5 LESSON 5.1 Writing Linear Equations from Situations and Graphs


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1 Writing Linear Equations? module You can use linear equations and their graphs to model realworld relationships involving constant rates of change. ESSENTIAL QUESTIN How can ou use linear equations to solve realworld problems? 5 LESSN 5.1 Writing Linear Equations from Situations and Graphs 8.F.4 LESSN 5.2 Writing Linear Equations from a Table 8.F.4 LESSN 5.3 Linear Relationships and Bivariate Data 8.SP.1, 8.SP.2, Houghton Mifflin Harcourt Publishing Compan Image Credits: Yellow Dog Productions/Gett Images 8.SP.3 RealWorld Video 125 Module 5 Linear equations can be used to describe man situations related to shopping. If a store advertised four books for $32.00, ou could write and solve a linear equation to find the price of each book. Math n the Spot Animated Math Go digital with our writein student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 125
2 D NT EDITChanges must be made through File info CorrectionKe=A Are You Read? Are YU Read? Assess Readiness Complete these eercises to review skills ou will need for this module. Use the assessment on this page to determine if students need intensive or strategic intervention for the module s prerequisite skills. Write Fractions as Decimals Write the fraction as a division problem. Write a decimal point and zeros in the dividend. Place a decimal point in the quotient. Divide as with whole numbers. Enrichment Write each fraction as a decimal. Access Are You Read? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment. nline Assessment and Intervention = _ Multipl the numerator and the denominator b a power of 10 so that the denominator is a whole number. Response to Intervention Intervention 0.5 =? 0.8 nline Practice and Help 1. _ Inverse perations nline and Print Resources EXAMPLE Skills Intervention worksheets Differentiated Instruction Skill 26 Write Fractions as Decimals Challenge worksheets Skill 57 Inverse perations Etend the Math PREAP Lesson Activities in TE 5n = 5n = 5 5 n=4 k+7=9 k+77=97 k=2 PREAP n is multiplied b 5. To solve the equation, use the inverse operation, division. 7 is added to k. To solve the equation, use the inverse operation, subtraction. Solve each equation using the inverse operation. 5. 7p = 28 RealWorld Video Viewing Guide 7. After students have watched the video, discuss the following: What does represent in the linear equation = + 40? number of games that cost $ What does the 40 represent in the linear equation = + 40? cost of the $40 game _ = c  8 = = m p=4 = 18 c=0 m = h  13 = 5 8. b + 9 = n = t = 55 Houghton Mifflin Harcourt Publishing Compan 3 EXAMPLE h = 18 b = 12 n = 4 t = 25 Unit 2 8_MCAAESE6984_U2M05.indd /05/13 1:01 PM PRFESSINAL DEVELPMENT VIDE Author Juli Dion models successful teaching practices as she eplores the concept of writing linear equations in an actual eighthgrade classroom. nline Teacher Edition Access a full suite of teaching resources online plan, present, and manage classes and assignments. Professional Development eplanner Easil plan our classes and access all our resources online. Interactive Answers and Solutions Customize answer kes to print or displa in the classroom. Choose to include answers onl or full solutions to all lesson eercises. Interactive Whiteboards Engage students with interactive whiteboardread lessons and activities. : nline Assessment and Intervention Assign automaticall graded homework, quizzes, tests, and intervention activities. Prepare our students with updated practice tests aligned with Common Core. Writing Linear Equations 126
3 Reading StartUp EL Have students complete the activities on this page b working alone or with others. Strategies for English Learners Each lesson in the TE contains specific strategies to help English Learners of all levels succeed. Emerging: Students at this level tpicall progress ver quickl, learning to use English for immediate needs as well as beginning to understand and use academic vocabular and other features of academic language. Epanding: Students at this level are challenged to increase their English skills in more contets, and learn a greater variet of vocabular and linguistic structures, appling their growing language skills in more sophisticated was appropriate to their age and grade level. Bridging: Students at this level continue to learn and appl a range of highlevel English language skills in a wide variet of contets, including comprehension and production of highl technical tets. Active Reading Integrating Language Arts Students can use these reading and notetaking strategies to help them organize and understand new concepts and vocabular. Additional Resources Differentiated Instruction Reading Strategies EL Houghton Mifflin Harcourt Publishing Compan Reading StartUp Visualize Vocabular Use the words to complete the diagram. You can put more than one word in each bubble. coordinate m slope Understand Vocabular Complete the sentences using the preview words. 