- Construct a consistent vocabulary of problems that can begin in Algebra 1 and extend through to Calculus, Statistics, and all courses in between.
- The problems must have the potential to be engaging to learners.
- The problems must hit at least 4 of the eight Mathematical Practices & high school math standards (CCSS).
My idea started with this idea for Algebra 2: Model the escape velocity of a rocket on the Moon and the Earth. ( PDF and Word DOCX) This ended up being a far more difficult task than I expected, mainly because the learners did not connect the idea of writing the equation of a line with the fact we had a function in front of us.
I Desmosed the project for a visual display, and we spent another day discussing it and achieved the goal. [Is it okay to use the name as a verb? I don’t care, I am doing it anyway.] It turned out great in the end, but it made me start thinking hard about how to connect Algebra 1 through Calc and Stats and make the ideas more real, more understandable, and more connected.
From there came the idea of using an “off the shelf” structure in a new or different manner to extend the lessons. Enter http://graphingstories.com . Dan Meyer started the Graphing Stories with a long time ago, and they are awesome. But they also fit the idea of using the video / graph combination to write the equations of lines and finding area under the curves.
With that in mind, I offer the following Desmos files:
- This uses the Graphing Story of water being poured into a graduated cylinder to create the graph. I took some points from the graph on screen, and wrote a function that goes through the point (0, 0) because we know it was empty at time 0.
- Notice that the line does not go through exactly all 4 points! That allows for discussion of variability and observation skills.
- I also used the (h, k) form to write the function f(x) because it is the easiest way to show the line.
- What does the slope MEAN? A standard AP Statistics interpretation is: As the time increases by 1 second, the water increases by 40.67ml.
- Next, find the area under the curve. Move the slider for “b” to the right and you see the area highlighted. Okay, standard triangle, ½ b*h, and you get 5205.33 ml*sec. ??? What does that even mean?
- It is called “absement” and it is the time-integral of displacement. Yes, we don’t need to discuss that for Algebra 1, but as teachers we should know it.
- The area is the sum of all the instantaneous moments of water before. With the Desmosed file, you can see and clearly communicate what it means. It means that you are adding up the area of the little triangle when b=1 with the larger triangle when b=1.5, and then with b=2, etc. Except the area is the sum of the instantaneous areas, not the discrete areas.
Notice that this one lesson required the learner to interpret a real life action, pouring water, into a graph, and then find the slope and write the equation of a line, and then interpret the slope, and then find the area under the curve.
These are all essential skills of the Calculus learner, done at the Algebra 1 level!
- Now we are removing cups from a scale. There are actually several questions that the video brought to my mind, like is this really a continuous line, or should it be more discrete? Time is continuous, but the weights really are stepped. But, I left it as is though because I wanted to not change it from what the video shows. That is a larger conversation in class.
- We now have a negative slope to calculate, which does not really make a huge difference for interpreting the slope: As the time increases by 1 second, the weight of the cups decreases by 3 grams.
- The fact the line only hits 1 point absolutely creates some conversation about which point to pick, variability, ect.
- The area gets fun, however.
- Notice that the FULL area is still a triangle. However, if you move the “b” slider across, you notice the partial areas, the area at 5 seconds, 8 seconds, etc, are trapezoids! Now the learner can be challenged and pushed to incorporate some extra questions of find the area of trapezoids.
- We still are doing and absement calculation and not a displacement calculation.
Finally, the Desmosed Lunar Modeling I started with:
It is far more complex and involved, but that is why it is an Algebra 2 lesson and not an Algebra 1 lesson.
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
I am soon to be embarking on a new direction / undertaking in my professional career, and in doing so will need to seriously delve into the realm of research and resources. Starting this summer, I am beginning a Ph.D. in Mathematics Education at the University of Reno, NV in the Math, Science, Technology and Society area of emphasis.
As a high school math teacher I need to be an expert in CCSSM. As a Ph.D. student I MUST, without question be an expert in CCSSM. To that end and because I have always been interested in research, I am going to compile lots of CCSSM resources and create a page here to house the ongoing collection.
The page can be found here: http://blog.mrwaddell.net/ccssm-resources or at the top of the page.
[I really need to return to blogging. My lack of focus on reflection has hampered me this semester, and I need to fix that. To that end, I am making a commitment to blog and to jog. Those are the foci this year of the ellipse that is my world.]
Yea, how often does that happen that a class gets excited about logs? It has not happened to me in several years, but this year I found a way. We started the second semester with graphing again. We have a standard list of things we look for, identify, and document on every single graph. The list is:
Asymptototes (vertical and / or slant):
End state behavior:
Every graph we do, we have to document all of these items. If we graph a line, most of the list is “none” but it creates the connection between all the graphs. Every graph has the same questions, it is just that some of the graphs / functions do not have those features.
So, I am doing this file on Desmos, and we are documenting. They have done all these as homework, so really we are checking answers and ensuring learning. Then weird things happen. They notice the symmetry of the inverses.
Then they ask to see the graph of the line of symmetry. Even nicer. THEN! OMG. We put the translation into the h-k form of the line, and we see the translation of the line of symmetry. [Okay, seriously. If you are not using the h-k forms to make connections, why not. See This post, or This post or any other of the several posts I have on this topic.]
And then I graph the exponential. …. …. They know there must be an inverse, but nothing we have done in class looks like that. …. And then, because I have the list of all the h-k forms on the board, someone asks, “Is that what the log thingy is for?”
And now they have a reason to learn logs. They are intrigued by logs. They are asking questions about logs. Because EVERYTHING in math has a forwards and a backwards, addition has subtraction, squares have square roots, and exponentials have logarithms.
They are interested and inquisitive about a topic that normally is not approached this way. I have done something good I think. Only time will tell if I can continue that on this topic.