Relationships in Two Variables

Relationships in Two Variables MDULE 1? ESSENTIAL QUESTIN How can ou use relationships in two variables to solve real-world problems? LESSN 1.1 Graphing on the Coordinate Plane 6.NS.6, 6.NS.6b LESSN 1. Independent and Dependent Variables in Tables and Graphs LESSN 1.3 Writing Equations from Tables Image Credits: Blickwinkel/Alam Real-World Video A two-variable equation can represent an animal s distance over time. A graph can displa the relationship between the variables. You can graph two or more animals data to visuall compare them. LESSN 1. Representing Algebraic Relationships in Tables and Graphs Math n the Spot Animated Math Personal Math Trainer Go digital with our write-in 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. 37

Are YU Read? Complete these eercises to review skills ou will need for this module. Multiplication Facts Personal Math Trainer nline Assessment and Intervention EXAMPLE 8 7 = Use a related fact ou know. 7 7 = 9 Think: 8 7 = (7 7) + 7 = 9 + 7 = 56 Multipl. 1. 7 6. 10 9 3. 13 1. 8 9 Write the rule for each table. 5. 1 3 7 1 1 8 6. 1 3 7 8 9 10 7. 1 3 8. 0 8 1 5 10 15 0 0 6 Graph rdered Pairs (First Quadrant) EXAMPLE 10 8 6 A 6 8 10 Graph each point on the coordinate grid above. Start at the origin. Move 9 units right. Then move 5 units up. Graph point A(9, 5). 9. B (0, 8) 10. C (, 3) 11. D (6, 7) 1. E (5, 0) 38 Unit 5

Reading Start-Up Visualize Vocabular Use the words to complete the chart. Definition A specific number whose value does not change A number that is multiplied b a variable in an algebraic epression A letter or smbol used to represent an unknown Active Reading Parts of the Algebraic Epression 1 + 3 Mathematical Representation Understand Vocabular Complete the sentences using the preview words. Laered Book Before beginning the module, create a laered book to help ou learn the concepts in this module. Label each flap with lesson titles from this module. As ou stud each lesson, write important ideas such as vocabular and formulas under the appropriate flap. Refer to our finished laered book as ou work on eercises from this module. 1 3 Review Word 1. The numbers in an ordered pair are.. A is formed b two number lines that intersect at right angles. Vocabular Review Words coefficient (coeficiente) constant (constante) equation (ecuación) negative number (número negativo) positive number (número positivo) scale (escala) variable (variable) Preview Words aes (ejes) coordinate plane (plano cartesiano) coordinates (coordenadas) dependent variable (variable dependiente) independent variable (variable independiente) ordered pair (par ordenado) origin (origen) quadrants (cuadrante) -ais (eje ) -coordinate (coordenada ) -ais (eje ) -coordinate (coordenada ) Module 1 39

MDULE 1 Unpacking the Standards 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 variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to epress one quantit, thought of as the dependent variable, in terms of the other quantit, thought of as the independent variable.... Ke Vocabular equation (ecuación) A mathematical sentence that shows that two epressions are equivalent. What It Means to You You will learn to write an equation that represents the relationship in a table. UNPACKING EXAMPLE Emil has a dog-walking service. She charges a dail fee of $7 to walk a dog twice a da. Create a table that shows how much Emil earns for walking 1, 6, 10, and 15 dogs. Write an equation that represents the situation. Dogs walked 1 6 10 15 Earnings ($) 7 70 105 Earnings is 7 times the number of dogs walked. Let the variable e represent earnings and the variable d represent the number of dogs walked. e = 7 d...analze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Ke Vocabular coordinate plane (plano cartesiano) A plane formed b the intersection of a horizontal number line called the -ais and a vertical number line called the -ais. Visit to see all the Common Core Standards unpacked. What It Means to You You can use words, a table, a graph, or an equation to model the same mathematical relationship. UNPACKING EXAMPLE The equation = represents the total cost for games of miniature golf. Make a table of values and a graph for this situation. Number of games, Total cost ($), 1 3 8 1 16 Total cost ($) 0 16 1 8 6 8 Number of games Image Credits: PhotoDisc/Gett Images 330 Unit 5

