FIU Modeling Workshop - Day 7

We began today with a series of demonstrations that together, will help students to conceptualize the "Normal" Force.  First up was a very nifty contraption which shows that even small forces do in fact, move a wall (Jon said that this even works with a brick wall!)

Jon attached a metal rod to the wall with modeling clay.  Between the table an the rod, he placed a T-pin his Biology teachers unknowningly provided to him.  Glued to the T-pin is a small piece of mirror.  A few feet away, they had a laser set up, which was pointed at the mirror.  As you push on the wall, the wall moves, which causes the bar to roll the mirror, which in turn changes the reflection of the laser.  Students can see the effect of you pushing on the wall by watching the laser dot on the opposite wall move up and down. {Hopefully that made sense.}  Here's a picture of the setup:

Next he suggested (they didn't find any springs until later in the day) to take the bowling ball and set it on top of a spring, which is itself on the table (you'll bring that part up later).  Ask the students what the spring is doing to the ball (to which they should reply pushing it up)

Then, set the ball on top of soft foam, and again ask what the foam is doing.  Then set it on some firm foam.  Next, set it on top of 2 meter sticks (elevated at each end by some blocks) so that the students can see the meter sticks flex in the middle.  Finally, place the ball on top of table by itself.  In each case, ask what the "base" is doing to the bowling ball.  If they still don't get it, ask what the table was doing to the spring at the beginning of the sequence.

At this point, ask the student to draw a FBD of the ball resting on the table.  At this point, now call the upward force of the table on the ball, the Normal force.  Ask what would happen to this force if the table surface was rotated (incline plane), and lead students to the fact that it is always perpendicular to the surface.

Jon then went on to describe how he uses surgical tubing ("borrowed" from the chem teacher) and student sitting/standing on a homemade hovercraft (here are the directions to make it*)(You can use on office chair if you don’t have hovercraft).  Here's a picture of the setup with an office chair {I guess Jon didn't want to bring his hovercraft from Minnesota, how rude}

*Modifications Jon made to the procedure:
Blue tarp works fine, don’t need that pattern of holes – he just put 30 small triangular holes throughout

Duct tape around between small disc and big disc

Use the biggest fender washer @home depot you can find instead of the coffee lid

Make the hole (at the very end) as close to the size of shopvac nozzle as you can (need a tight seal)

Take the class into the hallway, and ask for 2 volunteers.  One sits/stands on chair/hovercraft and holds a meterstick at his/her waist.  The second you tell to pull the rubber tubing to a fixed distance.  You tell the person to pull the other victim volunteer such that the distance the tubing is stretched does not change.  Let the carnage begin.  If you want to maintain some sense of safety have the other students line the hallway to help keep the demonstration moving down the hall instead of into doorways and other obstacles.

You can take the sequence to the next level by now asking what would happen if the person seated in the chair/standing on hovercraft were to throw a medicine ball?  (Demonstrate if you have one).  Now ask what would happen if you had a magic contraption that dropped unlimited medicine balls so you could constantly throw them?  Tell them, let's not imagine it. let's do it.  Grab a $CO_2$ fire extinguisher and release the trigger while sitting/standing.  (Jon said he removes any hose/nozzle, and that he worked out a deal with a local supply company to get an old extinguisher, and get ~$10 refills.  He said one full extinguisher will work for all his classes.)  At this point bring the class back inside and have them summarize what all has happened, using FBDs as needed.  Guide the students to the idea of Newton's 3rd Law: If "A" exerts a force on "B" to the "right," then "B" exerts a force on "A" to the "left."

From there, we moved on to individually, complete Unit IV wkst 3

(Jon told us that he doesn't use this sheet as written, but modifies it for his 1^st yt students)

{Has students do the FBD’s but modifies to do progressions in steps, not all at once}
{chris inserts a week or two of material from the math modeling curriculum to review trig concepts.}
{does math review before this unit not at beginning of the year like most teachers}

After completing the worksheet, we whiteboarded our results.
Notes from WB:

#4 – group made error on purpose – switching sine & cosine
(Acting as students saying that cos is always the  horizontal component of a vector)

Jon's series of questions: Which leg of the triangle is the longer leg, so which one should be bigger?

• What on the diagram will be equal to the Vertical leg? (Answer: weight)
• What will be equal to the Horizontal leg? (Answer: T1)
• Based on triangle, which should be the bigger force? (Since vert. leg>horizontal leg, Weight)
• Does you answer match that fact?

#8 – Jon – Giancoli has a great problem w/ lawn mower
{Which in looking through my copy looks like #26 in chapter 4}

One question they asked the group (mainly to have some fun at their expense)
If floor is frictionless, how does he push the broom?

