Wednesday, September 28, 2011

Modeling Unit 2

Unit II: Constant Velocity Particle Model
Using battery-powered buggies rolling across the table, we again worked through, What do you observe, What do can you measure, and what can you manipulate.  After going through this, we again developed our purpose (Chris led us to the procedure with Socratic dialogue) after starting with Jon's beginning statement (To determine the graphical and mathematical relationship between).  From there we were each given buggies (each a constant speed buggy, but each group's buggy traveled at a different speed), meter sticks and stopwatches (masking tape was also present if we wanted it).  The day drew to a close as most of the groups were finished plotting the data in  LoggerPro, and 3 groups shared their results on their whiteboards.

I'm guessing we'll finish our whiteboard discussion tomorrow.  One final think I'll add is that I've done a very similar lab at a 2 day physics workshop in Jacksonville.  However, the leader (another modeling guy with the exact same buggies) did it differently.  Each group was first given a blue buggy (all same, constant speed) and determine the relationship (found slope of d vs t graph).  When then had to turn that buggy in, and we were given a red buggy (again all red buggies were the same speed, all different speed than the blue buggies).  We again had to determine their speed.  At that point we had to turn in the red buggy as well.  The leader then asked us, using the mathematical models we had developed, to predict at what position the buggies would collide if a red stated at one end of the meter stick and the blue started at the other.  I'm not sure if we'll do that tomorrow, but I guess I'll found out then.

One other thing to point out, several of us asked if we should address plotting d vs t or t vs d with our students.  For the most part Jon and Chris were saying to make sure the students could justify why they were plotting it one or the other, and wait until later to broach that subject.  I'm not sure that we be as we progress through our whiteboard meeting or later in this cycle (or a future unit).

We started today by finishing our whiteboard discussion for our constant velocity lab (buggy lab).  As mentioned the previous post, Jon and Chris recommended to not worry about which variable was on a given axis (just make sure they had thought it through and had a reason).  However, they said as the whiteboard session is winding down, begin to force the discussion (with Socratic questions) to which graph ($d$ vs $t$ or $t$ vs $d$) gives more meaningful information (answer: $d$ vs $t$ since the slope is speed/velocity).  They also mentioned to direct the students to think about position and time interval rather than distance and elapsed time, as the former will help with the distinction of speed and velocity (which haven't been resolved as yet), and the concept of acceleration.

They also reminded us, that at this point in the year, the students "know" LoggerPro and scientific techniques learned in the first unit.  The supposedly know what slope means (but in reality they know how to calculate it, not what it means).

One important line of questioning to pose to the students is, "What does the slope of the $x$ vs $t$ graph represent?"

Slope is defined in algebra classes as the change in $y$ divided by the change in $x$.
$\large m= \frac{\Delta y}{\Delta x}$
Since the $y$-axis of a $x$ vs $t$ graph represents position, a change in position relative to a change in time, the slope represents the average speed over the elapsed time
$\large \overline{v}= \frac{\Delta x}{\Delta t}$
Since $\Delta x$ is a distance, it would have units $\textit m$.  $\Delta t$ is an elapsed time, so it would have units $\textit s$. Thus the slope of the x vs t graph should have units $\textit m/s$, which is consistent with the units for speed.  {Thanks Global Physics Department for introducing me to LaTex!}

Next on the agenda was using LoggerPro to create a $v$ vs $t$ graph for our data.  After which, we used the integral function to find the area under the "curve."  Again, we discussed the meaning of this area.  From math, we know:
$area=(base)(height)$
Since the base is time, $t$, and the height is velocity/speed, $v$, we can show that the area is displacement/distance.  In looking at my notes, one thing that I'll get clarification on, is when due we, the teacher, distinguish between speed/velocity and distance/displacement during this process.  Have we gotten to that point, and I forgot to note it, or are we not to that stage in the cycle.

Next on the agenda, we were asked to work on Unit II worksheets One & Two.  Each group was again  asked to present one part of this assignment on a whiteboard.  A few comments to note:
1. Jon mention that before beginning these experiments, he has the lab groups perform a vernier experiment using motion detectors and labquest mini interfaces to match their motion to given position vs time and velocity vs time graphs. (I mentioned that I do the same activity as a competition between lab groups, which I find gets the kids very excited.  I'll probably write about my "Physics Olympics" at some point in the near future.)
2. On wkst 1, question 2a, ask the group "How do you know they are the same?"  Meaning, get them to discover how the could determine the scales were the same given the limited information.
3. On wkst 1, question 2d, ask the group "Can two of your members enact the motion depicted in the graph?"
4. On wkst 2, question 6&7, use Socratic questioning to lead students to drawing dashed vertical lines at the points of discontinuity.  Someone asked about including open and closed dots to show where the object was at the point of discontinuity.  Chris answered that we don't know, nor do you need to get into that level of sophistication.
We ended the day by discussing the last tool in the modeling arsenal, Motion Maps:

The above examples show two different maps (the first above the red line, the second is below).  Some key features of the map are: the position vector, which shows the origin (X) and the direction of positive motion; the dot (which I couldn't get to work as a small dot); the arrow on the dot, which represents the velocity of the object at that location and time.

That's where we ended today.

We began today with work on worksheets four and five from Unit II.  While we were working on this, we got clarification of when to distinguish between vectors and scalars.  Jon said that you build it in slowly this unit and the next.  One trick that Jon recommended was to tell the students that scalar terms are shorter than their corresponding vector terms (speed/velocity, distance/displacement).  Remember, more letters in the word, more information.  Obviously, this won't help deepen the understanding of the concept, but it may help jog the memory for a student.

We all agreed that worksheet 5 does a great job of helping clarify the relationship between motion maps and and the other tools for modeling.  One thing someone mentioned is that they might have the students make basic, qualitative equations to bring that aspect into the fold.  Jon, while agreeing, also cautioned that we need to remember that motion maps are pseudo-quantitative at best.  Don't get too bogged down in the limits of the constant velocity model (what happens to the speed at the last instant shown on the graph?).  Chris mentioned that we need to make sure that the motion map does correctly depict the motion shown.  He illustrated this poignantly when one group had 3 points for their motion map.  One while it was moving away from the origin at constant speed, one while it was at rest, and a third while it was moving at constant speed toward the origin.  While at first trying to model to us how to lead the group with Socratic questioning, he saw the group presenting wasn't getting what he was selling. However, he kept at it and led the group to realize that they needed more than one point for each segment of the graph to show that the velocity was constant in a given section.

They also pointed out that a good convention to use is to put each type/segment of motion on a "different line," meaning if the object changes from one constant speed to another (stopping would constitute a new constant speed), put the first dot for the new motion slightly above (or below) the first set of points.  They also clarified to work from the displacement vector away (i.e.: if drawing above said reference, each segment is place higher).

To finish up the unit, Jon and Chris again elicited feedback.  We said we liked worksheet 5 (converting between the different tools of modeling), having students enact the motion shown in motion maps or velocity-time graphs, and the motion mapping activity with the vernier motion sensors.