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(1)Solving Systems of Linear Equations and Inequalities Chapter Overview and Pacing PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. Slope (pp. 256–262) • Find the slope of a line. • Use rate of change to solve problems.. 1. 1. 0.5. 0.5. Slope and Direct Variation (pp. 264–270) • Write and graph direct variation equations. • Solve problems involving direct variation.. 1. 1. 0.5. 0.5. 2 (with 5-3 Preview). 2 (with 5-3 Follow-Up). 1. 1 (with 5-3 Follow-Up). Writing Equations in Slope-Intercept Form (pp. 280–285) • Write an equation of a line given the slope and one point on a line. • Write an equation of a line given two points on the line.. 2. 2. 1. 1. Writing Equations in Point-Slope Form (pp. 286–291) • Write the equation of a line in point-slope form. • Write linear equations in different forms.. 1. 2. 0.5. 1 (with 5-4 Follow-Up). Geometry: Parallel and Perpendicular Lines (pp. 292–297) • Write an equation of the line that passes through a given point, parallel to a given line. • Write an equation of the line that passes through a given point, perpendicular to a given line.. 2. 1. 1. 0.5. 1. 1. Slope-Intercept Form (pp. 271–279) Preview: Use manipulatives to investigate slope-intercept form. • Write and graph linear equations in slope-intercept form. • Model real-world data with an equation in slope-intercept form. Follow-Up: Use a graphing calculator to identify families of linear graphs.. 3 2 (with 5-7 (with 5-7 Follow-Up) Follow-Up). Statistics: Scatter Plots and Lines of Fit (pp. 298–307) • Interpret points on a scatter plot. • Write equations for lines of fit. Follow-Up: Use a graphing calculator to find a median-fit line. Study Guide and Practice Test (pp. 308–313) Standardized Test Practice (pp. 314–315). 1. 1. Chapter Assessment. 1. 1. 0.5. 0.5. 14. 13. 7. 7. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 254A Chapter 5 Analyzing Linear Equations. 1 1 (with 5-7 (with 5-7 Follow-Up) Follow-Up).

(2) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 281–282. 283–284. 285. 286. 287–288. 289–290. 291. 292. 293–294. 295–296. 297. 298. 299–300. 301–302. 303. 304. 305–306. 307–308. 309. 310. 311–312. 313–314. 315. 316. 338. 317–318. 319–320. 321. 322. 338. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. 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 5 RESOURCE MASTERS. 39–40, 63–64 337. 337, 339. 29–30. See pages T12–T13.. Materials. 38. 5-1. 5-1. uncooked spaghetti, transparency showing coordinate plane. SM 41–44. 39. 5-2. 5-2. graphing calculator. SC 9. 40. 5-3. 5-3. (Preview: scissors, plastic sandwich bags, long rubber bands, tape, centimeter ruler, metal washers) (Follow-Up: graphing calculator). SC 10. 41. 5-4. 5-4. 11. GCS 31. 42. 5-5. 5-5. 12. 43. 5-6. 5-6. 13. 44. 5-7. 5-7. GCS 32, SM 51–56. 323–336, 340–342. grid paper, scissors, graphing calculator. 45. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 5 Analyzing Linear Equations 254B.

(3) Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 3, students learned to solve equations for a given variable (algebraic manipulation). In Chapter 4, students graphed and analyzed points that composed a relation or function. They learned that two points determine a specific line. They also plotted points that represent real-world data.. Slope The slope of a straight line is one of the most important characteristics of the line. The slope, a ratio of the vertical change in the line to the horizontal change, can be expressed in many ways. One common ris e , which can frequently be observed from definition is run graphed lines. On lines in which two points have been identified, you can also use the algebraic definition, y2 y1 , for which (x1, y1) and (x2, y2) represent the x2 x1. coordinates of points on that line. Rate of change describes how rapidly a line rises or falls. It also is used in the real-world context to express the relationship between two quantities, for example number of words typed in each minute.. Slope and Direct Variation. This Chapter Students closely examine the equations that represent the linear functions they graphed in Chapter 4. They learn to graph equations without finding two specific points. They use their skills in algebraic manipulation to rewrite linear equations in various forms. Students use their equation-writing skills to describe relationships in real-world data they have graphed.. The concept of direct variation grows from the meaning of ratio (Lesson 3-6). If the ratio of two variables is a constant, then direct variation is the way of expressing the relationship between the two variables. y. That is, x k, where y and x are variables and k is a constant (number). If you multiply each side of the equation by x, you get y kx. This represents an equation of a line and the k is the same value as the slope of the line. So when you graph a direct variation, you are graphing lines with slope k. All of these lines pass through the origin. In real-world applications, most direct variation graphs only occupy the first quadrant.. Slope-Intercept Form. Future Connections Slope is a key concept that spans mathematics through calculus and beyond. By knowing the characteristics of linear equations, students can determine what type(s) of solutions a system of equations might have. The concept of a best-fit line (or curve) is used again in Algebra 2 and Statistics courses.. 254C. Chapter 5 Analyzing Linear Equations. Slope-intercept form is y mx b, where m is the slope and b is the y-value where the line crosses the y-axis. The slope-intercept form offers two ways to graph a line. One can select two values for x and very easily calculate the corresponding values of y to create two ordered pairs that can be used to graph points on the line. Then the line is drawn that contains those two points. One can also use the slope and intercept to graph the line directly. The intercept gives a starting point on the y-axis. Use the slope to determine the distance and direction you go up/down and right/left to find another point on the line. Then draw the line..

(4) Writing Equations in Slope-Intercept Form It is important to understand what an equation represents and how to use it as a tool. The general expression for slope-intercept form is y mx b. This is the starting point for creating an equation from different types of information given. The goal is to use the given information to find values for m and b, so that you can rewrite the general form with x and y being the only unknowns.. Writing Equations in Point-Slope Form Point-slope form is derived from the definition of slope using the coordinates of two points on a line. Suppose one point is given as (x1, y1) and another point is unknown (x, y). Using the definition of slope, yy. Statistics: Scatter Plots and Lines of Fit A scatter plot includes graphs of ordered pairs that belong to a set in which the first coordinate represents one real-world measurement and the second coordinate represents another. Scatter plots can be used to visually identify trends, if they exist, and determine how strong that trend is. At this point in their studies, students do not have the mathematical background to attempt to write the equation of a best-fit line by using statistical formulas. So, they draw a line that seems characteristic of the data, select two points on that line, and then use those points to write an equation. Using this method, there are many correct best-fit lines that can be drawn. This should be understood so that students realize that predictions are totally dependent on the line drawn and have no factual rule for determining them.. 1 . Multiply each side by (x x1) you get m xx 1. and use the symmetric property of equality. You get y y1 m(x x1), which is the point-slope form of a linear equation. You can also manipulate equations in pointslope form and slope-intercept form to express them in standard form, Ax By C.. Geometry: Parallel and Perpendicular Lines This is a part of mathematics often called coordinate geometry or analytic geometry. In coordinate geometry, you use graphing and properties of graphs to prove geometric concepts. What properties, besides not intersecting, do parallel lines have? They lie in the same plane and have the same slope. Now consider perpendicular lines. We know they intersect, so they cannot have the same slope. Actually, they slope in opposite directions. That is, if one is vertical, the other is horizontal; if one slopes upward, the other slopes downward. A comparison of slopes of the two lines will lead you to discover that they are negative reciprocals of each other. To write the equation of a line that is parallel to or perpendicular to a given line, you must realize that you are still using the equation-writing skills presented in the previous lessons. You still need the slope and the coordinates of one of the points on the line to write the equation. Using the properties of parallel and perpendicular lines helps you to determine what slope you are using and the point is usually given to you.. www.algebra1.com/key_concepts Additional mathematical information and teaching notes are available in Glencoe’s Algebra 1 Key Concepts: Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows. • Slope (Lesson 7) • Writing Linear Equations in Slope-Intercept Form (Lesson 10) • Writing Linear Equations in Point-Slope and Standard Forms (Lesson 8) • Graphing Linear Equations (Lesson 12) • Intergration: Geometry/Parallel and Perpendicular Lines (Lesson 13) • Statistics: Scatter Plots and Best-Fit Lines (Lesson 9) Chapter 5 Analyzing Linear Equations 254D.