1. A set of data that is made up of two paired variables is bivariate data. 2. When the rate of change varies from point to point, the relationship is a nonlinear relationship. Active Reading = m + b slopeintercept form of an equation, linear equation TriFold Before beginning the module, create a trifold to help ou learn the concepts and vocabular in this module. Fold the paper into three sections. Label the columns What I Know, What I Need to Know, and What I Learned. Complete the first two columns before ou read. After studing the module, complete the third column. coordinate b intercept Vocabular Review Words linear equation (ecuación lineal) ordered pair (par ordenado) proportional relationship (relación proporcional) rate of change (tasa de cambio) slope (pendiente) slopeintercept form of an equation (forma de pendienteintersección) coordinate (coordenada ) coordinate (coordenada ) intercept (intersección con el eje ) Preview Words bivariate data (datos bivariados) nonlinear relationship (relación no lineal) Module Focus Coherence Rigor Tracking Your Learning Progression Before Students understand proportional and linear relationships: use tables and verbal descriptions to describe a linear relationship write and graph a linear relationship In this module Students represent and use linear relationships: write an equation in the form = m + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation After Students will connect that: there are various forms of linear equations in nonlinear relationships, the rate of change can var from point to point 127 Module 5
4 GETTING READY FR Writing Linear Equations GETTING READY FR Writing Linear Equations Understanding the standards and the vocabular terms in the standards will help ou know eactl what ou are epected to learn in this module. Use the eamples on this page to help students know eactl what the are epected to learn in this module. Content Areas CA Common Core Standards Functions 8.F Cluster Use functions to model relationships between quantities. Statistics and Probabilit 8.SP Cluster Investigate patterns of association in bivariate data. Go online to see a complete unpacking of the CA Common Core Standards. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Ke Vocabular rate of change (tasa de cambio) A ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. Use the equation of a linear model to solve problems in the contet of bivariate measurement data, interpreting the slope and intercept. Ke Vocabular bivariate data (datos bivariados) A set of data that is made up of two paired variables. 8.F.4 8.SP.3 Visit to see all CA Common Core Standards eplained. What It Means to You You will learn how to write an equation based on a situation that models a linear relationship. EXAMPLE 8.F.4 In 06 the fare for a taicab was an initial charge of $2.50 plus $0.30 per mile. Write an equation in slopeintercept form that can be used to calculate the total fare. The constant charge is $2.50. The rate of change is $0.30 per mile. The input variable,, is the number of miles driven. So 0.3 is the cost for the miles driven. The equation for the total fare,, is as follows: = What It Means to You You will see how to use a linear relationship between sets of data to make predictions. EXAMPLE 8.SP.3 The graph shows the temperatures in degrees Celsius inside the earth at certain depths in kilometers. Use the graph to write an equation and find the temperature at a depth of 12 km. The initial temperature is C. It increases at a rate of 10 C/km. The equation is t = 10d +. At a depth of 12 km, the temperature is 140 C. Temperature ( C) Temperature Inside Earth Depth (km) Houghton Mifflin Harcourt Publishing Compan 128 Unit 2 California Common Core Standards Lesson 5.1 Lesson 5.2 Lesson F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value. 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.2 For scatter plots that suggest a linear association, informall fit a straight line,. 8.SP.3 Use the equation of a linear model to solve problems in the contet of bivariate measurement data, interpreting the slope and intercept. Writing Linear Equations 128
5 LESSN 5.1 Writing Linear Equations from Situations and Graphs Lesson Support Content bjective Language bjective Students will learn how to write an equation to model a linear relationship given a graph or a description. Students will eplain how to write an equation to model a linear relationship given a graph or a description. California Common Core Standards 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. MP.2 Reason abstractl and quantitativel. Focus Coherence Rigor Building Background Eliciting Prior Knowledge Have students sketch the graph of = for 0. Then discuss the graph with the class. Ask students to eplain how to identif the intercept from the graph. Then discuss how to identif the slope b appling riseoverrun to the graph. Connect the intercept and the slope from the graph to the equation of the line in slopeintercept form (1, 5) (0, 2) Learning Progressions In this lesson, students write equations given a graph or a realworld description of a linear relationship. Important understandings for students include the following: Write an equation in slopeintercept form from a graph. Write an equation in slopeintercept form from a realworld description of a linear relationship. Students have graphed linear relationships using slopeintercept form and descriptions of reallife situations. In this lesson, the reverse the process and write the equation from the graph or description. The should start to understand the relationships among the equivalent forms and move easil between representations of linear relationships. Cluster Connections This lesson provides an ecellent opportunit to connect ideas in the cluster: Use functions to model relationships between quantities. Tell students that the points (1, 1) and (1, 7) lie on a line. Ask students to find the equation of the line. Discuss whether it is necessar to sketch the graph to find the equation of the line. = 43; It is not actuall necessar to sketch the graph. You can use the points to find the slope and then substitute one of the points in = m + b to find b. 129A
