• Observe and evaluate student performance during each discussion and activity. Determine whether students are recognizing the difference between input and output values for functions and whether they can identify what makes one relation a function and another relation not a function.
• Formally evaluate student understanding during the presentations of Activities 1, 2, and 3. Check incorrect responses to determine if the correct distinction is made between input and output.
• Evaluate individual responses using the Checkpoint Quiz. Ask students to make their own corrections and to describe their errors.

“We have discussed relations, looked at examples of relations, discussed functions, looked at examples of functions, connected relations to functions, connected various representations of linear functions to the idea of a constant rate of change, and briefly examined representations of nonlinear functions. Now we are going to take a particular linear function and represent it in as many ways as possible, using the knowledge we have gained.”

“Let’s take the linear function  and represent it in as many ways as possible. We will organize our representations using another table.”

Activity 1: Linear Representations

Part A: Distribute the Linear Representations Table (M-8-3-2_Linear Representations Table.docx and M-8-3-2_Linear Representations Table KEY.docx). Instruct students to view the table and discuss with a partner how they can tell that each representation matches the equation . They should then complete the “How it Matches ” column.

Part B: Provide student pairs with chart paper and markers. Randomly pass out a function slip to each pair. For the function, students will create as many other representations as they can in an organized table, similar to the Linear Representations Table from Part A. Ask each pair of students to make a poster for their function and be prepared to give a short presentation of their work.

Part 1: Writing Function Rules

“Working with your partner, you started with a function rule, or the equation of a function, and created several other representations for this function. Of course the function equation is not always provided. Sometimes we need to find it ourselves if we are given ordered pairs, a table, or a graph. That is what we will be learning and practicing next.”

Slope-Intercept Form

Explain that the standard way of writing a linear equation is called slope-intercept form. Slope-intercept form is defined as an equation in the form of y = mx + b, where m and b represent some significant values. Instruct students to view examples seen earlier in the lesson, such as
y = 5x − 8, and discuss what the 5 and −8’ might represent. (Really encourage students to think deeply about this! To help, ask students questions about how the representations of this function looked.)

After they have had ample time to think and discuss, lead students to discover that m is the slope, or constant rate of change, and b is the y-intercept as seen in a graph, and also the y-value as seen in the ordered pair (0, b).

“In order to write function rules, we need to know two pieces of information: the rate of change (slope, or m) and the y-intercept (or b). Look again at the previous example of
y = 5x − 8 to see the relationship between the y-intercept and the slope.”

Make sure students understand slope-intercept form and the meaning of the variables m and b before moving on.

“Suppose we have the following table and are asked to write the function rule.”

Show this table on the board.

“Can we verify that the data in the table represents a linear function?” (Yes, it has a constant rate of change. As days increase by 1, sales increase by $3.)

“What appears to be the rate of change?” (3)

“We can check this fact in two ways.” Go through the methods below:

1) Since the x-values increase by 1 each time, we can simply find the difference between consecutive y-values, and indicate that as our slope.

2) We can also find the slope using the ratio of change in y-values to the change in x-values. Choosing any two pairs of x- and y-values such as (1, 15) and (2, 18), we have:

“We must substitute slope into the linear equation   in order to find the
y-intercept and complete the function rule. Substituting the slope gives:

“The next step is to select any (x, y) pair and substitute those values into the equation to find the y-intercept. Let’s choose (2, 18). We now have:

“Solving this equation gives us the value of the y-intercept (b):

“Our final step is to substitute the slope and y-intercept into the equation y = mx + b to get .”

“We have now written a function rule to describe the relationship between the number of days and the dollar amount in sales. You can use this rule to predict the amount of sales for any number of days.”

Demonstrate finding the sales for some additional values, such as 9 days and 12 days. Also demonstrate how this function can be graphed using the y-intercept and slope if another representation of the function were needed.

“Suppose we have a table representing a linear function, but this time the input values are not consecutive numbers. Such an example is not intended to be more difficult. It is simply different. Let’s examine the following function and write a function rule.”

Show the table on the board.

“This time, our x-values do not increase by 1. Therefore, we cannot simply take the difference between consecutive y-values to find the slope. We must calculate the ratio of change in y-values per change in x-values. Choosing any two pairs of x- and y-values will work. We will choose (8, 61) and (10, 77). We find the slope to be:

“As before, we will substitute the slope value, 8, and the x- and y-values from one ordered pair (8, 61) to find b:

“Now by substituting 8 for m and −3 for b in the equation y = mx + b, we get . This rule can be used to determine the number of treats needed for any number of dogs.”

An accompanying skill is writing a function rule from a list of numbers in a pattern, or sequence. Since this lesson focused on looking at linear functions, arithmetic (not geometric) sequences will be used. It may be beneficial to review at this point the concept of arithmetic sequences with the class.

“An arithmetic sequence is a sequence with a constant rate of change, or a constant difference between values of terms. As we have already seen, arithmetic sequences are actually linear functions. They increase or decrease at a constant rate.”

“Earlier in the lesson, we looked at function rules and sequences representing the same linear function. We also learned to write a function rule based on a table. Our next step is to write rules based on a sequence. The process is essentially the same as what we just finished with the table form. However, we need to label the position of each value in the sequence by numbering them 1, 2, 3, and so on.”

“For example, suppose we have the arithmetic sequence: −7, −13, −19, −25, …”

“We can create a table using the natural numbers as the position values (input), and the actual values in the sequence as the output. We could also just place the position numbers directly above each value in the sequence list.”

Demonstrate both methods.