(5) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 255, 262, 270, 277, 285, 291, 297 Practice Quiz 1, p. 270 Practice Quiz 2, p. 297. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 29–30, 39–40, 63–64 Quizzes, CRM pp. 337–338 Mid-Chapter Test, CRM p. 339 Study Guide and Intervention, CRM pp. 281–282, 287–288, 293–294, 299–300, 305–306, 311–312, 317–318. Mixed Review. pp. 262, 270, 277, 285, 291, 297, 305. Cumulative Review, CRM p. 340. Error Analysis. Find the Error, pp. 259, 289 Common Misconceptions, p. 257. Find the Error, TWE pp. 259, 289 Unlocking Misconceptions, TWE p. 257 Tips for New Teachers, TWE pp. 262, 287. Standardized Test Practice. pp. 262, 269, 277, 281, 283, 285, 291, 297, 304, 313, 314–315. TWE pp. 314–315 Standardized Test Practice, CRM pp. 341–342. Open-Ended Assessment. Writing in Math, pp. 262, 269, 277, 285, 291, 297, 304 Open Ended, pp. 259, 267, 275, 283, 289, 291, 295, 301 Standardized Test, p. 315. Modeling: TWE pp. 262, 297 Speaking: TWE pp. 277, 285 Writing: TWE pp. 270, 291, 305 Open-Ended Assessment, CRM p. 335. Chapter Assessment. Study Guide, pp. 308–312 Practice Test, p. 313. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 323–328 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 329–334 Vocabulary Test/Review, CRM p. 336. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. 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.. 254E. Chapter 5 Analyzing Linear Equations.

(6) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson. AlgePASS Lesson. 5-4. 11 Finding x- and y-intercepts of Linear Equations. 5-5. 12 Writing Equations of Lines. 5-6. 13 Effects of Parameter Changes on Linear Functions. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 38–45 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 255 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 259, 267, 275, 283, 289, 291, 295, 301) • Reading Mathematics, p. 263 • Writing in Math questions in every lesson, pp. 262, 269, 277, 285, 291, 297, 304 • Reading Study Tip, p. 256 • WebQuest, p. 304 Teacher Wraparound Edition • Foldables Study Organizer, pp. 255, 308 • Study Notebook suggestions, pp. 259, 263, 267, 271, 275, 283, 289, 295, 301 • Modeling activities, pp. 262, 297 • Speaking activities, pp. 277, 285 • Writing activities, pp. 270, 291, 305 • Differentiated Instruction, (Verbal/Linguistic), p. 288 • ELL Resources, pp. 254, 261, 263, 268, 274, 276, 284, 288, 290, 296, 303, 308 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 5 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 5 Resource Masters, pp. 285, 291, 297, 303, 309, 315, 321) • 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 • Hot Words/Hot Topics Sections 1.5, 2.1, 2.4, 4.3, 6.3, 6.4, 6.7, 6.8 For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 5 Analyzing Linear Equations 254F.

(7) Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. 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.. Analyzing Linear Equations • Lesson 5-1 Find the slope of a line. • Lesson 5-2 Write direct variation equations. • Lessons 5-3 through 5-5 Write linear equations in slope-intercept and point-slope forms. • Lesson 5-6 Write equations for parallel and perpendicular lines. • Lesson 5-7 Draw a scatter plot and write the equations of a line of fit.. Key Vocabulary • • • • •. slope (p. 256) rate of change (p. 258) direct variation (p. 264) slope-intercept form (p. 272) point-slope form (p. 286). Linear equations are used to model a variety of real-world situations. The concept of slope allows you to analyze how a quantity changes over time. You can use a linear equation to model the cost of the space program. The United States began its exploration of space in January, 1958, when it launched its first satellite into orbit. In the 1970s, NASA developed the space shuttle to reduce costs by inventing the first reusable spacecraft. You will use a linear equation to model the cost of the space program in Lesson 5-7.. Lesson 5-1 5-2 5-3 Preview 5-3 5-3 Follow-Up 5-4 5-5 5-6 5-7 5-7 Follow-Up. NCTM Standards. Local Objectives. 2, 3, 4, 6, 7, 8, 9, 10 2, 4, 8, 9, 10 2, 3, 4, 8, 9, 10 2, 3, 4, 6, 8, 9, 10 2, 6, 7, 8, 10 2, 6, 8, 9, 10 2, 6, 8, 9, 10 2, 3, 6, 7, 8, 9, 10 2, 5, 6, 7, 8, 9, 10 2, 5, 7, 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 254. Chapter 5 Analyzing Linear Equations. 254 Chapter 5. Analyzing Linear Equations. 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 5 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 5 test..

(8) 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 5. For Lesson 5-1. This section provides a review of the basic concepts needed before beginning Chapter 5. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 29–30, 39–40, 63–64.. Simplify Fractions. Simplify. (For review, see pages 798 and 799.) 2 1 8 2 1. 2. 10 5 12 3 5 1 7 1 5. 6. 15 3 28 4. 1 4 4. 8 2 18 1 8. 1 12 2. 1 2 3. 8 4 9 7. 3 3. For Lesson 5-2. 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.. Evaluate Expressions. ab Evaluate for each set of values. cd. (For review, see Lesson 1-2.). 1 9. a 6, b 5, c 8, d 4 4 3 11. a 2, b 1, c 4, d 0 4 13. a 3, b 3, c 4, d 7 0. 10. a 5, b 1, c 2, d 1 2 12. a 8, b 2, c 1, d 1 5 1 1 3 14. a , b , c 7, d 9 2 2 2. For Lessons 5-3 through 5-7. Identify Points on a Coordinate Plane y. Write the ordered pair for each point.. J. (For review, see Lesson 4-1.). 15. J (1, 2). 16. K (3, 2). 17. L (2, 3). 18. M (0, 3). 19. N (2, 2). 20. P (3, 0). N P x. O. K L M. Prerequisite Skill. 5-2 5-3 5-4 5-5 5-6. Dividing Fractions (p. 262) Rewriting Equations (p. 270) Finding Slope (p. 277) Subtracting Integers (p. 285) Writing Multiplicative Inverses (p. 291) Slope-Intercept Form (p. 297). 5-7. Make this Foldable to help you organize information about writing linear equations. Begin with four sheets of grid paper. Fold and Cut. For Lesson. Staple Staple the eight half-sheets together to form a booklet.. Fold each sheet of grid paper in half along the width. Then cut along the crease.. Cut Tabs Cut seven lines from the bottom of the top sheet, six lines from the second sheet, and so on.. Label Label each of the tabs with a lesson number. The last tab is for the vocabulary.. 5-1 5-2 5-3 5-4 5-5 5-6 5-7 Vocabulary. As you read and study the chapter, use each page to write notes and to graph examples for each lesson.. Reading and Writing. Chapter 5. Analyzing Linear Equations. 255. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Descriptive Writing and Organizing Data After students make their Foldable, have them label a tab for each lesson in this chapter. At the end of each lesson, ask students to write a descriptive paragraph about their experiences with the concepts, computational skills, and the graphs presented. For example, students might write about how they felt when they were first asked to find the slope of a line or how the lesson appeared to them visually before they understood the concepts presented and how it appeared after mastery. Chapter 5 Analyzing Linear Equations 255.

(9) Lesson Notes. 1 Focus 5-Minute Check Transparency 5-1 Use as a quiz or review of Chapter 4.. Slope • Find the slope of a line. • Use rate of change to solve problems.. Vocabulary • slope • rate of change. Mathematical Background notes are available for this lesson on p. 254C.. is slope important in architecture? The slope of a roof describes how steep it is. It is the number of units the roof rises for each unit of run. In the photo, the roof rises 8 feet for each 12 feet of run. rise run 8 2 or 12 3. slope . Building on Prior Knowledge In Chapter 4, students learned that points on a line have coordinates that satisfy a given equation. In this lesson, they should recognize that there is another relationship that exists between any two points on a line. is slope important in architecture? Ask students: • What is the slope of the roof if 5 the rise is 10 and the run is 6? 3 • Which has a steeper slope, a roof whose rise is greater than the run or one whose run is greater than the rise? rise run • Geography The steepness of roofs on buildings is often associated with certain climates. Very steep roofs are used in rainy or snowy climates, while flatter roofs are often found in arid regions. What type of roof would be most common in our community? Answers may vary.. 12 ft run. Section of roof. 8 ft rise. FIND SLOPE The slope of a line is a number determined by any two points on the line. This number describes how steep the line is. The greater the absolute value of the slope, the steeper the line. Slope is the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move from one point to the other. The graph shows a line that passes through (1, 3) and (4, 5). rise run. y. slope . run: 4 1 3. change in y-coordinates. change in x-coordinates. (4, 5). 53 2 or 41 3. (1, 3). rise: 5 3 2 x. O. 2 So, the slope of the line is . 3. Slope of a Line Study Tip. • Words. The slope of a line is the ratio of the rise to the run.. • Symbols. The slope m of a nonvertical line through any two points, (x1, y1) and (x2, y2), can be found as follows.. Reading Math. In x1, the 1 is called a subscript. It is read x sub 1.. y y. 1 2 m x x 2. 256. 1. ← change in y ← change in x. • Model y x 2 x1 (x 2, y 2). y 2 y1 (x1, y1) O. Chapter 5 Analyzing Linear Equations. Resource Manager Workbook and Reproducible Masters Chapter 5 Resource Masters • Study Guide and Intervention, pp. 281–282 • Skills Practice, p. 283 • Practice, p. 284 • Reading to Learn Mathematics, p. 285 • Enrichment, p. 286. Parent and Student Study Guide Workbook, p. 38 Prerequisite Skills Workbook, pp. 39–40, 63–64. Transparencies 5-Minute Check Transparency 5-1 Answer Key Transparencies. Technology Interactive Chalkboard. x.

