• Assessment of student reports on the New Book Organization activity begins with how accurately each reporter identifies the classification scheme for the book organization. Al: subject; Betty: vowel in author’s first name; Carlos: alphabetical by title; Dion: alphabetical by author’s last name. Evaluate students’ justification for the chosen system. [IS.5 - All Students]
• Student performance on the Jigsaw activity will represent how well s/he understands the function notation, substitution, and simplifying. Evaluate student achievement on the Jigsaw activity by checking the accuracy of substitution for each function and simplifying to find f(x).

Activity 1: Organizing Books at a Library [IS.3 - Struggling Learners]

Tell students that in the first activity, New Book Organization (M-A2-6-1_New Book Organization.doc), they will pretend to work with a library that has received a large donation of new books and asked their class for help in shelving the new books so other students can easily find them. The worksheets for this activity explain that four librarian assistants (Al, Betty, Carlos, and Dion) have each come up with a different way to organize the new books. It is the students’ task to help figure out which is the best organizational system.

Arrange students in groups of four. [IS.4 - All Students] At each table, one student will work on Al’s system, one person on Betty’s system, and so on. Students should record which person at their table will be exploring each system.

Give each student a copy of the list of books as well as the sheet describing their system. Allow students to work for 10 minutes individually to sort the books using their assigned system.

After that, split students up based on which system they used—have all students who used Al’s system meet together, and so on. In these groups, students should discuss their successes and challenges organizing by this system.

After allowing students sufficient time to discuss the pros and cons of each system, have students return to their original groups and report to the group about each particular organizational system. After each group member has presented his/her system and described the pros and cons, the group should decide on a single organizational system and list reasons why that system was chosen.

Allow each group to share which system was chosen, along with the reasons. Connect the organizational systems to functions by using the following chart:

Now have the class proceed to a nearby vending machine (if one is not nearby or this is not practical, draw a picture of one on the board). On the machine, show the class the various labels and how they correspond to the items you will be purchasing. You push a button, and receive your desired object. Explain to the class how the button you push is the input to the vending machine and the item you receive is the output of the machine.

Example: Keypad buttons: A, B, C, D; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

B3 may correspond to a bag of potato chips and C4 may correspond to an apple. There may or may not be products available with labels B4 or C5. There is however, one and only one product for each letter–number pair. The vending machine may also offer the same product in different entry combinations. For example, D5 may also vend 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.

“Now, imagine what would happen if for each button on the machine, there were multiple possible items you could receive (but you still only got one). How would you respond if the machine did that? Because chaos would result and you would never know what you are getting if that happened, it is important that this vending machine give one and only one item for any given input. Mathematical functions are just like this machine, they must provide one and only one output for a given input. Otherwise, the function is not useful, just as a vending machine that gives multiple possible items for one input is not practical and is therefore not made.”

Students should already be familiar with various representations of linear functions (tables, graphs, equations, and stories/situations). Tell the class:

“Now, we’re going to look at different representations of both functions and nonfunctions and determine what makes something a function and what makes something not a function. Let’s start by looking at some real-life situations or stories.”

Make up some situations that represent functions and nonfunctions. Some suggestions are below.

As you go through these examples, keep to the verbal pattern of saying “each ______ has only one ______” for the functions and saying “some ______ could have more than one _______, or no ________” for the nonfunctions.

• The relation connecting each student in this room to his/her student ID number. (function)
• The relation connecting each student in this room to a vowel in his/her last name. (nonfunction)
• The relation connecting each student in this room to his/her race. (nonfunction) (Interesting side note: Until 1990, the Census bureau did treat race as a function: you could only choose one. Now the Census acknowledges the diversity of Americans, and race is no longer a function.)
• The relation connecting a type of food and the grocery store aisle it should be shelved in. (function) (Yes, if the items in a particular category are all found in one location; no, if they are found in multiple locations, e.g., cheese may be located in the dairy section and in the gourmet food section.)
• The relation connecting a song and the artist who sings it. (nonfunction) There are lots of album covers.
• The relation connecting a student and his/her first-hour teacher. (function)

After you give a few examples, ask students at their tables to describe two situations that can be represented by functions and one situation that cannot be represented by a function. Have groups share their situations with the rest of the class and have the rest of the class determine whether each situation represents a function or not.

Make up some tables of functions and nonfunctions: below are some suggestions, but there are many more that you could use.

Again, as you go through these examples, keep to the verbal pattern of saying “each ______ has only one ______” for the functions and saying “some ______ could have more than one _______, or no ______” for the nonfunctions.

