CA I.1 Bottle Graph Exploration Part I

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Activity Guide
: To recall linear relationships and the meaning of slope and vertical intercept.

Classroom Procedure: After an initial introduction to this activity students will work on the first two problems individually, then they will complete problem 3 by discussing their results with other students at their table. This is followed by a class discussion where we establish that these were linear relationships remembering y=mx+b and discussing what slope and intercept mean here. Students should work on the next class discussion questions individually first (if time permits) and then have a whole class discussion of what meaning they got from that equation. The Synthesis Questions should be completed individually as homework and students should be prepared to share their work on these problems.

Ideas this Activity Builds On: This is the first activity that students work on in Calculus I. It builds on their understanding of representing relationships graphically and linear relationships. If they don't already know that a linear function is y=mx+b then they pick it up during this activity in the class discussion.

Introduction/Motivation of the Activity: We don't usually give a whole lot of introduction or motivation for this activity specifically. However it does follow up our discussion of how students will be actively working during class and asked to think about questions that will help them to develop their understanding of concepts. So we do point out that we are starting this process right now with this activity.

Need to Establish by the End of Activity/Wrap-Up:
  • linearity of graph and what about container results in linearity - for instance discussing if 10 ml of water at the bottom of the glass cause an increase of 2 cm what will happen if I add 10ml near the top of the glass? Why?
  • difference in two lines -> slope - what does this mean for adding 10ml of water to each glass? Or ask if adding 10ml of water to the short fat glass causes an increase of 2cm then what can I conclude about adding 10ml of water to the fatter glass?
  • intercepts (and quick mention of proportionality):  Might ask is it really possible to have a truly proportional relationship between volume and height above table? Volume and Height are proportional if Volume/Height is always constant - how does this relate to the intercept?
  • With slope and intercept established (remembered), do they remember equation of line? How will this be different if the relationship was proportional?
Additional Notes:
  • We left synthesis questions on end of activity rather than pulling them off and separating - first day, don't want them to be confused about what they need to do before tomorrow. It is good to point out that they can answer these questions just with the work that we have done in class today and focusing on the meaning in the questions. Students often tell me that they can't "remember" how to do this from high school. I always tell them that we are not expecting them to "remember" anything other than what we have already reminded them of in the activity. 
  • We find proportional is a big misconception for these students. The majority think that proportional  means that if one variable is increasing the other is as well or something along those lines. This is one of the reasons we talk about it in the activity but even with that at least half of the students will get it wrong in the synthesis questions.

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Teacher Journal CA I.1

  • Bottle Graph Activity, Part I
    The changes we made definitely helped this activity. I think it's easier to make sure we hit everything now. I spent about 25 mins on it this time and it went really well. We had good conversations about linear functions and proportional - we'll see if any of it sticks! I had them start by reading the first two questions independently and then discussing with a partner what they needed to do - then they did it together - I actually like this just fine so the instructions could be changed to that. Then I chose two of those graphs (both linear - one proportional, one just linear) and put them both up on the document camera to have the discussion from there. We talked about how the graphs compared to the practical situation and why they made sense. I didn't choose any "incorrect ones" for timing reasons but don't think a lot was missed on that. I sent them home with finishing Page 2 for homework and we will talk about it at the beginning of the next class. The synthesis question they have to hand in two classes from now. Tomorrow will be CCI and then we'll get on to the second part of this activity.
    Posted Aug 22, 2012, 9:05 AM by Mairead Greene
  • Bottle Graph Activity, Part I
    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.
    Posted Jun 9, 2012, 1:37 PM by Paula Shorter
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Mairead Greene,
Aug 17, 2014, 7:11 PM
Mairead Greene,
Aug 17, 2014, 7:11 PM