This unit will introduce students to the concepts of y-intercept and slope. Students will: 
- understand and recognize linear function components including slope, y-intercept, domain, and range. 
- represent linear function as equations, tabular, and graphical forms. 
- relate linear functions to real world and comprehend the real-world context of the y-intercept. 

- How are relationships represented mathematically?
- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?
- How may data be arranged and represented to reveal the relationship between quantities?
- How can mathematics help to quantify, compare, depict, and model numbers?
 

- Linear Function: A function whose general equation is y = mx + b, where m and b stand for constants and m ≠ 0. A function in which the highest power associated with the independent variable is 1; a function that is represented by a line when graphed on a Cartesian plane. 
- Rate of Change: The limit of the ratio of an increment of the function value at the point to that of the independent variable as the increment of the variable approaches zero. Also referred to as “slope.” 
- Slope: The steepness of a line expressed as a ratio, using any two points on the line. A ratio of the rate at which the dependent variable is changing versus the rate at which the independent variable is changing; frequently expressed as \(rise \over run\) or  
- y-Intercept: The y-coordinate of the point at which the graph of a function crosses the y-axis.

- Vocabulary resource sheet (M-8-1-2_Vocabulary) 
- Components of a Linear Function (M-8-1-3_Components of a Linear Function) 
- Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY) 
- Cell Phone Scenario (M-8-1-3_Cell Phone Scenario and KEY) 
- chart paper 
- Variable T-Chart (M-8-1-3_Variable T-Chart) 
Independent Practice sheet (M-8-1-3_Independent Practice Template) 
- Lesson 3 Exit Ticket (M-8-1-3_Lesson 3 Exit Ticket and KEY) 
- Linear Concepts worksheet (M-8-1-3_Linear Concepts)

- Use the student's performance on the Cell Phone activity to measure their level of knowledge.
- Use the Lesson 3 Exit Ticket (M-8-1-3_Lesson 3 Exit Ticket and KEY) to assess mastery. 
- Assess the overall level of comprehension by looking at the PowerPoint presentations' accuracy and comprehensiveness.
 

Scaffolding, Active Engagement, Modeling, Formative Assessment 
W: Instruction will cover four components of a linear function: slope, y-intercept, domain, and range. Linear functions will also be studied in real-world scenarios to help students improve their conceptual grasp and recognize the importance of linear functions. 
H: The lesson takes an exploratory approach, teaching students how linear functions might be applied to real-life scenarios. 
E: Four activities are used to demonstrate aspects of linear functions, including y-intercept, domain, and range. Students should work on understanding the relationship between the concept and its real-world meaning, such as a y-intercept indicating a one-time charge or base fee for something, and a slope showing the rate for a service dependent on time or another unit. 
R: The exploration and generation of real-world connections to linear functions helps students gain a deeper knowledge of slope, y-intercept, and linear functions in general. 
E: Students' understanding is evaluated through questions, worksheets, and discussions during the class. 
T: To personalize the lesson to meet the needs of students, use the Small Group activity to reinforce important concepts or provide more practice, and use the Extension activity for those ready for a more challenging learning experience. 
O: The structure is exploratory, with a concentration on abstract thought as students acquire experience applying linear functions and examining their relationship to real-world contexts. 
 

Activity 1

"Now that we've explored the concept of slope, let's see if we can identify other linear function components such as y-intercept, domain, and range. We will continue to identify the slope and investigate each of these components using equations, tables, and graphs." Students should refer to the Vocabulary resource sheet (M-8-1-2_Vocabulary) as they continue to learn 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). Ask students to explain what they know based on the graph. Then, ask a volunteer to point to the y-intercept and explain what it represents. To help students understand the concept of the y-intercept, ask them questions like the ones mentioned below.

"Can two lines share the same y-intercept? Will the lines look alike? Explain your reasons. 
"Does the y-intercept have to be located on the y-axis? 
"Can the y-intercept go through the origin? 
"In an equation, where can you find the y-intercept? 
"Does the y-intercept have to be a whole number? 
"Can two y-intercepts share the same x-value? Explain your reasons. 
"Is it important to know the y-intercept in order to graph a line? Explain your reasons.
"The y-intercept also can be seen in real-world context." Post the Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY) for students to see. "This graph represents cell phone usage and cost." Ask students to examine the graph and make observations.

Use questions like the ones listed below to guide students' thinking. 

"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 from the Cell Phone Graph (M-8-1-3_Cell Phone Graph and KEY). While filling in the x/y chart, use the think-aloud approach to explain what the numbers indicate while referring to the graph. Ask volunteers to assist with this process.

Cell Phone Table 

To guide student thinking, use the Cell Phone chart above as a reference and ask questions like the ones listed below. 

"From the chart, what is the y-intercept? How do you know? 
"What pattern do you see in the y column? 
"Do you see a pattern? Is there a constant difference?
"Is this function linear? Explain your reasons. 
"Can you create a rule for this function?" 
"Why do we need 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 graph?
"Let's look at a similar real-world scenario with cell phone usage and cost." Post the Cell Phone Scenario (M-8-1-3_Cell Phone Scenario and KEY). Explain the real-world situation to students: "Suppose we have a cell phone package that indicates that even if we use our phone for 0 minutes in August, we will still owing $39. This is the amount owed for 0 minutes of talk time. Assume that 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?"