(10) Example 1 Positive Slope. 2 Teach. Find the slope of the line that passes through (1, 2) and (3, 4).. Study Tip. Let (1, 2) (x1, y1) and (3, 4) (x2, y2).. Common Misconception. y2 y1 m x2 x1 42 3 (1) 2 1 or 4 2. It may make your calculations easier to choose the point on the left as ( x1, y1). However, either point may be chosen as (x1, y1).. y. FIND SLOPE. rise run. Substitute. Simplify.. In-Class Examples. (3, 4). (1, 2 ). 1 Find the slope of the line that. x. O. Power Point®. passes through (3, 2) and 3 (5, 5). . 1 2. The slope is .. 8. 2 Find the slope of the line that passes through (3, 4) and (2, 8). 4. Example 2 Negative Slope Find the slope of the line that passes through (1, 2) and (4, 1). Let (1, 2) (x1, y1) and (4, 1) (x2, y2).. TEACHING TIP To verify this, have students rework each example, letting the other point be (x1, y1).. y2 y1 m x2 x1 1 (2) 4 (1) 3 3. or 1. 3 Find the slope of the line that. y. passes through (3, 4) and (4, 4). 0. (4, 1). rise run. x. O. Substitute.. Teaching Tip. Ask students how they would determine if two points lie on a horizontal line without graphing the points.. (1, 2). Simplify.. The slope is 1.. 4 Find the slope of the line that passes through (2, 4) and (2, 3). undefined. Example 3 Zero Slope Find the slope of the line that passes through (1, 2) and (1, 2). Let (1, 2) (x1, y1) and (1, 2) (x2, y2). y2 y1 m x2 x1 22 1 1. y. Concept Check. rise run. Slope Ask students to give the. Substitute.. (1, 2). slope of a very steep line and then the slope of one that is almost horizontal. Make sure students acknowledge that negative slopes are acceptable to meet these criteria. 2 2 Sample answers: 6, ; 5, 5 15. (1, 2). x. O. 0 2. or 0 Simplify. The slope is zero.. Example 4 Undefined Slope Find the slope of the line that passes through (1, 2) and (1, 3). Let (1, 2) (x1, y1) and (1, 3) (x2, y2). y2 y1 m x2 x1 3 (2) 11. y. rise run. 5 0. or . (1, 3 ). x. O. Since division by zero is undefined, the slope is undefined.. Interactive. (1, 2 ). Chalkboard PowerPoint® Presentations. www.algebra1.com/extra_examples. Lesson 5-1. Slope. 257. Unlocking Misconceptions • Computing Slope Many students automatically assume that the leftmost point has to be (x1, y1) and the point farther right is (x2, y2). The designation of (x1, y1) and (x2, y2) is arbitrary. However, one particular designation may make the subtraction easier than the other. • Improper Fractions Students should learn that slope is expressed as a fraction or integer. Make sure they understand that slope can be an improper fraction, but it is never expressed as a mixed number.. This CD-ROM is a customizable Microsoft® PowerPoint® presentation that includes: • Step-by-step, dynamic solutions of each In-Class Example from the Teacher Wraparound Edition • Additional, Your Turn exercises for each example • The 5-Minute Check Transparencies • Hot links to Glencoe Online Study Tools Lesson 5-1 Slope 257.

(11) In-Class Example. Classifying Lines. Power Point®. Positive Slope. Teaching Tip. Watch for students who try to find the cross product mentally and forget to multiply both 10 and r by 3.. Negative Slope. y. y. x. O. line through (6, 3) and (r, 2) 1 has a slope of . 4. Undefined Slope y. y horizontal line. line slopes down from left to right. line slopes up from left to right. 5 Find the value of r so that the. Slope of 0. x. O. vertical line. x. O. x. O. If you know the slope of a line and the coordinates of one of the points on a line, you can find the coordinates of other points on the line.. 2. Example 5 Find Coordinates Given Slope Let (r, 6) (x1, y1) and (10, 3) (x2, y2).. In-Class Example. y2 y1 m x2 x1. Power Point®. shows the number of U.S. passports issued in 1991, 1995, and 1999. Passports (millions). U.S. Passports Issued 7 6.7. 6. Study Tip. Substitute.. 3 2. 9 10 r. Subtract.. Look Back To review cross products, see Lesson 3-6.. 3(10 r) 2(9) 30 3r 18. 5.3. O. Find the cross products. Simplify.. 3r 12. Simplify. Divide each side by 3. Simplify.. 3.4. RATE OF CHANGE Slope can be used to describe a rate of change. The ’91. ’95 Year. ’99. rate of change tells, on average, how a quantity is changing over time.. Example 6 Find a Rate of Change. Source: U.S. State Department. a. Find the rates of change for 1991–1995 and 1995–1999. 475,000/yr; 350,000/yr b. Explain the meaning of the slope in each case. ’91–’95: The number of U.S. passports issued increased about 475,000 each year. ’95–’99: The number of U.S. passports issued increased about 350,000 each year. c. How are the different rates of change shown on the graph? There is a greater rate of change from ’91–’95 than from ’95–’99. So the ’91–’95 segment has the steeper slope.. Log on for: • Updated data • More activities on rate of change www.algebra1.com/ usa_today. DINING OUT The graph shows the amount spent on food and drink at U.S. restaurants in recent years. a. Find the rates of change for 1980–1990 and 1990–2000. Use the formula for slope.. USA TODAY Snapshots® Dining out. 1990: $239. change in quantity ← billion S| rise ← years run change in time. $300 $200 $100. 1980: $120. ’85. ’90. ’95. Source: National Restaurant Association By Hilary Wasson and Alejandro Gonzalez, USA TODAY. 258 Chapter 5 Analyzing Linear Equations. USA TODAY Education’s Online site offers resources and interactive features connected to each day’s newspaper. Experience TODAY, USA TODAY’s daily lesson plan, is available on the site and delivered daily to subscribers. This plan provides instruction for integrating USA TODAY graphics and key editorial features into your mathematics classroom. Log on to www.education.usatoday.com. Chapter 5 Analyzing Linear Equations. 2000: $376. Food and drink sales at U.S. restaurants by year (in billions):. Online Lesson Plans. 258. x. (10, 3). 3r 12 3 3. r4. 3. (r, 6). 30 3r 30 18 30 Add 30 to each side.. 4. 0. 3 6 10 r. . y. Slope formula. 3 2. . 6 TRAVEL The graph below. 5. 3 2. Find the value of r so that the line through (r, 6) and (10, 3) has a slope of .. RATE OF CHANGE.