6 Language Support EL PRFESSINAL DEVELPMENT California ELD Standards Emerging 2.I.5. Listening activel Demonstrate active listening in oral presentation activities b asking and answering basic questions with prompting and substantial support. Epanding 2.I.5. Listening activel Demonstrate active listening in oral presentation activities b asking and answering detailed questions with occasional prompting and moderate support. Bridging 2.I.5. Listening activel Demonstrate active listening in oral presentation activities b asking and answering detailed questions with minimal prompting and support. Linguistic Support EL Academic/Content Vocabular This lesson uses the terms dependent and independent variables. Students will need to understand the concept of dependence in order to grasp the meaning of these terms. In Guided Practice, in the problem of making the beaded necklaces, point out to students how the length of the necklace depends on the number of beads used. Use short comparative sentences to describe the relationship through contrasts. A short necklace uses fewer beads. A long necklace uses more beads. This means that the number of beads is the dependent variable. Background Knowledge There are several eamples in this lesson that are based on an understanding of rentals, memberships, and user fees. Use some dail life eperiences with rentals to frame more unfamiliar concepts and contets involving rental situations, such as office space or memberships in craft clubs. Discuss renting an apartment or house and wh those with more area (square footage) require more mone. The object is to make connections and understanding commonalities between students life eperiences and new situations presented in problems and eamples. Leveled Strategies for English Learners EL Emerging Use sentence frames to compare and contrast statements about eamples of independent and dependent variables. Have students respond to, or signif es/no or true/false to correct and incorrect statements about variables. Epanding Have students work in groups to create, compare, and contrast sentences to identif independent and dependent variables based on realworld problems or eamples. Bridging Have students verbalize their reasoning for identifing independent and dependent variables and for writing an equation based on a description of a realword problem. Math Talk The prompt focuses on the term through the origin of a line in a graph. Remember that this is not an ordinar use of the term origin. Post a graph that is labeled with the mathematical terms used in graphing for students to refer to visuall during discussions of the process of graphing and/or the values that graphs represent. Writing Linear Equations from Situations and Graphs 129B
7 L E S S N 5.1 CA Common Core Standards The student is epected to: Functions 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Mathematical Practices MP.2 Reasoning Writing Linear Equations from Situations and Graphs Engage ESSENTIAL QUESTIN How do ou write an equation to model a linear relationship given a graph or a description? Use pairs of values for input and output to determine the values for the slope m and the intercept b in the equation = m + b. Motivate the Lesson Ask: How can ou figure out the total cost for different plans for our cell phone when the fees and charges are not the same? What choices have ou made that involved comparing memberships or rentals? The Eplore Activit shows how to write an equation to help ou evaluate and compare costs. Eplore EXPLRE ACTIVITY Connect Vocabular EL Eplain the terms potter s wheel (a rotating disk used for shaping cla) and kiln (an oven for baking or dring cla or potter). Ask if an students have worked with cla. An such students ma be able to eplain these terms to the class. ADDITINAL EXAMPLE 1 A DJ charges a setup fee plus an hourl fee to provide music for a dance part. Use the graph to write an equation in slopeintercept form to represent the amount spent,, on hours of music. Amount spent ($) Time (h) = Interactive Whiteboard Interactive eample available online Eplain EXAMPLE 1 Questioning Strategies Mathematical Practices Wh do ou think that the points (0, 8) and (8, 18) were used to find the slope? Sample answer: These two points clearl lie at the intersection of vertical and horizontal grid lines. You can t be sure of the values of for the other points on the line. Describe the new graph if the membership fee were changed to $10. The intercept moves up to (0, 10); the slope stas the same. Engage with the Whiteboard Draw arrows showing the rise and run between the two points (0, 8) and (8, 18) on the graph. Ask students what the intercept, rise, and run represent in terms of the membership fee and rental fee. YUR TURN Avoid Common Errors Students ma be confused about which ais is which and about the meaning of the intercept. Verif that the understand that the value is zero at the intercept, and that the intercept is the value of the point where the graph crosses the ais. 129 Lesson 5.1
8 Houghton Mifflin Harcourt Publishing Compan? L E S S N 5.1 ESSENTIAL QUESTIN EXPLRE ACTIVITY Writing an Equation in SlopeIntercept Form Greta makes cla mugs and bowls as gifts at the Craft Studio. She pas a membership fee of $15 a month and an equipment fee of $3.00 an hour to use the potter s wheel, table, and kiln. Write an equation in the form = m + b that Greta can use to calculate her monthl costs. A What is the independent variable,, for this situation? the number of hours Greta uses the studio What is the dependent variable,, for this situation? the mone Greta pas the studio each month B During April, Greta does not use the equipment at all. What will be her number of hours () for April? 