For example 2, note that some x terms have more than one y: the x that is 1 has two y terms: either 1 or negative 1.

For example 6, note that some x terms have more than one y: the x that is 2 has two y terms: either 2 or 3.

After you give a few examples, ask students to create two tables that represent functions and one table that does not represent a function. Have groups share their tables with the rest of the class and have the rest of the class determine whether each table represents a function or not.

Make up graphs of functions and nonfunctions: below are some suggestions.

As you go through these examples, keep to the verbal pattern of saying “each x has only one y” for the functions and saying “some x could have more than one y” for the nonfunctions.

Another way to test if an equation represents a function is via the vertical line test. Explain to the student, using the graphs given as examples, that the vertical line test states that if we draw any vertical line on the graph, it can intersect the graph at most one time. If it intersects the graph more than one time, the graph is not a graph of a function. Demonstrate this test on the above graphs, being sure to emphasize that the test states it has to be true for any vertical line you can draw.

After you give a few examples, ask students to create two graphs that represent functions and one graph that does not represent a function. Have some members of the class share their graphs with the rest of the class and have the rest of the class determine whether each graph represents a function or not.

By this point, students have explored various representations of functions (and nonfunctions): real-world situations, tables, and graphs. Students are now ready to work on the Function or Not Worksheet (M-A2-6-1_Function or Not Worksheet.doc and M-A2-6-1_Function or Not Worksheet KEY.doc). The worksheet can be done in groups or individually.

While students are working, identify those who need extra help understanding the concept of functions in various representations and help those students verbalize their questions to their group to facilitate small group discussions and ensure understanding.

Introducing the concept of functions gives teachers and students an opportunity to write a short paragraph or journal entry based on one of the following questions:

• “Why do we need functions?”
• “Which representation of functions do you like best? Why?”
• “Draw two more functions and two nonfunctions. Write an explanation about the differences between the two groups: functions and nonfunctions.”
• “What is your own definition of function?”

While many algebra students easily create tables of x and y values from an equation, most balk at the f(x) notation. The purpose of this section of the lesson is to reinforce the use of the f(x) notation.

Give students 2 to 3 minutes at their tables to create a table for y = 2x + 1. Ask students to explain their method for doing this. Elicit discussion until all students understand the process.

Ask students:

“Is anyone tired of hearing ‘substitute … in for x’?”

“Aren’t we saying the same thing over and over again?”

Explain that to avoid saying the same lengthy phrase over and over again, mathematicians invented “function notation.”

Rewrite the table for y = 2x + 1 using function notation with input values less than −2 and greater than 2. Clearly explain where the x goes and where the y value goes. Create another table for a simple linear function using function notation.

Create another table for the function f(x) = x − 7, and clearly explain where the x goes and how you compute y to get the following table:

Now, provide students with the following information:

f (2) = 1; f (3) = 2; f (4) = 0; f (5) = 7

Help students work backwards and create a table to represent the same information. Create another set of four statements in function notation and have students work on their own to create a table to represent the information.

f (−2) = −3; f (−1) = −1; f (0) = 1; f (4) = 9

Explain to students that they are now able to move quickly between a function table and function notation. Remind students that a function table is just one of many ways to represent a function. For example, tell students that we can represent a function as f(x) = 2x + 3, or as y = 2x + 3. The notation f(x) and y are interchangeable; which one we use depends on context. Also, f(x) stands for f of x, not f times x. In addition, we can use other letters besides f to represent a function; for instance we could do the following: f(x) = 2x + 3, or g(x) = 2x + 3, or j(x) = 2x + 3, or if we felt like it, z(x) = 2x + 3. Any letter can be used to represent a function, though some are less confusing than others to use.

Graph-Go-Round Activity

Next, students will practice writing function notation from graphs. At your table, decide who will be A, B, C, D, and E, and designate a secretary. Each table should get one laminated graph from the Graph Go Round activity (M-A2-6-1_Graph Go Round.doc and M-A2-6-1_Graph Go Round KEY.doc).

