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UNIT 1 NUMERICAL RELATIONSHIPS Unit 1 Overview 1 Getting Ready 2

Activity 1 Investigating Patterns—Laws and Order 3 Lesson 1-1 Analyzing Sequences 3 Lesson 1-2 Analyzing More Sequences 7 Lesson 1-3 Increasing and Decreasing Sequences 11 Activity 1 Practice 15

Activity 2 Operations with Fractions—And the Beat Goes On 17 Lesson 2-1 Adding and Subtracting Fractions 17 Lesson 2-2 Multiplying and Dividing Fractions 23 Activity 2 Practice 29

Embedded Assessment 1 Patterns and Quantitative Reasoning—Game On 31

Activity 3 Powers and Roots—Squares and Cubes 33 Lesson 3-1 Area, Squares and Square Roots 33 Lesson 3-2 Volume, Cubes, and Cube Roots 39 Lesson 3-3 Exponents, Roots and Order of Operations 41 Activity 3 Practice 43

Activity 4 Rational Numbers—Know When to Fold ‘Em 45 Lesson 4-1 Modeling Fractions 45 Lesson 4-2 Rational Number Representations 50 Lesson 4-3 Comparing Rational Numbers 52 Activity 4 Practice 57

Activity 5 Rational and Irrational Numbers—Where Am I? 59 Lesson 5-1 Estimating Irrational Numbers 59 Lesson 5-2 Comparing Rational and Irrational Numbers 64 Activity 5 Practice 67

Embedded Assessment 2 Representing Rational and Irrational Numbers— Weather or Not? 69

Activity 6 Properties of Exponents—That’s a Lot of Cats 71 Lesson 6-1 Multiply and Divide with Exponents 71 Lesson 6-2 Negative Exponents 75 Lesson 6-3 Power of Zero and Power of Powers 78 Activity 6 Practice 81

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Activity 7 Scientific Notation—A Traveler’s Tale 83 Lesson 7-1 Scientific Notation vs. Standard Form 83 Lesson 7-2 Scientific Notation: Power of Zero, Negative Exponents, and Ordering 88 Activity 7 Practice 91

Activity 8 Operations with Scientific Notation—How Big is That Planet? 93 Lesson 8-1 Multiply and Divide in Scientific Notation 93 Lesson 8-2 Add and Subtract in Scientific Notation 97 Activity 8 Practice 99

Embedded Assessment 3 Exponents and Scientific Notation— Contagious Mathematics 101

UNIT 2 EQUATIONS Unit 2 Overview 103 Getting Ready 104

Activity 9 Writing Expressions—Pebbles in the Sand 105 Lesson 9-1 Representing Patterns 105 Lesson 9-2 Using Patterns to Write and Evaluate Expressions 111 Activity 9 Practice 117

Activity 10 Solving Equations—Cups and Cubes 119 Lesson 10-1 Solving Linear Equations with Models 119 Lesson 10-2 Solving Linear Equations Algebraically 124 Activity 10 Practice 129

Embedded Assessment 1 Expressions and Equations—What a Good Idea! 131

Activity 11 Exploring Slope—High Mountain Ratio 133 Lesson 11-1 Linear Equations and Slope 133 Lesson 11-2 More on Linear Equations and Slope 140 Activity 11 Practice 145

Activity 12 Slope-Intercept Form—Leaky Bottle 147 Lesson 12-1 Identifying Slope Using Tables and Graphs 147 Lesson 12-2 Comparing Slopes of Different Lines 151 Lesson 12-3 Linear Relationships Using Slope-Intercept Form 155 Activity 12 Practice 159

Activity 13 Proportional Relationships—Vary Interesting 161 Lesson 13-1 Linear Proportional Relationships 161 Lesson 13-2 Directly Proportional Relationships 166 Activity 13 Practice 172

Embedded Assessment 2 Linear Equations and Rates of Change—Who is That? 174

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Activity 14 Graphing Systems of Linear Equations—System of Trees 177 Lesson 14-1 Understanding Solutions to Linear Systems 177 Lesson 14-2 Solving Linear Systems by Graphing 182 Activity 14 Practice 187