(12) 1980–1990:. change in quantity 239 120 1990 1980 change in time 119 or 11.9 10. 3 Practice/Apply. Substitute. Simplify.. Spending on food and drink increased by $119 billion in a 10-year period for a rate of change of $11.9 billion per year. 1990–2000:. change in quantity 376 239 2000 1990 change in time 137 or 13.7 10. Study Notebook. Substitute. Simplify.. Over this 10-year period, spending increased by $137 billion, for a rate of change of $13.7 billion per year. b. Explain the meaning of the slope in each case. For 1980–1990, on average, $11.9 billion more was spent each year than the last. For 1990–2000, on average, $13.7 billion more was spent each year than the last. c. How are the different rates of change shown on the graph? There is a greater vertical change for 1990–2000 than for 1980–1990. Therefore, the section of the graph for 1990–2000 has a steeper slope.. Concept Check 1. Sample answer: Use (1, 3) as (x1, y1) and (3, 5) as (x2, y2) in the slope formula. 3. The difference in the x values is always 0, and division by 0 is undefined. 4. Carlos; Allison switched the order of the x-coordinates, resulting in an incorrect sign.. 1. Explain how you would find the slope of the line at the right.. y. b. negative slope. c. slope of 0. d. undefined slope. FIND THE ERROR If students are having difficulty with writing the coordinates in the correct order, have them complete the slope formula by filling in each ordered pair, instead of each pair of y or x values. Fill in the. x. O. 2. OPEN ENDED Draw the graph of a line having each slope. See students’ work. a. positive slope. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 5. • copy their drawings for Exercise 2 and write notes about each type of graph. • include any other item(s) that they find helpful in mastering the skills in this lesson.. (1, 3) (3, 5 ). 3. Explain why the formula for determining slope using the coordinates of two points does not apply to vertical lines.. y → 6 . x → 2. 4. FIND THE ERROR Carlos and Allison are finding the slope of the line that passes through (2, 6) and (5, 3).. first ordered pair, . Then fill in the second ordered. Carlos. Allison. 3 – 6 –3 = or –1 5–2 3. 6–3 3 = or 1 5–2 3. 63 25. pair, .. Who is correct? Explain your reasoning.. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 5–10 11, 12 13, 14. 1–4 5 6. Find the slope of the line that passes through each pair of points. 3 4 5. (1, 1), (3, 4) 6. (0, 0), (5, 4) 7. (2, 2), (1, 2) 4 2 3 5 8. (7, 4), (9, 1) 9. (3, 5), (2, 5) 0 10. (1, 3), (1, 0) 2. undefined. Find the value of r so the line that passes through each pair of points has the given slope. 11. (6, 2), (r, 6), m 4 5. 1 3. 12. (9, r), (6, 3), m 2 Lesson 5-1. Slope. 259. Teacher to Teacher Ruth Casey. Anderson County H.S., Lawrenceburg, KY. “I like to introduce the Greek letter ∆ (delta) to represent ‘change in’ with ∆y my students when studying slope. The definition of slope becomes m = , ∆x the change in the y-coordinate over the change in the x-coordinate.”. Lesson 5-1 Slope 259.

(13) Application Organization by Objective • Find Slope: 15–39, 41–48, 57 • Rate of Change: 40, 50–56 Odd/Even Assignments Exercises 15–36 and 41–48 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercise 56 involves research on the Internet or other reference materials.. Assignment Guide Basic: 15–29 odd, 37–43 odd, 50–55, 57–60, 63–85 Average: 15–49 odd, 50–55, 57–60, 63–85 (optional: 61, 62) Advanced: 16–48 even, 49, 53–76 (optional: 77–85). 13. 1.5 million subscribers per year. 60. 66 63 59. 50. 55 52. 40 0. ’90. ’92. ’94 Year. ’96. ’98. Practice and Apply Homework Help For Exercises. See Examples. 15–34 41–48 53–57. 1–4 5 6. Find the slope of the line that passes through each pair of points. 15.. y O. x. 3 4. 16.. 1 3. y. . (0, 3) (3, 2). ( 2, 1) (2, 4 ). Extra Practice. O. x. See page 831.. 17. (4, 1), (3, 3) 2 19. (2, 1), (2, 3) undefined 10 21. (5, 7), (2, 3) 73 23. (3, 4), (5, 1) 8 25. (5, 4), (5, 1) undefined 27. (2, 3), (8, 3) 0. 1 29. (8, 3), (6, 2) 2 15 ★ 31. (4.5, 1), (5.3, 2) 4 2 1 1 1 1 ★ 33. 22, 12, 2, 2 3. Karen’s Height Height (in.). 14. Without calculating, find a 2-year period that had a greater rate of change than 1990–1992. Explain your reasoning.. ★ indicates increased difficulty. 68 66 64. 18. (3, 3), (1, 3) 0 4 20. (2, 3), (9, 7) 7 2 22. (3, 6), (2, 4) 52 24. (2, 1), (5, 3) 3 26. (2, 6), (1, 3) 1 3 28. (3, 9), (7, 6) 4 1 30. (2, 0), (1, 1) 3 ★ 32. (0.75, 1), (0.75, 1) undefined 9 ★ 34. 34, 114, 12, 1 5. ARCHITECTURE Use a ruler to estimate the slope of each roof.. 62. 8 35. Sample answer: 35.. 60. 36.. 11 1 36. Sample answer: 3. 58 0. 70. 13. Find the rate of change for 1990–1992.. Sample answer: ’92–’94; steeper segment means greater rate of change.. Answer 50.. U.S. Cable TV Subscribers Number (millions). About the Exercises…. CABLE TV For Exercises 13 and 14, use the graph at the right.. 12 14 16 18 20 Age (years). s 37. Find the slope of the line that passes through the origin and (r, s). r 38. What is the slope of the line that passes through (a, b) and (a, b)? undefined. 39. PAINTING A ladder reaches a height of 16 feet on a wall. If the bottom of the ladder is placed 4 feet away from the wall, what is the slope of the ladder as a positive number? 4 260. Chapter 5 Analyzing Linear Equations. Differentiated Instruction Kinesthetic Use floor tiles as a grid or use masking tape on the floor to create a grid. Have students walk the path from one point to another on the floor, allowing only one horizontal and one vertical path. Ask them to describe their trip in terms of positive and negative movement and the number of squares traveled in each direction. Then have them write the description of their movement as the slope of the line connecting the two points. 260. Chapter 5 Analyzing Linear Equations.

