Objectives

This lesson explains the fundamental concept of a function in both mathematical and non-mathematical terms. Students will: - recognize functions and nonfunctions from situations, graphs, and tables. - use function notation to evaluate functions. - examine functions by looking at the graph of a function.

Core Questions

- What is a function, and how can we tell the difference between functions and nonfunctions when relationships are presented in a variety of ways?

Vocabulary

- Function: A relation in which every input value has a unique output value.  - Linear Function: An equation whose graph in a coordinate plane is a straight line.  - Domain: The set of valid numbers that can be input into the function.  - Range: The set of valid numbers that the function can give as output.  - Input: In functions, the independent variable.  - Output: In functions, the dependent variable.  - Independent Variable: Variable not under control by a researcher or experimenter.  - Dependent Variable: Variable that is under control by the researcher or experimenter.  - Vertical Line Test: No vertical line may intersect the graph of a function at more than one point.  - Relation: A set of ordered pairs.

Materials

- Function or Not Worksheet (M-A2-6-1_Function or Not Worksheet and KEY) - laminated copies of the Graph Go Round Worksheet (M-A2-6-1_Graph Go Round and M-A2-6-1_Graph Go Round KEY) - New Book Organization worksheet (M-A2-6-1_New Book Organization) - copies of the Jigsaw worksheet (M-A2-6-1_Jigsaw)

Assignment

- The accuracy with which each reporter identifies the book organization's classification scheme is the starting point for evaluating student reports on the New Book Organization activity. Al: subject; Betty: vowel in the author's first name; Carlos: alphabetical by title; Dion: alphabetical by author's surname. Assess the students' justification for the chosen system.  - The student's performance on the Jigsaw task will indicate how well s/he understands function notation, substitution, and simplification. Assess student performance on the Jigsaw activity by verifying the accuracy of substitution for each function and simplifying to determine f(x).

Supports

Active Engagement, Modeling, Explicit Instruction W: This lesson teaches students how to describe functions in various ways and distinguish between functional and non-functional interactions.  H: The New Book Organization task helps students understand how a function works in practice. In this activity, they will use the input/output method to construct a relationship between the independent and dependent variables.  E: The vending machine example illustrates function operations differently than book sorting and classification. This practice allows students to see the uniqueness of the outcome of each selection.  R: The six tasks on function/nonfunction discrimination encourage students to consider the uniqueness of the outcome for each input. Students must reaffirm their understanding that, while a given output can occur for several inputs, the relation is only a function if each input has only one outcome. This principle is highlighted graphically by the vertical line test.  E: The function tables starting with f(x) = 2x + 1 prepare students for the Graph Go Round activity, which involves translating between input/output values and x/y ordered pairs. They can assess their functions by matching the independent variables to the x-coordinate and the dependent variables to the y-coordinate.  T: The Jigsaw Activity accepts integers for functions ranging from -4 to 10. Start with simple functions like f(x) = 3x − 2, then advance to more complex functions using fractions, decimals, and parentheses.  O: The Jigsaw game allows students to express their ideas both verbally and in writing. Thus, it presents a challenge to all students. Furthermore, all students will hear various explanations, ensuring that at least one makes sense to them. Working in groups allows weaker students to observe and learn from a variety of different students. Mathematicians comprehend the fundamental nature of functions; practically all high school mathematics deals with functions. Learning to work with functions in general is an important component of advanced algebra, and calculus is essentially two additional things we can "do" to (or with) them. Students must have an intuitive awareness of the usefulness and utility of functions to progress in their advanced mathematical studies.

