• Use student performance on the Cell Phone activity to determine level of understanding.
• Use Lesson 3 Exit Ticket (M-8-1-3_Lesson 3 Exit Ticket and KEY.doc) for assessment of mastery.
• Evaluate the overall level of comprehension by assessing the accuracy and comprehensiveness of the PowerPoint presentations.

Activity 1

“Now that we’ve explored the concept of slope, let’s see if we can find other components of linear functions including the y-intercept, domain, and range. We will continue to identify the slope and examine each of these components in the following representations: equations, tables, and graphs.” Have students refer to the Vocabulary resource sheet (M-8-1-2_Vocabulary.docx) as they continue to explore the math vocabulary in this lesson.

“First, let’s look at y-intercept.” Post Components of a Linear Function (M-8-1-3_Components of a Linear Function.doc). Ask students to explain what they know by looking at the graph. Then ask a volunteer to point out the y-intercept and explain what it represents. To further expand the concept of the y-intercept, ask students questions similar to those listed below.

• “Can two lines have the same y-intercept? Will the lines look alike? Explain your reasoning.
• “Does the y-intercept have to be located on the y-axis?
• “Can the y-intercept go through the origin?
• “In an equation, where do you find the y-intercept?
• “Does the y-intercept have to be a whole number?
• “Can two y-intercepts have the same x-value? Explain your reasoning.
• “Is it necessary to know the y-intercept in order to graph a line? Explain your reasoning.

“The y-intercept also can be seen in real-world context.” Post Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY.doc) for students to see. “This graph represents cell phone usage and cost.” Ask students to look at the graph and make observations.

Guide student thinking using questions similar to those listed below.

• “Where is the y-intercept on this graph?” (0, 0)
• “What is a math term used to describe (0, 0) on a graph?” (origin)
• “If we talk for 0 minutes, how much do we owe?” ($0)
• “If we talk for 10 minutes, how much do we owe?” ($.70)
• “What is the slope or rate of this line? How do you know?” ($.07)
• “What would the slope represent in terms of this real-world context?” (rate of
  $.07 per minute)
• “Can you write a rule to represent this function?” (y = .07x)

Make an x/y chart on the board to represent the data shown in the Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY.doc). While filling in the x/y chart, use the think-aloud strategy to explain what the numbers represent while referring to the graph. Ask volunteers to help with this process.

Cell Phone Table

Using the Cell Phone chart above as a reference, pose questions similar to those listed below to guide student thinking.

• “Looking at the chart, what is the y-intercept? How do you know?
• “What pattern do you notice in the y column?
• “Do you see a pattern? Is it a constant difference?
• “Is this function linear? Explain your reasoning.
• “Can you write a rule for this function?
• “Why do we have to include zero in the chart?
• “Looking at the chart, what other values could be used for x and y?
• “What relationship do you see between the chart and the graph?

“Let’s look at a similar real-world context scenario related to cell phone usage and cost.” Post the Cell Phone Scenario (M-8-1-3_Cell Phone Scenario and KEY.doc). Explain this real-world context to students: “Suppose we have a cell phone package that states even if we talk on our cell phone for 0 minutes during the month of August, we still owe $39. This is the amount owed for 0 minutes of talk time. Suppose each additional minute of talk time adds $0.07.

“How is this real-world context similar to and different than the one we previously looked at?”

Model, think aloud, and question while completing the bottom half of the Cell Phone Scenario. “In this example, we can see how the y-intercept has real-world context. The graph does not start at the origin (0, 0) because there is a flat rate of $39 per month regardless of how many minutes are used during the month.”

Pose the following question to students: “Can you provide a description of a y-intercept in a real-world application?” If time permits, give students time to discuss with peers. On chart paper, have students in groups generate many, varied examples of a y-intercept in a real-world context. Have them share their ideas to strengthen the real-world connection of linear functions. Encourage students to create a quick sketch graph to clarify meaning.
“Y-intercept is the point where the line crosses the y-axis. Sometimes the line crosses at the origin (0, 0); sometimes it crosses at some other point. But what is y-intercept, really? It is the point at which the x-value is 0.

“We just spent time looking at y-intercept, which is one of the components of linear functions. Second, let’s look at the domain of a linear function, or any function for that matter. Domain is the set of input values, or x-values, of a function. In the previous example of the cell phone scenario, the domain would be (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).” Refer to the Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY.doc) so students can see the connection between domain and the values used for x. “Can you provide an example of domain? You can represent domain using an equation, table, or graph. You also can show domain as an example using a real-world scenario.” Give students time to discuss with a partner. Have students share their examples with the class. Clarify misunderstandings and highlight real-world context. “Domain is the set of input values, or
x-values, of a function.