0 What will be her cost () for April? $15 C What will be the intercept, b, in the equation? Greta spends 8 hours in Ma for a cost of $15 + 8($3) = 15 $39. In June, she spends 11 hours for a cost of $48. From Ma to June, the change in values is +3. From Ma to June, the change in values is +9. What will be the slope, m, in the equation? 3 D Writing Linear Equations from Situations and Graphs Use the values for m and b to write an equation for Greta s costs in the form = m + b: How do ou write an equation to model a linear relationship given a graph or a description? 8.F.4 = F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value. Interpret the rate of change and initial value. (For the full tet of the standard, see the table at the front of the book beginning on page CA2.) Math Talk Mathematical Practices What change could the studio make that would make a difference to the intercept of the equation? Changing the membership fee changes the intercept. Math n the Spot Math Talk Mathematical Practices If the graph of an equation is a line that goes through the origin, what is the value of the intercept? The value of the intercept is zero. nline Practice and Help Writing an Equation from a Graph You can use information presented in a graph to write an equation in slopeintercept form. EXAMPLE 1 A video club charges a onetime membership fee plus a rental fee for each DVD borrowed. Use the graph to write an equation in slopeintercept form to represent the amount spent,, on DVD rentals. STEP 1 STEP 2 STEP 3 Choose two points on the graph, ( 1, 1 ) and ( 2, 2 ), to find the slope. m = m = m = 10 8 = 1.25 Read the intercept from the graph. The intercept is 8. Dollars Amount spent ($) Amount on Gift Card 30 Video Club Costs Rentals Use our slope and intercept values to write an equation in slopeintercept form. = m + b = Reflect 1. What does the value of the slope represent in this contet? cost per DVD rental 2. Describe the meaning of the intercept. amount spent for 0 rentals, or the membership fee of $8 YUR TURN Find the change in values over the change in values. Substitute (0, 8) for ( 1, 1 ) and (8,18) for ( 2, 2 ). Simplif. Slopeintercept form Substitute 1.25 for m and 8 for. 3. The cash register subtracts $2.50 from a $25 Coffee Café gift card for ever medium coffee the customer bus. Use the graph to write an equation in slopeintercept form to represent this situation. = F Number of coffees Houghton Mifflin Harcourt Publishing Compan Lesson Unit 2 PRFESSINAL DEVELPMENT Integrate Mathematical Practices MP.2 This lesson provides an opportunit to address this Mathematical Practices standard. It calls for students to represent a situation smbolicall. Students read values from a graph and create a new representation of the linear relationship in the form of an equation. Math Background A constant rate of change can be shown b a linear graph. The rate of change is the slope of the line. Lines with positive slopes rise from left to right; lines with negative slopes go down. The magnitude of the slope describes the steepness. The line = makes a 45 angle with the ais and has a slope of 1. A line with a slope whose absolute value is between 0 and 1 is less steep than a 45 line. A line whose absolute value of the slope is greater than 1 is steeper than a 45 line. Writing Linear Equations from Situations and Graphs 130
9 ADDITINAL EXAMPLE 2 The cost for 25 square ards of carpet is $650 including deliver and installation. The cost for 40 square ards of installed carpet is $950. Write an equation in slopeintercept form for the cost of the installed carpet. = Interactive Whiteboard Interactive eample available online EXAMPLE 2 Questioning Strategies Mathematical Practices What does the rise over the run, or the slope, represent in the problem situation? change in rent over change in square feet, or rent per square foot If the equation for the rent were = , how would this change the rent? The rent 3 would be $75 more for an number of square feet. Focus on Critical Thinking In Eample 2, one student wrote this equation for the slope: m =. Eplain what error was made in this equation. The coordinates for the points must be in the same order in the numerator and in the denominator. YUR TURN Focus on Modeling Mathematical Practices Make sure that students understand that the number of chores Hari chooses to do is the independent, or input, variable, and the allowance he receives is the dependent, or output, variable. Integrating Language Arts EL Encourage English learners to take notes on new terms or concepts and to write them in familiar language. Elaborate Talk About It Summarize the Lesson Ask: In a linear relationship represented b = m + b, how do ou find m, the value for the constant rate of change, and the intercept, b? Use two sets of  and values to find m, the change in over the change in. Then substitute m,, and in = m + b and solve to find b. GUIDED PRACTICE Engage with the Whiteboard For Eercise 2, have two students label the graph with the coordinates of the two different points the choose to find the slope. Emphasize that using an two points on the line will result in the same slope. Avoid Common Errors Eercise 2 Ask students to predict whether the slope will be positive or negative before the do an calculations, and to eplain their answer before finding the slope. Students ma enter 1 and 2 in a different order than 1 and 2 and get the opposite slope. Eercise 3 Students ma use the wrong independent variable. Have them consider whether temperature is dependent upon chirps or chirps dependent upon temperature. 131 Lesson 5.1