Activity 15 Solving Systems of Linear Equations Algebraically— What’s the Point? 189 Lesson 15-1 Solving Linear System Algebraically 189 Lesson 15-2 Applications of Linear Systems 194 Activity 15 Practice 197

Embedded Assessment 3 Solving Systems of Linear Equations— Supply and Demand 199

UNIT 3 GEOMETRY Unit 3 Overview 201 Getting Ready 202

Activity 16 Angle-Pair Relationships—The Winning Angle 203 Lesson 16-1 Complementary and Supplementary Angles 203 Lesson 16-2 Angles Formed by Parallel Lines 208 Activity 16 Practice 215

Activity 17 Angles of Triangles and Quadrilaterals—The Parallel chute 217 Lesson 17-1 Angles in a Triangle 217 Lesson 17-2 Exterior Angles and Angles in Quadrilaterals 223 Activity 17 Practice 227

Embedded Assessment 1 Angle Measures—Light and Glass 229

Activity 18 Introduction to Transformations—Move It 231 Lesson 18-1 What is a Transformation? 231 Lesson 18-2 Translations and Coordinates 234 Lesson 18-3 Reflections and Coordinates 238 Lesson 18-4 Rotations and Coordinates 238 Activity 18 Practice 245

Activity 19 Rigid Transformations and Compositions—All the Right Moves 247 Lesson 19-1 Properties of Transformations 247 Lesson 19-2 Composition of Transformations 254 Activity 19 Practice 261

Embedded Assessment 2 Rigid Transformations—In Transformations We Trust 263

Activity 20 Similar Triangles—Mirrors and Shadows 265 Lesson 20-1 Exploring Similarity 265 Lesson 20-2 Properties and Conditions of Similar Triangles 271 Activity 20 Practice 277

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Activity 21 Dilations—Alice’s Adventures in Shrinking and Growing 279 Lesson 21-1 Stretching and Shrinking Geometric Figures 279 Lesson 21-2 Effects of Scale Factor 286 Activity 21 Practice 291

Embedded Assessment 3 Similarity and Dilations—Business as Usual 293

Activity 22 The Pythagorean Theorem—Stop the Presses 297 Lesson 22-1 Pythagorean Theorem: Squares of Lengths 297 Lesson 22-2 Pythagorean Theorem: Missing Lengths 301 Activity 22 Practice 305

Activity 23 Applying the Pythagorean Theorem—Diamond in the Rough 307 Lesson 23-1 The Pythagorean Theorem in Two and Three Dimensions 307 Lesson 23-2 The Pythagorean Theorem and the Coordinate Plane 311 Activity 23 Practice 315

Activity 24 The Converse of the Pythagorean Theorem—Paper Clips Chains 317 Lesson 24-1 The Converse of the Pythagorean Theorem 317 Lesson 24-2 Pythagorean Triples 321 Activity 24 Practice 323

Embedded Assessment 4 The Pythagorean Theorem—Camp Euclid 325

Activity 25 Surface Area—Greenhouse Gardens 327 Lesson 25-1 Lateral and Surface Areas of Prisms 327 Lesson 25-2 Lateral and Surface Areas of Cylinders 333 Activity 25 Practice 337

Activity 26 Volume of Solids—Castles in the Sand 339 Lesson 26-1 Volumes of Prisms and Pyramids 339 Lesson 26-2 Volumes of Cylinders, Cones, and Spheres 343 Lesson 26-3 Volumes of Composite Solids 347 Activity 26 Practice 351

Embedded Assessment 5 Surface Area and Volume—Air Dancing 353

UNIT 4 FUNCTIONS Unit 4 Overview 355 Getting Ready 356

Activity 27 Introduction to Functions—It’s All Related 357 Lesson 27-1 What is a Function? 357 Lesson 27-2 Mapping Inputs and Outputs 362 Lesson 27-3 Identifying Functions 366 Lesson 27-4 Graphs of Functions 370 Activity 27 Practice 377

