vertical change horizontal change 5 rise run The slope of a line is the same between any two points on the line. Using Rise and Run to Find Slope
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1 - Slope and -intercept - What You ll Learn To find the of a line To use -intercept form in graphing a linear equation... And Wh To solve real-world problems involving the incline of a ramp or the slant of a roof Find each difference (-) (-) GO for Help Lesson -6 New Vocabular -intercept -intercept form California Content Standards AF. Represent quantitative relationships graphicall. Introduce, Develop AF. Graph linear functions, noting that the vertical change per unit of horizontal change is alwas the same and know that the ratio is called the of a graph. Develop AF. Plot the values of quantities whose ratios are alwas the same. Fit a line to the plot and understand that the of the line equals the ratio of the quantities. Develop Finding the Slope of a Line Understanding Slope a. See back of book.. a. Graph =, =, and = on one coordinate plane. b. How does the graph of = k change as k, the coefficient of, increases? It becomes steeper. a. See back of book.. a. Graph = and =-on the same coordinate plane. b. How are the graphs of = and =-alike? Different? Answers ma var. Sample: The both are lines through the origin. From left to right, one rises while the other falls. The ratio that describes the tilt of a line is its. If a line slants upward from left to right, it has positive. If it slants downward, it has negative. To calculate, ou use this ratio. vertical change horizontal change rise run The of a line is the same between an two points on the line.. Plan California Content Standards AF. Represent quantitative relationships graphicall and interpret the meaning of a specific part of a graph in the situation represented b the graph. AF. Graph linear functions, noting that the vertical change (change in -value) per unit of horizontal change (change in -value) is alwas the same and know that the ratio ( rise over run ) is called the of a graph. AF. Plot the values of quantities whose ratios are alwas the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the of the line equals the ratio of the quantities. Using Rise and Run to Find Slope Find the of each line. a. b. run rise (, ) ( 6, ) 6 O rise run. What is the of the ski trail at the left? run (, ) rise (, ) O rise run California Math Background A formula for the of a nonvertical line is m =, where m is the and (, ) and (, ) are two points on the line. You ma choose an two points on the line to calculate the. More Math Background: p. C Lesson Planning and Resources See p. E for a list of the resources that support this lesson. Special Needs L Ask students to work with a partner to draw lines based on these clues: A line with a that is ; a line with a negative ; a line with a positive ; and a line with a of. Remind them that is m in the equation = m + b. - Slope and -intercept 99 Below Level L Have students measure the steepness, or, of various stairs and ramps around the school. For stairs, ou ma want to advise them on how to measure rise and run. CD, Online, or Transparencies Bell Ringer Practice Check Skills You'll Need Use student page, transparenc, or PowerPoint. California Standards Dail Review Use transparenc 6. 99
2 . Teach Activit Lab Activit Lab in Teaching Resources for use with the lesson. Guided Activit Proportional Linear Relationships For CA Standards Investigation Eercise (a), have students compare and contrast the linear equations =, =, and = for and proportionalit of - and - coordinates. Answers ma var. Sample: Each equation models a proportional linear equation because its graph is a line that goes through (, ). The ratio of - and -coordinates form proportions. The s of the lines represented b these equations are not the same. Teaching Tip To avoid confusion, suggest students alwas read the slant of lines from left to right, in the same wa the read tet from left to right. Remind students that if the alwas go from left to right, up is positive and down is negative. Alternative Method Invite students to develop visual or mnemonic devices to help distinguish between a of zero and an. One suggestion might be that a bicclist