In this unit, students will learn about concepts associated with linear functions. Students will: 
- examine the rate of change to determine whether the given patterns represent linear functions.
- determine whether given functions are linear by examining the function and looking at its graph. 
- identify the slope and y-intercept of an equation, table, or graph. 
- connect the constant rate of change and linearity.

- 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? 
 

- Function: A relation in which each value of an independent variable is associated with a unique value of the dependent variable. A relationship between variables that has one output for each and every input. 
- 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. 
- Pattern: The arrangement of a predetermined format of numbers, symbols, or objects. 
- 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. 

- Linear Function Table 1 (M-8-1-2_Linear Function Table 1) 
- Linear Function Table 2 (M-8-1-2_Linear Function Table 2) 
- Question Cards (M-8-1-2_Question Cards) 
- Graph Representing Linear Function Table 1 (M-8-1-2_Graph for Linear Function Table 1) 
- Visual of Slopes (M-8-1-2_Visual of Slopes) 
- Rise Over Run worksheet (M-8-1-2_Rise Over Run) 
- Vocabulary resource sheet (M-8-1-2_Vocabulary) 
- Is It Linear? worksheet (M-8-1-2_Is It Linear and KEY) 
- Function Exploration activity sheet (M-8-1-2_Function Exploration and KEY) 
- Lesson 2 Exit Ticket matching game (M-8-1-2_Lesson 2 Exit Ticket and KEY) 
- chart paper

- Student performance on the Is It Linear? worksheet (M-8-1-2_Is It Linear and KEY) during Activity 4 can be used to assess understanding.
- Observations made during Activity 2 (Linear Function Table) will help determine the student's level of comprehension. 
- Use the Lesson 2 Exit Ticket matching game (M-8-1-2_Lesson 2 Exit Ticket and KEY) to quickly assess students' grasp of the relationship between slope and the pictorial representation of it.
 

Active Engagement, Modeling, Explicit Instruction, Formative Assessment 
W: The lesson covers two linear function concepts: 1) determining linearity and 2) determining and interpreting components of a linear function. 
H: Students will investigate the unique connection between slope and constant rate of change in linear functions, as compared to the absence of such a connection with nonlinear functions. 
E: Students move from an abstract understanding of linearity to the actual specifics of the function. 
R: During the lesson, students have multiple opportunities to reflect, revisit, revise, and rethink through verification, exploration, creation of organizers, connection to the real world, and class discussions. 
E: During the lesson, students can self-evaluate through several activities. Whole class discussions also provide an opportunity to evaluate oneself. 
T: The class can be adjusted to match the needs of the students. Activities can be skipped or modified for classes or students that require less instruction, or for students who would benefit from more instruction or practice. Use the Extension section to change the lesson as needed. 
O: The abstract structure aims to develop both students' procedural and a conceptual understanding of linear functions. 

"How do patterns relate to the discussion of 'linear functions'? Linear functions actually represent patterns of numbers. A function is just a relationship with a single output for every input. A linear function has a constant rate of change (think of the pattern changing by a constant difference). This constant rate is also referred to as the slope of the function. The slope is the ratio of change in y-values per change in x-values, or \(rise \over run\)." Distribute the Vocabulary resource sheet (M-8-1-2_Vocabulary) to all students. Allow students to use this sheet as a reference throughout the lesson to familiarize themselves with math vocabulary. 

"Let's look at a table of values to illustrate what we're talking about." Post Linear Function Table 1 (M-8-1-2_Linear Function Table 1) for discussion with students.

Linear Function Table 1

Use questions like the ones mentioned below to guide student understanding and help students make the connection between the math vocabulary and the linear function table. Use the Vocabulary resource sheet (M-8-1-2_Vocabulary) to help students describe their answers and reasoning using math vocabulary. Refer to the table as the students share their answers. 

"How do we know this table represents a function?" (The table indicates that there is one output for each input.) 
"If we say it is a linear function, what does that mean?" (There is a constant rate of change. The pattern demonstrates a constant difference of change. The rule is: y = 2x + 1.) 
"What is one of our new vocabulary words that means the same thing as a constant rate?" (slope)
"What is another way to describe slope?" (The ratio of the change in y-values per change in x-values or rise over run.)
"How might the graph look like? Explain your reasoning." 
For additional practice, display Linear Function Table 2 (M-8-1-2_Linear Function Table 2) for students to see. Allow students to discuss the table with a partner, encouraging them to apply some of the new vocabulary. Keep track of student dialogues and interactions. Then, using the Question Cards (M-8-1-2_Question Cards), ask for five student volunteers. Hand one of the Question Cards to them. Allow these five students a few moments to read the question and formulate a mathematical response. Then, instruct each student to read the question and respond while referring to Linear Function Table 2.