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Activity 28 Comparing Functions—Which Car Wins? 379 Lesson 28-1 Representing Functions 379 Lesson 28-2 Analyzing Functions 391 Activity 28 Practice 393

Activity 29 Constructing Functions—Hold On to Your Hats 395 Lesson 29-1 Construct a Function 395 Lesson 29-2 Rate of Change and Initial Value 400 Activity 29 Practice 406

Embedded Assessment 1 Functions—Remember When? 408

Activity 30 Linear Functions—Get In Line 411 Lesson 30-1 Rate of Change 411 Lesson 30-2 Representing Linear Relationships 417 Activity 30 Practice 421

Activity 31 Linear and Non-Linear Functions—Measure Up 423 Lesson 31-1 Bean Experiment 423 Lesson 31-2 Bean Experiment Continued 429 Lesson 31-3 Scale Experiment 433 Activity 31 Practice 438

Embedded Assessment 2 Scatter Plots and Trend Lines—Remember When? 440

UNIT 5 PROBABILITY AND STATISTICS Unit 5 Overview 443 Getting Ready 444

Activity 32 Scatter Plots and Association—Cracker Snacker 445 Lesson32-1 Scatter Plots 445 Lesson 32-2 Association 449 Activity 32 Practice 453

Activity 33 Bivariate Data—Sue Swandive 455 Lesson 33-1 Collecting Data 455 Lesson 33-2 Trend Lines 459 Lesson 33-3 The Competition! 461 Activity 33 Practice 464

Embedded Assessment 1 Scatter Plots, Associations, and Trends—U.S. Census 465

Activity 34 Median-Median Line—Homework Help Line 467 Lesson 34-1 Finding the Median-Median Line 467 Lesson 34-2 Using the Median-Median Line 473 Activity 34 Practice 476

Activity 35 Two-Way Tables and Association—Student Opinions 477 Lesson 35-1 Two-Way Tables 477 Lesson 35-2 Investigating Association 481 Activity 35 Practice 484

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Embedded Assessment 2 Median-Median Line and Two-Way Tables— Mokher’s Measurements 485

UNIT 6 MANAGING MONEY Unit 6 Overview 487 Getting Ready 488

Activity 36 Managing Money—To Charge or Not 489 Lesson 36-1 The Cost of Borrowing 489 Lesson 36-2 Saving for the Future 494 Activity 36 Practice 498

RESOURCES 499 Formulas 500 Learning Strategies 504 Glossary 507 Academic Vocabulary Graphic Organizers 517

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To the StudentTo the StudentWelcome to the SpringBoard program. We hope you will discover how SpringBoard can help you achieve high academic standards, reach your learning goals, and prepare for success in future mathematics studies.

The program has been created with you in mind: the content you need to learn, the tools to help you learn, and the critical thinking skills that help you build confidence in your own knowledge of mathematics. The College Board publishes the SpringBoard program. It also publishes the PSAT/NMSQT, the SAT, and the Advanced Placement exams—all exams that you are likely to encounter in your student years. Preparing you to perform well on those exams and to develop the mathematics skills needed for high school success is the primary purpose of this program.

Standards-Based Mathematics LearningKnowledge of mathematics helps prepare you for future success in college, in work, and in your personal life. We all encounter some form of mathematics daily, from calculating the cost of groceries to determining the cost of materials and labor needed to build a new road. The SpringBoard program is based on learning standards that identify the mathematics skills and knowledge that you should master to succeed in high school and in future college-level work. In this course, the standards follow these broad areas of mathematics knowledge:

• Mathematical process standards• Number and operations• Proportionality• Expressions, equations, and relationships• Measurement and data• Personal financial literacy

Mathematical process standards guide your study of mathematics. They are actions you take to help you understand math rather than mathematical concepts. For example, the mathematical process standards suggest the following:

• Connect mathematics concepts to everyday life and situations around you.• Use models to solve problems and to justify solutions and their

reasonableness.• Choose tools, such as number lines, protractors, technology, or paper and

pencil to help you solve problems.• Communicate what you are learning both orally and in writing.• Create and use representations to communicate mathematical ideas.• Analyze mathematical relationships to connect ideas.• Learn and use precise mathematical language in both written and oral

communication.