Procedures

Activity 1: Organizing Books at a Library  Tell students that in the first activity, New Book Organization (M-A2-6-1_New Book Organization), they will pretend to work for a library that has received a significant donation of new books and has asked their class to assist with shelving the new books so that other students can easily find them. The worksheets for this project describe that four librarian assistants (Al, Betty, Carlos, and Dion) have each come up with a different method for organizing the new books. It is the students' responsibility to help determine which is the best organizational structure. Group students into fours. At each table, one student will work on Al's system, another on Betty's, and so on. Students should keep track of who at their table will be exploring certain systems. Give each student a copy of the book list as well as the form outlining their system. Allow students to work alone for 10 minutes to sort the books according to their assigned system. After that, divide students into groups based on the system they used—for example, all students who used Al's system should meet together. In these groups, students should discuss their successes and challenges organizing by the system. After giving students enough time to discuss the advantages and disadvantages of each system, have them return to their first groups and report to the group on each specific organizational structure. After each group member has presented his or her system and highlighted the benefits and drawbacks, the group should agree on a single organizational system and explain why it was chosen. Allow each group to share the system they chose, as well as their reasons. Connect organizational systems to functions using the chart below:<figure class="image"><img style="aspect-ratio:412/196;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_123.png" width="412" height="196"></figure>Now, direct the class to a nearby vending machine (if one is not nearby or this is not practical, draw a picture of one on the board). Show the students how the different labels on the machine correspond to the products you'll be ordering. You push a button to receive your preferred object. Explain to the class that the button you push is the input to the vending machine, and the item you receive is the machine's output. Example: Keypad buttons: A, B, C, D; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 B3 may represent a bag of potato chips, whereas C4 may represent an apple. There may or may not be products available with the labels B4 or C5. There is, however, only one product for each letter and number combination. The vending machine may also provide the same product in many entry combinations. For example, D5 may sell the same bag of potato chips as B3. However, the user is sure to get his or her selection of potato chips by choosing either B3 or D5. "Imagine what would happen if you could obtain many items for each button on the machine (but only one). How would you react if the machine did this? Because chaos would ensue and you would never know what you were getting, it is critical that this vending machine provide only one item for any given input. Mathematical functions, like this machine, must produce exactly one output for any given input. Otherwise, the function is not useful, just as a vending machine that offers several options for one input is impractical and is therefore not made." Students should be familiar with different representations of linear functions (tables, graphs, equations, and stories/situations). Tell the class: "Now, we'll look at several representations of both functions and nonfunctions to see what makes something a function and what make something not a function. Let's begin by looking at some real-life situations or stories." Make up some scenarios that depict functions and non-functions. Some suggestions are included below. As you go through these examples, remember to use the vocal pattern of saying "each __________ has only one __________" for functions and "some __________ could have more than one __________, or no __________" for nonfunctions. The relation between each student in this room and his or her student ID number. (function) The relation between each student in this room and a vowel in his or her last name. (nonfunction) The relation between each student in this room and his or her race. (nonfunction) (Interesting side note: Until 1990, the Census Bureau treated race as a function; you could only select one. The Census now recognizes the diversity of Americans, and race is no longer a function.) The relation between a type of food and the grocery store aisle where it should be shelved. (function) (Yes, if all of the items in a specific category are found in one location; no, if they are found in numerous locations, such as cheese in both the dairy and gourmet food sections.) The relation between a song and the musician who sings it. (nonfunction) There are plenty of album covers. The relation between a student and his or her first-hour teacher (function) After giving a few examples, ask the students at their tables to list two situations that functions can describe and one that they cannot. Allow groups to share their scenarios with the rest of the class, and let the rest of the class decide if each situation represents a function or not. Create some tables of functions and non-functions: The following are some suggestions, although there are many more that you might utilize: Again, when you go through these examples, use the verbal pattern of saying "each __________ has only one __________" for functions and "some __________ could have more than one __________, or no __________" for nonfunctions.<figure class="image"><img style="aspect-ratio:497/128;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_124.png" width="497" height="128"></figure>For example 2, note that some x terms include more than one y: the x that is 1 has two y terms: 1 or -1. For example 6, note that some x terms include more than one y: the x term with the value 2 has two y terms: 2 or 3. After providing a few examples, instruct students to create two tables representing functions and one table that does not reflect a function. Allow groups to share their tables with the rest of the class, and let the rest of the class decide whether each table represents a function or not. Make graphs of functions and nonfunctions: here are some ideas. As you work through these examples, stick to the vocal pattern of expressing "each x has only one y" for functions and "some x could have more than one y" for nonfunctions.