1 Students will explore linear functions through two activities. First, students will work in groups to connect a table, graph and equation that represent the same linear function. They will write a verbal description of the function and be asked to justify why the different representations relate to the same function. Then, students will compare two linear functions given a real life context. Suggested Grade Range: 6-9 Approximate Time: Two 45-minute lessons or one 90-minute lesson Common Core State Standards (CA): CCSS.Math.Content.8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). CCSS.Math.Content.8.F.A.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Lesson Content Objectives: Make connections among tables, graphs, equations and verbal descriptions of a linear function Determine when to use a specific representation (table, graph, equation or verbal description) of a linear equation Compare two linear functions each represented in a different way Materials Needed: Warm-up problem displayed on white-board or document camera. One copy per student of Representations of Linear Functions and Independent Practice handouts (included). Who Shares My Function? task (included). Make enough copies so that each student can be given a table, graph or equation handout. 1 An early version of this lesson was adapted and field-tested by Lauren Belcher, Michelle Castro, Cynthia Pulido, Thao Vo and Lindsey Worthy, participants in the California State University, Long Beach Foundational Level Mathematics/General Science Credential Program. STEM Activities for Middle and High School Students 1
Summary of Lesson Sequence: Provide the Warm-up problem (included) for students to complete independently. Introduce the lesson by connecting the warm-up problem to the lesson objectives. Model for students how a linear function can be represented by a verbal description, graph, table and equation using Representations of Linear Functions notes. Support students as they participate in Who Shares My Function Task by asking guiding questions that allow them to identify key aspects of a linear function represented by a table, graph, equation and verbal description. Provide the Independent Practice for students to compare two linear functions each represented in a different way (verbal description and equation). Close the lesson with a discussion of purposes of each representation of a linear function. Assumed Prior Knowledge: Prior to this lesson, students should be familiar with graphing points on a Cartesian plane. Students should have a basic definition of linear functions including that the graph of all linear functions is a straight line. Classroom Set-up: Students will be asked to participate in discussions and work in small groups for portions of this lesson. Lesson Description: Introduction Provide students with the following Warm-up problem: Maria needs to order shirts for the 8 th grade class at Cedar Middle School. The company she hires to print the shirts charges a $35.00 design fee and $6.00 for each shirt ordered. How much will Maria pay if she orders 5 shirts? 10 shirts? 50 shirts? x shirts? Students should first complete the word problem independently. Have students then share answers. If students have difficulty writing the equation for x shirts, model how one can use his/her strategy to determine the cost of 5, 10 and 50 shirts to determine the cost of x shirts. Record student answers in a table. Discuss with the class how the table represents distinct data points or costs (i.e. 5 shirts cost $65.00), whereas the rule they created for x shirts (Cost = 35 + 6x) generates an equation, that can be used to determine the cost of any amount of shirts. Ask: Is the relationship between cost and number of shirts ordered linear? How do we know? STEM Activities for Middle and High School Students 2
Tell students that they will be exploring multiple representations (tables, graphs, equations and verbal descriptions) of linear functions. Refer to the lesson objectives. Input and Model Provide students with Representations of Linear Functions handout. Representations of Linear Functions Verbal Description Graph John has $7 left over from his birthday. He is saving up for a new comic book subscription. Every week, John saves a dollar of his allowance for the comic book. Table Equation Model for students how to represent the verbal description as a table, graph, and equation: Verbal Description: The situation provides a starting value ($7 savings) and then a constant rate of change ($1 per week). Since this change is constant, this situation is an example of a linear function. Ask: Would the situation still be linear if John saved different amounts of money each week? Table: The table should represent John s savings over time. Each week, John saves more money. Demonstrate how to determine table values for 0, 1, and 2 weeks. Ask: STEM Activities for Middle and High School Students 3
How can we see the constant rate of change in the table? Where do we see the starting value? [Refer to the slope and y-intercept]. Given the coordinate pair (5, y), what are two ways we can get the y-value? What does this coordinate mean in the context of the verbal description? Equation: Using the starting value and constant rate of change, model how one can write an equation to represent the verbal description (y = 1x + 7). This is a linear equation because it is written in slope-intercept form where 1 represents the slope and 7 represents the y-intercept. Ask: What is the unit for the slope? For the y-intercept? What do the variables x and y represent in this equation? What are their units? How do we know? Graph: Model how one can graph the values in the table. Discuss with students how to appropriately label the x- and y-axes. Ask: How do we know that our graph represents a linear function? Where on the graph can we see our starting value and our constant rate of change? [Again, refer to the slope and y-intercept]. Guide Student Through Their Practice The Who Shares My Function? Task has four unique sets of table, graph and equation triplets. Provide each student with one handout that has a graph, table or equation. Ask students to walk around