We did
this activity on the same day as our discussion of the syllabus so we only
spent 30 minutes on this activity. I started them off without introducing it at
all  I just showed them the two cylinders (one taller than other) that they
would need for the activity. This turned out to be a mistake  I should have
mentioned something about pouring water into the cylinders as that was not
mentioned in the activity (we will fix this in the activity). They were
confused about how they should be considering the water  discrete or
continuous basically. Thinking about that now though it's actually not a bad
point of confusion and maybe something I should have talked to the whole class
about (I only had the conversation with one or two students as they were
working). As a result I think they probably spent too long on the very
first question. I let them work on it for about 10  15 minutes  asking them
to try it by themselves first and then to come together in small groups. I also
used this opportunity to set up how I wanted them to work in general. As
they were doing this I walked around the room and talked to various small
groups to see what they were thinking and how they were working. We then spent
15 minutes in class discussion  I pulled some sample graphs from around the
room and put them up on the document camera  on each one we discussed what was
good and what needed improvement. (When I took the graphs I told students if
there was something not quite right about their graph  not what it was  just
that there was something that needed to be adjusted and then asked them if they
were ok with me putting it up for the whole class to discuss.) We then settled
on a linear graph for each cylinder and discussed why. At this point I asked
them about the definition of proportional. I was running out of time so I gave
them the definition in terms of y=mx and then told them that they graphs we had
drawn (which went through the origin) were proportional and to think about why.
This is not my ideal way to do this but I think it ended up working ok. It was
really good to follow this up with the question in the synthesis questions on
proportionality which almost everyone answered incorrectly. I didn't get to
moving to another container on this first day but I think this is actually
better at the beginning of Part II anyway so we will move it there for next
time. Overall I think the activity went pretty well. One great thing about this
activity is that it is not what students are expecting and it tends to set
students who are worried about the class at ease. It also allows for lots of
time for the instructor to interact with the students in a relatively stress
free situation which is a good thing for the first day.

Calculus I and II > Calculus I > Teacher Journals > TJ: CA I.1 Bottle Graph Exploration, Part I >