10 Writing an Equation from a Description You can use information from a description of a linear relationship to find the slope and intercept and to write an equation. EXAMPLE 2 8.F.4 Math n the Spot nline Practice and Help YUR TURN 5. Hari s weekl allowance varies depending on the number of chores he does. He received $16 in allowance the week he did 12 chores, and $14 in allowance the week he did 8 chores. Write an equation for his allowance in slopeintercept form. = The rent charged for space in an office building is a linear relationship related to the size of the space rented. Write an equation in slopeintercept form for the rent at West Main Street ffice Rentals. Guided Practice STEP 1 STEP 2 Identif the independent and dependent variables. The independent variable is the square footage of floor space. The dependent variable is the monthl rent. West Main St. ffice Rentals ffices for rent at convenient locations. Monthl Rates: 600 square feet for $ square feet for $1150 Write the information given in the problem as ordered pairs. M Notes 1. Li is making beaded necklaces. For each necklace, she uses 27 spacer beads, plus 5 glass beads per inch of necklace length. Write an equation to find how man beads Li needs for each necklace. (Eplore Activit) a. independent variable: the length of the necklace in inches b. dependent variable: the total number of beads in the necklace c. equation: = STEP 3 STEP 4 The rent for 600 square feet of floor space is $750: (600, 750) The rent for 900 square feet of floor space is $1150: (900, 1150) Find the slope. m = = = = 4_ 3 Find the intercept. Use the slope and one of the ordered pairs. = m + b 750 = 4_ b Slopeintercept form Substitute for, m, and. 2. Kate is planning a trip to the beach. She estimates her average speed to graph her epected progress on the trip. Write an equation in slopeintercept form that represents the situation. (Eample 1) Choose two points on the graph to find the slope. m = = 50 = 300 = Read the intercept from the graph: b = 300 Use our slope and intercept values to write an equation in slopeintercept form. = Distance to beach (mi) M Beach Trip Driving time (h) Houghton Mifflin Harcourt Publishing Compan STEP = b 50 = b Substitute the slope and intercept. = m + b = 4_ 350 Multipl. Subtract 800 from both sides. Slopeintercept form Substitute 4 for m and 50 for b. 3 Reflect 4. Without graphing, tell whether the graph of this equation rises or falls from left to right. What does the sign of the slope mean in this contet? Slope is positive, so the graph rises from left to right. This means that the rent increases as the square footage increases.? 3. At 59 F, crickets chirp at a rate of 76 times per minute, and at 65 F, the chirp 100 times per minute. Write an equation in slopeintercept form that represents the situation. (Eample 2) Independent variable: temperature Dependent variable: chirps per minute m = = = 24 6 = 4 Use the slope and one of the ordered pairs in = m + b to find b. 100 = b; = b Write an equation in slopeintercept form. = ESSENTIAL QUESTIN CHECKIN 4. Eplain what m and b in the equation = m + b tell ou about the graph of the line with that equation. The slope of the graphed line is m, and the intercept is b. Houghton Mifflin Harcourt Publishing Compan Lesson Unit 2 DIFFERENTIATE INSTRUCTIN Multiple Representations Have students discuss the different was the know to describe the slope of a line. Have them write a list, such as this: rise over run, change in over change in, , slope, constant rate of change. Have them sketch a line that has a positive slope, another with a negative slope, and a line with a slope of zero (horizontal), and a line with an undefined slope (vertical). Visual Cues Draw a linear graph and mark two points on it in red. Then, on the same graph, mark two different points on it in blue. Ask students to discuss whether the slope of the line will be greater between the two red points or between the two blue points. Lead them to see that the slope of a given linear graph is alwas the same between an two points on the line. Additional Resources Differentiated Instruction includes: Reading Strategies Success for English Learners EL Reteach Challenge PREAP Writing Linear Equations from Situations and Graphs 132
11 5.1 LESSN QUIZ nline Assessment and Intervention nline homework assignment available 8.F.4 1. Lee charges $3 for a basket and $2.50 for each pound of fruit picked at the orchard. Write an equation in = m + b form for the total cost of pounds of fruit from the orchard. 2. A camp charges families a fee of $625 per month for one child and a certain amount more per month for each additional child. Use the graph to write an equation in slopeintercept form to represent the amount a famil with additional children would pa Evaluate GUIDED AND INDEPENDENT PRACTICE 8.F.4 Concepts & Skills Eplore Activit Writing an Equation in SlopeIntercept Form Eample 1 Writing an Equation from a Graph Eample 2 Writing an Equation from a Description Practice Eercise 1 Eercises 2, 7 9, Eercises 3, 5 6, Focus Coherence Rigor Eercise Depth of Knowledge (D..K.) Mathematical Practices Skills/Concepts MP.4 Modeling Skills/Concepts MP.6 Precision Skills/Concepts MP.4 Modeling Skills/Concepts MP.2 Reasoning Strategic Thinking MP.3 Logic Charge ($) Additional children Additional Resources Differentiated Instruction includes: Leveled Practice worksheets 3. Identif the intercept in question 2 above. Tell what the intercept means in this contet. 4. A driving range charges $4 to rent a golf club plus $2.75 for ever bucket of golf balls ou hit. Write an equation that shows the total cost c of hitting b buckets of golf balls. Lesson Quiz available online Answers 1. = = b = 625, the fied fee for one child 4. c = 2.75b Lesson 5.1
