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Unformatted text preview: Linear Relations and Functions Chapter Overview and Pacing PACING (days) Regular Block LESSON OBJECTIVES Basic/ Average Advanced Basic/ Average Advanced Relations and Functions (pp. 56–62) • Analyze and graph relations. • Find functional values. 1 optional 0.5 optional Linear Equations (pp. 63–67) • Identify linear equations and functions. • Write linear equations in standard form and graph them. 1 optional 0.5 optional Slope (pp. 68–74) • Find and use the slope of a line. • Graph parallel and perpendicular lines. 1 optional 0.5 optional Writing Linear Equations (pp. 75–80) • Write an equation of a line given the slope and a point on the line. • Write an equation of a line parallel or perpendicular to a given line. 1 optional 0.5 optional 2 (with 2-5 Follow-Up) optional 1 optional Special Functions (pp. 89–95) • Identify and graph step, constant, and identity functions. • Identify and graph absolute value and piecewise functions. 1 optional 0.5 optional Graphing Inequalities (pp. 96–99) • Graph linear inequalities. • Graph absolute value inequalities. 1 optional 0.5 optional Study Guide and Practice Test (pp. 100–105) Standardized Test Practice (pp. 106–107) 1 2 0.5 0.5 Chapter Assessment 1 1 0.5 0.5 10 3 5 1 Modeling Real-World Data: Using Scatter Plots (pp. 81–88) • Draw scatter plots. • Find and use prediction equations. Follow-Up: Lines of Regression TOTAL Pacing suggestions for the entire year can be found on pages T20–T21. 54A Chapter 2 Linear Relations and Functions Timesaving Tools ™ All-In-One Planner and Resource Center Chapter Resource Manager See pages T12–T13. 57–58 59–60 61 62 63–64 65–66 67 68 69–70 71–72 73 74 75–76 77–78 79 80 81–82 83–84 85 86 87–88 89–90 91 92 114 93–94 95–96 97 98 114 Materials 2-1 2-1 2-2 2-2 2-3 2-3 GCS 30, SC 3 2-4 2-4 SC 4 SM 97–102 2-5 2-5 tape measure, graph paper (Follow-Up: graphing calculator) GCS 29 2-6 2-6 graphing calculator, toothpicks 2-7 2-7 113 113, 115 5-M Tra inute nsp C are heck nci es Int e Cha racti lkb ve oar d Alg e2P A Plu SS: s (l T ess utori ons al ) Ap plic atio ns* Ass ess me nt Enr ich me nt S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice Ave rag e) Rea di Ma ng to the ma Learn tics CHAPTER 2 RESOURCE MASTERS 3 spaghetti graphing calculator, spaghetti 4 99–112, 116–118 *Key to Abbreviations: GCS  Graphing Calculator and Speadsheet Masters, SC  School-to-Career Masters, SM  Science and Mathematics Lab Manual Chapter 2 Linear Relations and Functions 54B Mathematical Connections and Background Continuity of Instruction Prior Knowledge In prior years students worked with coordinate systems, ordered pairs, and linear equations and functions. They manipulated and solved linear equations and inequalities algebraically. Also, they used graphs to represent two-variable data sets. This Chapter Students explore algebraic descriptions of linear functions, graphs of lines, and how to go back and forth between linear equations and graphed lines. They find the slope of a line containing two given points, relate the slope and y-intercept of a line to the values m and b in the slope-intercept form of an equation, and write equations for lines given two points or given one point and the slope. They find lines of fit for data and graph inequalities and special functions such as the greatest integer function and the absolute value function. Future Connections Students will continue to learn how algebraic expressions and the coordinate plane are related. At a simple level, they will learn how the “horizontal line test” identifies a one-toone function. As activities become more complex, they will use graphs to explore quadratic and other non-linear equations and inequalities and use graphs to represent and solve systems of equations and inequalities. 