FIU Modeling Workshop - Day 5

We started today with analyzing the graphs we made from the cart on an incline plane lab to derive the kinematic equations. Although this only takes about 10-15 minutes, it made this post too long.  So I made a separate post to show the process.  Although most texts give these equations, many omit the entire process.  For the sake of helping your students foster their connection between the graphs and the equations, Jon recommends spending the time to show these derivations.  {My guess is that you could either do this during whiteboarding, or as a mini-lecture (for those that aren't quite ready to give up the reins, and want to be a sage on the stage again).}

After showing all the derivations, we moved on to Lab Extension: Speeding Up and Slowing Down.  {As I've noted at least once before, we were given version 3 of this worksheet.  However, I'm only seeing version 2 on the modeling website.  I guess that's one more reason you need to go to the workshop and not just read my blog.}

Jon told us that he gives the students all the equipment except for the motion detector.  Jon said that after the individual groups show him the completed worksheet, he provides the motion detector.  After the students have all completed acquiring the data/graphs, he has them white board what they got for a given problem.  Chris does it a little differently.  He never gives them the detector, but rather has the students make the predictions for HW, and then the groups whiteboard their predictions at the beginning of class.  He then projects the actual results after the class has come to agreement for the each given problem. {My $0.02 on this is that I like Chris's approach better (sorry Jon).}

By the way, I had never seen motion maps that showed both velocity and acceleration at the same time.  For those like me, you plot the velocity above the displacement vector and acceleration below.  Have the points that represent the same time line up vertically.  I've tried to show what the map for #1 would look like below:



The blue vectors represent the velocity and the red vectors represent the acceleration for an object accelerating from rest. 
{I'm honestly not sure how to draw the first point for the acceleration portion, whether they should be inline with the arrow overlapping the second point, or as shown with the first point slightly above the second.  I'm guessing how I have it is correct.  And no, I didn't waste the time to make sure the arrows were to scale.  Remember motion maps are qualitative, not quantitative.}

A couple points made by Jon and Chris:

#3 is the first instance for the students where an object is speeding up even though it has a negative acceleration.  You need to socratically question the students (What is happening to the magnitude of the velocity?   What then is the sign of the acceleration?  Can a negative acceleration increase the velocity?).  According to both Chris and Jon, this is a confusing idea, since they are used to describing a negative acceleration as a deceleration (a term you should dissuade the students from using).

#4 is a similarly confusing example in that the acceleration is positive but the object is slowing down.  Again, use Socratic questioning to lead the students to this idea.

#6 Jon omits this problem as changing the origin doesn't really come up later in the curriculum.  He said that it's up to you and your students.  Do you want/have time to spend on this?
{My thoughts are that I might leave this out for standard level, buy include it for the honors level of my classes.  If I have more than 6 lab groups in honors (which I did this past year ('10-'11)), I might make additional problems with the adjusted origin so each group whiteboards their own problem.}

From there we worked on Worksheet 2,  Worksheet 2a, and a supplementary worksheet.

2a: #3 Jon mentioned that students tend to struggle with all the technical vocabulary in this problem.

2a:#5 Chris asked the group presenting: “I remember a problem from the earlier work, where the negative velocity and it was speeding up.  Why is this different?” {Your trying to get the kids to focus on the speeding up when acceleration is in the same direction as motion, (and slowing down when opposite) not based on +/- sign}

Wkst III:

1 c&d Jon mention that these problems are very tricky for students. 

We next moved on to another experiment using a Vernier Photogate and the Vernier Picket Fence. (note: you may need some of the accessories to attach the photogate to a ringstand).  We used the "picket fence" file provided by vernier.

We were asked to get one measurement of "g" for the picketfence by itself, and one value while a hanging weight was attached to the picketfence.  Jon and Chris made this a competition among groups to see who could get the closest value to the accepted (9.80665 $m/s^2$) for each set of measurements.  

{I'm not sure if I would tell my students the correct value or not.  I would probably just calculate the class average and then ask the students to explain our error.  One side note, one of my pet peeves is "human error."  To me that is a student being lazy and not wanting to think about what they did wrong.  I would push my students to say that the picket fence was rotated one way or another, photogate wasn't level, etc.}

From there, we then did another "competition" lab where we were provided the motion detectors, a rubber ball (similar to traditional dodgeball that could actually leave a mark, not the foam ones given now.)  {Don't get me started on that one.}, and a metal filing shelf (similar to this, only it was one level not two). The shelf was used over the top of the detector to help protect it from the ball.  The basic procedure was to toss the ball above the motion detector and have it fall towards the detector.  Again, the group with the closest value to "g" received a prize.  We used the "ball toss" file provided by vernier.