12 Name Class Date 5.1 Independent Practice 8.F.4 5. A dragonfl can beat its wings 30 times per second. Write an equation in slopeintercept form that shows the relationship between fling time in seconds and the number of times the dragonfl beats its wings. = A balloon is released from the top of a platform that is 50 meters tall. The balloon rises at the rate of 4 meters per second. Write an equation in slopeintercept form that tells the height of the balloon above the ground after a given number of seconds. = nline Practice and Help The graph shows the activit in a savings account. 12. What was the amount of the initial deposit that started this savings account? $ Find the slope and intercept of the graphed line. m = 500; b = 1000 Amount saved ($) 14. Write an equation in slopeintercept form for the activit in this savings account. = Eplain the meaning of the slope in this graph. The amount of mone in the savings account increases b $500 each month Savings Account Activit Months in plan Houghton Mifflin Harcourt Publishing Compan The graph shows a scuba diver s ascent over time. 7. Use the graph to find the slope of the line. Tell what the slope means in this contet. m = 0.125; the diver ascends at a rate of m/s Identif the intercept. Tell what the intercept means in this contet. 10; the diver starts 10 meters below the water s surface. 9. Write an equation in slopeintercept form that represents the diver s depth over time. = The formula for converting Celsius temperatures to Fahrenheit temperatures is a linear equation. Water freezes at 0 C, or 32 F, and it boils at 100 C, or 212 F. Find the slope and intercept for a graph that gives degrees Celsius on the horizontal ais and degrees Fahrenheit on the vertical ais. Then write an equation in slopeintercept form that converts degrees Celsius into degrees Fahrenheit. m = 9_ 5 ; b = 32; = 9_ + 32 where = F and = C The cost of renting a sailboat at a lake is $ per hour plus $12 for lifejackets. Write an equation in slopeintercept form that can be used to calculate the total amount ou would pa for using this sailboat. = + 12 Depth (m) 4 Scuba Diver s Ascent Time (s) FCUS N HIGHER RDER THINKING 16. Communicate Mathematical Ideas Eplain how ou decide which part of a problem will be represented b the variable, and which part will be represented b the variable in a graph of the situation. Eamine the problem and decide what is the thing ou start with and what is the thing ou are tring to find. Use what ou start with for and what ou are tring to find for. 17. Represent RealWorld Problems Describe what would be true about the rate of change in a situation that could not be represented b a graphed line and an equation in the form = m + b. The rate of change would not be constant. Using different pairs of points in the slope formula would give ou different results. 18. Draw Conclusions Must m, in the equation = m + b, alwas be a positive number? Eplain. No. A negative number for m means the dependent variable is decreasing as the independent variable increases, so the graph falls from left to right. Work Area Houghton Mifflin Harcourt Publishing Compan Lesson Unit 2 EXTEND THE MATH PREAP Activit available online Activit Give each pair of students a sheet of graph paper marked with the  and aes, and pencils or pieces of wire or spaghetti to use for the lines to make quick graphs. ne student calls out a slope (for eample, m = 0 or 1, or 0.5, or 5, or 1) and the other places the line on the graph in the approimate position going through the origin, (for eample: horizontal, bisecting QI, less steep than =, steeper than =, or bisecting QII). Then have one student call out a slope and a intercept and the other place the line. If students have trouble, have them place a line with the same slope through the origin and then move it to go through the intercept. Writing Linear Equations from Situations and Graphs 134
13 LESSN 5.2 Writing Linear Equations from a Table Lesson Support Content bjective Language bjective Students will learn how to write an equation to model a linear relationship given a table. Students will show how to write an equation to model a linear relationship given a table. California Common Core Standards 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. MP.4 Model with mathematics. Focus Coherence Rigor Building Background Visualizing Math Ask students to consider various tables of data that model linear relationships. Then have students work with partners to create four squares that characterize such data. Students can include definitions, equations, eamples, and noneamples. Discuss how their diagrams clarif the concept of linear relationships. proportional = m linear relationship nonproportional = m + b Learning Progressions In this lesson, students continue to write equations presented in different forms, specificall from a table. Important understandings for students include the following: Draw a graph from a table and then write an equation from the graph. Write an equation from a table. Students continue to construct functions that model linear relationships between two quantities. The determine the rate of change, or slope, and the initial value, or intercept, from a description of two values presented in a table. In the last lesson of this module, students will be read to make predictions using linear relationships and to contrast linear and nonlinear data. Cluster Connections This lesson provides an ecellent opportunit to connect ideas in the cluster: Use functions to model relationships between quantities. Have students refer to the following table Ask them if the data do or do not model a linear relationship and wh. Then have them write the equation in slopeintercept form. Sample answer: The data have a linear relationship since the rate of change is constant. = A