"Remember that a pattern with a constant difference is linear. The constant difference in a linear pattern is also the constant rate of change. It is represented by the coefficient in the formula representing the pattern. A pattern with a constant ratio is not linear."

Activity 1

Using a graph, demonstrate and explain how to find a constant rate of change or slope in the function, y = 2x + 1. Students can verify the function's linearity by looking at a graphic representation of it. "In the first part of this activity, we looked at linear function tables." Once again, display Linear Function Table 1 (M-8-1-2_Linear Function Table 1) for students to see. “Let’s look back at Linear Function Table 1.” Discuss with students. Then, show students the graph representing the linear function (M-8-1-2_Graph for Linear Function Table 1). 

Using the think-aloud strategy, demonstrate to students how data from the table is transferred to the graph. Use a graph to model and demonstrate. 

"Remember that when we plot coordinates on a graph, we need an x-value and a y-value. Looking at the table, I can use (1, 3) as my first point on the graph. To plot the point, I start at the origin and move 1 space to the right and 3 spaces up. I can return to the table to discover the next x-value and its corresponding y-value. That would be (2,5). To plot the point, I go from the origin 2 spaces to the right and 5 spaces up. I may continue the process with the other x-values and their corresponding y-values before drawing the line for the graph. This is one strategy." 

Continue to apply the think-aloud strategy and model using the actual graph (M-8-1-2_Graph for Linear Function Table 1). "If we look closer at the graph, we can see that the line crosses the y-axis at 1. That relates to the equation: y = 2x + 1. The 1 represents the y-intercept." (At this point, just start connecting the equation used to describe the rule of the function table, rather than teaching students the slope-intercept form.) "When the line is drawn, we can go to a point on the line and move up and over to get to the next point." Model the graph for students to see. Begin at (0, 1), which is a point on the line. "This is the y-intercept because it is the point on the y-axis at which the function crosses the y-axis." Then move up 2 and over to the right 1, which is another point on the line. It also appears as a table coordinate (1, 3). Repeat the process for students to see. Explain to them that this represents the slope of the line. It, too, is connected to the equation y = 2x+1. "The slope is represented by the coefficient in the equation, which in this case is 2." Continue making the connection to the equation used to describe the function table rule, rather than teaching students the slope-intercept form. 

Repeat the process by modeling and visually describing the slope as rise over run. "Notice how I can land on the line each time by going up 2 and over to the right, up 2 over 1, ... This represents a pattern. This rise over run pattern called a slope. Slope is calculated by dividing the rise (vertical change) by the run (horizontal change). In this example, the rise is a positive vertical change of 2, while the run is a positive horizontal change of 1. So the slope would be \(+2 \over +1\), which equals 2. Slope describes the rate of change. The slope of a line determines how steep or shallow a line is.” Use the Rise Over Run activity sheet (M-8-1-2_Rise Over Run) to help students visualize and understand the idea of slope. Make sure every student has a copy. Using the think-aloud strategy, demonstrate how the example on the page was completed so that students may see slope and hear math vocabulary associated with slope. Allow students to work with a partner. Monitor students' performance and offer assistance as needed. Go over the answers as a class. "When learning about slope it is important to understand the components of a line: slope and y-intercept."

Show students examples of steep and shallow slopes. Also, note that "up 2, right 1" has the same slope as "down 2, left 1," because \(+2 \over +1\) = \(-2 \over -1\) = 2. Visual of Slopes (M-8-1-2_Visual of Slopes) features pictures of various real-world slopes. Use questions like the ones below to help students see and conceptualize slope. 

"How is slope represented in these pictures?
"What are some of the similarities between the slopes in the pictures? 
"What are the differences between the slopes in the pictures? 
"Are all the slopes in the same direction? Explain your reasons.
"Which picture represents the steepest slope? Explain your reasons. 
"Where do you most often see steep slopes?
"Which picture represents the shallowest slope? Explain your reasons.
"Where do you typically see shallow slopes? 
"Can you think of any other real-world examples of slope?"

The following websites can be used to demonstrate to students how the steepness of a line changes when the slope changes: 

http://www.shodor.org/interactivate/activities/SlopeSlider/ 
http://zonalandeducation.com/mmts/functionInstitute/linearFunctions/lsif.html. 
When using the second website, click the "You Control" button to manually adjust the slope. Ask students to make observations on what they notice when the slope changes. After completing several examples, have students work with a partner to develop a generalization about slope and its relation to the steepness of a line. Post students' generalizations on chart paper. Clarify any misunderstandings and provide more examples if necessary.