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 activity, you may need to help students explicitly. Each student should find the point labeled with his/her letter on the graph. Starting with A, have students read aloud the point on the graph and then read it as function notation. For example, A may say, “A is at (1, 5) so f of 1 is 5 and the written function notation for it is f(1) = 5.” Students need to practice not only reading graphs but also saying each function notation correctly. It is also important to tell students that correctly spoken function notation will help them remember how to use the words they say to solve the problems. As each student reads the coordinates of his/her point and the point using function notation, have the secretary record both ways to express 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 work through the graphs until they are passed the graph they began with. Ask students if there were any graphs that were more difficult than others. Answer student questions and allow the secretaries to use the activity’s key (M-A2-6-1_Graph Go Round KEY.doc) to record the correct answers if a group had incorrect or incomplete answers. Then collect the secretary’s work.

Now that students have learned how to write function notation when beginning from tables and from graphs, explain to students that another way functions are often expressed is as equations.

Begin with a function like: f(x) = −2x + 3. Remind students that f (x) represents the y-coordinate. Help the class work out f (1), f (0), and f (−5). Do additional examples as necessary, reinforcing that whatever value takes the place of x inside the parentheses is substituted in for x on the right-hand side of the equation, and then order of operations is followed to obtain the y-value.

Jigsaw Activity

Arrange students for a jigsaw activity. Begin by separating students into groups of four. In each group, one student should choose a letter (W, X, Y, Z). Hand out one copy of the Jigsaw worksheet (M-A2-6-1_Jigsaw.doc) to each student. Students should circle their own letter in the top right and write the names of the other members of their group.

Provide a function for the first part of the Jigsaw worksheet on the board. To get students started, begin with a simple linear function, such as p(x) = 3x −2. After they understand the process and can work successfully, raise the difficulty by increasing the complexity of the function (include square roots, quadratics, fractions, etc.). Have students work on the first part of the Jigsaw worksheet silently for 30 seconds to 1 minute. After that, have students rearrange their grouping.

All the students who are designated W students should meet in one corner of the room; similarly with X, Y, and Z students. In each group, students should compare their answers and work together to come up with a consensus answer, ensuring that each student in each group understands how to get the correct answer. Make sure to check in with each group at this point to be sure they have correct answers and that each group has achieved full understanding.

After all the groups have come up with consensus answers, have students get back into their original groups and have the A students begin as “teachers.” They should teach the other students at their table how to solve their problem. Students in each group should continue teaching each other until each student has all four problems solved correctly.

Once all students have completed the first third of the Jigsaw worksheet, put a second function up on the board and, after that, a third function. The functions should get progressively more complicated, but always based on how the class is performing. Make sure students understand linear functions before moving to quadratics or square root functions.

Review the various representations of functions (tables, graphs, and equations) and how to move from one to another, particularly from graphs to tables, tables to graphs, and equations to tables. Depending on the class, you can provide points, expressed as traditional ordered pairs or in function notation, and have students draw a graph that contains those points or even write an equation for that graph. (For the latter, make sure to start 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 important to be able to move efficiently and fluidly from one representation of a function to another because functions are expressed in various forms, depending on the situation and the task at hand.

Ask students which representation of functions makes more sense to them. Students will have different opinions, so ask them to explain their thinking to the class. During this conversation, have students focus on how easy (or difficult) it is to take each representation of a function and translate it to another representation. (For example, students may say, “I prefer seeing it as a table.” Ask why they think it is easier to work with a table. Possible responses might mention how easy it is to move from a table to a graph by plotting points.)

Also briefly mention to the class another way of listing a function. Show the class the function y = 2x − 5, and say that we can list this function in another way, as (x, y) coordinates. We call this method 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. In a sense, it is like listing in a table, except we now use (x, y) instead of table headings to tell us what is going on.

Also graph on the board the following functions for the class y = x² and y = 2x. Show the class that the function x² has many cases where we can get the same output with two different inputs. For instance, notice that if we provide the function y = x² with x = 3 and x = −3, both will return 9 as the output. However, for the function y = 2x, this never happens, each input provides a unique output. These functions where each input provides a unique output are called one to one, and a quick way to tell if a function is one-to-one is if any horizontal line you draw intersects the graph more than once, then it is not one-to-one. Demonstrate this by drawing a horizontal line on the graph of y = x² through the points (−3, 9) and (3, 9) and show the class how it intersects the graph more than once there.

Extension:

• Number sequences may also find representation as functions. For example, the sequence −3, 1, 5, 9, 13, … has as its rule: add 4. In this way, each input value gives its output as the next element in the sequence: (−3, 1), (1, 5), (5, 9), (9, 13). We may represent this sequence as a function, f(x) = x + 4. Assign the following sequences for representation as ordered pairs and as a function.

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]

Point out to students that each successive input value does not have to be the most recent output value. For example, in the previous 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, …