? LESSN 1.1 ESSENTIAL QUESTIN Graphing on the Coordinate Plane 6.NS.6c find and position pairs of integers and other rational numbers on a coordinate plane. Also 6.NS.6, 6.NS.6b, 6.NS.8 How do ou locate and name points in the coordinate plane? Naming Points in the Coordinate Plane A coordinate plane is formed b two number lines that intersect at right angles. The point of intersection is 0 on each number line. The two number lines are called the aes. The horizontal ais is called the -ais. The vertical ais is called the -ais. The point where the aes intersect is called the origin. The two aes divide the coordinate plane into four quadrants. -6 Quadrant II - -ais - 6 - Quadrant III - -6 Quadrant I -ais 6 rigin Quadrant IV An ordered pair is a pair of numbers that gives the location of a point on a coordinate plane. The first number tells how far to the right (positive) or left (negative) the point is located from the origin. The second number tells how far up (positive) or down (negative) the point is located from the origin. Math n the Spot The numbers in an ordered pair are called coordinates. The first number is the -coordinate and the second number is the -coordinate. EXAMPLE 1 Identif the coordinates of each point. Name the quadrant where each point is located. Point A is 1 unit left of the origin, and 5 units down. It has -coordinate -1 and -coordinate -5, written (-1, -5). It is located in Quadrant III. - B - 6.NS.6c, 6.NS.6b Point B is units right of the origin, and 3 units up. It has -coordinate and -coordinate 3, written (, 3). It is located in Quadrant I. A - - Lesson 1.1 331

Reflect 1. If both coordinates of a point are negative, in which quadrant is the point located?. Describe the coordinates of all points in Quadrant I. 3. Communicate Mathematical Ideas Eplain wh (-3, 5) represents a different location than (3, 5). YUR TURN Personal Math Trainer nline Assessment and Intervention Identif the coordinates of each point. Name the quadrant where each point is located.. G E E F - - - 5. F H - G H Math n the Spot Graphing Points in the Coordinate Plane Points that are located on the aes are not located in an quadrant. Points on the -ais have a -coordinate of 0, and points on the -ais have an -coordinate of 0. EXAMPLE 6.NS.6c, 6.NS.6 Graph and label each point on the coordinate plane. A(-5, ), B(3, 1.5), C(0, -3) Point A is 5 units left and units up from the origin. Point B is 3 units right and 1.5 units up from the origin. Graph the point halfwa between (3, 1) and (3, ). Point C is 3 units down from the origin. Graph the point on the -ais. A - - - B - C 33 Unit 5

YUR TURN Graph and label each point on the coordinate plane. 6. P(-, ) 7. Q(3,.5) - - Personal Math Trainer nline Assessment and Intervention 8. R(-.5, -5) - 9. S(, -5) - 10. T(-.5, 0) Reading Scales on Aes The scale of an ais is the number of units that each grid line represents. So far, the graphs in this lesson have a scale of 1 unit, but graphs frequentl use other units. EXAMPLE 3 6.NS.8 Math n the Spot The graph shows the location of a cit. It also shows the location of Gar s and Jen s houses. The scale on each ais represents miles. A Use the scale to describe Gar s location relative to the cit. Gar 0 10 Jen Cit -0-10 10 0 Each grid square is 5 miles on a side. -10 B Gar s house is at (-5, 15), which is 5 miles west and 15 miles north of the cit. -0 Describe the location of Jen s house relative to Gar s house. Jen s house is located 6 grid squares to the right of Gar s house. Since each grid square is 5 miles on a side, her house is 6 5 = 30 miles from Gar s. YUR TURN W N S E Math Talk Mathematical Practices How are north, south, east, and west represented on the graph in Eample 3? Use the graph in the Eample. 11. Ted lives 0 miles south and 0 miles west of the cit represented on the graph in Eample 3. His brother Ned lives 50 miles north of Ted s house. Give the coordinates of each brother s house. Personal Math Trainer nline Assessment and Intervention Lesson 1.1 333