“The third component that we are going to look at is range. In the context of linear functions, range is the set of output values, or y-values, of a function. In the previous example of the cell phone scenario, the range would be (0, 0.07, 0.14, 0.21, 0.28, 0.35, 0.42, 0.49, 0.56, 0.63, 0.70).” Refer to the Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY.doc) so students see the connection between the range and the values used for y. “Can you provide an example of range? You can represent range using an equation, table, or graph. You also can show range as an example via real-world scenario.” Give students time to discuss with a partner. Have students share their examples with the class. Clarify misunderstandings and highlight real-world context. “Range is the set of output values, or y-values, of a function.

“We have just explored the components of linear functions including y-intercept, domain, and range.” For students having difficulty, explain that the domain is x, and the range is y. Remember, d comes before r, and x comes before y. Also students can be given the Variable
T-Chart (M-8-1-3_Variable T-Chart.doc) to help visualize how the terms can be categorized.

Activity 2

“Let’s consider the linear function, y = 2x + 1. We are going to identify four components of a linear function.”

Display the graph below.

Ask students to say everything they know about this line. Record student responses on chart paper. Then as a class find the slope, y-intercept, domain, and range of the graphed line.

With a partner, have students create a linear function and connect it to real-world context if possible. Have them represent the linear function in equation, tabular, and graphical forms on chart paper. Then have students identify the slope, y-intercept, domain, and range. Monitor performance and provide necessary support. Functions, representations, and identified components can be shared during a class discussion or as a carousel walk. A carousel walk is an activity in which students post their work around the room and rotate from station to station observing other students’ work. Debrief at the end.

Activity 3

For independent practice have students consider a graph similar to the one below. Give each student a copy of the Independent Practice sheet (M-8-1-3_Independent Practice Template.doc) to complete using a similar graph of a linear function.

Monitor student performance and provide support if necessary. If time permits, have students share a real-world problem situation that could be illustrated by the given linear function. The variety of problem situations that students observe will help each student see the breadth of problems that could be realized from this function.

Activity 4

Give real-world context and accompanying data in a table and ask students to find the slope and y-intercept. Explain the reasonable domain and range to be included in the analysis. Analyze the real-world implication represented by the linear function. Use a real-world context similar to the one listed below.

A new resident to Wichita Falls, Texas, recently purchased a new home. The table below reveals the predicted value of the home, after depreciation over a 15-year span.

There are several activities that can be used as a review for this lesson:

• Use Lesson 3 Exit Ticket (M-8-1-3_Lesson 3 Exit Ticket and KEY.doc).
• To get students more comfortable with the math vocabulary in this lesson use one of the following vocabulary activities. Be sure students have the Vocabulary resource sheet available to use (M-8-1-2_Vocabulary.docx). Select the words you want your students to practice at this time.
  • Option 1: Cut apart the cards and have students categorize the word cards. Students then have to explain the similarities between the words in each category. Set criteria like minimum number in each category or minimum number of categories. The categorization must be based on math context.
  • Option 2: Connect 2. Have students choose two words and explain how the words are related. For example, with domain and range, both are necessary parts of a function.
  • Option 3: Students can demonstrate their understanding of the vocabulary by completing a quick RAFT writing activity using the vocabulary words introduced in the unit. A RAFT is a writing strategy that can be used to integrate writing and math vocabulary. The R stands for the role the writer will take; the A stands for the audience the writer is writing to; the F stands for the format of the writing; and the T stands for the topic to be written about. Have students choose one of the following options and demonstrate their understanding using as many math terms in context as possible.

Extension:

• Routine: As real-world linear situations occur during the school year, quiz students about the slope or rate of change, domain, range, and how they know the relationship is linear.
• Small Group: Give students a copy of the Linear Concepts worksheet (M-8-1-3_Linear Concepts.doc). Have them work individually, then compare answers. Students should discuss any problems over which they disagree. This is a great opportunity for students to help explain concepts to other students.
  • Expansion:

1. Have students create a short PowerPoint presentation with 10–12 slides illustrating the overall concept of linear function. Students should be encouraged to reveal ingenuity. The PowerPoint presentations can be shown and viewed by the class in a peer review format. Discussion and debate should follow.
2. Given a slope of −2 and y-intercept of 30, have students write a possible (and appropriate) real-world scenario that could be modeled using this linear function information.
3. Before beginning the next activity, provide students with tools to explore various real-world situations. Videos and situations, described by Linear Functions—Real Life Data Web page by PBS, give students a pivotal point from which to learn about linear functions in the real world. From exposure to these scenarios, students will be prepared to brainstorm more real-world linear functions on their own.