Fill out the bottom half of the Cell Phone Scenario by modeling, thinking aloud, and asking questions. "In this case, we can see how the y-intercept has a real-world context. The graph does not begin at the origin (0, 0) because there is a flat rate of $39 per month regardless of the number of minutes used during the month."

Ask students, "Can you provide a description of a y-intercept in a real-world application?" Allow students to discuss with their peers if they have time. On chart paper, have students work in groups to produce a variety of y-intercept examples in a real-world situation. Allow them to share their ideas for improving the real-world connection of linear functions. Encourage students to produce a quick sketch graph to clarify the meaning.

"The y-intercept is the point at which the line crosses the y-axis. Sometimes the line crosses at the origin (0, 0), and sometimes it crosses at some other point. But what is the y-intercept? It is the point at which the x-value is 0. 

"We recently spent time looking at the y-intercept, which is a component of linear functions. Second, let's consider 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) to help students understand the relationship between domain and the values used for x. "Could you give an example of a domain? Domain can be represented using an equation, a table, or a graph. You can also demonstrate domain as an example with a real-world scenario." Allow students time to discuss with a partner. Ask students to share their examples with the class. Clarify misunderstandings and emphasize real-world context. "Domain is the set of input values, or x-values, of a function. 

"The third component that we'll look at is range. In the context of linear functions, range refers to the set of output values, y-values, of a function. In the previous cell phone example, 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) to help students understand the relationship between the range and the values used for y. "Could you give an example of range? Range can be represented using an equation, a table, or a graph. You may also demonstrate range using a real-world scenario." Allow students time to discuss with a partner. Ask students to share their examples with the class. Clarify misunderstandings and emphasize 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." If students are having difficulty, explain that the domain is x and the range is y. Remember that d comes before r, and x comes before y. Students can use the Variable T-Chart (M-8-1-3_Variable T-Chart) to help visualize how the terms can be categorized.

Activity 2

"Let's consider the linear function y = 2x + 1. We will identify four components of a linear function." 

Show the graph below.

Ask students to explain everything they know about this line. Use chart paper to record student responses. Then, as a class, determine the slope, y-intercept, domain, and range of the graphed line.

Students should work with a partner to create a linear function and, if possible, connect it to a real-world context. On chart paper, have them represent the linear function as an equation, tabular, and a graph. Then ask students to identify the slope, y-intercept, domain, and range. Keep track of performance and offer assistance as needed. Functions, representations, and identified components can be shared during a class discussion or through a carousel walk. A carousel walk is an activity in which students place their work around the room and rotate from station to station, observing the work of other students. Debriefing at the end.

Activity 3

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

Monitor student performance and offer assistance if necessary. If time allows, ask students to relate a real-world problem scenario that can be illustrated by the given linear function. The variety of problem situations that students observe will assist each student understand the breadth of problems that can be solved using this function.

Activity 4

Provide real-world context and accompanying data in a table, and ask students to calculate the slope and y-intercept. Explain the reasonable domain and range to be included in the analysis. Examine the real-world implications indicated by the linear function. Use a real-world scenario such as the one given below. 

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

Several activities can be used to review this lesson:

Use the Lesson 3 Exit Ticket (M-8-1-3_Lesson 3 Exit Ticket and KEY). 
To help students become more comfortable with the math vocabulary in this lesson, do one of the vocabulary activities below. Ensure that students have access to the Vocabulary resource sheet (M-8-1-2_Vocabulary). Choose whatever terms you want your students to practice at this time. 
Option 1: Cut the cards apart and have students categorize the word cards. Students are then asked to explain the similarities between the terms in each category. Set criteria, such as the minimum number in each category or the minimum number of categories. The categorization must be based on mathematical context.
Option 2: Connect 2. Ask students to choose two words and explain how they are related. For example, domain and range are both necessary components of a function.
Option 3: Students can demonstrate their grasp of the vocabulary by completing a quick RAFT writing activity that includes the vocabulary words covered in the unit. A RAFT is a writing method that helps to connect writing and math vocabulary. The R represents the role the writer will take; the A represents the audience the writer is writing to; the F represents the format of the writing; and the T represents the topic to be written about. Ask students to select one of the following options and demonstrate their understanding by using as many math terms in context as possible.

Extension:

Routine: As real-world linear scenarios arise during the school year, quiz students on the slope or rate of change, domain, range, and how they can tell the relationship is linear.

Small Group: Give students a copy of the Linear Concepts Worksheet (M-8-1-3_Linear Concepts). Allow them to work individually first, then compare their answers. Students should discuss any problems on which they disagree. This is an excellent opportunity for students to help explain concepts to other students.

Expansion:
1. Students should develop a short PowerPoint presentation with 10-12 slides that demonstrates 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. With a slope of -2 and a y-intercept of 30, students should create a possible (and appropriate) real-world scenario that can be described using this linear function information.
3. Before moving on to the next activity, provide students the tools they need to examine numerous real-world scenarios. PBS's Linear Functions—Real Life Data Web page describes videos and circumstances that provide students with an important starting point for learning about linear functions in the real world. Students will be better prepared to brainstorm more real-world linear functions after experiencing these scenarios.