Guided Practice Identif the coordinates of each point in the coordinate plane. Name the quadrant where each point is located. (Eample 1) 1. Point A is 5 units of the origin and 1 unit from the origin. Its coordinates are. It is in quadrant.. Point B is units right of the origin and units down from the origin. Its coordinates are. It is in quadrant. A - - - - B Graph and label each point on the coordinate plane above. (Eample ) 3. Point C at (-3.5, 3). Point D at (5, 0) For 5 7, use the coordinate plane shown. (Eample 3) 5. Describe the scale of the graph. 6. Plot point A at ( - 1_, ). 7. Plot point B at ( 1_, - ). 8. Vocabular Describe how an ordered pair represents a point on a coordinate plane. Include the terms -coordinate, -coordinate, and origin in our answer. - -1 1 1-1 -? ESSENTIAL QUESTIN CHECK-IN 9. Give the coordinates of one point in each of the four quadrants, one point on the -ais, and one point on the -ais. 33 Unit 5

Name Class Date 1.1 Independent Practice 6.NS.6, 6.NS.6b, 6.NS.6c, 6.NS.8 Personal Math Trainer nline Assessment and Intervention For 10 13, use the coordinate plane shown. Each unit represents 1 kilometer. 10. Write the ordered pairs that represent the location of Sam and the theater. Theater 11. Describe Sam s location relative to the theater. - - Sam - 1. Sam wants to meet his friend Beth at a restaurant before the go to the theater. The restaurant is 9 km south of the theater. Plot and label a point representing the restaurant. What are the coordinates of the point? - W N S E 13. Beth describes her current location: I m directl south of the theater, halfwa to the restaurant. Plot and label a point representing Beth s location. What are the coordinates of the point? For 1 15, use the coordinate plane shown. 1. Find the coordinates of points T, U, and V. 15. Points T, U, and V are the vertices of a rectangle. Point W is the fourth verte. Plot point W and give its coordinates. 16. Eplain the Error Janine tells her friend that ordered pairs that have an -coordinate of 0 lie on the -ais. She uses the origin as an eample. Describe Janine s error. Use a countereample to eplain wh Janine s statement is false. V U 1.0 0.5-1.0-0.5 0.5 1.0-0.5-1.0 T Lesson 1.1 335

FCUS N HIGHER RDER THINKING Work Area 17. Critical Thinking Choose scales for the coordinate plane shown so that ou can graph the points J(, 0), K(3, 10), L(3, -0), M(-, 50), and N(-5, -50). Eplain wh ou chose the scale for each ais. 18. Communicate Mathematical Ideas Edgar wants to plot the ordered pair (1.8, -1.) on a coordinate plane. n each ais, one grid square equals 0.1. Starting at the origin, how can Edgar find (1.8, -1.)? 19. Represent Real-World Problems Zach graphs some ordered pairs in the coordinate plane. The -values of the ordered pairs represent the number of hours since noon, and the -values represent the temperature at that time. a. In which quadrants could Zach graph points? Eplain our thinking. b. In what part of the world and at what time of ear might Zach collect data so that the points he plots are in Quadrant IV? 336 Unit 5

LESSN 1. Independent and Dependent Variables in Tables and Graphs Use variables to represent two quantities in a real-world problem that change in relationship to one another; Analze the relationship between the dependent and independent variables.? ESSENTIAL QUESTIN How can ou identif independent and dependent quantities from tables and graphs? EXPLRE ACTIVITY 1 Identifing Independent and Dependent Quantities from a Table Man real-world situations involve two variable quantities in which one quantit depends on the other. The quantit that depends on the other quantit is called the dependent variable, and the quantit it depends on is called the independent variable. A freight train moves at a constant speed. The distance in miles that the train has traveled after hours is shown in the table. Time (h) 0 1 3 Distance (mi) 0 50 100 150 A What are the two quantities in this situation? B Which of these quantities depends on the other? What is the independent variable? What is the dependent variable? How far does the train travel each hour? The relationship between the distance traveled b the train and the time in hours can be represented b an equation in two variables. Distance traveled (miles) = Distance traveled per hour Time (hours) = 50 Lesson 1. 337

EXPLRE ACTIVITY (cont d) Reflect 1. Analze Relationships Describe how the value of the independent variable is related to the value of the dependent variable. Is the relationship additive or multiplicative?. What are the units of the independent variable and of the dependent variable? 3. A rate is used in the equation. What is the rate? EXPLRE ACTIVITY Identifing Independent and Dependent Variables from a Graph In Eplore Activit 1, ou used a table to represent a relationship between an independent variable (time) and a dependent variable (distance). You can also use a graph to show this relationship. An art teacher has 0 pounds of cla but wants to bu more cla for her class. The amount of cla purchased b the teacher and the amount of cla available for the class are shown on the graph. A If the teacher bus 10 more pounds of cla, how man pounds will be available for the art class? If the art class has a total of 50 pounds of cla available, how man pounds of cla did the teacher bu? How can ou use the graph to find this information? lb Cla available for classes (Ib) 80 60 0 0 Cla Used in Art Class 0 0 60 80 Cla bought b teacher (Ib) Image Credits: Hill Street Studios/Corbis 338 Unit 5

