# CA I.1 Bottle Graph Exploration Part I

**Activity Guide**

**Purpose**: 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 the end of the 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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