(14) 40. PART-TIME JOBS In 1991, the federal minimum wage rate was $4.25 per hour. In 1997, it was increased to $5.15. Find the annual rate of change in the federal minimum wage rate from 1991 to 1997. $0.15 per year. NAME ______________________________________________ DATE. ____________ PERIOD _____. Study Guide andIntervention Intervention, 5-1 Study Guide and p. 281 Slope (shown) and p. 282. Find Slope y y x 2 x1. rise run. 2 1 m or m , where (x1, y1) and (x2, y2) are the coordinates. Slope of a Line. of any two points on a nonvertical line. 41. (6, 2), (9, r), m 1 1. 42. (4, 5), (3, r), m 8 13. 4 3. 4. 1 ★ 45. 12, 14 , r, 54 , m 4 4 ★ 47. (4, r), (r, 2), m 53 7. . . 2 ★ 48. (r, 5), (2, r), m 9. Example 2 Find the value of r so that the line through (10, r) and (3, 4) has a 2 7. slope of .. Let (3, 5) (x1, y1) and (4, 2) (x2, y2). y y. 44. (2, 7), (r, 3), m 5 3 2 1 1 1 46. , r , 1, , m ★ 3 2 2 3. 43. (5, r), (2, 3), m 1. . Example 1 Find the slope of the line that passes through (3, 5) and (4, 2).. y y. 2 1 m x x 2. 2 1 m x2 x1. Slope formula. 1. 4r 2 3 10 7 4r 2 7 7. Slope formula. 2 5 4 (3) 7 7. . y2 2, y1 5, x2 4, x1 3. 2 m 7 , y2 4, y1 r, x2 3, x1 10. Simplify.. 2(7) 7(4 r) 14 28 7r 14 7r 2r. Simplify.. 1. 7. Lesson 5-1. Find the value of r so the line that passes through each pair of points has the given slope.. Cross multiply. Distributive Property Subtract 28 from each side. Divide each side by 7.. Exercises Find the slope of the line that passes through each pair of points.. ★ 49. CRITICAL THINKING Explain how you know that the slope of the line through 49. (4, 5) is (4, 5) and (4, 5) is positive without calculating. in Quadrant III and (4, 5) is in Quadrant I. The segment HEALTH For Exercises 50–52, use the table that shows Karen’s height connecting them from age 12 to age 20. goes from lower left to upper right, which Age (years) 12 14 16 18 20 is a positive slope. Height (inches). 60. 64. 66. 67. 67. 2. (4, 1), (2, 5) 2. 1. (4, 9), (1, 6) 1. 8. (2, 5), (6, 2) . 3. 10. (6, 8), (r, 2), m 1 4. 11. (1, 3), (7, r), m 3 4. 13. (7, 5), (6, r), m 0 5. 14. (r, 4), (7, 1), m 11 4. 15. (7, 5), (r, 9), m 6 . 17. (10, 4), (2, r), m 0.5. 1 18. (r, 3), (7, r), m 5. 1.. 55. Explain the meaning of the part of the graph with a negative slope. a decline in enrollment. 2.. y. x. 4 5. 0. 4. (6, 3), (7, 4) 7. 5. (9, 3), (7, 5) 1. 6. (6, 2), (5, 4) 2. 7. (7, 4), (4, 8) 4 9. (5, 9), (3, 9) 0. 1 7. 12.5. 1 5. 11. (3, 9), (2, 8) . 13 12. (2, 5), (7, 8) 9. 13. (12, 10), (12, 5) undefined. 14. (0.2, 0.9), (0.5, 0.9) 0. 15. , , , . 73 34 . 1 2 3 3. . 1 4. Find the value of r so the line that passes through each pair of points has the given slope.. 11.3. 1 2. 1 4. 16. (2, r), (6, 7), m 3. ’70 ’75 ’80 ’85 ’90 ’95 ’00 Year. 17. (4, 3), (r, 5), m 4 9 2. 7 6. 18. (3, 4), (5, r), m 5. 19. (5, r), (1, 3), m 4. 20. (1, 4), (r, 5), m undefined 1. 21. (7, 2), (8, r), m 5 7. 1 5. 22. (r, 7), (11, 8), m 16. 56. RESEARCH Use the Internet or other reference to find the population of your city or town in 1930, 1940, . . . , 2000. For which decade was the rate of change the greatest? See students’ work.. 23. (r, 2), (5, r), m 0 2. 24. ROOFING The pitch of a roof is the number of feet the roof rises for each 12 feet horizontally. If a roof has a pitch of 8, what is its slope expressed as a positive number?. 2 3 25. SALES A daily newspaper had 12,125 subscribers when it began publication. Five years later it had 10,100 subscribers. What is the average yearly rate of change in the number of subscribers for the five-year period? 405 subscribers per year NAME ______________________________________________ DATE. 57. CONSTRUCTION The slope of a stairway determines how easy it is to climb the stairs. Suppose the vertical distance between two floors is 8 feet 9 inches. Find the total run of the ideal stairway in feet and inches. 13 ft 9 in.. ____________ PERIOD _____. Reading 5-1 Readingto to Learn Learn Mathematics Mathematics, p. 285 Slope. tread (ideal 11 in.). x. O. 10. (15, 2), (6, 5) . 13.3. (3, 3). x. (–2, –3). 8. (7, 8), (7, 5) undefined. 13. 10 0. y. (3, 1). 3. 13.5. 13.2. 12.4. 3.. y. O. 14.3. 11. ____________ PERIOD _____. (–2, 3). U.S. Public School Enrollment Grades 9–12. 12. 2. (–2, 3). O. Number (millions). 54. Find the rate of change from 1985 to 1990. 0.22. 10. (–1, 0). 14. 23 3. Find the slope of the line that passes through each pair of points.. There was no change in height.. 15. 12. (2, 8), (r, 4) m 3 6. 3. NAME ______________________________________________ DATE. 52. Explain the meaning of the horizontal section of the graph.. 16. 9. (4, 3.5), (4, 3.5) 0. Determine the value of r so the line that passes through each pair of points has the given slope.. Skills Practice, p. 283 and 5-1 Practice (Average) Practice, p. 284 (shown) Slope. 51. Use the graph to determine the two-year period when Karen grew the fastest. Explain your reasoning. 12–14; steepest part of the graph. 53. For which 5-year period was the rate of change the greatest? When was the rate of change the least? ’90–’95; ’80–’85. 6. (4, 3), (8, 3) 0. 3 4. 4 5. 7. (1, 2), (6, 2) . 2. SCHOOL For Exercises 53–55, use the graph that shows public school enrollment.. undefined. 5. (14, 8), (7, 6) . 2 16. (10, r), (3, 4), m 7. 50. Make a broken-line graph of the data. See margin.. 3. (4, 1), (4, 5). 2 7. 4 3. 4. (2, 1), (8, 9) . Pre-Activity. ELL. Why is slope important in architecture?. Read the introduction to Lesson 5-1 at the top of page 260 in your textbook. Then complete the definition of slope and fill in the boxes on the graph with the words rise and run.. riser (ideal 7 in.). rise slope run. y. run rise. In this graph, the rise is. 3. units, and the run is. 3 units 5 units. 5. units.. 3 5. x. O. Lesson 5-1. Thus, the slope of this line is or .. Reading the Lesson 1. Describe each type of slope and include a sketch. Type of Slope. www.algebra1.com/self_check_quiz. Lesson 5-1. Slope. Description of Graph. Sketch. The graph rises as you go from left to right.. positive. 261. y. The graph falls as you go from left to right.. negative. x. O. y. x. O. NAME ______________________________________________ DATE. 5-1 Enrichment Enrichment,. ____________ PERIOD _____. zero. The graph is a horizontal line.. undefined. The graph is a vertical line.. p. 286. y x. O. Treasure Hunt with Slopes. y O. Using the definition of slope, draw lines with the slopes listed below. A correct solution will trace the route to the treasure.. x. 2. Describe how each expression is related to slope. Treasure. y y x2 x1. 2 1 a. difference of y-coordinates divided by difference of. rise run. b. . corresponding x-coordinates. how far up or down as compared to how far left or right. $52,000 increase in spending 26 months. c. slope used as rate of change. Helping You Remember 3. The word rise is usually associated with going up. Sometimes going from one point on the graph does not involve a rise and a run but a fall and a run. Describe how you could select points so that it is always a rise from the first point to the second point.. Sample answer: If the slope is negative, choose the second point so that its x-coordinate is less than that of the first point.. Lesson 5-1 Slope 261.

(15) 58. WRITING IN MATH. 4 Assess. Why is slope important in architecture?. Open-Ended Assessment Modeling Use a transparency of a coordinate plane and a piece of thin spaghetti to create a “line” on the overhead projector. Ask students to determine whether the slope of that line is positive, negative, zero, or undefined and then calculate the actual slope. Repeat until all types of slope have been addressed.. Intervention If there is any doubt whether your students thoroughly understand slope, consider spending an extra day on this lesson. Use the Extra Practice on p. 831, the Study Guide and Intervention masters, or the Practice masters in the Chapter 5 Resource Masters to reinforce this concept.. Answer the question that was posed at the beginning of the lesson.. Include the following in your answer: See margin. • an explanation of how to find the slope of a roof, and • a comparison of the appearance of roofs with different slopes.. Standardized Test Practice. 59. The slope of the line passing through (5, 4) and (5, 10) is D A. positive.. negative.. B. zero.. C. D. undefined.. D. ac . bd. 60. The slope of the line passing through (a, b) and (c, d) is B A. Extending the Lesson. New. dc . ba. bd . ac. B. db . ac. C. 61. Choose four different pairs of points from those labeled on the graph. Find the slope of the line using the coordinates of each pair of points. Describe your findings. 1 ; The slope is the same 3. y. (4, 0). x. O (2, 2). regardless of points chosen.. (1, 1). (5, 3). 62. MAKE A CONJECTURE Determine whether Q(2, 3), R(1, 1), and S(4, 2) lie on the same line. Explain your reasoning. See margin.. Maintain Your Skills Mixed Review. Write an equation for each relation. 63.. x. 1. 2. 3. 4. 5. f(x). 5. 10. 15. 20. 25. (Lesson 4-6). 64.. f(x) 5x. x. 2. 1. 1. 2. 4. f(x). 13. 12. 10. 9. 7. f(x) 11 x. Determine whether each relation is a function. (Lesson 4-5). Getting Ready for Lesson 5-2 PREREQUISITE SKILL Lesson 5-2 presents direct variation in which students must find quotients of numbers to determine the constant of variation. Exercises 77–85 should be used to determine your students’ familiarity with finding quotients involving fractions.. Answers 58. Sample answer: Analysis of the slope of a roof might help to determine the materials of which it should be made and its functionality. Answers should include the following. • To find the slope of the roof, find a vertical line that passes through the peak of the roof and a horizontal line that passes through the eave. Find the distances from the intersection of those two lines to the peak and to the eave. Use those measures as the rise and run to calculate the slope. 262. Chapter 5 Analyzing Linear Equations. 65. y 15 yes. 66. x 5 no. 67. {(1, 0), (1, 4), (1, 1)} no. 68. {(6, 3), (5, 2), (2, 3)} yes. 69. Graph x y 0. (Lesson 4-4) See margin. 70. What number is 40% of 37.5?. (Lesson 3-4). 15. Find each product. (Lesson 2-4) 71. 7(3) 21 74. (8)(3.7) 29.6. Getting Ready for the Next Lesson. 72. (4)(2) 8 75.. . . 7 1 8 3. 7 24. 73. (9)(4) 36 3 1 1 76. (14) 1 4 2 4. PREREQUISITE SKILL Find each quotient. (To review dividing fractions, see pages 800 and 801.). 2 3 1 1 1 80. 1 2 3 2 4 7 83. 18 20 8 7. 77. 6 9. 1 4 3 1 81. 4 6 3 2 84. 8 5. 78. 12 48 1 2 15 16. 4. 2 3 79. 10 26 8 3 1 3 82. 6 4 8 2 2 1 85. 2 10 3 4 3. 262 Chapter 5 Analyzing Linear Equations. • A roof that is steeper than one with a rise of 6 and a run of 12 would be one with a rise greater than 6 and the same run. A roof with a steeper slope appears taller than one with a less steep slope.. 62. No, they do not. Slope of 4 is and slope of RS QR 3 1 is . If they lie on the 3. same line, the slopes should be the same.. 69.. y x y 0. O. x.