<figure class="image"><img style="aspect-ratio:489/307;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_125.png" width="489" height="307"></figure>The vertical line test is another method for determining whether a given equation is a function. Explain to the students, using the graphs as examples, that the vertical line test states that any vertical line drawn on a graph can only intersect the graph once. If it intersects the graph more than once, it is not a graph of a function. Demonstrate this test on the graphs above, emphasizing that the test must be true for any vertical line that may be drawn. After providing a few examples, instruct students to design two graphs representing functions and one graph that does not represent a function. Allow certain students to share their graphs with the rest of the class, and let the remainder of the class decide whether or not each graph represents a function. Students have now studied a variety of function (and nonfunction) representations, including real-world scenarios, tables, and graphs. Students are now prepared to complete the Function or Not Worksheet (M-A2-6-1_Function or Not Worksheet and KEY). The worksheet can be completed in groups or separately. While students are working, identify those who require more assistance comprehending the concept of functions in various representations and help those students verbalize their questions to their group to foster small group discussions and ensure comprehension. Introducing the notion of functions allows teachers and students to write a short essay or journal entry about one of the following questions: "Why do we need functions?" "Which representation of functions do you prefer? Why?"  "Draw two additional functions and two non-functions. Write an explanation of the distinctions between the two categories: functions and nonfunctions."  "What is your own definition of function?"  While many algebra students can easily generate tables of x and y values from an equation, most struggle with the f(x) notation. This purpose of this section of the lesson aims to reinforce the use of the f(x) notation.  Allow students two to three minutes at their tables to construct a table for y = 2x + 1. Ask the students to explain how they did this. Encourage conversation until all students comprehend the method.<figure class="image"><img style="aspect-ratio:149/446;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_126.png" width="149" height="446"></figure>Ask students: "Is anyone tired of hearing 'substitute... in for x'?" "Aren't we saying the same thing over and over again?" Explain how mathematicians devised "function notation" to avoid repeating the same long sentence. Rewrite the table for y = 2x + 1 in function notation with input values less than -2 and greater than 2. Clearly explain where the x and y values go. Create a new table for a basic linear function using function notation.<figure class="image"><img style="aspect-ratio:422/448;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_127.png" width="422" height="448"></figure>Make another table for the function f(x) = x − 7, and describe how to compute y and where x goes in detail to obtain the following table:<figure class="image"><img style="aspect-ratio:595/408;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_128.png" width="595" height="408"></figure>Present the following information to the students now: f(2) = 1; f(3) = 2; f(4) = 0; f(5) = 7 Help students work backwards to develop a table that represents the same information. Create another set of four statements in function notation, and have students work independently to develop a table to display the information.<figure class="image"><img style="aspect-ratio:147/384;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_129.png" width="147" height="384"></figure>f(−2) = −3; f(−1) = −1; f(0) = 1; f(4) = 9<figure class="image"><img style="aspect-ratio:147/386;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_130.png" width="147" height="386"></figure>Explain to students that they may now quickly switch between a function table and function notation. Remind students that a function table is only one of several methods to express a function. For example, teach them that a function can be represented as f(x) = 2x + 3 or y = 2x + 3. The notations f(x) and y are interchangeable; the one we use depends on the context. Also, f(x) refers to f of x, not f times x. Furthermore, we can describe a function using letters other than f; for example, we could write f(x) = 2x + 3, g(x) = 2x + 3, j(x) = 2x + 3, or, if we felt like it, z(x) = 2x + 3. Any letter can be used to represent a function; however, some are easier to use than others. Graph-Go-Round Activity Then, students will practice writing function notation on graphs. Determine who will be A, B, C, D, and E at your table, as well as who will serve as secretary. Each table should receive a laminated graph from the Graph Go Round activity (M-A2-6-1_Graph Go Round and M-A2-6-1_Graph Go Round KEY). Ask students, "Does the graph you have represent a function?" (yes) "For each x-value, is there only one y-value?" (yes) For the first round of the task, you may need to assist students explicitly. Each student should find the point associated with his or her letter on the graph. Beginning with A, have students read aloud the point on the graph and then read it in function notation. For example, A could state, "A is at (1, 5), so f of 1 is 5, and the written function notation for it is f(1) = 5." Students should practice not only reading graphs but also accurately pronouncing each function notation. It is also critical to tell students that correctly speaking function notation will help them recall how to apply the words they say to solve the problems. As each student reads the coordinates of his or her point and uses function notation, have the secretary record both ways of expressing the point. When all groups are done with round 1, provide any additional clarification to ensure that groups understand the task at hand. At this point, tell students they will get approximately 1 minute with each graph and then will have to pass it to the next group; make sure each group knows which group they will pass to and receive from. Have students move through the graphs until they reach the one they started with. Ask students if any graphs were more difficult