{I think I might introduce video analysis at this point, either have the students do it in their groups, or run this as a demo, videotaping the students tossing the ball.  Then I would show on the smartboard how to use video analysis.  I would probably use the tool in LoggerPro, however, seeing Rhett Allain use VideoTracker throughout his blog, makes me think it might be worth it to have the students download and use that program.  However it might be worth leaving video analysis until we get to 2D motion.}

We ended the day by whiteboarding sections of the assigned reading from the previous night.  Instead of giving a summary of the summary, I would just say we read Aron's book and discussed 2.7 - 2.16 (excluding 2.14).  It builds off most of the concepts already discussed yesterday.  For those that haven't read it, it's a great text that explains many of the students' misconceptions and strategies to help overcome them.

Derivation of Kinematic Equations

To begin the derivation of the kinematic equations, first start with constant velocity (tumble buggie lab).  The more of this you can get the students to do, the better.

Figure 1 shows a plot of position vs time:




With the graph, guide the students to explain what the slope of the line represents.  You need to get the students to say the slope represents how much the position of the object changes every second, not just speed/velocity (we're trying to build a concept/model, not memorize a word).  Continue to dialogue by asking questions such as "what does it mean if the line is higher/lower/negative," and "what does it mean if the y-intercept is higher/lower" will help clarify the concept of velocity.

You can then go on to have the student explain how to determine the average velocity by adding an initial and final time (and corresponding position at said times) as seen in Figure 2.
 


Have the students start from the the beginning again:

$\large \overline{v}=\frac{\Delta x}{\Delta t}$

$\large \overline {v} = \frac {x_f - x_i}{\Delta t}$ 

Rewriting in $y=mx+b$ format, you get:

$ x_f = \overline{v}\Delta t + x_i$

Also remind the students about the velocity vs time graph for constant motion, as seen in Figure 3.



From there, you can guide the students to the idea that the area under the line represents the displacement, shown in Figure 4:



Get students to explain that the shaded area is a rectangle:

$Area = base*height$

Since the "base" has units of time and the "height" has units of velocity, then:

$\large s*\frac{m}{s}=m$ 

So, the area has units of distance.  Since that area could be above or below the x-axis (thus positive or negative), the area is the displacement:

$\Delta x=\Delta t * v_o$

From this point, we can now engage the students with the velocity vs time graph for the cart on the inclined track (Figure 5).  Again, lead the students to say that this graph now shows that the velocity is changing with time.  Repeat the same questions about what the slope represents, what does it mean if line is steaper, etc.
{It may seem redundant, but a major misconception is what slope actually is. Most students only think of the definition/equation, not the true, physical concept}




Once again, lead the students to the definition of acceleration {I'll omit the derivation to save time, I'm surprised you're still reading this far}

$\overline {a} = \frac {\Delta v}{\Delta t}$

Which, again, can be rearranged into a "y = mx + b" format:

$v_f = v_i + \overline{a}\Delta t$

Which I'll call Equation 1
{At this point, I'll let you know that I circle the equation with a red marker in my class.  The only time I use a red marker is for a fundamental equation.  I picked this up from a college professor.  This fits in with the ABC Gum Rule talked about in Day 2.}

From here, we again lead the students to explain what the area under the velocity vs time graph represents.  The catch here is for them to recognize that the area under the curve can be broken into two geometric shapes as seen in Figure 6:

The area for this shape would therefore be the area of the triangle plus the area of the rectangle:

$\large \Delta x = v_o \Delta t + \frac {1}{2}\Delta v \Delta t$

Which, after distributing the second term for $v-v_o$ and combining like terms, simplifies to :

$\large \Delta x = \frac{1}{2}\left(v_f + v_i \right)\Delta t$

Which I'll call Equation 2, which also gets the red encirclement.

From there we can algebraically combine Equations 1 and 2, to get equations which are easier to use in common situations.

If you know acceleration, but not final velocity, rearrange Equation 1 so it is explicit for $\Delta v$ and substitute that into the unsimplified form of  Equation 2.  That looks something like this:

$\Delta v=\overline {a}\Delta t$

$\large \Delta x = v_o \Delta t + \frac {1}{2}\Delta v \Delta t$

 $\large \Delta x=v_o\Delta t+\frac{1}{2}\left(\overline{a}\Delta t\right)\Delta t$

$\large \Delta x = v_o \Delta t + \frac {1}{2}\overline {a}\left( \Delta t \right)^2$

Which is circled in red and called Equation 3.

If the time interval is not known, you can rearrange Equation 1, so that it is explicit for $\Delta t$ and then substitute into Equation 2.

