Using Two Variables to Represent a Relationship LAUNCH (8 MIN) Before What is generally meant by the phrase think outside the box? During How is it possible that two cardboard boxes can be the same size but different weights? After Suppose the tape, staples, and/or glue that hold the box together are very light compared to the rest of the box. Do they still relate to its weight? PART 1 (7 MIN) During the Intro The quantities are described as related and unknown. How do you know that the quantities have some affect on each other and that we do not know their values? Before solving the problem What are some ways to place an order? If 50 orders were placed by phone, how many would have been placed online? If more than 50 orders were placed by phone, would the number of orders placed online increase or decrease? After completing the solution Why would the fruit basket business care how the orders are related? PART 2 (8 MIN) Xiao Says (Screen 1) Use the Xiao Says button to point out that variables should have a clear connection to the quantities they represent. Why would t not be a good choice of variable for time spent doing warm-ups? After completing the solution Why doesn t it matter whether you write w or d first in the equation? KEY CONCEPT (2 MIN) Students are introduced to the terms independent variable and dependent variable in order to qualify which variable quantity in a relationship affects the other. Xiao Says Use the Xiao Says button to explain that some variables, like the box in the Launch, depend on several independent variables. PART 3 (8 MIN) Which variable is easier for you to identify? Do you agree or disagree with the following statement? The independent variable doesn t change. After solving the problem How would you classify the quantities population of a state and number of senators from that state? CLOSE AND CHECK (8 MIN) Why can you not solve the equations in this lesson? Besides equations, how else can you model equality?
Using Two Variables to Represent a Relationship LESSON OBJECTIVES 1. Use variables to represent two quantities in a real-world problem that change in relationship to one another. 2. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. FOCUS QUESTION What does it mean for one quantity to depend on another? How do you represent such a relationship? MATH BACKGROUND Prior to this lesson, students learned about equations and inequalities with one variable. They learned to keep equations balanced by doing the same thing to each side and applied that concept using inverse operations to solve equations and inequalities. In some cases, they wrote and solved equations and inequalities that modeled real-world problems. In this lesson, students learn to use variables to represent two related and unknown quantities. They learn to identify the variables as dependent or independent and to represent these relationships with equations. They come to understand that a change in one variable affects the value of the other variable, so they cannot solve twovariable equations without knowing the value of one of the variables. In future lessons, students will use tables and graphs to help identify patterns and then represent those relationships with equations. In Grades 7 and 8, students will use twovariable equations to describe proportions in the topics Proportions and Proportional Relationships, Lines, and Linear Equations and solve systems of equations in the topic Systems of Two Linear Equations. LAUNCH (8 MIN) Objective: Identify independent and dependent real-world relationships. Students identify quantities that affect another quantity, as well as quantities that do not. They begin to explore whether a relationship exists between two quantities and what that relationship is. They prepare to formally learn about independent and dependent variables and how to write an equation to show the relationship between two quantities using variables. Before What is generally meant by the phrase think outside the box? [Sample answer: thinking in nontraditional and unexpected ways or having a more creative outlook and approach] During How is it possible that two cardboard boxes can be the same size but different weights? [Sample answer: They may be made of different thicknesses of cardboard.]