(16) Reading Mathematics. Mathematical Words and Everyday Words. Getting Started. You may have noticed that many words used in mathematics are also used in everyday language. You can use the everyday meaning of these words to better understand their mathematical meaning. The table shows two mathematical words along with their everyday and mathematical meanings.. Word. Everyday Meaning. Mathematical Meaning. expression. 1. something that expresses or communicates in words, art, music, or movement 2. the manner in which one expresses oneself, especially in speaking, depicting, or performing. one or more numbers or variables along with one or more arithmetic operations. function. 1. the action for which one is particularly fitted or employed 2. an official ceremony or a formal social occasion 3. something closely related to another thing and dependent on it for its existence, value, or significance. a relationship in which the output depends upon the input. Source: The American Heritage Dictionary of the English Language. Notice that the mathematical meaning is more specific, but related to the everyday meaning. For example, the mathematical meaning of expression is closely related to the first everyday definition. In mathematics, an expression communicates using symbols.. Reading to Learn. 1. Sample answer: The mathematical meaning of function meaning? is most closely 2. RESEARCH Use the Internet or other reference to find the everyday meaning of related to the third each word below. How might these words apply to mathematics? Make a table definition in the like the one above and note the mathematical meanings that you learn as you everyday meanings. study Chapter 5. a–c. See pp. 315A–315B for sample answers. 1. How does the mathematical meaning of function compare to the everyday. a. slope. b. intercept. c. parallel. Reading Mathematics. Before using this page, ask students if there are any words they know that have more than one meaning, depending on how they are used. Some examples might be: bolt: a fastener; a roll of cloth measured to a specified length bow: a decorative knot formed by a ribbon or piece of cloth; a weapon made of curved material and a cord row: a line of seats or objects; using a paddle to move a boat through water. Teach Word Association Explain to students that if they can relate a word they are trying to learn to something with which they are already familiar, it makes it easier to remember what that word means. This is a technique taught to business people to improve their recollection of names and business contacts. By relating mathematical terms to everyday things, they can recall their meanings more readily.. Assess Study Notebook. Investigating Slope-Intercept Form 263 Mathematical Words and Everyday Words 263. Ask students to summarize what they have learned about mathematical words and everyday words.. ELL English Language Learners may benefit from writing key concepts from this activity in their Study Notebooks in their native language and then in English. Reading Mathematics Mathematical Words and Everyday Words 263.

(17) Lesson Notes. Slope and Direct Variation • Write and graph direct variation equations.. 1 Focus. • Solve problems involving direct variation.. Vocabulary • • • •. direct variation constant of variation family of graphs parent graph. is slope related to your shower? A standard showerhead uses about 6 gallons of water per minute. If you graph the ordered pairs from the table, the slope of the line is 6.. Mathematical Background notes are available for this lesson on p. 254C. is slope related to your shower? Ask students: • What is the value of y if x 2.5? 15 • What is the value of x if y 30? 5 • Plumbing What could you do to change the value of the slope in the water use equation? Turn the faucets to increase or decrease the water flow.. y (gallons). 0. 0. 1. 6. 25. 2. 12. 20. 3. 18. 4. 24. Gallons of Water Used in a Shower y. 1. 0. 2 3 Minutes. x. Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points.. In part b, remind students that since the line slopes downward from left to right, the slope is negative.. In Chapter 3, students studied proportions. Direct variation is also known as direct proportion.. y. b.. (1, 3). x. O. x (1, 2). The constant of variation is 2. y y x2 x1. Slope formula. 2 1 m. m 10. (x1, y1) = (0, 0) (x2, y2) = (1, 3). 2 0 (x1, y1) = (0, 0) m . m3. The slope is 3.. m 2. 30. x2 y2 x1 y1 . Both of these last two. (0, 0) O. y 3x. y2 y1 m x2 x1. y2. y. y 2x. (0, 0). The constant of variation is 3.. If y kx, then x1 x2 and. 10. Slope formula. (x2, y2) = (1, 2). The slope is 2.. Compare the constant of variation with the slope of the graph for each example. Notice that the slope of the graph of y kx is k.. equations are known as direct proportions. 264. Chapter 5 Analyzing Linear Equations. Resource Manager Workbook and Reproducible Masters Chapter 5 Resource Masters • Study Guide and Intervention, pp. 287–288 • Skills Practice, p. 289 • Practice, p. 290 • Reading to Learn Mathematics, p. 291 • Enrichment, p. 292 • Assessment, p. 337. 4. Example 1 Slope and Constant of Variation. TEACHING TIP. Building on Prior Knowledge. 10. DIRECT VARIATION A direct variation is described by an equation of the form y kx, where k 0. We say that y varies directly with x or y varies directly as x. In the equation y kx, k is the constant of variation.. a.. DIRECT VARIATION. 15 5. The equation is y 6x. The number of gallons of water y depends directly on the amount of time in the shower x.. 2 Teach. y1. x (minutes). Gallons. 5-Minute Check Transparency 5-2 Use as a quiz or review of Lesson 5-1.. Parent and Student Study Guide Workbook, p. 39 Prerequisite Skills Workbook, pp. 29–30 Science and Mathematics Lab Manual, pp. 41–44. Transparencies 5-Minute Check Transparency 5-2 Answer Key Transparencies. Technology Interactive Chalkboard.

(18) The ordered pair (0, 0) is a solution of y kx. Therefore, the graph of y kx passes through the origin. You can use this information to graph direct variation equations.. In-Class Examples. Example 2 Direct Variation with k 0. 1 Name the constant of variation for each equation. Then find the slope of the line that passes through each pair of points.. Graph y 4x.. TEACHING TIP If students have difficulty using the slope to graph, remind them that they can also make a table of values.. y. Step 1 Write the slope as a ratio. 4 1. 4 . rise run. a.. Step 2. Graph (0, 0).. Step 3. From the point (0, 0), move up 4 units and right 1 unit. Draw a dot.. Power Point®. y. y 4x O. x. (1, 2) (0, 0). Step 4 Draw a line containing the points.. x. O. y 2x. Example 3 Direct Variation with k 0 1 3. Graph y x. Step 1. 1 3. 1 3. b.. y. Write the slope as a ratio. . Step 2. constant of variation: 2; slope: 2 rise run. y. y 13 x. Graph (0, 0).. Step 3. From the point (0, 0), move down 1 unit and right 3 units. Draw a dot.. Step 4. Draw a line containing the points.. (0, 0). x. O. x. O. y –4x (1, –4). A family of graphs includes graphs and equations of graphs that have at least one characteristic in common. The parent graph is the simplest graph in a family.. constant of variation: 4; slope: 4. Teaching Tip. Point out in Example 2 that an integer such as 4 can be written as a ratio with a denominator of 1.. Family of Graphs. 1. All the graphs pass through the origin. 2. None of the graphs have the same slope. 3. Sample answer: y 5x; See students’ graphs. 4. Sample answer: 3 2. y x 5. This family of graphs has a y-intercept of 0. Their slopes are all different.. The calculator screen shows the graphs of y x, y 2x, and y 4x.. 2 Graph y x.. Think and Discuss. yx. y 2x. y. 1. Describe any similarities among the graphs. y 4x 2. Describe any differences among the graphs. 3. Write an equation whose graph has a [10, 10] scl: 1 by [10, 10] scl: 1 steeper slope than y 4x. Check your answer by graphing y 4x and your equation. 4. Write an equation whose graph lies between the graphs of y x and y 2x. Check your answer by graphing the equations. 5. Write a description of this family of graphs. What characteristics do the graphs have in common? How are they different? 6. The equations whose graphs are in this family are all of the form y mx. How does the graph change as the absolute value of m increases?. yx. Teaching Tip. As |m| increases, the graph becomes more steep.. www.algebra1.com/extra_examples. Lesson 5-2 Slope and Direct Variation. x. O. 265. Point out that in Example 3 students can either use rise 1 and run 3, or rise 1 and run 3.. 3 Graph y 32 x. y. Family of Graphs Graphing calculators are ideal for studying families of graphs. The Y= screen allows students to enter many functions so they can experiment while investigating Questions 3 and 4.. O. x y 32 x. Lesson 5-2 Direct Variation 265.

