In this lesson, students will investigate linear relationships, leading to connections between various representations of linear functions. Students will: 
- calculate the missing values for a specified function or function pattern. 
- determine the rule of a linear function from a table, graph, or list of values. 
- describe the similarities and differences between displaying a function in descriptive, rule, table, or graph form, as well as the advantages of each. 
- recognize and move between different representations of linear functions. 
- match numerous descriptions (words, rules, tables, or graphs) that represent the same linear function.

- How are relationships expressed mathematically? 
- How may data be arranged and portrayed to reveal the connection between quantities? 
- How are expressions, equations, and inequalities utilized to quantify, solve, model, and/or analyze mathematical problems? 
- How can mathematics help us communicate more effectively? 
- How can we utilize probability and data analysis to make predictions? 
- How does the type of data affect the display method? 
- How can mathematics help to measure, compare, depict, and model numbers? 
 

- Function: A relation where each input value is mapped/related to one and only one output value. In other words, for each input value, there is exactly one output value.
- Linear Function: A function that has a constant rate of change, or slope.
- Mapping: The “matching” of an input value to an output value.
- Nonlinear Function: A function with a degree of two or higher. For example,
f(x) = 3\(x^2\) – 1 is a nonlinear function because the degree of the independent variable is 2.
- Rate of Change: Of a function, the slope of the tangent to the graph of the function.
- Relation: Ordered pairs that relate an input value and an output value.
- Slope: The measure of the steepness of a line. The slope of a line is calculated by finding the ratio in the change of the y-values to the change in the x-values.

- Vocabulary Journal page (M-8-3-1_Vocabulary Journal)
- Linear Representations Table (M-8-3-2_Linear Representations Table and KEY)
- Checkpoint Quiz (M-8-3-2_Checkpoint Quiz)
- Matching worksheet (M-8-3-2_Matching and KEY)
- linear function slips (see Activity 2)
- envelopes and small plastic bags (see Activity 3)
- three-section spinner marked 1, 2, and 3 or standard number cube
- linear situation cards
- grid paper
- whiteboard with coordinate grid markings (optional)

- Observe and assess student performance during each discussion and activity. 
- Determine if students understand the difference between input and output values for functions, as well as whether they can identify what makes one relation a function and another relation not a function. 
- Formally assess students' comprehension during the presentations of Activities 1, 2, and 3. Check incorrect responses to see if the input and output are correctly distinguished. 
- Use the Checkpoint Quiz to evaluate individual responses. Ask students to make their own adjustments and explain their mistakes.

 

Scaffolding, Active Engagement, Modeling, Explicit Instruction
W: Students analyze several examples of linear functions and predict values based on a limited amount of data. Students will learn about the various representations of linear functions, including how to express them as equations, tables, sequences, and graphs. Finally, students will discover how understanding and interpreting functions can help them solve problems. 
H: The opening discussion encourages students to broaden their understanding of linear functions by learning how to represent them in other formats. 
E: Students create linear functions, establish function rules, and compare different representations. The lesson covers both abstract and concrete pieces, with the ultimate purpose of improving procedural and conceptual comprehension of linear functions. 
R: Students evaluate their learning by providing justifications, debating positions, and applying function knowledge. Throughout the lesson, students are invited to provide solutions and justifications. Both the students presenting and the audience are encouraged to think about and revise their solutions. 
E: Students analyze and reflect on their understanding through open-ended activities and class discussions. Students may be evaluated informally during class discussions and work time. Formal evaluation can occur during student presentations of numerous activities, as well as with the Checkpoint Quiz, which will be given around halfway through the lesson. 
T: Use the Extension ideas to personalize the lesson to the requirements of your students. The routine is appropriate for any student. The small-group activity is appropriate for students who require extra practice, and it may be done with the entire class, whereas the expansion and station activities are appropriate for students who have demonstrated proficiency and are ready for a challenge. 
O: This lesson teaches students how to represent linear functions in numerous ways. First, they'll practice writing function rules in slope, y-intercept, and slope-intercept form. The students then develop and present functions using a variety of approaches. Finally, students combine different representations of the same function and justify their decisions. Students end the class by creating their own matching activity to quiz other students. The lecture is somewhat abstract in nature, with some actual modeling incorporated.

"We discussed relations, looked at examples of relations, discussed functions, looked at examples of functions, connected relations to functions, linked various representations of linear functions to the concept of a constant rate of change, and briefly examined representations of nonlinear functions." Now we'll take a specific linear function and describe it in as many different ways as possible, using everything we've learned."

