cumulative Review for Chapters 1–3

141

iv   Contents 

Chapter 4: Inequalities, Absolute Value, Piecewise and Step Functions 4.1 Graphing Linear Inequalities

145

A-CED.3; A-REI.12

4.2 Absolute Value Inequalities and Graphing on the Number Line A-CED.3; A-REI.3 4.3 Graphing Piecewise and Step Functions

144

150 154

F-IF.7b

Piecewise Functions Step Functions

154 159

4.4 Graphing Absolute Value Functions

163 163 164

F-IF.7b; F-BF.3

Key Features of the Graph Transformations of Absolute Value Functions

4.5 Solving Absolute Value Equations Algebraically A-REI.3

171

Chapter 4 Review

174

Cumulative Review for Chapters 1–4

180

Chapter 5: S  ystems of Linear Equations and Inequalities 5.1 Solving Systems of Linear Equations by Graphing

A-REI.6; A-REI.11

184 185

5.2 Solving Systems of Linear Equations by Substitution

A-REI.6

190

5.3 Solving Systems of Linear Equations by Elimination

A-REI.5; A-REI.6

194

5.4 Solving Systems of Linear Inequalities by Graphing

A-CED.3; A-REI.12 199

Chapter 5 Review

203

Cumulative Review for Chapters 1–5

207

Chapter 6: Operations with Polynomials 6.1 Adding and Subtracting Polynomials

210

A-SSE.2; A-APR.1

Adding or Subtracting Monomials with Like Terms Standard Form of Polynomials Adding and Subtracting Polynomials

6.2 Multiplying a Monomial by a Monomial A-APR.1 Multiplying Powers with Like Bases Multiplying a Monomial by a Monomial

211 211 212 214 216 216 218

6.3 Multiplying a Polynomial by a Monomial A-APR.1

219

6.4 Multiplying a Polynomial by a Polynomial

222 222 223 223

A-APR.1

Using the Distributive Property for Multiplying Polynomials Using the FOIL Method for Multiplying Polynomials Using the Box Method for Multiplying Polynomials

Contents    v

6.5 Special Products of Binomials A-APR.1

225

6.6 Negative Integers as Exponents A-SSE.2

228

6.7 Dividing Polynomials A-SSE.2; A-APR.1

230 230 232

Dividing a Monomial by a Monomial Dividing a Polynomial by a Monomial

Chapter 6 Review

234

Cumulative Review for Chapters 1–6

236

Chapter 7: Special Products and Factoring

240

7.1 Greatest Common Factors A-SSE.2

241

7.2 Factoring the Difference of Two Squares A-SSE.2

244

7.3 Factoring Trinomials

246 246 247 249 251

A-SSE.2

Perfect Square Trinomials Special Quadratic Trinomials of the Form x2 + bx + c General Quadratic Trinomials of the Form ax2 + bx + c, where a fi 1 Factoring by Grouping

7.4 Factoring Completely A-SSE.2

253

Chapter 7 Review

254

Cumulative Review for Chapters 1–7

255

Chapter 8: Quadratic Equations and Functions

258

8.1 Standard Form of the Quadratic Equation A-SSE.2

259

8.2 Solving Quadratic Equations Algebraically

260 260 265 267 271

A-SSE.3a; A-REI.1; A-REI.4b; A-CED.2

Solving Quadratic Equations Using the Zero Product Property Solving Cubic Equations Algebraically Incomplete Quadratic Equations Fractional Quadratic Equations

8.3 Solving Quadratic Equations by Completing the Square A-REI.4a; A-REI.4b

273

8.4 Solving Quadratic Equations from the Graph A-APR.3; A-REI.10

276

8.5 Graphing Quadratic Functions

280 280 281

A-APR.3

Key Features of the Graph Graphing Quadratic Functions of the Form f(x) = ax2 + bx + c

8.6 Quadratic Functions in Vertex Form 8.7 Transformations of Quadratic Functions

A-SSE.2; A-REI.4a; A-REI.4b; F-IF.8a F-BF.3

8.8 The Quadratic Formula and the Discriminant A-REI.4a; A-REI.4b Quadratic Formula The Discriminant

vi   Contents 

285 288 290 290 293

8.9 Modeling with Quadratic Equations

A-SSE.1a; A-SSE.1b; A-SSE.3a; A-SSE.3b; A-CED.2; A-CED.3; A-REI.4a; A-REI.4b; F-IF.4; F-IF.5; F-IF.6; F-IF.7a; F-IF.8a

8.10 Solving Quadratic-Linear Systems of Equations A-REI.7; A-REI.11 Graphic Solutions to Quadratic-Linear Systems Solving Quadratic-Linear Pairs Algebraically