the room to find two other people that have the corresponding graph, table or equation. For example, if a student is given the linear equation y = 5x + 7, he/she needs to find two people with a corresponding table and graph that represent that equation. Once they have found their group of three, ask students to record the missing representations on their handout. Then have students work together to write three justifications for why the table, graph and equation represent the same function. Bring the class back together. Explain to students that they will first write a verbal description of their equation using the idea of a savings account: Each week a person adds or subtracts the same amount of money from the account. The amount of money they have over time depends on how much money they started in the account and the constant rate of change each week. Ask students to use this scenario to write a verbal description of the equation. Have students share their verbal descriptions with the class. STEM Activities for Middle and High School Students 4
[If splitting the lesson into two shorter lessons, this would be an appropriate place to stop]. Ask students to write a second verbal description representing the function that is not about a savings account. If students are struggling, provide them with additional linear relationship contexts that may work. After groups have completed the task, bring whole class together to share findings. Check for Understanding Check for student understanding while they are working on the guided practice by asking the following questions: What similarities did you find in all three representations? Connect each of these similarities to your verbal description. [Starting value/yintercept; Constant rate of change/slope] What information can you gain from the graph that you can t see in the table and equation? Similarly, what can be seen in the table that is not easily seen in the equation and graph? What about the equation? How can you use your equation to generate more points in the table? Independent Practice Closure Provide students with the Independent Practice task to complete on their own. Students must compare two different linear functions that are represented in different ways. Bring the whole group together and ask students to share findings. Make sure students provide mathematical evidence to support their answers. A table, graph, equation or verbal description can represent a linear function. Though the constant rate of change (slope) and starting value (y-intercept) can be found in each of these representations, they each provide different information as well. A table gives distinct data points, whereas the equation describes the relationship between any (x, y) pair satisfying the linear function. A verbal description connects the linear function to a real life situation. A graph gives a visual representation of the linear function. STEM Activities for Middle and High School Students 5
The task students completed during independent practice asked them to compare two linear functions. Students may have had an initial thought as to which plan Marcus should choose. As they moved from the equation and verbal description to generate tables and graphs they created more evidence to support or refute that initial thought. The plan Marcus chooses depends on how many months he plans to keep it. Students should see that the graphical representation best illuminates the answer, even though it can be seen in the other representations. Suggestions for Differentiation and Extension The teacher can provide a specific Who Share s My Function? handout to students who may struggle with the concept. Students may feel more confident if they are given a table of values to match to a graph and equation, rather than if they are given the other two representations first. Additionally, for classes that do not divide evenly into groups of three, a student who struggles can be paired with a student who has a stronger understanding of the concept before they are asked to find the other group members. After students have completed the independent practice, they can be asked to compare a third plan that is represented by a nonlinear function. (For example, a third plan requires no sign up fee but the amount Marcus pays doubles each month. He must pay $5 the first month.) Students can be asked to complete an additional matching task, asking them to match the table, graph and equations of both linear and nonlinear functions. STEM Activities for Middle and High School Students 6
Representations of Linear Functions Verbal Description Graph John has $7 left over from his birthday. He is saving up for a new comic book subscription. Every week, John saves a dollar of his allowance for the comic book. Table Equation STEM Activities for Middle and High School Students 7
x y 1 12 2 17 4 27 7 42 STEM Activities for Middle and High School Students 8
y = 5x + 7 STEM Activities for Middle and High School Students 9
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x y 2 10 3 15 5 25 6 30 STEM Activities for Middle and High School Students 11
y = 5x STEM Activities for Middle and High School Students 12
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x y 1 12 2 19 4 33 7 54 STEM Activities for Middle and High School Students 14
y = 7x + 5 STEM Activities for Middle and High School Students 15
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x y 0 7 1 2 3-8 4-13 STEM Activities for Middle and High School Students 17
y = -5x + 7 STEM Activities for Middle and High School Students 18
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Independent Practice Marcus is deciding between two cell phone plans. He is given the following information about the two plans: Plan 1 This plan requires a $20 enrollment fee. Then Marcus will have to pay $65 each month he continues the plan. Plan 2 The total cost y Marcus will pay after x months is given by the equation y = 45x + 80. Create a table to represent each plan: Graph the two plans on the same coordinate grid: Which plan should Marcus choose? Justify your answer using evidence from the tables and graphs. STEM Activities for Middle and High School Students 20