Using Two Variables to Represent a Relationship continued After Suppose the tape, staples, and/or glue that hold the box together are very light compared to the rest of the box. Do they still relate to its weight? [Yes; no matter how insignificant, the box materials relate to its weight.] Solution Notes In addition to the box itself, students may list the size of the box, the materials that hold the box together, the thickness of the box, or the contents of the box. Encourage students to choose relevant things that do not relate to the weight of the box. The color of the box is a relevant suggestion, while the color of the sky is not related to the box in the first place. Connect Your Learning Move to the Connect Your Learning screen. In the Launch, students considered what factors affect the weight of a box. Flip this relationship around and discuss which factors the weight of the box depends on. Use the Focus Question to start a discussion about quantities that have a relationship and quantities that do not. PART 1 (7 MIN) Objective: Identify two quantities in a real-world problem that change in relationship to one another. Students find the relationship between two quantities whose values are variable but whose sum is constant. During the Intro The quantities are described as related and unknown. How do you know that the quantities have some affect on each other and that we do not know their values? [Sample answer: You know there is a total of 12 pieces of fruit, but you do not know the exact number of apples or pears. As one quantity increases, the other decreases.] Before solving the problem What are some ways to place an order? [Sample answers: in person; online; over the phone; by mail] How can you find the number of online orders? [Sample answer: You cannot find the number of online orders because you do not know the number of orders placed over the phone.] If 50 orders were placed by phone, how many would have been placed online? If more than 50 orders were placed by phone, would the number of orders placed online increase or decrease? [53 orders placed online; the number would decrease] After completing the solution Why would the fruit basket business care how the orders are related? [Sample answer: They can plan how to advertise fruit baskets or decide how much phone coverage they need.] Solution Notes You can use the Addition mode of the Place-Value Blocks tool to show different ways to split up the two types of orders. Model how increasing one means decreasing the other.
Got It Notes Using Two Variables to Represent a Relationship continued Students may ask why the number of orders is different. Tell them that this number can be for a different business or could be the same business for a different month. Emphasize that the variables have changed, so you are investigating a different aspect of the business either way. You can discuss with students why this information is important. A business should know its customers. For example, suppose most of the customers were buying baskets at the last minute, for example. The company should charge more for overnight shipping to encourage people to purchase baskets ahead of time. If you show answer choices, consider the following possible student errors: Students who select B or D have not used information from the problem statement. An answer of B may mean that the student is using information about the number of orders from the Example. For choice D, the students may be taking information from the Intro. Although an answer of C shows two related quantities, the total is not an unknown quantity. PART 2 (8 MIN) Objective: Use variables to represent two quantities in a real-world problem (with only one operation) that change in relationship to one another. Students learn to represent related and unknown quantities with variables and to use those variables in an equation that models the relationship. Instructional Design Use a Words-to-Equation organizer to show students how to write an equation from a two-variable relationship. Stress that variables are numbers, so a represents the number of apples, not apples. Move to Screen 2 to present a similar problem that can be modeled with an equation. Use the blank Know-Need-Plan organizer provided to help students identify the variable quantities and the relationship. Xiao Says (Screen 1) Use the Xiao Says button to point out that variables should have a clear connection to the quantities they represent. Why would t not be a good choice of variable for time spent doing warm-ups? [t could also stand for time doing drills, or possibly even total time.] After solving the problem Why doesn t it matter whether you write w or d first in the equation? [The operation is addition, so you can apply the Commutative Property.]