14 Language Support EL PRFESSINAL DEVELPMENT California ELD Standards Emerging 2.I.8. Analzing language choices Eplain how phrasing or different common words with similar meanings produce different effects on the audience. Epanding 2.I.8. Analzing language choices Eplain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience. Bridging 2.I.8. Analzing language choices Eplain how phrasing or different words with similar meanings or figurative language produce shades of meaning, nuances, and different effects on the audience. Linguistic Support EL Academic/Content Vocabular The terms increase and decrease are used frequentl in this lesson. Students need other common words for these terms, such as gets bigger or gets smaller to substitute to clarif the meaning. Also note that the terms can be used as verbs and nouns. To increase or to decrease an amount is a verb. An increase or a decrease is a noun. Background Knowledge Several eamples in Guided Practice use the concept of a form of pament, such as a prepaid cell phone or bus pass, in which paments are deducted per usage to produce a balance on the card. Discuss with students actual eamples of this form of pament from students realworld eperiences. Leveled Strategies for English Learners EL Emerging Post a chart with common epressions that mean the same as increase and decrease. Use sentence frames for students to complete with a common epression to review the meaning of the terms. Epanding Have students use sentence frames to describe the increase or decrease of quantities represented in a table. Bridging Have several groups of students create narratives to describe the use of a pament card from their practical realworld eperiences. Have them echange scenarios to create a table and graph to represent the decreases per usage of the pament card. Math Talk The prompt refers to a variable as represented in an equation. Be sure to identif the dependent variable and independent variable in the realworld problem with a word or short phrase for English learners. Writing Linear Equations from a Table 135B
15 L E S S N 5.2 CA Common Core Standards The student is epected to: Functions 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Mathematical Practices MP.4 Modeling Writing Linear Equations from a Table Engage ESSENTIAL QUESTIN How do ou write an equation to model a linear relationship given a table? Use two data points from the table to determine the slope m and the intercept b in the equation = m + b. Motivate the Lesson Ask: Have ou ever recorded measurements in a table? Tables are often used in science labs to record temperatures, weights, and other measures. Graphing the information in a table provides another wa to look at the data. Eplore Have a student provide a number between 10 and. Then have students make a table with integer values from 1 through 10. Fill in the values so that and for each column add to the selected number. Ask students to write an equation for the relationship. Eplain ADDITINAL EXAMPLE 1 The Daile famil uses maple sap to make srup. The table shows the temperature of the sap as it heats. Graph the data, and find the slope and intercept from the graph. Then write the equation for the graph in slopeintercept form. Time (h) Temp ( F) Temperature ( F) F = 45h + 38 Srup Temperature Time (h) Interactive Whiteboard Interactive eample available online EXAMPLE 1 Questioning Strategies Mathematical Practices Describe informall what is happening in this eperiment. The temperature of a fish tank is measured at the beginning as 82 degrees Fahrenheit. This temperature graduall falls over 5 hours at a stead rate until it reaches 72 degrees Fahrenheit. Could ou find the equation from the table without drawing a graph? Eplain. Yes; choose two points from the table to find the slope, and use (0, 82) to find the intercept. Engage with the Whiteboard Have students draw lines to etend the ais and the graphed line. Where will the line meet the ais if the rate of cooling stas the same? Students should check their prediction b substituting 0 for in the equation and solving for (41 hours). YUR TURN Avoid Common Errors Remind students to refer to the slope formula on the previous page when calculating the slope. The should write out the formula and substitute the coordinates of two points. EXAMPLE 2 Questioning Strategies Mathematical Practices How can ou find the intercept when it is not given as one of the points in the table? Substitute m and an point in the table as (, ) to solve for b in = m + b. What does the intercept represent in this situation? the base price without minutes 135 Lesson 5.2
16 5.2? Writing Linear Equations from a Table ESSENTIAL QUESTIN 8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. YUR TURN nline Practice and Help How do ou write an equation to model a linear relationship given a table? Math Talk Mathematical Practices Graphing from a Table to Write an Equation Which variable in the equation = m + b tells ou the volume of water released ever second from Hoover Dam? You can use information from a table to draw a graph of a linear relationship and to write an equation for the graphed line. Math n the Spot EXAMPL 1 EXAMPLE 8.F Temperature ( F) Graph the ordered pairs from the table (time, temperature). STEP 2 Draw a line through the points. STEP 3 Choose two points on the graph to find the slope: for eample, choose (0, 82) and (1, 80).  STEP 4 STEP 5 0, , Time (s) 5 75, , , ,000 m = 15,000; b = 0; = 15,000 The information from a table can also help ou to write the equation that represents a given situation without drawing the graph. Math n the Spot EXAMPLE 2 8.F.4 Elizabeth s cell phone plan lets her choose how man minutes are included each month. The table shows the plan s monthl cost for a given number of included minutes. Write an equation in slopeintercept form to represent the situation Animated Math Minutes included, Cost of plan ($), STEP m = 21 Use the slope formula m = Substitute (0, 82) for (1, 1) and (1, 80) for (2, 2). 