"Let's represent the linear function y = 5x - 8 in as many ways as possible. We shall organize our representations in another table."

Activity 1: Linear Representations

Part A: Hand out the Linear Representations Table (M-8-3-2_Linear Representations Table and KEY). Instruct students to look at the table and discuss with a partner how they can tell which representations meet the equation y = 5x - 8. They should then fill out the "How it Matches y = 5x - 8" column.

Part B: Provide student pairs with chart paper and markers. Distribute function slips at random to each pair. Students will generate as many alternative representations as possible in an organized table, similar to Part A's Linear Representations Table. Ask each pair of students to create a poster for their function and prepare to give a brief presentation about their work.

Part 1: Writing Function Rules

"Working with your partner, you began with a function rule, or an equation for a function, and then created several other representations for this function. Of course, the function equation is not always available. When given ordered pairs, a table, or a graph, we may need to find it on our own. This is what we will learn and practice next."

Slope-Intercept Form

Explain that the traditional approach to write a linear equation is in slope-intercept form. The slope-intercept form is described as an equation in the form of y = mx + b, where m and b represent significant values. Instruct students to analyze examples from the lesson, such as y = 5x - 8, and consider what the 5 and -8' could represent. (Encourage students to think carefully about this! To help, ask students questions about how the representations of this function looked.)

After students have had adequate time to think and discuss, lead them to discover that m is the slope, or constant rate of change, and b is the graph's y-intercept as well as the ordered pair's y-value (0, b).

"To write function rules, we require two pieces of information: the rate of change (slope, or m), and the y-intercept (or b). Refer to the preceding example of y = 5x - 8 to understand the relationship between the y-intercept and slope."

Before proceeding, ensure that students grasp the slope-intercept form as well as the meaning of the variables m and b.

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

Display 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 the number of days increases by 1, sales increase by 3 dollars.)

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

"We can check this fact in two ways." Review the methods listed below:

1) Because the x-values increase by 1 each time, we can calculate the difference between consecutive y-values and use that as our slope.

2) The slope can also be calculated using the change in y-values divided by the change in x-values. Using any two pairs of x- and y-values, such as (1, 15) and (2, 18), we have:

"In order to get the value of the variable "y", we need to replace the slope in the linear equation y = mx + b. The slope can be substituted to get the following:

y = 3x + b 

"The subsequent procedure involves choosing a (x, y) pair and substituting those values into the equation to determine the y-intercept. Let's select the point (2, 18). We now have:

18 = 3(2) + b

"By solving this equation, we may determine the value of the y-intercept (b):"

18 = 6 + b
18 = 6 - b
12 = b

“Our final step is to replace the slope and y-intercept in the equation y = mx + b to get y = 3x + 12."

"We have now created a function rule to express the relationship between the number of days and the dollar amount of sales. You can use this rule to forecast the amount of sales for any number of days."

Show how to calculate sales for additional values, like 9 days and 12 days. Show how this function can be graphed using the y-intercept and slope if a different representation of the function is required.

"Suppose we have a table that represents a linear function, but this time the input values are not consecutive numbers. This example is not intended to be more challenging. It's simply different. Let's look at the following function and create a function rule."

Display the table on the board.

"This time, our x-values do not increase by 1. As a result, we cannot calculate the slope simply by subtracting consecutive y-values. We must determine the ratio of change in y-values to change in x-values. Choosing any two pairs of x- and y-values will work. We will select (8, 61) and (10, 77). We determine 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 determine b:

61 = 8(8) + b
61 - 64 = b
-3 = b

"Substituting 8 for m and -3 for b in the the equation y = mx + b, we get y = 8x - 3. This rule can be used to calculate the number of treats required for any number of dogs."

A related skill is creating a function rule from a list of numbers in a pattern or sequence. Since this lesson is about linear functions, arithmetic (rather than geometric) sequences will be used. It may be useful at this point to go over the concept of arithmetic sequences with the class.

"An arithmetic sequence is a sequence with a constant rate of change or has a constant difference between values of terms. As we've already shown, arithmetic sequences are linear functions. They increase or decrease at a constant rate."

"Earlier in the lesson, we discussed function rules and sequences that reflect the same linear function. We also learned how to create a function rule based on a table. The next step is to create rules based on a sequence. The procedure is substantially identical to what we just completed with the table form. However, we must identify 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 natural numbers as position values (input) and the actual values in the sequence as output. We may also simply place the position numbers immediately above each value in the sequence list."