54C Chapter 2 Linear Relations and Functions Relations and Functions This lesson begins an exploration of two important themes of algebra. One theme is how algebraic equations and the Cartesian coordinate system are related. The other theme is the relationships between equations that represent entire lines and numbers that represent points on or properties of that line. For this lesson the central idea is ordered pairs. First, ordered pairs are explored as names for points in a coordinate plane. Second, several ordered pairs are used to describe how the set of the first elements (the domain) can be related to the set of the second elements (the range). Third, for equations that represent a line or a curve, ordered pairs are used to determine the graph that represents that equation in the coordinate plane. In the lesson the relation between functions and relations is explored in two ways. Mappings of domain elements to range elements are used to identify functions that are one to one, functions that are not one to one, and relations that are not functions. In the coordinate plane, the vertical line test is used to distinguish a relation from a function. Relationships between ordered pairs and functions are explored in two ways. First, students are given an equation (for a curve or for a line) and make a table of ordered pairs for the equation. Second, students are given a function and a domain value, and evaluate the function to find the range value. Linear Equations In this lesson students deal with linear functions and intercepts. Linear functions and equations can be written in slope-intercept form, f(x)  mx  b or y  mx + b, or in standard form, Ax  By  C. The graph of a linear function or equation is always a line. Slope Slope is a fundamental concept in algebra and higher mathematics. In this lesson, students calculate the slope of a line given two points on the line and explore the slopes of families or pairs of lines that are parallel and the slopes of pairs of lines that are perpendicular. Students graph a line given two points or given one point and the slope. In the coordinate plane, students associate lines with slopes that are positive, negative, zero, or undefined. Writing Linear Equations This lesson focuses on the slope and y-intercept of a linear equation. In the slope-intercept form of a linear equation, y  mx  b, m represents the slope of the line and b is the y-intercept. Students use two forms of a linear equation, the slope-intercept form and the point-slope form, to write an equation given two points, given a point and the slope, or given a point and the equation of a parallel or a perpendicular line. Special Functions In this lesson, students explore special functions. The identity and constant functions are special linear functions. The graph of a step function is a series of line segments. An absolute value function has a V-shaped graph made up of portions of two lines. A piecewise function is a function written using two or more algebraic expressions. Graphing Inequalities Modeling Real-World Data: Using Scatter Plots This lesson explores equations that approximate the relation between domain values and range values, extending the idea of using an algebraic equation to represent a set of points in a plane. Starting with a scatter plot of data, students mentally picture a line through the data. After selecting two points on that line, they calculate the slope and y-intercept of that line. The equation, called a line of fit or a prediction equation, may be used to calculate the value of one variable given a value of the other. Activities in this lesson require three steps: given a set of ordered pairs, students identify a line that represents a set of ordered pairs; then they select two ordered pairs that lie on the line; and finally they calculate the slope and y-intercept for that line. In this lesson the graph of an equation is seen as the boundary between two regions of the coordinate plane. An inequality is a description of one of the two regions, and whether the boundary is part of that region depends on the inequality symbol that is used. Students explore how inequalities, including absolute value inequalities, are modeled by points in the coordinate plane, and vice versa. Additional mathematical information and teaching notes are available in Glencoe’s Algebra 2 Key Concepts: Mathematical Background and Teaching Notes, which is available at . The lessons appropriate for this chapter are as follows. • Linear Relations and Functions (Lesson 5) • Graphing Linear Equations (Lessons 6 and 12) • Slope (Lesson 7) • Writing Linear Equations in Point-Slope and Standard Forms (Lesson 8) • Writing Linear Equations in Slope-Intercept Form (Lesson 10) • Integration: Geometry/Parallel and Perpendicular Lines (Lesson 13) • Statistics: Scatter Plots and Best-Line Fits (Lesson 9) • Graphing Inequalities in Two Variables (Lesson 17) Chapter 2 Linear Relations and Functions 54D and Assessment ASSESSMENT INTERVENTION Type Student Edition Teacher