(19) In-Class Example. Direct Variation Graphs. Power Point®. • Direct variation equations are of the form y kx, where k 0. • The graph of y kx always passes through the origin.. Teaching Tip. Be sure students do not interchange the values of x and y when substituting values into an equation.. • The slope can be positive. k. 0. • The slope can be negative. k. y. y. y kx. O. 4 Suppose y varies directly as. x. O. 0. y kx x. x, and y 9 when x 3.. a. Write a direct variation equation that relates x and y. y 3x. If you know that y varies directly as x, you can write a direct variation equation that relates the two quantities.. Example 4 Write and Solve a Direct Variation Equation. b. Use the direct variation equation to find x when y 15. 5. Suppose y varies directly as x, and y 28 when x 7. a. Write a direct variation equation that relates x and y. Find the value of k. y kx Direct variation formula 28 k(7) Replace y with 28 and x with 7.. SOLVE PROBLEMS. In-Class Example. 28 k(7) 7 7. Power Point®. 4k Simplify. Therefore, y 4x.. 5 TRAVEL The Ramirez family. b. Use the direct variation equation to find x when y 52.. is driving cross-country on vacation. They drive 330 miles in 5.5 hours.. y 4x 52 4x 52 4x 4 4. a. Write a direct variation equation to find the distance driven for any number of hours. d 60t. Direct variation equation Replace y with 52. Divide each side by 4.. 13 x Simplify. Therefore, x 13 when y 52.. SOLVE PROBLEMS One of the most common uses of direct variation is the. b. Graph the equation. Travel Time. More About . . .. d. formula for distance, d rt. In the formula, distance d varies directly as time t, and the rate r is the constant of variation.. Example 5 Direct Variation Equation. 500 400. BIOLOGY A flock of snow geese migrated 375 miles in 7.5 hours.. 300. a. Write a direct variation equation for the distance flown in any time.. 200. Words. 100. Variables Let r rate. t. c. Estimate how many hours it would take to drive 600 miles. 10 h. Distance. Biology Snow geese migrate more than 3000 miles from their winter home in the southwest United States to their summer home in the Canadian arctic. Source: Audubon Society. 266. Equation. equals. rate. times. time.. . 6. . 2 3 4 5 Time (hours). . 1. . 0. The distance traveled is 375 miles, and the time is 7.5 hours.. . Distance (miles). Divide each side by 7.. 375 mi. . r. . 7.5 h. Solve for the rate. 375 r(7.5) Original equation 375 r(7.5) 7.5 7.5. Divide each side by 7.5.. 50 r Simplify. Therefore, the direct variation equation is d 50t.. Chapter 5 Analyzing Linear Equations. Differentiated Instruction Interpersonal Have small groups of students use a triple-beam balance and 4 stacks of identical washers. Each stack should contain a different number of washers tied together so students cannot weigh just one washer. Record the number of washers n in each stack. Have students weigh one stack and then predict the weights W of the other stacks. How do they think this relates to the equation W kn? See students’ work. What does k represent? the weight of each washer 266. Chapter 5 Analyzing Linear Equations.

(20) b. Graph the equation.. (7.5, 375). 400. rise run. c. Estimate how many hours of flying time it would take the geese to migrate 3000 miles. d 50t 3000 50t. d Distance (miles). 50 1. m . 3 Practice/Apply. Migration of Snow Geese. The graph of d 50t passes through the origin with slope 50.. 300. d 50t. 200. Study Notebook. 100. Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 5. • include the explanation from Exercise 3 to show how direct variation and slope are related. • include any other item(s) that they find helpful in mastering the skills in this lesson.. 0 1. Original equation. 2. 3 4 5 6 Time (hours). 7. 8. t. Replace d with 3000.. 3000 50t 50 50. Divide each side by 50.. t 60 Simplify. At this rate, it will take 60 hours of flying time to migrate 3000 miles.. Concept Check 2. b, constant of variation 4; c, constant 1 of variation 3. Guided Practice GUIDED PRACTICE KEY Exercises. 1. OPEN ENDED Write a general equation for y varies directly as x. y kx 2. Choose the equations that represent direct variations. Then find the constant of variation for each direct variation. a. 15 rs. 9 t. d. s . c. z x. 3. Explain how the constant of variation and the slope are related in a direct variation equation. They are equal.. About the Exercises…. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 4.. 1; 1. y. 3. Examples. 4–8 9–11 12–14. 1 3. b. 4a b. 5.. 3. (3, 1). 1 –3 4 5. Organization by Objective • Direct Variation: 15–42, 47 • Solve Problems: 43–46, 48–55. 1; 1. y. (2, 2). (0, 0). O. x. O. (0, 0). y 13 x. Odd/Even Assignments Exercises 15–42 are structured so that students practice the same concepts whether they are assigned odd or even problems. Alert! Exercises 59–62 require the use of a graphing calculator.. x. yx. Graph each equation. 6–8. See margin. 6. y 2x. 10 9. 10. y x; 8.1. Application. 1 2. 7. y 3x. 8. y x. Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve. 9 9. If y 27 when x 6, find x when y 45. y x; 10 2 10. If y 10 when x 9, find x when y 9. 1 11. If y 7 when x 14, find y when x 20. y x; 10 2. Assignment Guide Basic: 15–51 odd, 52, 53, 56–58, 63–78 Average: 15–51 odd, 54–58, 63–78 (optional: 59–62) Advanced: 16–46 even, 47, 48, 50, 54–72 (optional: 73–78) All: Practice Quiz 1 (1–10). JOBS For Exercises 12–14, use the following information. Suppose you work at a job where your pay varies directly as the number of hours you work. Your pay for 7.5 hours is $45. 12. Write a direct variation equation relating your pay to the hours worked. y 6x 13. Graph the equation. See margin. 14. Find your pay if you work 30 hours. $180 Lesson 5-2 Slope and Direct Variation. 267. Answers 6.. 7.. y. 8.. y. y. y 12 x. y 3 x O. 13.. y. y 6x. y 2x. O. x. O. x. x O. x. Lesson 5-2 Direct Variation 267.