B What are the two quantities in this situation? Which of these quantities depends on the other? What is the independent variable? What is the dependent variable? C D The relationship between the amount of cla purchased b the teacher and the amount of cla available to the class can be represented b an equation in two variables. Amount of cla available (pounds) = Current amount of cla (pounds) = 0 + + Amount of cla purchased (pounds) Describe in words how the value of the independent variable is related to the value of the dependent variable. Reflect. In this situation, the same units are used for the independent and dependent variables. How is this different from the situation involving the train in the first Eplore? 5. Analze Relationships Tell whether the relationship between the independent variable and the dependent variable is a multiplicative or an additive relationship. 6. What are the units of the independent variable, and what are the units of the dependent variable? independent variable: ; dependent variable: Lesson 1. 339

Math n the Spot Describing Relationships Between Independent and Dependent Variables Thinking about how one quantit depends on another helps ou identif which quantit is the independent variable and which quantit is the dependent variable. In a graph, the independent variable is usuall shown on the horizontal ais and the dependent variable on the vertical ais. EXAMPLE 1 A The table shows a relationship between two variables, and. Describe a possible situation the table could represent. Describe the independent and dependent variables in the situation. Independent variable, 0 1 3 Dependent variable, 10 11 1 13 As increases b 1, increases b 1. The relationship is additive. The value of is alwas 10 units greater than the value of. The table could represent Jina s savings if she starts with $10 and adds $1 to her savings ever da. The independent variable,, is the number of das she has been adding mone to her savings. The dependent variable,, is her savings after das. B The graph shows a relationship between two variables. Describe a possible situation that the graph could represent. Describe the independent and dependent variables. As increases b 1, increases b 1. The relationship is multiplicative. The value of is alwas 1 times the value of. 36 1 6 The graph could represent the number of eggs in cartons that each hold 1 eggs. The independent variable,, is the number of cartons. The dependent variable,, is the total number of eggs. Reflect 7. What are other possible situations that the table and graph in the Eamples could represent? 30 Unit 5

YUR TURN Describe real-world values that the variables could represent. Describe the relationship between the independent and dependent variables. 8. 0 1 3 15 16 17 18 Personal Math Trainer nline Assessment and Intervention 9. 0 1 3 0 16 3 8 6 10. 18 1 6 6 Lesson 1. 31

Guided Practice 1. A boat rental shop rents paddleboats for a fee plus an additional cost per hour. The cost of renting for different numbers of hours is shown in the table. Time (hours) 0 1 3 Cost ($) 10 11 1 13 What is the independent variable, and what is the dependent variable? How do ou know? (Eplore Activit 1). A car travels at a constant rate of 60 miles per hour. (Eplore Activit 1) a. Complete the table. Time (h) 0 1 3 Distance (mi) b. What is the independent variable, and what is the dependent? c. Describe how the value of the independent variable is related to the value of the dependent variable. Use the graph to answer the questions. 3. Describe in words how the value of the independent variable is related to the value of the dependent variable. (Eplore Activit ) 30 0?. Describe a real-world situation that the graph could represent. (Eample 1) ESSENTIAL QUESTIN CHECK-IN 5. How can ou identif the dependent and independent variables in a real-world situation modeled b a graph? 10 6 3 Unit 5

Name Class Date 1. Independent Practice 6. The graph shows the relationship between the hours a soccer team practiced after the season started and their total practice time for the ear. a. How man hours did the soccer team practice before the season began? b. What are the two quantities in this situation? c. What are the dependent and independent variables? Total practice time for ear (hours) Personal Math Trainer 10 nline Assessment and Intervention 8 6 Practice time during the season (hours) d. Is the relationship between the variables additive or multiplicative? Eplain. e. Analze Relationships Describe the relationship between the quantities in words. 7. Multistep Teresa is buing glitter markers to put in gift bags. The table shows the relationship between the number of gift bags and the number of glitter markers she needs to bu. a. What is the dependent variable? b. What is the independent variable? c. Is the relationship additive or multiplicative? Eplain. Number of gift bags, 0 1 3 Number of markers, 0 5 10 15 d. Describe the relationship between the quantities in words. Lesson 1. 33