Activity 2

Refer to Linear Function Table 1 (M-8-1-2_Linear Function Table 1) and its corresponding graph (M-8-1-2_Graph for Linear Function Table 1). As a think-aloud exercise, summarize the function using the terms listed below. Then, to enhance knowledge, have students work with a pair to summarize the function using the listed terms: 

Linear 
Constant pace of change. 
Slope of 2 
"Consider the following question: Does a linear function with a greater slope rise more quickly or less quickly than a linear function with a smaller slope? Can you give some examples? What happens to a linear function whose slope is negative? Remember the visuals we saw of real-world examples of slope." Show students the Visual of Slopes (M-8-1-2_Visual of Slopes).

Allow students time to think about examples, draw them, and regroup. Ask students to provide deductions and explanations. The text below is an example of what type of confirmation you can provide. 

"Note that a linear function with a greater slope rises more quickly than a linear function with a smaller slope. A linear function with a slope of 3 (e.g., y = 3x + 2) has a steeper line than a linear function with a slope of 2 (e.g., y = 2x - 1). A linear function with a greater absolute value will fall more quickly than one with a smaller absolute value. A linear function with a slope of −2 (e.g., y = −2x + 1) will not be as steep or fall as quickly as a linear function with a slope of −3 (e.g., y = −3x + 2). So, in general, a linear function with a slope of greater absolute value rises or falls more quickly than a linear function with a slope of smaller absolute values. This rule would apply to both positive and negative slopes. 

"In this example, what is the rate of change in the output values? Is there a constant rate of change?" (There is a constant rate of change of 2 over 1, which is just 2.)

"How does our input value change? In other words, what happens to each input value before we receive the output value? Remember that we're looking at a function here." (Each input value, or x-value, is multiplied by 2, with 1 added to the product, resulting, i.e., 2x + 1.) 

"The constant rate of change we found is the same as the slope." 

Activity 3

To help students become more comfortable with the math vocabulary in this lesson, try one of the vocabulary activities listed below. Make sure that students have a copy of the Vocabulary resource sheet (M-8-1-2_Vocabulary). You can specify which terms your students should practice at this time.

Cut apart the vocabulary cards and have students match the word to its definition.
Cut apart the vocabulary cards and have students play Concentration with a partner. Turn the word cards face down on one side and the definition cards face down on the opposite side. The first player flips over two cards: one word and one definition card. If the cards match, the player keeps them and tries again. If not, the player returns the two cards to their original positions, face down, and the next player takes their turn. Play continues until all cards are matched.
Cut apart the vocabulary cards and have students categorize the word cards. Students are then asked to explain the similarities between the terms in each categorize. Set requirements such as minimum number in each category, minimum number of categories, and categorization based on math context. 
Connect 2. Ask students to choose two words and explain how they are related. For example, domain and range are both necessary parts of a function. 

Activity 4

Have students complete the Is It Linear? Worksheet (M-8-1-2_Is Is Linear and KEY)

Extension:

Use the following suggestions to personalize the class to the requirements of the students.

Routine: Remind students a few times a week how to find slope on the graph of a line. Whether counting rise over run or just asking about steepness or positive/negative rise, students require numerous opportunities to master the concept of slope. 
Check that the slope of this linear function is the same as the constant rate of change.
(The text below can be used to reinforce students' ideas throughout the class discussion.) 

"When given two points, (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)), the slope can be calculated using the following formula:

\({(y_2) - (y_1)} \over {(x_2) - (x_1)}\)

"So, subtracting the first two y-values yields (5 - 3 = 2). If we subtract the first two x-values, we get (2 - 1 = 1). The ratio of \(2 \over 1\) = 2. Thus, the slope is 2. Our constant rate of change was 2. This illustration demonstrates that the slope is the constant rate of change. 

"The presence of a constant rate of change verifies that this function is linear." The ability of students to recognize the relationship between constant rate of change and linearity is important.

"We can also determine that the function is linear and thus has a constant rate of change by looking at the function equation. Note that the highest exponent of the equation y = 2x + 1 is 1, indicating a linear function, and, consequently, indicating the presence of a slope, or constant rate of change. When a linear function is expressed as an equation in this format, the coefficient of x represents the slope of the line. 

"Can we tell if a function has a constant rate of change (and is linear) by looking at its graph? Certainly." 

Small Groups: Students that require further practice can be given the Function Exploration activity sheet (M-8-1-2_Function Exploration and KEY) to complete. This will allow them to see the differences between linear and nonlinear relationships in a graphical sense. 
Students may also be assigned to complete the activities on the following website: 
http://www.watertown.k12.ma.us/wms/math/math_help/gradeeight/moving/msa.html 

Expansion: Challenge students who are ready for a challenge beyond the requirements of the standards to work on problems from the pdf document accessible at: 
http://www.mindset.co.za/resources/0000062453/0000135697/0000138942/MATHS%20Gr%2010%20Session%2011%20LN.pdf.