Prior to class, write three or four function representations on the board. Some examples are:

• the cost of paying for a gym membership if there is an initial membership fee.
• the cost for renting movies if there is no membership fee.
• the profit for selling cookies if there is a cost for baking supplies and a specific selling price.

Represent one situation as a description, one as a table, and one as a graph.

Ask students to predict values that are present within the data and also beyond the provided data. (Students may have initial difficulty with some of these predictions, which is fine at this point.)

“The situations we just looked at are called relations and functions. This lesson will help you represent situations such as these in many different forms, and also help you understand how to interpret them.”

“Let’s begin by defining the term ‘relation.’ Can you explain what a mathematical relation is in your own words?” (Sample answer: how long it takes to travel between two points, depending on how fast you are traveling)

Accept ideas, rewording them slightly as necessary to correct errors. After student responses have been shared, offer a formal summary. Provide students with a Vocabulary Journal Page (M-8-3-1_Vocabulary Journal.docx) to record the definition. Encourage students to use this page whenever a new term is introduced throughout the unit. Keep a supply of journal pages available in the room for students to use when they have completed the first page.

“A relation relates two things. A relation might relate two numbers, two symbols, two objects, or two names. In mathematics, a relation links an input value and an output value. A relation does not have restrictions on the output values for any given input value. A relation can be represented in many different ways, including, but not limited to, a list of ordered pairs, a sequence, an equation, a table, or a graph. Often, a relation is drawn using two sets, with values or elements mapped to one another. Based on this description of a relation, can you think of some examples of different relations?”

Provide students time to write down various relations. Call on several students to come to the board and share their examples. Try to include a variety of different representations.

Below are several possible examples of relations. Be sure to include at least one of each of these if they are not shared by a student. Have students add examples to their own list.

• Example 1:      (0, −3), (2, 1), (4, 8), (−7, −2), (0, 1)
• Example 2:      4, 10, 16, 22, …
• Example 3:      y = –2x – 3     

• Example 4:

• Example 5:

“Of the examples we just visited, as well as others you created, some were simply relations, whereas others were both relations and functions.”

At this time, help students understand the difference between a relation and a function. Students should be able to determine the difference and also understand that a function is both a relation and a function.

“As you have determined, functions are related to the topic of relations. Here are two important questions to ponder.”

Write the following questions on the board.

1. Are all relations also functions? (no)
2. Are all functions also relations? (yes)

Allow students to make conjectures. Make a classroom tally chart for students who said “yes” to question 1 and those who said “yes” to question 2.

“As you may have guessed from our discussion of questions 1 and 2, we must develop an understanding of functions in order to accurately answer the questions. To start, we need to define a function. Try to write down a definition on scratch paper or in your Vocabulary Journal. If you have difficulty putting the definition in words, try to illustrate your thoughts with a table, graph, or other representation.”

Give students time to represent the concept of a function, in word form or in some other manner. Ask students to share their ideas with the class.

“A function is actually a type of relation. Remember that relations do not have any restrictions on which input values can be mapped to different output values. One major difference with functions is that they have a very important restriction. A function is a relation, whereby each input value is mapped, or related to, one and only one output value. In other words, for each input value, there is exactly one output value. This restriction only goes one way though, as an output value may be mapped to several input values.”

Go through the examples below, explaining the input and output relations which make the first and third examples functions (and relations), but the second example just a relation.

“Considering this definition, we can say that all functions are relations, but not all relations are functions. This statement marks a very important distinction.”

Before moving on, revisit students’ definitions and/or representations of functions. Provide time for discussion and debate regarding the accuracy of the definitions and representations. If your discussion leads to an example that is not a function, have students describe how it could be changed into a function.

“As with relations, we will look at various representations of functions. Before doing so, let’s revisit some relations from earlier in the lesson and determine which ones are also functions.”

Activity 1: Identifying Functions

Put the following examples on the board or point out where they still may be located from earlier in the lesson. Students will determine which relations are functions. Ask students to provide a brief justification. Answers should not be given to students until after the activity. (Answers are provided in italics.)

• Example 1:

(0, −3), (2, 1), (4, 8), (−7, −2), (0, 1)               Not a function; 0 is mapped to −3 and 1.

• Example 2:

4, 10, 16, 22, …                                              Is a function; each input value is mapped to only one output value; the input values are the natural numbers of 1, 2, 3,…

• Example 3:

y = −2x + 3                                                    Is a function; for each different x that is substituted into the equation, a unique output for the value of y is created.

• Example 4:

 Is a function; each input value is mapped to only one output value; It does not matter that 1 appears as an output for more than one input value. The key here is that an input value is not mapped to more than one output value.

• Example 5:

Provide discussion time for students to share findings and ask questions.

Linear Functions

“In this part of the lesson, we are going to focus on a subset of functions called linear functions. You have already seen some linear functions and likely created some of your own during our activities today. Can anyone define a linear function?” Allow students to provide definitions, descriptions, or sketches that help describe linear functions before sharing the formal definition.