might have zero difficult riding on a flat road, but riding up a vertical wall would be impossible (). Error Prevention It is natural to write the difference of the -coordinates in the numerator since the -coordinates are written first in ordered pairs. Students might remember to write the difference of the -coordinates in the numerator b noting the rhming of rise and s, or if the circle the -coordinates. Vocabular Tip You ma sa that a vertical line has no. But be sure that ou do not confuse no with. nline Visit: PHSchool.com Web Code: awe-77 Chapter Linear Functions and Graphing If ou know two points of a line, ou can find the of the line using the following formula. difference in coordinates = difference in coordinates The -coordinate ou use first in the numerator must correspond to the -coordinate ou use first in the denominator. Horizontal and vertical lines are special cases for. () Slope is for a horizontal line. Division b zero is. Slope is for a vertical line. Find the of the line through C(, 6) and D(, ).. Find the of the line through V(, -), Q(, -7). The ratios between the quantities in some real-world relations are equal. The of the line representing the relation equals the ratio. (, ) difference in coordinates difference in coordinates () 6 6 Cost (cents) (, ) O 6 (, ) (, ) () Using Coordinates to Find Slope Interpreting Slope in a Real-World Situation O A store sells sugar in bulk for cents per pound. Graph the relation (pounds of sugar, cost). Compare the of the line through the points on our graph to the ratio cost. pounds of sugar Use two points to find the. 7 = 7 cost pounds of sugar 6 The equals the ratio. Pounds of Sugar. Find the of the line determined b the relation (inches, feet). Advanced Learners L Challenge students to find the and -intercept of the equation + = 6. ; English Learners EL Point out that the rise goes up as ou move to the right when the is positive, and down when the is negative. In Eample, have them trace the rise with a finger, and then trace the run, alwas flat and horizontal.
3 Using Slope to Graph Linear Equations Here is the graph of =. The of the line is, or. (, ) The -intercept of the line is the rise (, ) point where the line crosses the -ais. run The constant in the equation is the O -intercept. Additional Eamples. (, ) (, ) O -intercept You can use -intercept form to help ou graph an equation. Slope-Intercept Form The equation = m + b is the -intercept form. In this form, m is the of the line, and b is the -intercept. Graphing a Linear Equation A ramp s from a warehouse door down to a street. The equation models the ramp, where is the horizontal distance in feet from the bottom of the door and is the height in feet above the street. Graph the equation. Step Since the -intercept is, graph (, ). Step Since the is or, (, ) move unit down from (, ). (, ) Then move units right to graph a second point. Step Draw a line through the points. O. Graph each equation. a b. See right. a. = - b. =- + Closure Vocabular Tip a. The word intercept sounds like intersect, which means to cross. Think of the - intercept as where the line crosses the -ais. Have students draw the graph of a line with positive. Check students graphs. Have students select two points on their line and find the. Repeat for negative, zero, and s. O O = For: Writing Equations Activit Use: Interactive Tetbook, - b. = - Slope and -intercept Additional Eamples b. Find the of the line through each pair of points. a. E(7, ), F(-, ) 9 b. A(-, -), B(, -) c. R(-, -), S(-, ) A store sells almonds in bulk for $6 per pound. Graph the relation (pounds of almonds, cost). Find the of the line through the points on our graph. Compare the to cost the ratio pounds of almonds. Cost (Dollars) CD, Online, or Transparencies Find the of each line. a. (, ) O (, ) (, ) O (, 6) 6 6 Pounds of Almonds The is 6, which is the same as the ratio cost pounds of almonds. A ramp s from a warehouse door down to a street. The function =- + models the ramp, where is the horizontal distance in feet from the bottom of the door and is the height in feet above the street. Graph the equation. See left. Resources Dail Notetaking Guide - L Dail Notetaking Guide - Adapted Version L
