- 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.