Solution Notes Using Two Variables to Represent a Relationship continued You can compare your Know-Need-Plan organizer to the provided solution. Make sure students understand the three distinct parts of the solution: assigning variables, writing the relationship, and modeling the relationship with an equation. Students may want to solve this equation, so remind them that they only know how to solve equations with one variable. Emphasize that you do not need to know the values of the variables because the problem only asks you to write an equation. Differentiated Instruction For struggling students: You can divide the class into pairs and have each student write an explanation of a real-world situation that involves two related and unknown quantities, such as the number of shirts purchased and the amount of money spent or the height of a plant and the number of days it has been growing. Have each student define variables and write an equation that describes their partner s situation. For advanced students: Ask students to use what they know about balancing equations to show that w 90 d is also a possible solution. Got It Notes While the two variables in the Example were both parts of a total, one of the variables in this problem is the total. This is the first time that the number shown in the problem is not the total. Note the use of a capital T as a variable in the provided solution. You can use the Words to Equation organizer in the provided solution to help students connect the wording of the problem statement to the equation. Got It 2 Notes Make sure students justify their answer of either Yes or No. You might list all the equations students came up with for the previous Got It problem about gift-wrapping. KEY CONCEPT (2 MIN) Teaching Tips for the Key Concept Students are introduced to the terms independent variable and dependent variable in order to qualify which variable quantity in a relationship affects the other. Xiao Says Use the Xiao Says button to explain that some variables, like the box in the Launch, depend on several independent variables. In this example, the amount a plant grows is dependent on several factors, or quantities. Why it Works Explain to students that independent and dependent have the same meanings in mathematics as they do otherwise. Ask students what those meanings are. Help students remember the terms by connecting the everyday and mathematical meanings: the dependent variable needs to depend on other variables for a value, and the independent variable is so independent that it doesn t need to rely on other variables. PART 3 (8 MIN) Objective: Analyze the relationship between the dependent and independent variables. ELL Support On the Student Companion page for the Part 3 Got It, there are three tasks for students to complete and discuss:
Using Two Variables to Represent a Relationship continued What do you know about buying flowers that helped you with this problem? Which action, picking out the flowers or finding the cost of the flowers, happens first? How does this help you determine the dependent variable? Beginning Have students act out the problem. Then have students discuss the actions and the importance of the order of the actions. Intermediate Have students discuss how they applied their background knowledge to answer this problem. Did they have personal experience and knowledge of buying flowers to form a bouquet? Advanced Have students explain why the order matters by changing the order of the actions and discussing the result. For each statement, students identify two related variables and label one as independent and the other dependent. They also practice describing the relationship using words. Instructional Design You can point out the use of color to highlight the quantities in each statement. Call on students to choose each statement and identify the independent and dependent variable. Which variable is easier for you to identify? [Sample answers: Independent variable; it is the quantity that can affect other quantities. Dependent variable; it relies on the other quantity highlighted.] Do you agree or disagree with the following statement? The independent variable doesn t change. [Sample answer: I disagree; the independent variable does change, but its value is not affected by the other variable. For example, the number of miles you bike can change from day to day, but the number of calories you burn does not affect the number of miles you bike.] After solving the problem How would you classify the quantities population of a state and number of senators from that state? [The number of senators is not dependent on the population of a state. There are two Senators for each state.] Solution Notes Being able to state the solutions in sentence form indicates a strong understanding of the relationship between the dependent and independent variables. Got It Notes If you show answer choices, consider the following possible student errors: Students who choose A have found the independent variable. Tell students who choose C or D that a quantity cannot be the dependent variable if it is not mentioned in the problem statement, even if it may be a relevant variable.
Using Two Variables to Represent a Relationship continued CLOSE AND CHECK (7 MIN) Focus Question Sample Answer A quantity depends on another quantity if a change in the second quantity affects the first quantity. You represent such a relationship with a two-variable equation. Focus Question Notes Excellent responses to the Focus Question should mention the terms independent variable and dependent variable. Students may or may not realize that having two variables is not a guarantee that a relationship exists. They may also use the Launch as an example of a real-world quantity that depends on more than one variable. Students may notice that some problems in this lesson did not necessarily have an independent and dependent variable. For example, the Intro of Part 1 involves two variables that affect each other, but either could be the independent variable depending on the situation. Emphasize that you can use an equation to show the value of one variable in terms of the other. There are many ways to write this equation, but encourage students to write the dependent variable in terms of the independent variable. Essential Question Connection Previously, students worked with equations that had one variable and could be solved. Now students are writing equations involving two variables and looking at the relationship between them. The Essential Question asks, How are two-variable relationships different from one-variable relationships? When do you need two variables? Use the questions that follow to help students begin to answer the Essential Question. Why can you not solve the equations in this lesson? [Sample answer: You can solve an equation if you can isolate the variable so that the other side of the equation is a numerical expression. A two-variable equation still has a variable on the other side, so you need to know the value of one variable to find the other.] Besides equations, how else can you model equality? [Sample answer: You can use a table, a graph, or words to show that two quantities are equal.]