Reflection: Introduction to Functions  Section 4: Closing
Student responses to the exit ticket "What will you remember about how a function machine works?" were very interesting to me and followed a couple of themes.
First, two students remembered that the input is not necessarily limited to numbers. In the opening of today's class, we used words and other examples to get students to understand the concept of a function, rather than the focus on a mathematical rule. I found it interesting that this was a big take away for students. I think it is sometimes helpful for students to step away from the more traditional idea of math in order to have deep understanding of a new concept. One student commented that this was "great," which I thought was cute, but also reflected on how he was thinking positively about function machines.
Next, two students had somewhat different understandings about some work we did with the difference between a pattern and a function. In one of the In/Out tables we looked at, one student noticed a repeating pattern: take the third letter of the input for the first input, and then the fourth letter for the second input, then the third letter, then the fourth letter, etc. Other students noticed (as the text wanted them to) that the Output was the second vowel of the input. This led us to a nice discussion about how a function always has to give the same result for the same Input. That is, a pattern could take the third letter the first time and the fourth the second, but a function would always have to have the same Output for a given Input; it couldn't switch around. In class, I was careful to compliment the student who had found a cool pattern that was not actually a function. I think this lead to some confusion though, based on some of the Exit Tickets. One student wrote, "... patterns can't be a function." In the next class, I will ask him to say more about this point to check for understanding. Another student wrote, "...patterns can't always be function." When I asked him to say more, he wrote, "A consistent output." I would like to open class in our next lesson and have him explain his thinking. I like this phrase "a consistent output" and want students to revisit the idea that a function always has to give the same Output for the same Inputs.
Lastly, another student wrote, "That the in & out have to be somewhat the same." When I asked her more about this statement at the very end of class, I was trying to understand what she meant. I said to her, but can't we put in a word and get out a number? She said yes, but they have to be related. I think this is a brilliant idea and a point that may be missing from this lesson. The relationship between the domain and the range.
I plan to open my next lesson with these exit tickets typed up and bring out these key points.
Introduction to Functions
Lesson 6 of 14
Objective: SWBAT use tables as one representation of a function. SWBAT understand and use the terms input and output and correlate them to the domain and range of a function. SWBAT give verbal statements of rules.
Opening
Begin class with a big image of a function machine on the board (like the homemade one in the Resources section). Tell students you have this machine, and something is happening inside of it. Begin to put numbers into the machine. For example, tell students you are putting in the number 3. Then, make some of kind of noises or silliness to how them that something is happening to the 3 inside the machine. Ask them if they can guess what will come out of the machine after this something happens. Keep your first example simple. For example, you might choose to multiply each Input by 2. Let students guess until they have a 6 come out of the machine and then write the number 6 somewhere near where it says Output. Continue on with more numerical examples for the same rule. Make sure you emphasize that the same thing is happening to each Input once inside the machine. It shouldn't take long for students to realize that each number is getting multiplied by 2 inside of the machine.
Next, tell students the machine will now do something different inside. I like to move on to a nonnumerical example next so students get the idea of a what a function is without being overly concerned with actual numbers. You might do something like put in the name of a student as the Input, and have the second letter of the students name come out as an Output. Students love examples that involve them! Continue on until you're sure students have the idea that a value goes into the machine as an Input, something happens inside the machine, and a corresponding value comes out of the machine as an Output.
Have fun with your examples. If you have longer than a 60 minute block, you could also let students come up and present their own function machines as a challenge to the class.
Now, show students that you have made a big mess of your board trying to keep track of what this machine is doing. Ask them if they can think of a way to organize this information that won't be so messy. Try to elicit the idea for a table, or more specifically, an In/Out table. Show students they can keep track of what goes into the machine on the In side of the table, and they can keep track of their Outputs on the Out side of the table. Additionally, you can have students write a rule in words for one table as an example (perhaps under the table). Encourage students to use In/Out language. You might ask them, "In this table, what do you do to the In to get the Out?" Look for a response like "You multiply the In by 2 in order to get the out." Be sure to emphasize again with students that the Out value depends on what goes In to the function machine. This is a key idea about functions and will also help students later with graphing dependent variables.