8. T borrowed $500 from his parents. The graph shows how much he owes them each month if he pas back a certain amount each month. a. Describe the relationship between the number of months and the amount T owes. Identif an independent and dependent variable and eplain our thinking. Amount T owes (dollars) 500 00 300 00 100 T s Loan Paments 6 8 10 Months b. How long will it take T to pa back his parents? FCUS N HIGHER RDER THINKING Work Area 9. Error Analsis A discount store has a special: 8 cans of juice for a dollar. A shopper decides that since the number of cans purchased is 8 times the number of dollars spent, the cost is the independent variable and the number of cans is the dependent variable. Do ou agree? Eplain. 10. Analze Relationships Provide an eample of a real-world relationship where there is no clear independent or dependent variable. Eplain. 3 Unit 5

? LESSN 1.3 ESSENTIAL QUESTIN Writing Equations from Tables How can ou use an equation to show a relationship between two variables? write an equation to epress one quantit, thought of as the dependent variable, in terms of the other quantit, thought of as the independent variable. EXPLRE ACTIVITY Writing an Equation to Represent a Real-World Relationship Man real-world situations involve two variable quantities in which one quantit depends on the other. This tpe of relationship can be represented b a table. You can also use an equation to model the relationship. The table shows how much Amanda earns for walking 1,, or 3 dogs. Use the table to determine how much Amanda earns per dog. Then write an equation that models the relationship between number of dogs walked and earnings. Use our equation to complete the table. Dogs walked 1 3 5 10 0 Earnings $8 $16 $ A For each column, compare the number of dogs walked and earnings. What is the pattern? For 1 dog, Amanda earns 1 8 = $8. For dogs, she earns 8 = $16. B C D E Based on the pattern, Amanda earns $ for each dog she walks. Write an equation that relates the number of dogs Amanda walks to the amount she earns. Let e represent earnings and d represent dogs. Use our equation to complete the table for 5, 10, and 0 walked dogs. Amanda s earnings depend on. Reflect 1. What If? If Amanda changed the amount earned per dog to $11, what equation could ou write to model the relationship between number of dogs walked and earnings? Lesson 1.3 35

Math n the Spot Writing an Equation Based on a Table The relationship between two variables where one variable depends on the other can be represented in a table or b an equation. An equation epresses the dependent variable in terms of the independent variable. When there is no real-world situation to consider, we usuall sa is the independent variable and is the dependent variable. The value of depends on the value of. EXAMPLE 1 Animated Math Write an equation that epresses in terms of. A 1 3 5 0.5 1 1.5.5 STEP 1 STEP Compare the - and -values to find a pattern. Each -value is 1, or 0.5 times, the corresponding -value. Use the pattern to write an equation epressing in terms of. = 0.5 B 6 8 10 5 7 9 11 13 STEP 1 Compare the - and -values to find a pattern. Each -value is 3 more than the corresponding -value. Math Talk Mathematical Practices How can ou check that our equations are correct? STEP YUR TURN Use the pattern to write an equation epressing in terms of. = + 3 For each table, write an equation that epresses in terms of.. 1 11 10 3. 10 9 8 10 1 1 5 30 35 Personal Math Trainer nline Assessment and Intervention. 5 3 5. 10 9 8 0 1 0 36 Unit 5

