### CA I.3 Introduction to Functions

Activity Guide

Purpose
: To define functions and inverse functions and learn how to work with these definitions. (Also domain, range and functional notation are introduced.)

Classroom Procedure: We begin this activity by defining functions during a class discussion. I usually use something like "A function is a way of describing a relationship between two variables - one is the input variable and the other is the output variable - and the function must associate each value of the input variable with exactly one value of the output variable".  After this definition is established students are asked to try each  of the problems that follow on their own first and then compare their answers with classmates nearby.  While they are working on these problems the instructor is walking around the room monitoring progress, encouraging students to try the problems and answering any questions they might have (without telling them how to do the problem). After they have done this we discuss the problems as a class. This class discussion might happen after each problem or after they have worked on a couple of problems if thinks seem to be going fairly well.

Ideas this Activity Builds On: This is the first activity we have that follows up from the introductory bottle graph activities. In those activities we have dealt with functions informally, we have mentioned linear relationships because this is something that most students are familiar with. We have talked about slope and vertical intercept for the linear relationships and general descriptions of growth and decay (including concavity) for the non-linear relationships. Now we are introducing the formal definition of functions and inverses functions.

Introduction/Motivation of the Activity: During the first two activities we have actually been working on functions that we never explicitly talked about. After we have discussed the definition of a function as a class, we can connect back to the bottle graphs to ask if they were functions and why. Then continue on to the rest of the activity.

Need to Establish by the End of Activity/Wrap-Up: By the end of this activity students need to understand the definition of function well enough to identify whether any given relationship (in any form) is a function or not. They need to be able to use the definition of inverse function to determine the inverse of a function given in any form.

Additional Notes: The synthesis questions for this activity push students to understanding how recursive notation can be used to describe the manner in which a function is growing or decaying. Students find this very difficult but it is a great activity in reading a mathematical statement and making practical sense of it. We find that our students struggle with reading throughout this class and this is a good first place to start talking to them about how important it is that they are proficient readers of mathematics by the end of the semester. (As many application problems require students to be proficient readers to even start the problems.)

#### Teacher Journal CA I.3

• Introduction to Functions  Students do run into trouble on the recursive notation in the synthesis questions. I have added a note that explains that this is a reading comprehension exercise so I'm not sure if that will change the way they approach the question or not but we'll see!
Posted Sep 11, 2012, 10:28 AM by Mairead Greene
• Introduction to Functions  This activity took two days in class. It went pretty much as expected I think. The first day we worked on the first 3 questions and then I sent them away with 4 and 5. I talked to them a little about 5 because I thought the wording was still a little unclear. During our discussion the next day though it was nice that with the current wording this was a function and that students had a graph that they couldn't use the vertical line test on if they had just graphed a circle of radius 4 around the origin. That was unexpectedly good! Other than that they were a little disconcerted at having to fix something to get a function but they got used to that. There were some good discussions around domain and range. The second day was our work on inverse functions. During this discussion they have to be forced time and again to go back to the beginning definition and justify their reasoning from there. We believe this is very important as we come back to inverse relationships a number of times in our Calc I, Calc II sequence and each time we want them to reason from that definition. I encouraged them to think about how all their other ways of thinking about inverse functions come from that definition - switching x and y roles, arrows going from range to domain instead of from domain to range, flipping a graph over y=x, etc. They did pretty well on this overall I thought!  The graphical example is nice because they have to catch that it's not a function - this tripped up the "best" students in my class more than the students who were thinking through a lot of this stuff for the first time. It's a good point for students who have had calculus before to realize that they still need to be paying attention and being careful with what is being asked. The synthesis questions for this activity are really difficult for them as it uses recursive notation with very little intro so I think next time I would like to talk to them about those synthesis questions before they leave class and highlight that I realize this is hard thinking but that it will be very important for their learning to do that thinking without an example.
Posted Jun 9, 2012, 1:08 PM by Paula Shorter
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