2 = 2 m = 1 Simplif Time (h) Read the intercept from the table or graph. b = ,000 Writing an Equation from a Table Tank Temperature Temperature ( F) Houghton Mifflin Harcourt Publishing Compan Image Credits: Imagebroker/Alam Images Time (h) The table shows the volume of water released b Hoover Dam over a certain period of time. Graph the data, and find the slope and intercept from the graph. Then write the equation for the graph in slopeintercept form. Water Released Water Released from Hoover Dam from Hoover Dam Time (s) Volume of water (m3) the slope, or m The table shows the temperature of a fish tank during an eperiment. Graph the data, and find the slope and intercept from the graph. Then write the equation for the graph in slopeintercept form. STEP 1 1. The intercept is the value when time is 0. Use these slope and intercept values to write an equation in slopeintercept form. = m + b = _MCABESE6984_U2M05L2.indd 135 change in (  14) 6 m = = = = (0100) change in Math Talk STEP 2 Mathematical Practices The variable b, or the intercept, shows the initial temperature /1/13 2:50 AM Find the intercept. Use the slope and an point from the table. = m + b 14 = b 14 = 6 + b 8=b Which variable in the equation = m + b shows the initial temperature of the fish tank at the beginning of the eperiment? Lesson 5.2 Notice that the change in cost is the same for each increase of 100 minutes. So, the relationship is linear. Choose an two ordered pairs from the table to find the slope. STEP 3 Substitute for, m, and. Multipl. Subtract 6 from both sides. Substitute the slope and intercept. = m + b = Slopeintercept form Houghton Mifflin Harcourt Publishing Compan LESSN D NT EDITChanges must be made through File info CorrectionKe=A Volume (m3) D NT EDITChanges must be made through File info CorrectionKe=B Slopeintercept form Substitute 0.06 for m and 8 for b. Unit 2 8_MCAAESE6984_U2M05L2.indd /04/13 8:23 PM PRFESSINAL DEVELPMENT Integrate Mathematical Practices MP.4 This lesson provides an opportunit to address this Mathematical Practices standard. It calls for students to appl mathematics to problems arising in everda life, societ, and the workplace. Students appl what the know about linear relationships to problems arising from an eperiment measuring changes in temperature, measuring the flow of water, and eamining the cost of a cellphone plan. The relate details of everda relationships to the formal summar of a linear equation in mathematical terms. Math Background A table of values generated from measurements such as temperature will often not have a constant rate of change and cannot be precisel described b a linear relationship. In man cases, however, the ordered pairs can be plotted and a line of best fit can be drawn through them. The slope and intercept of the line of best fit can be determined and used to write an equation that describes the relationship. Writing Linear Equations from a Table 136
17 ADDITINAL EXAMPLE 2 Zara made an initial deposit to a bank account and then added a fied amount ever week. The table shows the mone in her account. Write an equation in slopeintercept form to represent the situation. Number of weeks, Balance ($), = + 1 Animated Math Write Equations for Lines Students eplore changing the slope and intercept of a line in a dnamic graphing tool. Interactive Whiteboard Interactive eample available online Focus on Reasoning How can ou tell, without drawing the graph, where the line of the graph would start and whether it rises or falls from left to right? Would the line be more or less steep than the graph for =? Eplain. The intercept tells where the line starts (0, 8); the positive slope of 0.06 tells that the line rises. Since the slope is less than one, the graph is less steep than =. Talk About It Check for Understanding Ask: If the base rate increased from $8 to $12 but the per minute rate staed the same at $0.06 per minute, does the slope change? The intercept? Write a new equation for a base fee of $12. no; es; = YUR TURN Connect to Dail Life Students ma not understand how sales commission works. Give one or more eamples, using points from the table to make this clear, eplain: The salesperson is paid $250 per week regardless of how man computers are sold. If 10 computers are sold, the total pa is $250 plus $75 for each of the 10 computers. Elaborate Talk About It Summarize the Lesson Ask: How can ou use values given in a table of a linear relationship to draw a graph and write an equation in = m + b form? Use values from the table to draw a graph. Use two points to find the slope. To find the intercept, either read it from the graph or substitute m and an point in the table as (, ) to solve for b in = m + b. GUIDED PRACTICE Engage with the Whiteboard For Eercise 1, have students mark on the graph several points that are not in the table, such as the points that tell how much is left on the pass after 3 rides and after 14 rides. Label the points with their coordinates. Focus on Communication Have students informall describe the information that is shown in the table for Eercises 2 5, and what it might mean for someone who is climbing mountains. It gets colder the higher ou climb. Avoid Common Errors Eercise 5 To avoid calculating an incorrect value, remind students that the can use the table to predict, or check, the reasonableness of the temperature. The should see that the temperature must be between 35 F and 43 F. 137 Lesson 5.2
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