“A linear function can be defined in several ways. Simply defined, a linear function is a function that has a constant rate of change. This definition applies to all representations and covers all other sorts of definitions. For example, we could say that a linear function is graphed as a line. This statement is certainly true. In fact, the root word of linear is line. What makes a line special? A line illustrates a constant rate of change, or a constant slope. All nonvertical straight lines represent functions. A line also pairs each domain value (x) with exactly one range value (y).”

Describe the concept of slope by demonstrating the constant rate of change. Show how it can be seen and calculated from a graph and by using coordinate pairs. Be sure to indicate that slope can be positive or negative, depending on whether the situation (or graphed line) is increasing from left to right or decreasing.

The slope is found by comparing the change in the rise and the change in the run from point to point. In this case, the “rise over run” (or slope) is 4/3.

Explain how to find the constant rate of change or slope, using the coordinates instead. For example, if two points were (1, 3) and (−2, −1), the slope would be calculated as shown below.

It is important for students to understand that for both methods of finding slope, any two points along the line will give you the same slope. If time permits, demonstrate this by calculating the slope in this example using a completely different pair of points.

Several examples of linear patterns are listed below. Reminding students that all linear patterns have a constant rate of change should help them with these patterns. Ask students to consider each example and try to fill in the missing values or extend the pattern. As they finish, call on several students to share their responses with the class.

• −4, −1, 2, 5, 8, ___, ____, …

(11 and 14, the values are increasing by 3 repeatedly)

• 4, 9, 14, ___, 24, …

(19, the values are increasing by 5 repeatedly)

• 98, 92, 86, 80, ____, ____, …

(74 and 68, the values are decreasing by 6 repeatedly)

• 3, 12, ___, 30, …

(21, the values are increasing by 9 repeatedly)

• 4, −1, −6, ____, ____, −21, …

(−11, −16, the values are decreasing by 5 repeatedly)

Linear Representations

There are a variety of ways to represent any function, including a linear function. Explain that the forms range from phrases and sentences, to lists of numbers or ordered pairs, to tables and graphs. Emphasize the importance of being able to identify a constant rate of change in any of these forms in order to determine whether or not the situation is linear.

“While viewing the linear functions below, let’s stop and think about the definition of a linear function and how it can be applied to various representations of functions.”

Ask students to consider real-life examples with a constant rate of change. Use an example such as Joe earns money mowing lawns. Each week he puts $10 in his savings account. Discuss how the balance will increase each week at a constant rate of $10. If graphed, this would be a line with a positive slope of  , or 10. Continue the discussion with questions, such as:

• “How do the linear functions we just looked at support the definition of a linear function having a constant rate of change?” (They are graphed as line; each input has exactly one output; the difference between each data point is the same each time; etc.)
• “What is a constant rate of change? What does that really mean?” (when something changes by the same exact amount each time)
• “If a function does not have a constant rate of change, what might its graph look like?” (in discrete parts with multiple slopes, vertical line, quadratic, cubic, etc.)
• “What is a rate?” (A rate compares two units or two variables, i.e., x and y. Rate is often discussed in terms of distance compared to time. Constant simply means “the same.” Thus, a constant rate is the same rate applied across the function, no matter what part of the graphed line, table, or situation you consider.)
• “What other phrase seems to be synonymous with ‘constant rate’?” (Slope is the rate of change of y-values per corresponding x-values. Slope can be determined by finding the ratio of change in y-values, divided by change in x-values. The phrase rise over run is sometimes used to describe this ratio based on the changes seen in the graphed line. We can calculate the slope by counting the changes on a graph or by looking at any two ordered pairs and using the formula: )
• “Can you think of another real-life example that is linear in nature?” (Answers will vary.)

“A constant slope means the same slope is found throughout the entire function. We see this type of slope only with linear functions. In our next activity we will examine several representations of functions and determine which are linear and why.”

Activity 2: Linear or Not?

Provide each student with a copy of the Linear or Not? worksheet (M-8-3-1_Linear or Not.docx and M-8-3-1_Linear or Not KEY.docx). Ask students to take each function example and consider the linear function definition. Write the definition on the board:

“A linear function is a function with a constant rate of change.”

Students will write “yes” or “no” in the “Constant Rate of Change?” column. A constant rate of change is also known as constant slope. Students will provide an explanation under the “Explain” column for all examples, stating why each is or is not linear. Allow students to work with a partner. Let students know that you will select a student pair to present each example when the work time ends. Move about the room, assisting students with yes/no answers and explanations. Use guiding questions to direct students to a logical path if their thinking is incorrect. Encourage students to consider each presentation, and add to or revise the ideas they have listed on their papers. Discuss any lingering questions before moving on to the next part of the lesson.

Although not the intent or focus of this lesson, it is important to compare representations of linear functions to those of functions that are not linear.

“What definition could you give for a nonlinear function?” (A nonlinear function is a function that is not linear; a nonlinear function is a function that does not have a constant rate of change, or constant slope.)

Listed below are some examples of nonlinear functions to discuss with the class. It is not important that students learn the names or equations for these functions. They should be able to see that these functions do not have a constant rate of change, which means they are nonlinear. Assist students with strategies to determine whether the rate of change is constant.