Using Tables and Equations to Solve Problems You can use tables and equations to solve real-world problems. EXAMPLE Problem Solving Math n the Spot A certain percent of the sale price of paintings at a galler will be donated to charit. The donation will be $50 if a painting sells for $00. The donation will be $75 if a painting sells for $300. Find the amount of the donation if a painting sells for $1,00. Analze Information You know the donation amount when the sale price of a painting is $00 and $300. You need to find the donation amount if a painting sells for $1,00. Formulate a Plan You can make a table to help ou determine the relationship between sale price and donation amount. Then ou can write an equation that models the relationship. Use the equation to find the unknown donation amount. Justif Solve and Evaluate Make a table. Sale price ($) 00 300 Donation amount ($) 50 75 50 00 = 50 00 = 5 100 = 5% 75 300 = 75 3 300 3 = 5 100 = 5% Write an equation. Let p represent the sale price of the painting. Let d represent the donation amount to charit. The donation amount is equal to 5% of the sale price. d = 0.5 p Find the donation amount when the sale price is $1,00. d = 0.5 p d = 0.5 1,00 d = 300 Simplif to find the donation amount. When the sale price is $1,00, the donation to charit is $300. Justif and Evaluate Substitute $1,00 for the sale price of the painting. Substitute values from the table for p and d to check that the are solutions of the equation d = 0.5 p. Then check our answer of $300 b substituting for d and solving for p. d = 0.5 p d = 0.5 p d = 0.5 p d = 0.5 00 d = 0.5 300 300 = 0.5 p d = 50 d = 75 p = 1,00 ne wa to determine the relationship between sale price and donation amount is to find the percent. p is the independent variable; its value does not depend on an other value. d is the dependent variable; its value depends on the price of the painting. Lesson 1.3 37

YUR TURN Personal Math Trainer nline Assessment and Intervention 6. When Ran is 10, his brother Kle is 15. When Ran is 16, Kle will be 1. When Ran is 1, Kle will be 6. Write and solve an equation to find Kle s age when Ran is 5. Guided Practice Write an equation to epress in terms of. (Eplore Activit, Eample 1) 1. 10 0 30 0. 6 16 6 36 0 1 3 0 8 1 3. 6 8 10. 7 9 11 13 1 36 8 6 8 5. Jameson downloaded one digital song for $1.35, two digital songs for $.70, and 5 digital songs for $6.75. Write and solve an equation to find the cost to download 5 digital songs. (Eample ) Songs downloaded 1 5 10? Total cost ($) 1.35 Number of songs = n; Cost = The total cost of 5 songs is ESSENTIAL QUESTIN CHECK-IN 6. Eplain how to use a table to write an equation that represents the relationship in the table. 38 Unit 5

Name Class Date 1.3 Independent Practice Personal Math Trainer nline Assessment and Intervention 7. Vocabular What does it mean for an equation to epress in terms of? 8. The length of a rectangle is inches more than twice its width. Write an equation relating the length l of the rectangle to its width w. 9. Look for a Pattern Compare the -values in the table to the corresponding -values. What pattern do ou see? How is this pattern used to write an equation that represents the relationship between the - and -values? 0 8 3 5 6 7 8 10. Eplain the Error A student modeled the relationship in the table with the equation =. Eplain the student s error. Write an equation that correctl models the relationship. 6 8 8 16 3 11. Multistep Marvin earns $8.5 per hour at his summer job. He wants to bu a video game sstem that costs $06.5. a. Write an equation to model the relationship between number of hours worked h and amount earned e. b. Solve our equation to find the number of hours Marvin needs to work in order to afford the video game sstem. Lesson 1.3 39

1. Communicate Mathematical Ideas For ever hour that Noah studies, his test score goes up 3 points. Eplain which is the independent variable and which is the dependent variable. Write an equation modeling the relationship between hours studied h and the increase in Noah s test score s. FCUS N HIGHER RDER THINKING Work Area 13. Make a Conjecture Compare the -values in the table to the corresponding -values. Determine whether there is an additive relationship or a multiplicative relationship between and. If possible, write an equation modeling the relationship. If, not eplain wh. 1 3 5 7 3 6 8 1 1. Represent Real-World Problems Describe a real-world situation in which there is an additive or multiplicative relationship between two quantities. Make a table that includes at least three pairs of values. Then write an equation that models the relationship between the quantities. 15. Critical Thinking Georgia knows that there is either an additive or multiplicative relationship between and. She onl knows a single pair of data values. Eplain whether Georgia has enough information to write an equation that models the relationship between and. 350 Unit 5