Resources Ongoing Prerequisite Skills, pp. 55, 62, 67, 74, 80, 86, 95 Practice Quiz 1, p. 74 Practice Quiz 2, p. 95 5-Minute Check Transparencies Quizzes, CRM pp. 113–114 Mid-Chapter Test, CRM p. 115 Study Guide and Intervention, CRM pp. 57–58, 63–64, 69–70, 75–76, 81–82, 87–88, 93–94 Mixed Review pp. 62, 67, 74, 80, 86, 95, 99 Cumulative Review, CRM p. 116 Error Analysis Find the Error, pp. 60, 71 Find the Error, TWE pp. 60, 71 Unlocking Misconceptions, TWE p. 58 Tips for New Teachers, TWE pp. 62, 74, 90 Standardized Test Practice pp. 62, 67, 74, 76, 78, 80, 86, 95, 99, 105, 106–107 TWE p. 76 Standardized Test Practice, CRM pp. 117–118 Open-Ended Assessment Writing in Math, pp. 62, 67, 73, 80, 86, 94, 99 Open Ended, pp. 60, 65, 71, 78, 83, 92, 98 Modeling: TWE pp. 67, 74, 95 Speaking: TWE pp. 62, 98 Writing: TWE pp. 80, 86 Open-Ended Assessment, CRM p. 111 Chapter Assessment Study Guide, pp. 100–104 Practice Test, p. 105 Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 99–104 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 105–110 Vocabulary Test/Review, CRM p. 112 Technology/Internet Alge2PASS: Tutorial Plus Standardized Test Practice CD-ROM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabulary_review Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests. 54E Chapter 2 Linear Relations and Functions Reading and Writing in Mathematics Intervention Technology Alge2PASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 2 Lesson Alge2PASS Lesson 2-2 3 Graphing Linear Equations on the Coordinate Plane 2-7 4 Graphing Linear Inequalities on the Coordinate Plane ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at . Glencoe Algebra 2 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 55 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 60, 65, 71, 78, 83, 92, 98, 100) • Writing in Math questions in every lesson, pp. 62, 67, 73, 80, 86, 94, 99 • Reading Study Tip, pp. 56, 59, 71, 82 • WebQuest, p. 84 Teacher Wraparound Edition Intervention at Home Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. • For chapter review, there is vocabulary review, test practice, and standardized test practice. For more information on Intervention and Assessment, see pp. T8–T11. • Foldables Study Organizer, pp. 55, 100 • Study Notebook suggestions, pp. 60, 65, 71, 78, 83, 93, 97 • Modeling activities, pp. 67, 74, 95 • Speaking activities, pp. 62, 98 • Writing activities, pp. 80, 86 • Differentiated Instruction, (Verbal/Linguistic), p. 92 • ELL Resources, pp. 54, 61, 66, 73, 79, 85, 92, 94, 99, 100 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 2 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 2 Resource Masters, pp. 61, 67, 73, 79, 85, 91, 97) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 2 Linear Relations and Functions 54F Linear Relations and Functions Notes Have students read over the list of objectives and make a list of any words with which they are not familiar. • Lesson 2-1 Analyze relations and functions. • Lessons 2-2 and 2-4 Identify, graph, and write linear equations. • Lesson 2-3 Find the slope of a line. • Lesson 2-5 Draw scatter plots and find prediction equations. • Lessons 2-6 and 2-7 Graph special functions, linear inequalities, and absolute value inequalities. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson. Key Vocabulary • • • • • linear equation (p. 63) linear function (p. 63) slope (p. 68) slope-intercept form (p. 75) point-slope form (p. 76) Linear equations can be used to model relationships between many real-world quantities. One of the most common uses of a linear model is to make predictions. Most hot springs are the result of groundwater passing through or near recently formed, hot, igneous rocks. Iceland, Yellowstone Park in the United States, and North Island of New Zealand are noted for their hot springs. You will use a linear equation to find the temperature of underground rocks in Lesson 2-2. Lesson 2-1 2-2 2-3 2-4 2-5 2-5 Follow-Up 2-6 2-7 NCTM Standards Local Objectives 1, 2, 7, 9, 10 1, 2, 4, 6, 8, 9 1, 2, 4, 6, 7, 8, 9, 10 1, 2, 6, 8, 9, 10 1, 2, 5, 6, 8, 9, 10 1, 2, 4, 5, 6, 9, 10 1, 2, 5, 6, 8, 9, 10 1, 2, 6, 8, 9, 10 Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 54 54 Chapter 2 Linear Relations and Functions Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 2 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 2 test. Chapter 2 Linear Relations and Functions Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 2. For Lesson 2-1 Identify Points on a Coordinate Plane Write the ordered pair for each point. 