4 . Practice Assignment Guide Check Your Understanding Go over Eercises in class before assigning the Standards Practice. California Standards Practice A Practice b Eample B Appl Your Skills C Challenge Multiple Choice Practice 6 Mied Review 7 Homework Quick Check To check students understanding of ke skills and concepts, go over Eercises 6,,,, and. Adapted Practice Practice - Find the of the line through each pair of points.. A(, ), B(6, ). J(, 6), K(, ). P(, 7), Q(, 7). M(7, ), N(, ) Complete the table. Equation Equation in Slope -intercept Slope-Intercept Form Find the of each line. 7.. O O L Slope and -intercept L C Check Your Understanding B A O 6. Yes, the between an two points is constant. A GO Practice b Eample for Help Eample (page 99) Eample (page ) Use the graph at the left for Eercises.. Cop and complete the solution below to find the of the graph using points A and B. j = j 6 j = j. Use Points B and C to find the of the graph.. Based on our answers to Eercises and, does the graph represent a linear equation? Eplain. See left. Standards Practice AF., AF., AF. For more eercises, see Etra Skills and Word Problem Practice. Find the of each line.. (, ). (, ) (, ) O Find the of the line through each pair of points A(, 6), B(, ) 7. E(, -), F(, -). N(-, ), Q(, -) 9. G(, ), H(6, ) (, ) O 6 (, ) (7, ) Graph each equation. 9.. Eample (page ) Find the of the line determined b each relation.. (time in hours, distance traveled at mi/h) O O. (number of pens, cost at $.6/pen).6 - Guided Problem Solving GPS GPS Eercise Find the of the line. (, ) (, ) (, ) O (, ) L Eample (page ). The student could have calculated difference in coordinates. difference in coordinates Identif the and -intercept of the graph of each equation. Then graph the equation.. See back of book.. = 7 +. =-. = -. = + 6. = = -. = 9. =- +. = + 6 Understand the Problem. What are ou asked to find?. What four points are given on the line? Make and Carr Out a Plan. What is the formula for finding of a line using the coordinates of two points on the line? B Appl Your Skills. Error Analsis A student said that the of the line through (, ) and (, ) is. What error could this student have made? See left.. Choose two points on the line. Write a ratio to show the difference in the -coordinates over the difference in the -coordinates.. Subtract the -coordinates and the -coordinates. Chapter Linear Functions and Graphing 6. Simplif. What is the of the line? Check the Answer 7. To check our answer, find the of the line b using the other two points labeled on the line. Solve Another Problem. Find the of the line. O
5 Visit: PHSchool.com Web Code: awe-. The upper roof has the steeper pitch because it has the greater. 7 ft ft 9 ft ft b. The lines are parallel. Eplanations ma var. Sample: Their rise run ratios are the same, so the never meet. C Challenge. Open-Ended Write equations for five different lines that intersect at (, ). Answers ma var. Sample:, ±, ±, ±, ±. Find the of each line.. (, ). GPS (, ) (, ) O O (, ) (, ) (, ). The of a roof is its pitch. You indicate the pitch of a roof b a ratio a i b, where a is the number of feet of rise for ever b feet of run. In the photos at the left, which house has the roof with a steeper pitch? Eplain. See above left. 6. Multiple Choice Find the of the line through (, -) and (, -). C Find the of the line through each pair of points. 7. C Q, R, DQ, R. L(7, -6.), M(, -.) 9. J(., ), K(.,.). A Q, R, B Q, R Solve each equation for. Then graph the equation. 9. See margin.. - =. + =. + =. + =. - = 6. + = 7. = -. - = =. Does the point (-, ) lie on the graph of =- +? Eplain. no; u ( ) ±. Does the point (-, -) lie on the graph of - 6 =? Eplain. es; ( ) 6( ). Graph the line with no that passes through (, -). See back of book.. a. Graph the groups of equations on three coordinate planes. See back of book. Group Group Group = - =- - =-6 = =- = = + =- + =. b. How are the lines in each group related to each other? Eplain. See left. c. Reasoning What is the coefficient of in the equation of a graph that has?. The of a road is its grade. What do ou think it means for rise the grade of a road to be %? run. As ou run ft horizontall, ou rise ft verticall. PHSchool.com, Web Code: awa- Rate of Change and Slope-intercept Form Eercise (a) For each equation in Groups and, have the students find the rate of change. Then ask: What is the relationship between the rate of change, the of the line m, and the -intercept b? Answers ma var. Sample: In Group, the rate of change for all three equations is. In Group, the rate of change for all three equations is. In both groups, all three equations are in -intercept form, = m ± b. The rate of change is equal to the of the line m. The rate of change is not affected b the value of the -intercept b.. ± O. O. ± O. ± O ± O O O 6 6 O 6 = 6 O
