• Student performance on the Is It Linear? worksheet (M-8-1-2_Is It Linear and KEY.doc) during Activity 4 may be used to gauge level of understanding.
• Observations during Activity 2 (Linear Function Table) will help in determining student level of comprehension.
• Use the Lesson 2 Exit Ticket matching game (M-8-1-2_Lesson 2 Exit Ticket and KEY.doc) to provide a quick evaluation of student understanding of the relationship between slope and the graphical representation of slope.

“How do patterns lead into the discussion of ‘linear functions’? Linear functions actually represent patterns of numbers. A function is simply a relationship that has one output for each and every input. A linear function is a function that has a constant rate of change (think of the pattern as changing by a constant difference). This constant rate is also known as the slope of the function. The slope is the ratio of the change in y-values per change in
x-values, or .” Hand out the Vocabulary resource sheet (M-8-1-2_Vocabulary.docx) to each student. Allow students to use this sheet as a reference during the lesson to familiarize themselves with the math vocabulary.

“Let’s look at a table of values to illustrate what we’re talking about.” Post the Linear Function Table 1 (M-8-1-2_Linear Function Table 1.doc) to discuss with students.

Linear Function Table 1

Use questions similar to those listed below to guide student understanding and help students make the connection between the math vocabulary and the linear function table. Have students refer to the Vocabulary resource sheet (M-8-1-2_Vocabulary.docx) to help them use the math vocabulary to explain their answers and reasoning. Refer to the table as students are sharing their answers.

• “How do we know this table represents a function?” (The table shows 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 shows 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.)
• “What might the graph look like? Explain your reasoning.”

For additional practice, post the Linear Function Table 2 (M-8-1-2_Linear Function Table 2.doc) for students to see. Have students discuss the table with a partner encouraging them to use some of the new vocabulary. Monitor student dialogue and interaction. Then using the Question Cards (M-8-1-2_Question Cards.doc), ask for five student volunteers. Hand out one of the Question Cards to them. Give these five students a few moments to read the question and conjure up a mathematical response. Then have each student read the question and respond while referring to the Linear Function Table 2.

“Remember 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, show and explain how you can identify a constant rate of change or slope in the function, y = 2x + 1. By seeing a visual representation of the function students can verify the linearity of the function. “In the first part of this activity, we looked at linear function tables.” Once again, post the Linear Function Table 1 (M-8-1-2_Linear Function Table 1.doc) for students to see. “Let’s look back at Linear Function Table 1.” Discuss with students. Then show students the graph that represents the linear function (M-8-1-2_Graph for Linear Function Table 1.doc).

Using the think-aloud strategy explain to students how the data from the table is transferred to the graph. Model and show using the graph.

“Remember 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. Starting from the origin, I go over to the right 1 space, and up 3 spaces to plot the point. I can go back to the table and find the next x-value and its corresponding y-value. That would be (2, 5). From the origin, I go over to the right 2 spaces and up 5 spaces to plot the point. I can continue this process using the other x-values and their corresponding y-values, and then draw the line for the graph. This is one strategy.”

Continue to use the think-aloud strategy and model using the actual graph (M-8-1-2_Graph for Linear Function Table 1.doc). “If we look closer at the graph, we can see that the line crosses the y-axis at 1. That is related to the equation: y = 2x + 1. The 1 represents the y-intercept.” (At this point just start making the connection to the equation used to describe the rule of the function table and not necessarily 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 using the graph for students to see. Start at (0, 1) since that 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 go up 2 and over to the right 1, which is another point on the line. It also is a coordinate in the table (1, 3). Repeat the process for students to see. Explain to students that this shows the slope of the line. It, too, is related 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 rule of the function table and not necessarily teaching students the slope-intercept form.

Repeat the process again by modeling and visually explaining slope as rise over run. “Notice how I can land on the line each time if I go up 2 and over to the right 1, up 2 over 1, …. This is a pattern. This rise over run pattern is slope. Slope is determined by the rise (vertical change) divided by the run (horizontal change). Notice in this example the rise is a positive vertical change of 2, and the run is a positive horizontal change of 1. So the slope would be , which equals 2. Slope explains the rate of change. The slope of a line affects how steep or shallow a line is.” Use the Rise Over Run activity sheet (M-8-1-2_Rise Over Run.docx) to help students visualize and conceptualize what is meant by slope. Be sure each student has a copy. Using the think-aloud strategy, model how the example on the page was completed so students can visualize slope and hear the math vocabulary related to slope. Allow students to work with a partner. Monitor student performance and provide necessary support. Go over the answers together 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 slopes and shallow slopes. Also point out that “up 2, right 1” produces the same slope as “down 2, left 1,” because . Pictures of different real-world slopes can be seen on Visual of Slopes (M-8-1-2_Visual of Slopes.doc). Use questions similar to those listed below to help students visualize and conceptualize what is meant by slope.