$\large \Delta t = \frac{\Delta v}{\overline{a}}$

$\large \Delta x = \frac{1}{2}\left(v_f + v_i \right)\Delta t$

$\large \Delta x =\frac{1}{2}\left(v_i +v_f \right)\left(\frac{v_f-v_i}{\overline{a}}\right)$

$\large \Delta x=\frac{1}{2\overline{a}}\left(v_f +v_i\right)\left(v_f - v_i\right)$

$\large \Delta x = \frac{v_f^2 - v_i^2}{2 \overline{a}}$

Which would be Equation 4, and also get the red box.  With that, you have the four equations of kinematics:

 $v_f = v_i + \overline{a}\Delta t$

 $\large \Delta x = \frac{1}{2}\left(v_f + v_i \right)\Delta t$

$\large \Delta x = v_o \Delta t + \frac {1}{2}\overline {a}\left( \Delta t \right)^2$

$\large \Delta x = \frac{v_f^2 - v_i^2}{2 \overline{a}}$ 

FIU Modeling Workshop - Day 4

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.

From there, we moved onto summarizing our HW from the night before.  We were asked to read the first half of Arons text "Teaching Introductory Physics."  Each group was asked to whiteboard a summary of one part of the reading.  (We did a similar activity yesterday for ch 1.  I left it out since, to me, it enhances the workshop and the mindset of modeling, but I would recommend reading it for yourself.  However, one member from my group had to leave so I'm including it here for her.)

Section 2.1 is the introduction to the chapter on Rectilinear Kinematics (not sure why he can't just call it 1D Kinematics, but who am I to criticize).  One of the main things the group said that jumped out at me was to remember that mankind's development of this concept/model took of 1500 years, so it's OK if our students struggle a little.  Some of the greatest minds in history couldn't understand it. 

Sect. 2.2: In this section, Arons begins to explain why there is so much difficulty with understanding 1D motion.  He says part of the difficulty is that we, the teachers, aren't consistent with our terminology when we teach it.  Instead of focusing on time and distance, we need say "instantaneous time" for clock reading, "time interval" for elapsed time, and position.  In doing this, we will more easily flow into the more complicated concepts found later in the course.

Sect. 2.3: One of those concepts that we need to begin to foreshadow is that of "event."  By focusing on position and clock reading as truly instantaneous, the concept of "event" will make more sense when reference frames come up in relativity later in the year.  It also is important that we have our students verbalize these terms, and explain them, not just us use them over and over. 

Sect. 2.4: This section build on the last to develop the concept of instantaneous position.  One other important idea is to say some happened at a moment in time, not for a moment of time, as the latter suggest a time interval. 

Sect. 2.5: This section began discussing average velocity, by say that we should not use the term "average velocity" until after the students have been exposed to several experiences with motion.  By using the term average, our students often think that we mean "not complicated" and can become discouraged when they struggle with the "simple" concept.  Instead we should focus on the ratio of $\frac {\Delta s}{\Delta t}$ over the name (Arons uses $s$ to represent any direction instead of the more common $x$ more commonly used today).  Have  the students verbalize that this represents how fast the object is moving: higher ratio, quicker motion.  Also to have students explain the inverse ratio $\frac {\Delta t}{\Delta s}$ to represent the slowness of the motion.

Sect. 2.6 Discusses graphs of position vs clock readings.  The keys points from the group we that to understand motion, we need to make sure that the students are relating the graphs to actual motion.  By having the students show the motion depicted in a graph with their hands, both the teacher and the student will insure that the concept is understood.  One other point they made was to tell the students to think of the motion activity, especially when trying to comprehend motion maps.  They said to think of the origin as the motion detector itself.  One other suggestion that came up was to modify the motion mapping activity by slowing the sampling rate and then showing the data points instead of the line connecting the points.  They thought this might help students connect the graphs they were creating with their motion.


From there, we moved on to Unit III, and began the "Cart on an Incline Plane Lab."  After talking to Jon and Chris about Brian Frank's Blog during our lunch break, Jon and Chris altered the first question to begin the cycle by asking "What do you see/notice?"  Again, we moved through "What can you measure?" and "What can you manipulate?" before arriving at the purpose: "to determine the mathematical and graphical relationships between position and time for a cart on an incline plane at a fixed angle.  As a group, we seemed to struggle with this process this time.  I'm not sure if we are taking our role in "student mode" too seriously and confusing ourselves in the process.  We didn't seem satisfied to look at this relationship and many wanted to add more quantities such as mass and angle into the procedure.  One good suggestion was to focus the students back on the first question, "What do you see?"  By using what is already on the board, we can steer the students to the necessary target of this unit.  I might add that we should include some help to our students to focus on the actual demonstration we are doing (cart rolling down a fixed ramp) and try to dissuade them from altering the set up.  Possibly reminding them that our job is to create the experience that will teach them the necessary component of physics, and we chose this exact one for a reason. 