than others. Answer student questions and allow secretaries to utilize the activity's key (M-A2-6-1_Graph Go Round KEY) to record accurate responses if a group provided erroneous or missing information. Then, gather the secretary's work. Explain to students that functions can also be stated as equations now that they have learned how to write function notation when starting with tables and graphs. Begin with a function like f(x) = −2x+3. Remind students that f(x) denotes the y-coordinate. Assist the class in determining f(1), f(0), and f(-5) values. Perform more examples as needed, emphasizing that whichever value replaces x inside the parenthesis is replaced for x on the right-hand side of the equation, and the order of operations is followed to achieve the y-value. Jigsaw Activity Prepare students for a jigsaw activity. Begin by dividing students into groups of four. One student from each group should choose a letter (W, X, Y, or Z). Each student will receive one copy of the Jigsaw worksheet (M-A2-6-1_Jigsaw). Students should circle their own letter in the upper right corner and write the names of the other members of their group. Create a function for the first section of the Jigsaw worksheet on the board. To get students started, begin with a basic linear function, like p(x) = 3x - 2. Once they grasp the method and can operate successfully, increase the difficulty by making the function more complex. Students should work silently on the first half of the Jigsaw assignment for 30 seconds to 1 minute. After that, have the students rearrange their groups. All designated W students should gather in one corner of the room, as should X, Y, and Z students. Students in each group should compare their responses and collaborate to reach a consensus, ensuring that each student knows how to arrive at the proper solution. Check-in with each group at this point to ensure they have correct answers and that each group has achieved full understanding. After all of the groups have reached a consensus, have the students return to their original groups, with the A students serving as "teachers." They should show the other students at their table how to solve the problem. Students in each group should continue teaching each other until each student has correctly solved all four problems. Once all students have finished the first third of the Jigsaw worksheet, display a second function on the board, followed by a third. The functions should become progressively more complicated, but always in response to how the class performs. Prior to going on to quadratics or square root functions, ensure that students grasp linear functions. Examine the various representations of functions (tables, graphs, and equations) and how to move between them, focusing on graphs to tables, tables to graphs, and equations to tables. Depending on the lesson, you can give students points stated as standard ordered pairs or in function notation and ask them to construct a graph with those points or even write an equation for that graph. (For the latter, begin with points that all lie on the same line, such as f(2) = 1; f(3) = −1; f(4) = -3.) Remind students that it is critical to be able to move efficiently and fluidly from one representation of a function to another because functions are stated in a variety of ways depending on the situation and the task at hand. Ask students which model of functions makes the most sense to them. Students will have differing opinions, so ask them to explain their reasoning to the class. During this discussion, students should consider how easy (or difficult) it is to transform each representation of a function into another. (For example, students may respond, "I prefer seeing it as a table." Ask why they think using a table is more convenient. Possible comments can include how simple it is to transition from a table to a graph by plotting points.) Also, quickly explain to the class another method of listing a function. Show the class the function y = 2x - 5, and explain that it can also be written as (x, y) coordinates. We refer to this strategy as listing by ordered pairs. So for this function, some of the ordered pairs are: (0, −5), (1, −3), (2, −1), (3, 1), (4, 3), and so on. It's similar to listing in a table, but instead of table headings, we use (x, y) to indicate what's going on. Graph on the board the following functions for the class: y = x² and y = 2x. Show the class how the function x² can get the same result with two different inputs. For example, the functions y = x² with x = 3 and x = -3 both produce 9. However, with the function y = 2x, this never occurs; each input yields a different result. These functions, in which each input generates a unique output, are referred to as one-to-one. If any horizontal line you draw intersects the graph more than once, the function is not one-to-one. Draw a horizontal line on the graph of y = x² across the points (−3, 9) and (3, 9), demonstrating how it intersects the graph multiple times. Extension: Number sequences may also be represented as functions. For example, the sequence −3, 1, 5, 9, 13, … has the rule: add 4. In this approach, each input value delivers its output as the following element in the sequence: (-3, 1), (1, 5), (5, 9), and (9, 13). We may write this sequence as a function, f(x) = x + 4. Assign the sequences listed below for representation as ordered pairs and functions. 37, 27, 17, 7, −3, …                                        [(37, 27), …(7, −3); f(x) = x − 10] 0, 1, 3, 7, 15, 31, …                                         [(0, 1), …(15, 31); f(x) = 2x + 1]   Explain to students that each successive input value does not have to be the most recent output value. For example, in the prior sequence, the ordered pairs (−1, −1) are appropriate because 2(−1) + 1 = −1. 0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, 6, …                    [(0,1),…..(10,6); f(x) = \(1 \over 2\)x + 1]

Layla Nguyen

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The Definition of a Function (M-A2-6-1)

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Layla Nguyen

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Description

This lesson explains the fundamental concept of a function in both mathematical and non-mathematical terms. Students will:
- recognize functions and nonfunctions from situations, graphs, and tables.
- use function notation to evaluate functions.
- examine functions by looking at the graph of a function.

Grade

Grade 11 Grade 12