6 . Assess & Reteach CD, Online, or Transparencies Lesson Quiz Find the of the line through each pair of points.. A(, ), B(-, -). F(-, ), G(, -9). Identif the and -intercept of =- +. Then graph the line. ; O GO Multiple Choice Practice and Mied Review for Help NS., MR. MG. Lesson 7- For California Standards Tutorials, visit PHSchool.com. Web Code: awq-9. Bert bus a -subject notebook and a -subject notebook at the prices shown in the sign. If the ta rate is %, what is the total cost? NOTEBOOKS & PAPER -subject: $. -subject: $. -subject: $. $.9 $7. $7.6 $. 6. Bessie wants to run to train tracks from the couch to a table. The distance from the couch to the table is meters. Each track is centimeters long. So far, she has put down 9 tracks. How much space is left? A. m. m m. m Solve and graph each inequalit See back of book. 7. +, 7. # C Use this Checkpoint Quiz to check students understanding of the skills and concepts of Lessons - through -. Resources Teaching Resources Checkpoint Quiz Lesson 6-7 Lesson 6-6 Find each percent of change. Tell whether the change is an increase or a decrease.. from to 9. from to. from to 9 % decrease % increase % decrease. During the season, New York theater goers bought. million tickets for a total of $79. million. Theater goers spent a total of.% more than the ear before. What was the total amount spent during? about $7. million Enrichment - Linear Relationships Use Reteaching and -intercept - to see relationships among lines. Graph the given equations on the coordinate aes.. Find a. the of the line.. a. Find two points on the line whose coordinates are eas to b. read, like (, ) and (, ). b. rise run c.. c. The is. O d. d. You could also find the from just the coordinates (, ) and (, ). Slope and -intercept O O rise up three units run over four units difference in coordinates. What kind difference of lines coordinates did ou draw () in Eercises and? Since we wrote from (, ) first parallel in the numerator, lines we must write first in the denominator. We could put and first.. Complete: If two lines have the same but different -intercepts, then the lines are Find the parallel of each line.. Graph. the given equations on the coordinate aes... a. 6. a. b. b. O O O O 7. Find What the kind of of lines the did line ou through draw in each pair set? of points.. A(, ), B(, ) perpendicular. L(, 6), M(, ). Find the product of the s in Eercise 7 ; In Eercise 6.. J(6, ), K(, ) 6. P(, ), Q(, ) L L. Yes; there is one range value for each domain value.. Answers ma var. Sample: If ever vertical line passes through just one graphed point, then the relation is a function.. Find three solutions of 9 - =. Answers ma var. Sample: (, 9), (, ), (, ). Graph - = on a coordinate plane. See back of book.. Is {(-, ), (-, ), (, -), (, -)} a function? Eplain. See left.. Eplain how to use the vertical-line test to determine whether a relation is a function. See left. Find the of the line through the given points.. A(, ), B(, ) 6. D(-, -), F(, -6) 7. G(-, ), H(-, -6). J(., ), K (-, -) 9. What are the and the -intercept of =- +?, 9. Complete: If the product of the s of two lines is, then the lines are 7. S(, ), T( 9, ). G(, 7), H(, ) perpendicular. Chapter Linear Functions and Graphing California Standards Resources California Dail Review Transparencies California Review and Practice Workbook Progress Monitoring Assessments Math Intervention Skills Review and Practice Workbook Alternative Assessment Give each student a laminated coordinate plane and a piece of uncooked spaghetti. On the chalkboard, write equations for lines (in -intercept form) whose -intercepts are integers. Have students use the spaghetti to graph each equation.
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