LESSN 1. Representing Algebraic Relationships in Tables and Graphs...Analze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.? ESSENTIAL QUESTIN How can ou use verbal descriptions, tables, and graphs to represent algebraic relationships? EXPLRE ACTIVITY 1 Representing Algebraic Relationships Angie s walking speed is 5 kilometers per hour, and Ma s is kilometers per hour. Show how the distance each girl walks is related to time. A B For each girl, make a table comparing time and distance. Time (h) 0 1 3 Angie s distance (km) 0 5 10 Time (h) 0 1 3 Ma s distance (km) 0 8 For each girl, make a graph showing her distance as it depends on time. Plot points from the table and connect them with a line. Write an equation for each girl that relates distance to time. For ever hour Angie walks, she travels 5 km. For ever hour Ma walks, she travels km. Distance (km) 0 16 1 8 Angie 1 3 5 Time (h) Angie s equation: Distance (km) 0 16 1 8 Ma s equation: Ma 1 3 5 Time (h) Reflect 1. Analze Relationships How can ou use the tables to determine which girl is walking faster? How can ou use the graphs? Math Talk Mathematical Practices Wh does it make sense to connect the points in each graph? Lesson 1. 351

EXPLRE ACTIVITY Writing an Equation from a Graph Cherise pas the entrance fee to visit a museum, then bus souvenirs at the gift shop. The graph shows the relationship between the total amount she spends at the museum and the amount she spends at the gift shop. Write an equation to represent the relationship. A Read the ordered pairs from the graph. Use them to complete a table comparing total spent to amount spent at the gift shop. B C Reflect Gift shop amount ($) 0 5 10 15 Total amount ($) 5 10 What is the pattern in the table? Write an equation that epresses the total amount spent,, in terms of the amount spent at the gift shop,.. Communicate Mathematical Ideas Identif the dependent and independent quantities in this situation. 3. Multiple Representations Draw a line through the points on the graph. Find the point that represents Cherise spending $18 at the gift shop. Use this point to find the total she would spend if she spent $18 at the gift shop. Then use our equation from C to verif our answer. Total amount ($) 3 8 0 16 1 8 8 1 16 0 Gift shop amount ($) Image Credits: Thinkstock/ Corbis 35 Unit 5

Graphing an Equation An ordered pair (, ) that makes an equation like = + 1 true is called a solution of the equation. The graph of an equation represents all the ordered pairs that are solutions. EXAMPLE 1 Math n the Spot Graph each equation. A = + 1 STEP 1 STEP STEP 3 Make a table of values. Choose some values for and use the equation to find the corresponding values for. Plot the ordered pairs from the table. Draw a line through the plotted points to represent all of the ordered pair solutions of the equation. + 1 = (, ) 1 1 + 1 = (1, ) + 1 = 3 (, 3) 3 3 + 1 = (3, ) + 1 = 5 (, 5) 5 5 + 1 = 6 (5, 6) 10 8 6 1 3 5 Math Talk Mathematical Practices Is the ordered pair (3.5,.5) a solution of the equation = + 1? Eplain. B = STEP 1 STEP STEP 3 Make a table of values. Choose some values for and use the equation to find the corresponding values for. Plot the ordered pairs from the table. Draw a line through the plotted points to represent all of the ordered pair solutions of the equation. = (, ) 1 1 = (1, ) = (, ) 3 3 = 6 (3, 6) = 8 (, 8) 5 5 = 10 (5, 10) 10 8 6 1 3 5 Lesson 1. 353

YUR TURN Personal Math Trainer nline Assessment and Intervention. Graph = +.5. +.5 = (, ) 10 8 6 1 3 5 Guided Practice Frank mows lawns in the summer to earn etra mone. He can mow 3 lawns ever hour he works. (Eplore Activit 1 and Eplore Activit ) 1. Make a table to show the relationship between the number of hours Frank works,, and the number of lawns he mows,. Graph the relationship and write an equation. Label the aes of our graph. Hours worked 0 1 Lawns mowed 10 8 6 1 3 5 Graph = 1.5. (Eample 1)?. Make a table to show the relationship. 3. Plot the points and draw a line through them. ESSENTIAL QUESTIN CHECK-IN 5 3 1 1. How can a table represent an algebraic relationship between two variables? 3 5 35 Unit 5