8.11 Graphing Cubic and Root Functions

A-APR.3; A-REI.10; F-IF.1; F-IF.7b

Graphing Cubic Functions Graphing Square Root Functions Graphing Cube Root Functions

295 302 302 307 310 310 312 313

Chapter 8 Review

314

Cumulative Review for Chapters 1-8

317

Chapter 9: Exponents and Exponential Functions 9.1 Rational Exponents and Radicals N-RN.1; N-RN.2; N-RN.3 9.2 Graphing Exponential Functions

321

A-SSE.3c; A-REI.10; F-IF.6; F-IF.7e;

324

F-IF.8b; F-IF.9; F-BF.3; F-LE.1a; F-LE.1c; F-LE.3

Key Features of the Graph Transformations of Exponential Functions

324 326

9.3 Modeling Exponential Growth and Decay

N-Q.3; A-SSE.1b; A-SSE.3c; A-CED.2;

334

A-CED.3; A-REI.11; F-IF.4; F-IF.5; F-IF.6; F-IF.7e; F-IF.8b; F-LE.5

9.4 Sequences and Arithmetic Sequences

F-IF.3; F-BF.1a; F-BF.2; F-LE.2

Sequences Recursive Formula for Arithmetic Sequences General Rule for Arithmetic Sequences

9.5 Geometric Sequences

320

F-IF.3; F-BF.1a; F-BF.2; F-LE.1a; F-LE.1c; F-LE.2

Recursive Formula for Geometric Sequences General Rule for Geometric Sequences

341 341 342 343 348 348 348

Chapter 9 Review

353

Cumulative Review for Chapters 1–9

358

Chapter 10: Interpreting Quantitative and Categorical Data 10.1 Simple Single-Count Statistics S-ID.1; S-ID.3 Histograms Dot Plots

10.2 Measures of Central Tendency

S-ID.2; S-ID.3

10.3 Single-Count Statistics with Dispersion S-ID.1; S-ID.2; S-ID.3 Interquartile Range and the Five-Number Summary Box-and-Whisker Plots Sample Variance and Standard Deviation

364 365 366 366 370 372 372 374 376 Contents    vii

10.4 Two-Valued Statistics for Linear Behavior

S-ID.6a; S-ID.6b; S-ID.6c; S-ID.7;

381 381 382 384 387

S-ID.8; S-ID.9

Scatter Plots Linear Regression Residuals and Residual Plots Common Regression Analysis Errors

10.5 Two-Valued Statistics for Non-Linear Behavior S-ID.6a; S-ID.6b

390

10.6 Analyzing Bivariate Categorical Data

392 393 393

S-ID.5

Two-Way Relative Frequency Tables and Conditional Relative Frequency Association in Categorical Data

Chapter 10 Review

396

Cumulative Review for Chapters 1–10

399

Digital Activities

402

Appendix: Using Your Graphing Calculator

418

Glossary

420

Index

428

viii   Contents 

Getting Started About This Book Algebra 1 is written to help students understand and explore the concepts of algebra as well as prepare them for new statewide, end-of-course assessments based on the traditional pathway for Algebra 1. All instruction, model problems, and practice items were developed to support the Common Core Learning Standards (CCLS). Each chapter opens with lesson-by-lesson alignment with the standards. The eight Mathematical Practice Standards are embedded throughout in selected Model Problems, extensive practice problem sets, and the comprehensive Chapter and Cumulative Reviews. In Algebra 1, students will explore many types of functions including linear, quadratic, exponential, square root, cube root, piecewise, step, and absolute value. In addition, they will create, compare, and graph functions, and learn how various transformations affect these functions. Students will also gain experience using descriptive statistics (categorical and quantitative data) to model a context and draw meaningful conclusions. Each chapter incorporates multiple PARCC and Smarter Balanced-type performance tasks that measure the ability of students to think critically and apply their knowledge in real-world situations. In addition, teachers have access to a full range of digital simulations, electronic whiteboard lessons, videos, interactive problems, and applications to stimulate conceptual understanding. Careful and consistent use of this text and the supporting materials will both prepare students for end-of-course exams and ensure they are becoming college and career ready.

