I am trying to come up with ways to connect the ideas of Statistics and Calculus down through the curriculum in all levels, even into Algebra 1. I think I have a way to do it that makes partial sense and can be done with reasonable effectiveness, but I have an interpretation problem. If anyone can see a way out of it at the end, please let me know in the comments.
First off, I started with http://graphingstories.com and selected the water volume exercise. My reasons are focused and simple. 1. It is linear, so it works well in the algebra 1 course during the first semester. 2. Finding the slope of the line is straightforward, and requires some transfer of skills, but not a lot of transfer. 3. The area part of it is just a triangle, so the math is not complex.
I am trying to build from simple through the more complex, so starting off simple is helping me wrap my brain around it.
So, I start with this video:
You end up with a graph that looks like this (click to enlarge):
Great, now for some math. Find the rate of change of the line: Best guess: (610ml – 0ml)/(15 sec-0sec) = 40.67 ml/sec.
What does that mean? As the seconds increase by 1, the water volume increases by 40.67 ml.
The equation of the line is milliliters= 40.67 ml/sec * seconds + 0.
So far so good. The interpretation of everything so far makes complete sense. It is attainable for algebra 1 or geometry, and has meaning in context of the video.
So I want to inject a little geometry into the problem and I find the area under the curve. No big deal, it is a triangle, so it is 1/2*610 ml * 15sec = 4575 ml*sec.
What does that even mean? Does it have a meaning? Am I doing something that should not be done? What does a value of 4575 ml*sec even mean? It is not the total volume, that is 610 ml. It is not the sum from 1 to 610, that is 186,355 according to wolframalpha.
In calculus, we find the areas under curves all the time, but they are specific types of curves. We find the area and it is the displacement, or it is the total distance. But does this area have a context or meaning?
I really would like to figure out a way to make sense of this, but if there is not a way, then I will have to go back to the drawing board.
Edit: The word for what I am looking for is called absement. Links to resources are:
http://forums.xkcd.com/viewtopic.php?f=18&t=34744 [read all the way to the bottom]
And a pic that makes the idea of absement more clear: http://ow.ly/i/4HEwh