Monday, June 30, 2014

Constant Velocity Model -- Summary and Implementation Reflection

My current impression is that the constant velocity model is a tool for describing and solving problems related to objects moving with a constant velocity.  Two common examples of such motion would be much of our motion in cars between stop lights and falling objects that have reached a terminal velocity.
  It includes several complimentary representations of motion, from which the speed, direction, and (often) starting point of the motion can be determined:

  •  a verbal description of the motion (the object moves a constant distance in each unit of time, in a constant direction)
  • a diagram of the motion in the form of a motion map with equally spaced position dots at each "snapshot" of equal time intervals and equal length velocity arrows starting at each position dot and pointing in the direction of the motion
  • a graph of position vs. time, with the vertical-intercept representing the starting position of the object (relative to a designated reference point) and the slope representing the speed of the motion through its steepness and the direction of the motion through its sign
  • a graph of velocity vs. time, which by defintion of this being a "constant velocity model" will be a constant value whose magnitude and sign correspond to the slope of the position-time graph
  • an equation that comes from the position-time graph:  x = vt + x0, where x is the position relative to the reference point, v is the velocity of the object, t is the time elapsed since the beginning of the motion, and x0 is the starting position of the motion.
It can be developed by a inquiry activity in which students track the position of a constant-velocity vehicle from a variety of starting points as a function of elapsed time, graph that position-time data, consider the significance of various features of the graphs and equations.  It is then, within the modeling instruction materials, additionally translated to the motion maps and velocity-time graph representations through Socratic small group and large group discussions.  (Is this an operational definition of creating operational definitions?)

I have previously done parts of this modeling-building process with both my ninth grade Physics I and my advanced 11th grade Physics course.  In general, my students seem to get comfortable with the position-time graphs quite quickly but struggle much more with the velocity-time graphs and figuring out how to go backwards from the v-t to x-t graphs.  And the idea that the area under the v-t graph tells the displacement, and the difference between average velocity and average speed are fairly fuzzy to many of them, at least in part because I'm not as careful about differentiating position, distance, and displacement up front as I could be.  Based on my experience in this summer's modeling workshop, I will be strongly considering the following alterations next year:
  • I will try my best to steal a (possibly shorter version of) Laura's discussion leading to operational definitions of position, reference point, distance, and displacement.  
  • I've done the Buggy Lab with both my courses.  However, each time I prescribed that everyone do two identical runs, one forward from the reference point (ish) and one back towards the reference point from high positions.  I loved the combination of similar first runs with a broad variety of second runs for different groups (different starting positions, speeds, and directions, in various combinations) to bring interest and richness to the succeeding whiteboard discussion of the results.  I will definitely do that next year.  
  • I haven't really used Predicted Graphs before -- I like the idea of asking students try to think through the relationships between factors graphically.  I have traditionally had them make "If, then, because" hypotheses verbally, and I wonder if you lose anything by not having students write out the verbal explanation of their thinking.
  • I also finally feel like I understand the 5% rule enough to try introducing it to my classes, although I remain concerned that it will creep into my IB students' externally moderated lab reports in places it doesn't belong.
  • I wonder about alternatives to reserving one of my school's two laptop carts every time we do a lab and the students want to be able to fit lines for 5 or 10 minutes per group.  I think I should educate myself on how they can do it with the graphing calculators many of them bring, the Open Office distribution on the linux desktop in my room, and maybe the free Logger Pro Lite software I could have the students with laptops download.  (Does the free version do curve fits?  I'll have to check.)
  • It was very helpful to see the model development and summary through the unit.  I will be more conscious about explicitly generalizing and publishing the consensus of the classes on the graph and the associated equation after the buggy lab discussion next year, and adding to it as we add further representations (motion maps, v-t graphs).
  • I've explained motion maps by first asking students to try to come up with their own system for diagraming motion, and then guiding them into invention something similar to motion maps, illustrated with the motion maps reading.  I feel like my introduction of that was relatively effective this spring, although I love the blinking open-close kinesthetic experience.  I asked my students to image a flash photo taken in the dark once per second, and that worked pretty well for them, but I think adding the blinking will help clarify it.
  • I loved "walking the graph" (and think it's also helpful for walking the motion map, and probably for walking the v-t curve, too) -- wish I'd thought of this sooner.  I think it will really help make the graphs and motion maps more concrete for my students.
  • I've used most of the worksheets in Unit 1 with my students, but I've been very guilty of underutilizing their discussion and conception clarification potential.  I think I often take for granted that my students are using language the way I hope they will, and don't ask for additional explanation often enough to realize where they're not as clear as they seem.  I've also been super guilty of glossing over the highly useful ambiguities in the worksheets ("Is away from the detector always in the positive direction?", "What happens between the dots on a position - time graph we're given without actually seeing the motion for ourselves?").
  • I try to whiteboard most of the worksheets in some form, but I usually let student groups choose which problems they do, and usually each problem is done only by one group.  I'm still getting used to the "assign two problems per group" strategy that seems to be much more common in the workshop, but I can see some advantages of it.  I'm willing to try it, although it will require getting a lot more whiteboards than I have now.
  • I've already mentioned that I have a huge crush on the "Movie Shot" end of unit lab.  I will be stealing this.  

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