Name Class Date 1. Independent Practice Personal Math Trainer nline Assessment and Intervention Students at Mills Middle School are required to work a certain number of communit service hours. The table shows the numbers of additional hours several students worked beond their required hours, as well as the total numbers of hours worked. 5. Read the ordered pairs from the graph to make a table. Additional hours Total hours 6. Write an equation that epresses the total hours in terms of the additional hours. Total (h) 50 0 30 0 10 10 0 30 0 50 Additional (h) 7. Analze Relationships How man communit service hours are students required to work? Eplain. Beth is using a map. Let represent a distance in centimeters on the map. To find an actual distance in kilometers, Beth uses the equation = 8. 8. Make a table comparing a distance on the map to the actual distance. Map distance (cm) Actual distance (km) 9. Make a graph that compares the map distance to the actual distance. Label the aes of the graph. 10. Critical Thinking The actual distance between Town A and Town B is 6 kilometers. What is the distance on Beth s map? Did ou use the graph or the equation to find the answer? Wh? 50 0 30 0 10 1 3 5 Lesson 1. 355

11. Multistep The equation = 9 represents the total cost for movie tickets. Label the aes of the graph. a. Make a table and a graph to represent the relationship between and. Number of tickets, 50 Total cost ($), 0 30 b. Critical Thinking In this situation, which quantit is dependent and which is independent? Justif our answer. 0 10 1 3 5 c. Multiple Representations Eight friends want to go see a movie. Would ou prefer to use an equation, a table, or a graph to find the cost of 8 movie tickets? Eplain how ou would use our chosen method to find the cost. FCUS N HIGHER RDER THINKING Work Area 1. Critical Thinking Suppose ou graph = 5 and = + 500 on the same coordinate plane. Which line will be steeper? Wh? 13. Persevere in Problem Solving Marcus plotted the points (0, 0), (6, ), (18, 6), and (1, 7) on a graph. He wrote an equation for the relationship. Find another ordered pair that could be a solution of Marcus s equation. Justif our answer. 1. Error Analsis The cost of a personal pizza is $. A drink costs $1. Anna wrote the equation = + 1 to represent the relationship between total cost of buing meals that include one personal pizza and one drink. Describe Anna s error and write the correct equation. 356 Unit 5

MDULE QUIZ Read 1.1 Graphing on the Coordinate Plane Graph each point on the coordinate plane. Personal Math Trainer nline Assessment and Intervention 1. A(-, ). B(3, 5) 3. C(6, -). D(-3, -5) 5. E(7, ) 6. F(-, 6) 6-6 - 6 1. Independent and Dependent Variables in Tables and Graphs 7. Jon bus packages of pens for $5 each. Identif the independent and dependent variables in the situation. -6 1.3 Writing Equations from Tables Write an equation that represents the data in the table. 8. 3 5 8 10 9. 5 10 15 0 1 35 56 70 17 7 3 1. Representing Algebraic Relationships in Tables and Graphs Graph each equation. 10. = + 3 11. = 5 8 6 6 8 0 30 0 10 6 8 ESSENTIAL QUESTIN 1. How can ou write an equation in two variables to solve a problem? Module 1 357

MDULE 1 MIXED REVIEW Assessment Readiness Personal Math Trainer nline Assessment and Intervention Selected Response 1. What are the coordinates of point G on the coordinate grid below? - - - G - A (, 3) C (-, 3) B (, -3) D (-, -3). A point is located in quadrant II of a coordinate plane. Which of the following could be the coordinates of that point? A (-5, -7) C (-5, 7) B (5, 7) D (5, -7) 3. Matt had 5 librar books. He checked 1 additional book out ever week without returning an books. Which equation describes the number of books he has,, after weeks? A = 5 B = 5 - C = 1 + 5 D = 5 +. Stewart is plaing a video game. He earns the same number of points for each prize he captures. He earned 1,00 points for 6 prizes,,000 points for 10 prizes, and,600 points for 13 prizes. Which is the dependent variable in the situation? 358 Unit 5 A the number of prizes captured B the number of points earned C the number of hours D the number of prizes available 5. Which point is not on the graph of the equation = 10 +? A (0, 10) C (8, ) B (3, 13) D (5, 15) 6. Am gets paid b the hour. Her sister helps. As shown, Am gives her sister part of her earnings. Which equation represents Am s pa when her sister s pa is $13? Mini-Task Am s pa in dollars Sister s pa in dollars 16 1 8 8 1 16 10 0 30 0 6 8 A = 13 5 C 5 = 13 B 13 = 5 D 13 = 5 7. The table compares the ages, in ears, of two cousins. Ann s age, 8 1 Tom s age, 8 1 16 a. Write an equation that compares Tom s and Ann s ages. b. Draw a graph to represent the equation.