Buy one for $5 and get the rest for $2! (While Supplies Last)

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1 Model Linear Functions with Graphs, Tables, and Equations Student Probe Erica saw the advertisement below for a store that was having a back to school sale on notebooks. She knew it could be modeled by a linear function. Write a linear function that describes the cost of the notebooks as a function of the number of notebooks purchased. Back to School Notebook Sale Buy one for $5 and get the rest for $2! (While Supplies Last) Answer: Either f(x) = 2(x 1) + 5 ($2 for every notebook except the first one plus the five dollars for the first one) or f(x) = 2x + 3 ($2 for every one plus $3 extra for the first one). Lesson Description In this lesson, students will use linear, real world scenarios to model functions with tables, graphs, and equations using function notation. Rationale Understanding problem situations and describing them mathematically is a crucial skill for students as they At a Glance What: Models real life linear functions using tables, graphs, and equations. Common Core State Standard: CC.9 12.F.IF.4. For a function that models a relationship between two qualities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Matched Arkansas Standard: AR.9 12.LF.TM.1.1 (LF.1.TM.1) Identify a linear relationship represented by a table, by a graph, and by symbolic forms Mathematical Practices: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Who: Students who cannot create models of linear functions. Grade Level: Algebra I Prerequisite Vocabulary: independent variable, dependent variable, slope Prerequisite Skills: Create tables and graphs Delivery Format: Individual, pairs, small group, whole group prepare for their work futures, as well as their future mathematics courses. The ability to differentiate among the types of functions (linear, quadratic, exponential, etc.) and their representations helps students as they describe rates of growth, future values, and reasonableness of calculations. Students need to be able to move fluently among representations, gleaning important information regardless of the presentation of the function. Preparation Review Tables, Graphs, and Equations of Linear Functions. It may be a more appropriate lesson for students who have difficulty understanding linear functions.

2 Lesson The teacher says or does Expect students to say or do If students do not, then the teacher says or does 1. The parking rate for a Little Hours Cost in $ Model creating the table. Rock company is an initial (x) (y) charge of $2 and then $1 for each hour. Construct a table with hours as the independent variable (x, up to 8) and the cost as the dependent variable (y). 2. Construct a graph using the data in the table What is the cost when the car is first parked? What is the charge after one hour?... Model for students. 3. Is this an example of a linear, quadratic, or exponential function? How do you know? 4. How do you know this relationship is linear? 5. What is the constant rate of change, or slope? How did you figure that out? (Students should be able to determine the slope from both the table and the graph.) What does the slope mean in Linear, because it graphs as a line. It increases at a constant rate. OR It has a constant rate of change. The slope is 1. Answers may vary, but listen for, I looked at the graph and the rise over run is one, and I looked at the table and the cost increases by one dollar for every hour. $2 per hour Refer to Tables, Graphs, and Equations of Linear Functions. Review the definition of slope. Model on the table and on the graph.

3 the context of the problem? The teacher says or does Expect students to say or do If students do not, then the teacher says or does 6. What is the y intercept? How do you know? (Students should be able to determine the y intercept from both the table and the graph.) 2 Answers may vary, but listen for, I looked at the graph and the line crosses the y axis at 2, and I looked at the table and the charge at 0 hours is Review the definition of y intercept. Model on the graph and on the table. 7. Write an equation of the line we graphed. 8. Write the equation using function notation. 9. Let s consider another situation. Amanda works at a fast food restaurant and earns $7.25 per hour. Construct a table with her hours worked as the independent variable and her pay as the dependent variable. Use 0 to 8 hours worked. 10. Construct a graph using the data in the table. $2. y x 2 Remember the equation for lines: y mx b. What is m? What is b? x Review the definition of function notation. 2 f x Hours Cost in $ (x) (y) $ $ $ $ $36 6 $ $ $57.75 Model creating the table. How much would she earn if she worked 0 hours? If she worked 1 hour?... Model for students. 11. Write an equation of the line we graphed in function notation. 12. How is this equation like the one we wrote earlier? 7.25 f x x What is the rate of change (slope)? What is the y intercept? Answers will vary, but listen Prompt students. for, linear and constant rate of change.

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5 The teacher says or does Expect students to say or do If students do not, then the teacher says or does 13. How are the equations different? Prompt students. 14. Repeat the lessons with other linear scenarios, if necessary. Answers will vary, but listen for, the slopes are different, and the y intercepts are different. Teacher Notes 1. The use of graphing calculators is suggested for this lesson. 2. Student may require numerous opportunities over time to work with mathematical models before they grasp the important concepts. Several short sessions over several weeks may be more effective use of instructional time than a single extended session. 3. As students become proficient in creating and using multiple representations of linear functions, include a mix of quadratic and/or exponential functions and their representations. Variations Other scenarios might include the following. 1. A salesperson makes $4000 per month and 6% commission. 2. The distance sound travels per second can be modeled as y = 0.21t where y is the distance traveled and t is the time in seconds. 3. Compare two linear equations to represent cell phone rates or taxi fares and picking the best plan. Formative Assessment A taxi fare is calculated the following way: passengers pay a flat $5 fee and an additional $1 per mile driven. Construct a table for taxi rides ranging from 0 to 8 miles. Construct a graph using the data in the table. Write a function that models the fare as a function of distance. Answer: f x x 5

6 References Algebra I, Integration, Applications, Connections; Glencoe/McGraw Hill, 1998 Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on May 12, 2011.