Eight Standards for Mathematical Practice The mathematical practices are a common thread for students to think about and understand math as they progress from Kindergarten through high school. Students should use the mathematical practices as a method to break down concepts and solve problems, including representing problems logically, justifying conclusions, applying mathematics to practical situations, explaining the mathematics accurately to other students, or deviating from a known procedure to find a shortcut. MP 1 Make sense of problems and persevere in solving them. Attack new problems by analyzing what students already know. Students should understand that many different strategies can work. Ask leading questions to direct the discussion. Take time to think. • explain the meaning of the problem • analyze given information, constraints, and relationships • plan a solution route • try simpler forms of the initial problem • use concrete objects to help conceptualize • monitor progress and change course, if needed • continually ask, “Does this make sense?”

Getting Started

1

MP 2 Reason abstractly and quantitatively. Represent problems with symbols and/or pictures. • make sense of quantities and their relationships • decontextualize—represent a situation symbolically and contextualize—consider what given symbols represent • create a clear representation of the problem • consider the units involved • attend to the meaning of numbers and variables, not just how to compute them • use properties of operations and objects MP 3 Construct viable arguments and critique the reasoning

of others.

Ask questions, defend answers, and/or make speculations using correct math vocabulary. • use assumptions, definitions, and previously established results • make conjectures and build a valid progression of statements • use counterexamples • justify conclusions and communicate them to others • determine whether the arguments of others seem right MP 4 Model with mathematics. Show the relevance of math by solving real-world problems. Look for opportunities to use math for current situations in and outside of school in all subject areas. • apply mathematics to solve everyday problems • analyze and chart relationships using diagrams, two-way tables, graphs, flowcharts, and formulas to draw conclusions • apply knowledge to simplify a complicated situation • interpret results and consider whether answers make sense MP 5 Use appropriate tools strategically. Provide an assortment of tools for students and let them decide which ones to use. • choose appropriately from existing tools (pencil and paper, concrete models, ruler, protractor, calculator, spreadsheet, dynamic geometry software, etc.) when solving mathematical problems • detect possible errors by using estimation or other mathematical knowledge • use technology to explore and compare predictions and deepen understanding of concepts MP 6 Attend to precision. Use precise and detailed language in math. Instead of saying “I don’t get it,” students should be able to elaborate on where they lost the connection. Students should specify units in their answers and correctly label diagrams. • speak and write precisely using correct mathematical language • state the meaning of symbols and use them properly • specify units of measure and label axes appropriately • calculate precisely and efficiently • express answers with the proper degree of accuracy MP 7 Look for and make use of structure. See patterns and the significance of given information and objects. Use these to solve more complex problems. • see the big picture

2    Getting Started 

• • • • •

discern a pattern or structure recognize the significance of given aspects apply strategies to similar problems step back for an overview and shift perspective see complicated things as being composed of several objects

MP 8 Look for and express regularity in repeated reasoning. Understand why a process works so students can apply it to new situations. • notice repeated calculations and look for both general methods and shortcuts • maintain oversight of the process while paying attention to the details • evaluate the reasonableness of intermediate results • create generalizations founded on observations

Test-Taking Strategies General Strategies • Become familiar with the directions and format of the test ahead of time. There will be both multiple-choice and extended response questions where you must show the steps you used to solve a problem, including formulas, diagrams, graphs, charts, and so on, where appropriate. • Pace yourself. Do not race to
answer every question immediately. On the other hand, do not linger over any
question too long. Keep in mind that you will need more time to complete the
extended response questions than to complete the multiple-choice questions. • Speed comes from practice. The more you practice, the faster you will become and the more comfortable you will be with the material. Practice as often as you
can.

Specific Strategies • Always scan the answer choices before beginning to work on a multiple-choice question. This will help you to focus on the kind of answer that is required. Are you looking for fractions, decimals, percents, integers, squares, cubes, and so
on? Eliminate choices that clearly do not answer the question asked. • Do not assume that your answer is correct just because it appears among the choices. The wrong choices are usually there because they represent common student errors. After you find an answer, always reread the problem to make sure you have chosen the answer to the question that is asked, not the question you have in your mind. • Sub-in. To sub-in means to substitute. You can sub-in friendly numbers for the variables to find a pattern and determine the solution to the problem. • Backfill. If a problem is simple enough and you want to avoid doing the more complex algebra, or if a problem presents a phrase such as x 5 ?, then just fill in
the answer choices that are given in the problem until you find the one that works. • Do the math. This is the ultimate strategy. Don’t go wild searching in your mind for tricks, gimmicks, or math magic to solve every problem. Most of the time the best way to get the right answer is to do the math and solve the problem.

Getting Started

3