### TJ: CA I.3 Introduction to Functions

#### 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! |

#### 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. |

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