In the middle school years, your study of mathematics begins with a basic understanding of fractions and the operations performed with them. Your study continues with the development of a deep understanding of the rational numbers, their different representations, and the connections between these numbers and other number systems and operations. You will need a broad understanding of addition, subtraction, and multiplication with rational numbers, along with computational fluency with whole-number operations.

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As you continue your studies, you will examine ratios and rates, which will allow you to make comparisons between numbers. Ratio and rates represent proportionality. Understanding the concept of proportionality is critical to future success in your study of algebra and the rest of the high school mathematics curriculum.

See pages xiii–xvi for a complete list of the Texas Essential Knowledge and Skills for this course.

Strategies for Learning MathematicsSome tools to help you learn are built into every activity. At the beginning of each activity, you will see suggested learning strategies. Each of these strategies is explained in full in the Resources section of your book. As you learn to use each strategy, you’ll have the opportunity to decide which strategies work best for you. Suggested learning strategies include:

• Reading strategies, which help you learn to look at problem descriptions in different ways, from marking the text to highlight key information to turning problem information into questions that help you break the problem down into its separate parts.

• Writing strategies, which help you focus on your purpose for writing and what you’re writing about.

• Problem-solving strategies, which give you multiple ways to approach the problem, from learning to identify the tasks within a problem to looking for patterns or working backward to see how the problem is set up.

• Collaborative strategies, which you’ll use with your classmates to explore concepts and problems in group discussions and working with partners.

Building Mathematics Knowledge and SkillsWhether it is mathematics or sports or cooking, one way we learn something really well is by practice and repetition. To help you learn mathematics, the SpringBoard program is built around problem solving, reasoning and justification, communication, connections between concepts and ideas, and visual representation of mathematical concepts.

Problem Solving Many of the problems in this book are based on real-life situations that require you to analyze the situation and the information in the problem, make decisions, determine the strategies you’ll use to solve the problem, and justify your solution. Having a real-world focus helps you see how mathematics is used in everyday life.

Reasoning and Justification One part of learning mathematics, or any subject, is learning not only how to solve problems but also why you solved them the way you did. You will have many opportunities to predict possible solutions and then to verify solutions. You will be asked to explain the reasoning behind how you solved the problem, the mathematics concepts involved, and why your approach was appropriate for solving the problem.

Communication When learning a language, saying words out loud helps you learn to pronounce the words and to remember them. Communicating about mathematics, orally and in writing, with your classmates and teachers helps

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you organize your learning and explain mathematics concepts and problem-solving strategies more precisely. Sharing your ideas and thoughts allows you and your classmates to build on each other’s ideas and expand your own understanding.

Mathematics Connections As you study mathematics, you will learn many different concepts and ways of solving problems. Reading the problem descriptions will take you into the real-life applications of mathematics. As you develop your mathematics knowledge, you will see the many connections between mathematics concepts and between mathematics and your own life.

Representations Artists create representations through drawings and paintings. In mathematics, representations can take many forms, such as numeric, verbal, graphic, or symbolic. In this course, you are encouraged to use representations to organize problem information, present possible solutions, and communicate your reasoning. Creating representations is a tool you can use to gain understanding of concepts and communicate that understanding to others.

We hope you enjoy your study of mathematics using the SpringBoard program. We, the writers, are all classroom teachers, and we created this program because we love mathematics. We wanted to inspire you to learn mathematics and build confidence that you can be successful in your math studies and in using mathematics in daily life.

Grade 88.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

8.1.A apply mathematics to problems arising in everyday life, society, and the workplace;

8.1.B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

8.1.C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

8.1.D communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

8.1.E create and use representations to organize, record, and communicate mathematical ideas;

8.1.F analyze mathematical relationships to connect and communicate mathematical ideas; and

8.1.G display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Texas Essential Knowledge and SkillsTexas Essential Knowledge and Skills

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