After we were settle on the debate, Jon told us to use the motion detector to acquire "clean data" for the cart.  When asked what he meant, he said, "You'll see what I mean."  He put on the board for us to then manually calculate 10 values of "average speed" over the range of our data (he originally put instantaneous, but later corrected it).  He also remembered later that he usually has the file "01a Graph matching.cmbl" from the physics file loaded onto the computers, so that the velocity will not be automatically calculated for the students.  For the average velocity, he tells the students to use the position and clock readings from just before and after the point we are using to calculate the average velocity.  Jon also mentioned that during the acquisition of data, and the subsequent creation of the whiteboards, he would talk to the groups about the significance of the data produced (if you started the cart after the motion detector, what do the initial data points mean?  points after the cart hit the bottom of the track?). 

At the end of the day, most of the groups had presented their whiteboards, with 1 or 2 left for tomorrow. 

FIU Modeling Workshop - Day 3

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.

FIU Modeling Workshop - Day 2

To start of today, we finished up the "Board Meeting" with the groups that studied length vs period.  For physics teachers, this is obviously the group that was able to show an actual correlation.  One of the most interesting parts of the discussion, to me, was when Jon and Chris recommended not worrying about linearization yet.  They told us to now worry about that battle, as it will come up as you move into the next phase of the cycle.  Just let the kids use LoggerPro to get the mathematical relationship.  They did recommend spending some time to discuss whether or not the data should go through the origin.  In the course of that discussion then mentioned what they called the "5% Rule" which basically states that if the y-intercept is less than 5% of the biggest measured value in the data for the y axis, assume that it goes through the origin.

After we finished that discussion, we then moved into the next phase of the modeling cycle in which we worked on linearizing data using LoggerPro.  The worksheet had 6 data sets (we had version 3 of this worksheet, I'll add that link if I find it), and we had to plot the data and determine how to manipulate the data to create a linear graph that went through the origin.  Jon and Chris mention that they only used the first four problems (which I think are the 4 in version 2) with their classes as they have found that they are sufficient to get the students acclimated to the process. Jon and Chris did recommend to have the students write the regressed equation rather than the proportion shown (ie: equation with slope and y-intercept, not y is proportional to 1/x).

After we had linearized the data, each group was assigned a different problem to put on a whiteboard to share with the cohort.  Again, we were able to get a greater feel for how the whiteboarding process works, and able to ask questions as to how to moderate, when to step in and when to let the conversation go.

After a brief break, we then moved on to discuss our HW from the previous night.  To do that, each group was assigned a different section of the reading and asked to provide a synopsis on a whiteboard.  To sum up, we had a very lengthy discussion on the discrepancy between what a teacher thinks he/she is teaching and what the student is learning.  I didn't bring it up, but this made me thing of Frank Noschese's blog on Pseudoteaching. In our discussion, we talked about how as we, as teachers, think we are helping our students understand a concept through example problems, our students, for the most part, are fixating on the equations produced.  The problem with that is students mistakenly think they can apply the derived equation to any problem dealing with the same concept.  I alluded to Rhett Allain's post by describing an "ABC Gum Rule."  (I didn't really have a name for this concept until I read Rhett's post, but I would always tell my kids that they had to always start from the basic equations, they could not ever start with derived equations.  Thanks Rhett!)  The rule, as I pointed out, is that you never want to eat Already Been Chewed Gum, rather, you always want a new piece.  Same thing for physics, you should always start a problem from the beginning, not an equation that was made for some other situation (which may or not be the same).

After that, Jon and Chris asked for feedback as to how we thought the first unit went.  It's amazing how well these modeling people all act as I remember Frank Noschese blogging about getting feedback from students more often than just the end of the year (read the post here).

They asked what worked and what didn't?  To the first we said, we liked learning: how to use LoggerPro (especially for linearization), the linearization summary sheet, breaking up the pendulum lab to finish the lab in less time (made groups take more ownership of work since others were depending on them to get it right), and using inductive reasoning to determine relationship instead of the teacher just telling "us" the answer.  What we didn't like: some wanted more explicit explanation of the relationships between independent and dependent variables (hopefully I'm remembering that correctly), and some wanted the workshop to move a little faster (I think she was referring to limiting some of the discussion, but Chris rephrased it as getting started quicker/more punctual coming out of breaks.  I'm not sure which was what she meant).

From there were moved onto 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).