(21) NAME ______________________________________________ DATE. ____________ PERIOD _____. Study Guide andIntervention Intervention, 5-2 Study Guide and. Practice and Apply. p. 287 (shown) and p. 288 Slope and Direct Variation. Homework Help. Direct Variation A direct variation is described by an equation of the form y kx, where k 0. We say that y varies directly as x. In the equation y kx, k is the constant of variation. Example 1. Example 2. Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points.. Suppose y varies directly as x, and y 30 when x 5.. y 21 x. (2, 1). O (0, 0). x. 1. 1. For y x, the constant of variation is . 2 2 y y. 2 1 m x2 x1. b. Use the direct variation equation to find x when y 18. y 6x Direct variation equation 18 6x Replace y with 18. 3x Divide each side by 6. Therefore, x 3 when y 18.. Slope formula. 10 20. (x1, y1) (0, 0), (x2, y2) (2, 1). 1. 2. Simplify.. Lesson 5-2. a. Write a direct variation equation that relates x and y. Find the value of k. y kx Direct variation equation 30 k(5) Replace y with 30 and x with 5. 6k Divide each side by 5. Therefore, the equation is y 6x.. y. For Exercises. See Examples. 15–32 33–42 43–46, 52–55. 1–3 4. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 16.. O (0, 0) x. See page 831.. 4; 4. 2; 2 3.. y. 18.. (0, 0). y –2x. (0, 0). 2; 2. O. 3 3 ; 2 2. Write a direct variation equation that relates x to y. Assume that y varies directly as x. Then solve. 5. If y 9 when x 3, find x when y 6. y 3x; 2. 3 32. 3. 7. If y when x , find x when y . y 2x; 4 8 16 NAME ______________________________________________ DATE. p. 289 and Practice, p. 290 (shown) Slope and Direct Variation y. y 43 x. 2.. y (3, 4). (–2, 5). (0, 0). x. O. y 25 x. y 34 x. (0, 0). (0, 0). x. O. x. O. 5 5 ; 2 2. y. 3.. (4, 3). y 32 x x. (0, 0). y 14 x. 1 ; 4 1 4. Graph each equation. 21–32. See pp. 315A–315B.. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.. 4 4 ; 3 3. (4, 1) x. ____________ PERIOD _____. Skills Practice, 5-2 Practice (Average) 3 3 ; 4 4. x (2, 2). 3 3 ; 2 2. 1; 1. 6. If y 4.8 when x 1.6, find x when y 24. y 3x; 8 1. (0, 0). O. 4. If y 4 when x 2, find y when x 16. y 2x; 32. 1. y. O. (2, 3). (0, 0). x. 3; 3. 20.. y. x. O (–2, –3). O. ( 2, 1). y x. y 23 x. (0, 0). (1, 3). x. O. 19.. y. y. y 3x. (–1, 2). 1 ; 2 1 2. x. (0, 0). x. O. y 4x. O. Exercises. 1.. (1, 4). Extra Practice. Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points. 2.. y. y 12 x. (0, 0). (2, 4). y 2x. 1. y. 17.. y. 5. The slope is . 2. 1.. y. 15.. 21. y x. 22. y 3x. 23. y x. 24. y 4x. 1 25. y x 4 1 29. y x 5. 3 26. y x 5 2 30. y x 3. 5 27. y x 2 4 31. y x 3. 28. y x. 7 5. 9 2. 32. y x. Graph each equation. 4. y 2x. 6 5. 5. y x. y. O. 5 3. 6. y x. y. y. Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve.. x O. O. x. x. 33. If y 8 when x 4, find y when x 5. y 2x; 10. Write a direct variation equation that relates x and y. Assume that y varies directly as x. Then solve.. 34. If y 36 when x 6, find x when y 42. y 6x; 7. 7. If y 7.5 when x 0.5, find y when x 0.3. y 15x; 4.5 8. If y 80 when x 32, find x when y 100. y 2.5x; 40. 1 32. 3 4. 35. If y 16 when x 4, find x when y 20. y 4x; 5. 3 8. 9. If y when x 24, find y when x 12. y x; Write a direct variation equation that relates the variables. Then graph the equation. 10. MEASURE The width W of a rectangle is two thirds of the length .. 2 W . C 4.50t. Rectangle Dimensions. Cost of Tickets. W. C. 10. 25. 8. 20. Cost ($). Width. 3. 36. If y 18 when x 6, find x when y 6. y 3x; 2 1 37. If y 4 when x 12, find y when x 24. y x; 8 3 4 38. If y 12 when x 15, find x when y 21. y x; 26.25 5 39. If y 2.5 when x 0.5, find y when x 20. y 5x; 100 2 40. If y 6.6 when x 9.9, find y when x 6.6. y x; 4.4 3 32 2 1 1 41. If y 2 when x , find y when x 1. y x; 12 3 4 8 3 4 2 42. If y 6 when x , find x when y 12. y 9x; 3 3. 11. TICKETS The total cost C of tickets is $4.50 times the number of tickets t.. 6 4. 15 10. 2. 5. 0. 2. 4. 6 8 10 12 Length. 0. 1. 2. 3 4 5 Tickets. 6. t. 12. PRODUCE The cost of bananas varies directly with their weight. Miguel bought 1 2. 3 pounds of bananas for $1.12. Write an equation that relates the cost of the bananas 1 4. to their weight. Then find the cost of 4 pounds of bananas. C 0.32p; $1.36 NAME ______________________________________________ DATE. ____________ PERIOD _____. Reading 5-2 Readingto to Learn Learn Mathematics Mathematics, p. 291 Slope and Direct Variation. Pre-Activity. ELL. How is slope related to your shower? Read the introduction to Lesson 5-2 at the top of page 268 in your textbook. • How do the numbers in the table relate to the graph shown?. They are the coordinates of the points on the graph.. • Think about the first sentence. What does it mean to say that a standard showerhead uses about 6 gallons of water per minute?. Sample answer: For each minute the shower runs, 6 gallons of water come out. So, if the shower ran 10 minutes, that would be 60 gallons.. 43–46. See margin for graphs.. Write a direct variation equation that relates the variables. Then graph the equation. 43. GEOMETRY The circumference C of a circle is about 3.14 times the diameter d. 44. GEOMETRY The perimeter P of a square is 4 times the length of a side s. P 4s. 43. C 3.14d 45. C 0.99n 46. C 14.49p. 45. SEWING The total cost is C for n yards of ribbon priced at $0.99 per yard. 46. RETAIL Kona coffee beans are $14.49 per pound. The total cost of p pounds is C.. Reading the Lesson 1. What is the form of a direct variation equation? y kx. 268. Chapter 5 Analyzing Linear Equations. 2. How is the constant of variation related to slope? The constant of variation has. the same value as the slope of the graph of the equation. 3. The expression “y varies directly as x” can be written as the equation y kx. How would you write an equation for “w varies directly as the square of t”? w kt 2 4. For each situation, write an equation with the proper constant of variation.. NAME ______________________________________________ DATE. 5-2 Enrichment Enrichment,. ____________ PERIOD _____. p. 292. a. The distance d varies directly as time t, and a cheetah can travel 88 feet in 1 second.. d 88t. b. The perimeter p of a pentagon with all sides of equal length varies directly as the length s of a side of the pentagon. A pentagon has 5 sides. p 5s c. The wages W earned by an employee vary directly with the number of hours h that are worked. Enrique earned $172.50 for 23 hours of work. W $7.50h. Helping You Remember 5. Look up the word constant in a dictionary. How does this definition relate to the term constant of variation? Sample answer: Something unchanging; the constant. of variation relates x and y in the same value every time, and that relationship never changes.. nth Power Variation An equation of the form y kxn, where k 0, describes an nth power variation. The variable n can be replaced by 2 to indicate the second power of x (the square of x) or by 3 to indicate the third power of x (the cube of x). Assume that the weight of a person of average build varies directly as the cube of that person’s height. The equation of variation has the form w kh3. The weight that a person’s legs will support is proportional to the cross-sectional area of the leg bones. This area varies directly as the square of the person’s height. The equation of variation has the form s kh2. Answer each question. 1. For a person 6 feet tall who weighs 200 pounds, find a value for k in the equation w kh3.. k 0.93 2. Use your answer from Exercise 1 to predict the weight of a person who is 5 feet tall. about 116 pounds. 268. Chapter 5 Analyzing Linear Equations.

(22) 47. It also doubles. y If k, and x is. 47. CRITICAL THINKING Suppose y varies directly as x. If the value of x is doubled, what happens to the value of y? Explain.. multiplied by 2, y must also be multiplied by 2 to maintain the value of k.. BIOLOGY Which line in the graph represents the sprinting speeds of each animal?. Answers. x. 43. C Sprinting Speeds. Distance (miles). 48. elephant, 25 mph 4 49. reindeer, 32 mph 2 50. lion, 50 mph 1 51. grizzly bear, 30 mph 3. 80. 2. 1. C 3.14 d. 60 40. 0. 4. 3. 20. 1 2 Time (hours). 0. d. 44. P. SPACE For Exercises 52 and 53, use the following information. The weight of an object on the moon varies directly with its weight on Earth. With all of his equipment, astronaut Neil Armstrong weighed 360 pounds on Earth, but weighed only 60 pounds on the moon. 52. m 1e 6. 52. Write an equation that relates weight on the moon m with weight on Earth e.. P 4s. 53. Suppose you weigh 138 pounds on Earth. What would you weigh on the moon?. 23 lb. Veterinary Medicine Veterinarians compare the age of an animal to the age of a human on the basis of bone and tooth growth.. Online Research For information about a career as a veterinarian, visit: www.algebra1.com/ careers. Standardized Test Practice 60. They all pass through (0, 0) and have negative slope, but each has a different slope.. Graphing Calculator. ANIMALS For Exercises 54 and 55, use the following information. Most animals age more rapidly than humans do. The chart shows equivalent ages for horses and humans.. Horse age (x). 0. 1. 2. 3. 4. 5. Human age (y). 0. 3. 6. 9. 12. 15. s. 0. 45. C. 54. Write an equation that relates human age to horse age. y 3x 55. Find the equivalent horse age for a human who is 16 years old. 5 yr 4 mo 56. WRITING IN MATH. C 0.99 n. Answer the question that was posed at the beginning of the lesson. See margin.. How is slope related to your shower? Include the following in your answer: • an equation that relates the number of gallons y to the time spent in the shower x for a low-flow showerhead that uses only 2.5 gallons of water per minute, and • a comparison of the steepness of the graph of this equation to the graph at the top of page 268.. n. 0. 46.. C 40. 57. Which equation best describes the graph at the right? D A. y 2x. C. y x. 1 2. B. y 2x. D. y x. y. 30 20. 1 2. x. O. 10. 58. Which equation does not model a direct variation? C A. y 4x. B. y 22x. C. y 3x 1. D. y x. p 2. 1 2. FAMILIES OF GRAPHS For Exercises 59–62, use the graphs of y 1x, y 2x, and y 4x which form a family of graphs. 59. Graph y 1x, y 2x, and y 4x on the same screen. See margin. 60. How are these graphs similar to the graphs in the Graphing Calculator Investigation on page 265? How are they different?. www.algebra1.com/self_check_quiz. Lesson 5-2 Slope and Direct Variation. 269. 4. 6. 8. 56. The slope of the equation that relates time and water use is the number of gallons used per minute in the shower. Answers should include the following. • y 2.5x • Less steep; the slope is less than the slope of the graph on page 268. 59.. Lesson 5-2 Direct Variation 269.

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