• “How is slope represented in these pictures?
• “What are some similarities between the slopes in the pictures?
• “What are some differences between the slopes in the pictures?
• “Are all the slopes going in the same direction? Explain your reasoning.
• “Which picture shows the steepest slope? Explain your reasoning.
• “Where do you most often see steep slopes?
• “Which picture shows the shallowest slope? Explain your reasoning.
• “Where do you most often see shallow slopes?
• “Can you think of other real-world examples of slope?”

The following Web sites can be used to show 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 Web site, click on the button “You Control” to manually change the slope. Have students make observations about what they notice when the slope changes. After completing multiple examples, have students work with a partner to create a generalization about slope and its relation to the steepness of a line. Post students’ generalizations on chart paper. Clarify any misunderstandings and model with more examples if necessary.

Activity 2

Refer back to the Linear Function Table 1 (M-8-1-2_Linear Function Table 1.doc) and the corresponding graph (M-8-1-2_Graph for Linear Function Table 1.doc). Summarize the function using the following terms as a think-aloud exercise. Then have students turn to a partner and summarize the function using the listed terms to reinforce understanding:

• Linear
• Constant rate of change
• Slope of 2

“Here is a question to consider: Does a linear function with a greater slope rise more quickly or less quickly than a linear function with a smaller slope? Can you provide a couple examples? What happens to a linear function with a negative slope? Think back to the visuals we saw of real-world examples of slope.” Display for students the Visual of Slopes (M-8-1-2_Visual of Slopes.doc).

Allow students time to think of examples, draw these, and regroup. Ask students to offer deductions and explanations. The text below is a sample of the type of confirmation you can give.

“Note that a linear function with a greater slope rises more quickly than a linear function with a smaller slope. For example, 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 slope of greater absolute value will fall more quickly than one with a smaller absolute value. For example, 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, we could generalize that 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 between 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 get the output value? Remember, we are looking at a function here.” (Each input value, or x-value, is multiplied by 2, with 1 added to that product, i.e., 2x + 1.)

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

Activity 3

To get students more comfortable with the math vocabulary in this lesson, use one of the following vocabulary activities. Be sure students have a copy of the Vocabulary resource sheet (M-8-1-2_Vocabulary.docx). You can tailor which words you want your students to practice at this time.

• Cut apart the vocabulary cards and have students match the word with the 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 other side. The first player turns over two cards—one word card and one definition card. If it is a match, the player keeps the cards and tries again. If not, the player returns the two cards to their original spots face down, and it is the next player’s turn. Play continues until all cards have been matched.
• Cut apart the vocabulary 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, minimum number of categories, and the categorization must be based on math context.
• Connect 2. Have students choose two words and explain how the words are related. For example: domain and range—both are necessary parts of a function.

Activity 4

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

Extension:

Use the suggestions below to tailor the lesson to meet the needs of the students.

• Routine: A few times a week, refresh students’ memories about how slope can be found on the graph of a line. Whether counting rise over run or just asking about steepness or positive/negative rise, students need multiple opportunities to master the concept of slope.

Verify that the constant rate of change is the same as the slope in this linear function.

(The following text can be used to reiterate students’ thinking during the class discussion.)

“When given two points, (x₁, y₁) and (x₂, y₂), the slope can be found by using the following formula:

“So, if we subtract the first two y-values, we get (5 − 3 = 2). If we subtract the first two x-values, we get (2 − 1 = 1). The ratio of . Thus, the slope is 2. Our constant rate of change was 2. This illustration shows that the constant rate of change is the slope.

“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 very important.

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

“Can we determine whether a function has a constant rate of change (and is linear) by looking at a graph? Certainly.”

• Small Groups: Students who may benefit from additional practice can be given the Function Exploration activity sheet (M-8-1-2_Function Exploration and KEY.docx) to work on. This will help them to see the differences between linear relationships and nonlinear relationships in a graphical sense.

Students may also be assigned to work their way through the activities on the following Web site:

http://www.watertown.k12.ma.us/wms/math/math_help/gradeeight/moving/msa.html

• Expansion: Assign students who are ready for a challenge beyond the requirements of the standard to work on the problems available in the pdf document found here:

http://www.mindset.co.za/resources/0000062453/0000135697/0000138942/MATHS%20Gr%2010%20Session%2011%20LN.pdf