Resources (1)
Resources (1)
Resources
Investigation
Now students have the opportunity to jump in and try to figure out the rules for some prewritten tables. IMP has some creative In/Out tables in their activity Inside Out, on page 10 of their Year 1 textbook. You can use any In/Out tables however, or create your own. What I like about the IMP examples is that out of five tables, two involve numbers, two involve patterns found in words (the number of letters in a word minus 1, for example) and one table involves pictures of faces (the pattern has to do with the number of eyes on the face). If you have students who need more differentiation to access this content, the TERC EMPower curriculum has a great series of tables that may be more accessible to your students on pages 10  14 of their Seeking Patterns, Building Rules textbook.
I like to let students work in pairs or small groups on these tables. I find they enjoy the challenge of trying to find the pattern. As you circulate around the room, be sure students are writing a rule in words for each table using the language you described in the opening. For example, make sure students write something like "To get the Out, you take the second vowel in the word" rather than just "the second vowel in the word." Also, students are likely to get stumped on a few of these tables. That's ok! Let them struggle through. You might remind them that sometimes it helps to step back from a particular problem and revisit it later. Some general tips I give to students when working with In/Out tables are:
 Well, these Out values are all bigger than the In values. What can you do to a number to make it bigger? You said you can add or multiply to make a number of bigger. What would happen if you did both operations?
 Does the pattern you're using work all the way through the table? Did you check it with all the values?
 If you know the Out value in a table, how can you find the In value?
Question 6 asks students to create two of their own In/Out tables There is lots of opportunity here for sharing student work. In the past, I have had students post their tables around the school with a little envelope for students not in the class to submit their patterns. You can even make a contest of it! See the my video in the Resources section to hear more about how I use this activity schoolwide. I like to encourage students to try and stump their fellow classmates. You can encourage students to use a lot of creativity when they write their tables. Again, they do not need to use only numbers!
Resources (1)
Resources (1)
Resources
Discussion
Ask students to share out the rules they found for the five tables. Again, insist on students using language like "To get the out... " or "the Out is..." If some groups had trouble finding rules, as the group presenting to be explicit about how they found a rule that worked. For example, in Question 2, to get the Out you multiply the In by 3 and add 1. Ask students how they found this rule. Some guiding questions might be:
 How did you start?
 How did you know to multiply by 3? (If there is time here, you might want to discuss the pattern of the Out values increasing by 3 each time as an indication to multiply by 3. Students would have had to add consecutive In values to the table in order to see that. Applaud them if they did so! You can raise this idea as a strategy for other tables.
 When the Out value was 76, how did you use your rule to find the In?
When you get to Question 5, there is an Out value that simply says "Can't be done." Ask students how they came up with an In value that would not work inside the machine. They would have to find a word that did NOT have a second vowel.
Now is a good time to introduce the word domain as the set of things that can go into the machine as Inputs. You can go back to the first numerical tables as ask students what kinds of "things" can go into those two machines. Students should realize that the Ins in this case must be a number. In Questions 3 and 5, the Ins must be a word and in Question 4, the In must be a picture of a face.
You tell students that the set of outputs for a function machine is called the range. Point out that the range depends on what the domain is. For example, in Question 5, the range would be all vowels because the rule is to take the second vowel of a word.
Lastly, you'll want to be clear with students that not all In/Out tables are functions. You might show students a table that has the same Input twice with two different Outputs. You can go back to your function machine graphic and ask students what's happening here. You will want to try to elicit that the function machine would have to be broken in order to give two different Outputs for the same Input. Emphasize that the function machine has to give the same Output each time the same Input is given (because the same thing is happening each time inside).
Closing
You have covered a lot of new concepts with students today. Give students time to reflect on what they've learned and absorb some new content. You can close class with an Exit Ticket activity about functions. You can ask students: What will you remember about how a function machine works?

Note: This lesson is derived from the Interactive Mathematics Program.
Program, I. (2008, June 3). Inside Out. Retrieved from the Connexions Web site: http://cnx.org/content/m15960/1.3/
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