“Since the position numbers increase by 1 and are indeed consecutive, we can find the slope by finding the difference between any two consecutive term values. The difference is −6 because −13 – (−7) = −6. We can check the slope by finding the ratio of the change in at least one other pair of values. Take the term value for the larger consecutive number minus the term value for the smaller consecutive number.

“Our slope is correct. Now, we need to find the y-intercept. We will substitute our slope, –6, and any (x, y) pair, such as (4, –25), into the y = mx + b form for equations of a line.

“Thus, our function rule for this arithmetic sequence can be written as” (write on the board):

Distribute the Checkpoint Quiz (M-8-3-2_Checkpoint Quiz.docx and M-8-3-2_Checkpoint Quiz KEY.docx). Students will be asked to write function rules for tables and arithmetic sequences. Use the results to determine if more work on function rules is needed before moving on with the lesson. Results can also be used to determine whether remediation or enrichment would be appropriate for individual students or small groups at the end of the lesson.

Part 2: Graphing Linear Functions

“There are advantages to seeing the table or list of values that represent a function. Both of these forms allow us to see exact values that make up the pattern of change within the function. Sometimes it is beneficial to see a visual representation of a function. This can help us easily determine whether a function is increasing or decreasing, changing slowly or quickly, and changing at a constant or varied rate. We can gain a great deal of information quickly by looking at the graph of a function. In this portion of the lesson we will be graphing linear functions using an x and y table. Doing so involves plotting x-values and
y-values as ordered pairs.”

With students, work through graphing the linear function represented by the values in the table below. Provide students with grid paper or personal white boards with grid markings so they can work through examples with you.

Demonstrate adding another column to write down the specific ordered pairs that the x and y values represent.

Plot all of the points as a class.

“Now need to determine whether or not we should connect the points that fall within our line. Should we connect the points? Why or why not?” Provide students with an opportunity to provide conjectures. Shown below is a sample excerpt from a classroom discussion, facilitated by the teacher.

“The data we have represented by this function is continuous. There are infinitely many real numbers that can be used as input values (x-values), with infinitely many real number output values (y-values). Although the data points in the table are discrete, the table only shows a small portion of the possible points that represent that function. Discrete data involves specific values, without any values included between those other values. If, for example, we were not representing a function here, and simply needed to plot x- and y-values, those values would be discrete. They would not be a part of something bigger, or more encompassing. Discrete values have nothing more to show than the values themselves. With a function, however, there are infinitely many ordered pairs (or points on the line) to consider. Thus, the data for that function is continuous and should be connected to represent all of the additional values.”

Instruct students to determine the equation of the line they have graphed using the methods introduced earlier this lesson. (The equation for this function is  .)

Activity 2: Linear Functions in the Real World

Give all students function slips. For each function they will create a table of values and a graph. They must also develop a real-world context to accompany the linear function and use the graph to answer at least one question posed by the real-world problem. Students will present their work. During presentations, allow the audience to ask questions of the presenter and assist the student in correcting any errors.

Summarize the advantages and disadvantages of using the different representations of functions, especially when they are in real-world contexts. Also address any questions students still have or any common errors noted throughout the presentations.

Part 3: Matching Linear Function Representations

“This portion of the lesson will help assess your ability to match different representations of the same function.”

Distribute the Matching worksheet (M-8-3-2_Matching.docx and M-8-3-2_Matching KEY.docx). The worksheet provides a sample set of functions, but more may be added. Instruct students to complete the worksheet.

As they finish, discuss the correctly-matched representations and possible strategies used to determine the matches. Ask students to present justifications for their matching.

Activity 3: Making Matches

Instruct each student to create sets of 3 matched representations for 5 different linear functions (equation, table, and graph for each). Students should cut out these 15 representations. Instruct students to number the 15 representations in a random order and to make a key that groups the sets of matching representations for each of the 5 functions. Provide paper, grid paper, and/or index cards for students to use. Each student will also need a small bag, envelope, or paper clip to secure the 15 representations.

When students have finished creating their matched representations, they should mix up the cards and trade piles with a partner. Give the partners time to match each other’s linear function representations. The student who created the set will use his/her own key to check his/her partner’s work. Discuss any difficulties.

Extension:

• Routine: Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ Vocabulary Journals: arithmetic sequence, continuous, discrete, function, linear function, nonlinear function, rate of change, relation, and slope. Keep a supply of Vocabulary Journal pages on hand so students can add pages as needed. Bring up instances of functions, constant rate of change, and slope as seen throughout the school year. Ask students to bring function graphs and examples that they see outside of class and discuss the use and meaning in each particular context. Always require students to use appropriate vocabulary in both verbal and written responses.
• Small Group, Review: The class should be arranged in groups of two to four students. Ask each student to create a set of five questions (and answers), with at least one in each of the following categories: function representations, function rules, and graphing functions.

Each member of the small group will ask the rest of the group questions. Hold a discussion related to any difficulties or concerns.

• Station, Exploring Linearity: Place several linear situation cards at the station. Each of these cards should contain a real-life linear situation described in words. Groups of three students are assigned to work together at this station. Explain to students the following instructions (and post a copy at each station if necessary):

1. Randomly select a linear situation card.
2. Have two students spin the spinner until they get different values (1, 2, or 3); the last student gets the remaining number.

1. The student who spins 1 must make a table of values for the situation.
2. The student who spins 2 must make a graph of the table of values.
3. The student who spins 3 must use the table, graph, and description to write a function rule.

3. Make a poster of the representations and attach the linear situation card to the upper left corner.

• Expansion, Connecting It: Ask students to provide examples of discrete and continuous data. Have students provide an example of when points would and would not be connected on a graph. You may also ask students to represent their functions with mapping.