Illustrate both techniques.

"Because the position numbers increase by 1 and are consecutive, we can calculate the slope by taking the difference between any two consecutive term values. The difference is -6 because -13 - (-7) = -6. We can determine the slope by calculating the ratio of change in at least one other pair of values. Calculate the term value for the larger consecutive number minus the term value for the smaller consecutive number.

"The 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 equation for a line.

y = mx + b
-25 = -6(4) + b
-25 + 24 = b
-1 = b

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

y = - 6x - 1 

Distribute the Checkpoint Quiz (M-8-3-2_Checkpoint Quiz). Students will be instructed to write function rules for tables and arithmetic sequences. Use the results to evaluate whether extra practice on function rules is required before continuing with the lesson. At the end of the class, the results can be used to determine if individual students or small groups would benefit from remediation or enrichment activities.


Part 2: Graphing Linear Functions

"There are benefits to seeing the table or list of data that represent a function. Both of these forms show us the specific values that make up the pattern of change within the function. It can be useful to see a visual representation of a function. This allows us to easily determine whether a function is increasing or decreasing, changing slowly or quickly, and changing at a constant or varied rate. Looking at a function's graph allows us to quickly learn a lot of information. In this section of the class, we will graph linear functions using an x and y table. This involves plotting x- and y-values as ordered pairs.

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

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

Plot the points as a class.

"We now need to decide whether or not to connect the points that fall within our line. Should we connect the points? Why, or why not?" Give students the opportunity to make conjectures. The following is an excerpt from a classroom discussion facilitated by the teacher.

"The data represented by this function is continuous. There are infinitely many real numbers that can be utilized as input values (x-values) with infinitely many real numbers output values (y-values). Although the data points in the table are discrete, they only reflect a small portion of possible points for that function. Discrete data consists of particular values without any values include between those other values. If we were not modeling a function here and only required to plot x- and y-values, those values would be discrete. They would not be a part of something bigger or more inclusive. Discrete values just reveal the values themselves. With a function, however, there are infinitely many ordered pairs (or points on the line) to examine. Thus, the data for that function is continuous and should be connected to represent all of the additional values."

Instruct students to use the methods given previously in the class to get the equation of the line they graphed. (The equation for this function is \(2 \over 3\)x - 1).

 
Activity 2: Linear Functions in the Real World

Give each student a function slip. Each function will be represented by a table of values and a graph. They must also create a real-world context for 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 the presenter questions and aid the student in correcting any errors.

Summarize the benefits and drawbacks of using various representations of functions, particularly in real-world contexts. Also, address any remaining questions or any common errors identified during 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 and KEY). The worksheet includes a sample set of functions; however, more can be added. Instruct students to finish the worksheet.

As they finish, talk about the appropriately matched representations and possible strategy for determining the matches. Request that students present justifications for their matching.

 

Activity 3: Making Matches

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

Once students have completed their matched representations, they should mix the cards and trade piles with a partner. Allow the partners time to match their linear function representations. The student who developed the set will use his or her own key to verify his or her partner's work. Discuss any difficulties.

Extension:

Routine: Discuss the significance of understanding and using the appropriate vocabulary words to communicate mathematical ideas clearly. During this lesson, students should record the following terms in their Vocabulary Journals: arithmetic sequence, continuous, discrete, function, linear function, nonlinear function, rate of change, relation, slope. Keep a supply of Vocabulary Journal pages on hand so that students can add them as needed. Bring up examples of functions, constant rate of change, and slope as seen throughout the school year. Ask students to bring function graphs and examples from outside the classroom to discuss their use and meaning in each situation. Always require students to use appropriate vocabulary in both verbal and written responses.

Small Group, Review: The class should be organized into groups of two to four students. Request that each student prepare a set of five questions (and answers) containing at least one from each of the following categories: function representations, function rules, and graphing functions.
Each member of the small group will ask the other members questions. Hold a discussion about 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 stated in words. Students are assigned to work in groups of three at this station. Explain to students the following instructions (and post a copy at each station if needed):
1. Choose one of the linear circumstance cards at random.
2. Allow two students to spin the spinner until they get different values (1, 2, or 3); the last student will get the remaining number.
a. The student who spins 1 must make a table of values for the situation.
b. The student who spins 2 must make a graph using the table of values.
c. The student who spins 3 must use the table, graph, and a description to create a function rule.
3. Create a poster of the representations and place the linear situation card in the upper left corner.

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