1. A (3, 3) 2. B (2, 3) 3. C (3, 1) 4. D (2, 0) 5. E (0, 4) 6. F (3, 2) y A B D C x O F E For Lesson 2-1 Evaluate Expressions Evaluate each expression if a  1, b  3, c  2, and d  0. 7. c  d 2 8. 4c  b 11 9. a2  5a  3 9 ac 12.  3 ab 11.  2 10. 2b2  b  7 28 (For review, see Lesson 1-1.) bc cd For Lesson 2-4 Simplify Expressions Simplify each expression. (For review, see Lesson 1-2.) 13. x  (1) x  1 14. x  (5) x  5 1 1 17. [x  (4)] x  2 2 2 16. 4[x  (2)] 4x  8 For Lessons 2-6 and 2-7 This section provides a review of the basic concepts needed before beginning Chapter 2. Page references are included for additional student help. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson. For Lesson 2-2 2-4 2-5 2-6 2-7 Prerequisite Skill Solving Equations (p. 62) Solving Equations (p. 74) Finding a Median (p. 80) Absolute Value (p. 86) Inequalities (p. 95) 15. 2[x  (3)] 2x  6 1 1 18. [x  (6)] x  2 3 3 Evaluate Expressions with Absolute Value Evaluate each expression if x = 3, y = 4, and z = 4.5. (For review, see Lesson 1-4.) 19. x 3 20. y 4 21. 5x 15 22. 2z 9 23. 5y  z 2.5 24. 3x  y  x  z 10.5 Make this Foldable to help you organize information about relations and functions. Begin with two sheets of grid paper. Fold Cut and Label Fold in half along the width and staple along the fold. Cut the top three sheets and label as shown. Graphing Graphing Linear Linear Relations Functions Reading and Writing As you read and study the chapter, write notes, examples, and graphs under the tabs. Chapter 2 Linear Relations and Functions 55 TM For more information about Foldables, see Teaching Mathematics with Foldables. Organization of Data: Annotating As students read and work their way through the chapter, have them make annotations under the appropriate tabs of their Foldable. Explain to them that annotations are usually notes taken in the margins of books, which we own, to organize the text for review or studying. Annotations often include questions that arise, reader comments and reactions, short summaries, steps or data numbered by the reader, and key points highlighted or underlined. Chapter 2 Linear Relations and Functions 55 Lesson Notes 5-Minute Check Transparency 2-1 Use as a quiz or review of Chapter 1. Mathematical Background notes are available for this lesson on p. 54C. do relations and functions apply to biology? Ask students: • What is the difference between average lifetime and maximum lifetime? The average lifetime is a representative number of years for any animal of that type, while the maximum lifetime is the greatest age ever attained by an animal of that type. • Why can you be sure that the second number in the ordered pairs for this data is always greater than or equal to the first? For each animal, the maximum age will always equal or exceed the average age. Vocabulary • ordered pair • Cartesian coordinate plane • quadrant • relation • domain • range • function • mapping • one-to-one function • vertical line test • independent variable • dependent variable • functional notation do relations and functions apply to biology? The table shows the average lifetime and maximum lifetime for some animals. The data can also be represented as ordered pairs . The ordered pairs for the data are (12, 28), (15, 30), (8, 20), (12, 20), and (20, 50). The first number in each ordered pair is the average lifetime, and the second number is the maximum lifetime. (12, 28) average lifetime ← In Chapter 1, students solved equations and inequalities. In this lesson, students relate equations to functions and relations, as well as to their graphs. • Find functional values. ← Building on Prior Knowledge • Analyze and graph relations. Average Lifetime (years) Maximum Lifetime (years) Cat 12 28 Cow 15 30 Deer 8 20 Dog 12 20 Horse 20 50 Source: The World Almanac GRAPH RELATIONS You can graph the ordered pairs above by creating a coordinate system with two axes. Each point represents one of the ordered pairs above. Remember that each point in the coordinate plane can be named by exactly one ordered pair and that every ordered pair names exactly one point in the coordinate plane. The graph of the animal lifetime data lies in only one part of the Cartesian coordinate plane—the part with all positive numbers. The Cartes...
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