At that point someone mentioned this Cartoon over at xkcd
               http://xkcd.com/669/

Next we Whiteboarded sections of Hake “Socratic Pedagogy in the intro phys lab”

Due to time constraints, Jon showed us a trick if we ever need to move things along:
If running short on time – have all students display boards, then ask if anyone has questions.
  Address the questions as needed, and move on.

Here a link to SDI labs as provided by Chris

From there we moved on to another Demo to continue to explain Newtons $3^{rd}$ Law:

Equipment: 2 spring scales & 2 volunteers

Scales attached between the 2 people, 1 person pulls while the other just holds on, then they switch, lastly both pull on the scales.

(If you don't have large spring scales, use 2 bathroom scales/ or vernier force plates)

For bathroom scales (have a "reader" looks over each shoulder & call out values)

Next they set up 2 vernier carts each w/ force sensor attached, on  cart track track
(Jon mentioned that Steiner (sp?) has variations of worksheets in the modeling website, probably under password wall for those that attended the workshop)

(Before you begin, zero the sensors and make sure one has direction flipped, or you won't see both sets of data in the plot)

            1^st Trial- both cars moving with equal mass & approx same speed

            2^nd Trial - add standard masses to one car, so the collision has uneven mass

            3^rd Trial - One stationary vs one moving

            4^th Trial - One moving fast, the other slow

            5^th Trial - Cars start together and explosion with cart “spring”
{Obviously (?) you could keep going if you feel the need 

Next, they took the sensors off the cars, and attached them at the hooks, and plotted real-time data of the students pulling the sensors apart.

Unit IV: Worksheet 4

(Due to the complexity, Jon has his students first just answer the A/B/C part of the problems and has students whiteboard their answers. Then he has them draw the FBDs, however, they only need to depict the interactions of block A on B and B on A (no other forces yet), and again, they quickly whiteboard their answers.  He then walks them through one or two of the problems, and assigns the rest for homework.  Whiteboard results at the start of the next class)

We again finished the unit we feedback. Chris said they were going to limit the discussion to 15 minutes.

What we liked:

Wkst 3 – we liked FBD  & crunching numbers (we're physics teachers, what do you expect)

Wkst 4 – we also liked how this helped to solidify Newt’s 3^rd law

We liked the progression of demos for 3^rd law

Especially the Laser reflection based on pushing the wall

For the most part, talking about Forces with little math (have yet to bring up $a=\frac{F}{m}$)

What we didn't like:

Would like for this unit to have more lab and less time on complicated worksheets
(Demos are good, but students are watching not doing)
{someone mentioned possibly using force table labs to introduce 2D/trig}

We felt that Worksheet 1 would be too big of a jump our students and would have like to see what Chris did
      to get his kids ready for it.

 A few had concerns that their students would never be able to ever do some of this work

          

Chris also said that for his AP class he has a summer assignment, which is primarily a math review

Unit V: "Atwood Machine" with vernier track
From there, we started the next unit.  Here's what each end of the track looked like (the middle is just a track)

Jon changed the first question slightly:
What factors will effect the motion? (between letting go of the cart and hanging mass hitting ground)

What factor effects the cart’s acceleration?

Hanging mass

Mass of car

Friction: {adjust tilt of track until cart rolls at constant speed}

(pasco hanger approx. applies force to balance friction)

Mass of pulley

Mass of the earth/gravity

Angle of the track - ?

Starting speed -?

{Chris showed us a quicker way of working through the process by guiding us to eliminate the factors mentioned that cannot be adjusted (Mass of Earth) or that could be removed with creative lab design.  The last two options we left open, that depending on your class, you may or may not want to divide and conquer.}

Purpose: What is the graphical & mathematical relationship that exist between the mass of the cart and the force that is accelerating it.

Before getting started, we talked about multiple variations to this experiment

• Keeping the mass of the hanger while adding mass to the car
• Using photogate(s) above the track instead of motion detector
• Variation of this option is to attach picket fence to cart the cart and use vernier program
• Using kinematic equations and measure total time with stopwatch for total distance measured
• Having the students predict the mass of the system from data, and then, after showing prediction to the teacher, measuring the mass and comparing results to predictions {I like this!}

Equipment

Attach right angle to cart

Pulley at end of track

Hanger

String

Motion detector

Standard masses

FIU Modeling Workshop - Day 6

We started today by finishing our whiteboard summaries of Ch 2 from Aron's book.  Since I didn't go into detail on the Day 5 post, I'll omit them here as well.

From there, we wrapped up Unit III with some feedback to Jon and Chris:

What worked:

• The worksheet "stacks of kinematic graphs" - we felt that it was a great tool for helping students convert from one type of kinematic graph to another.  Chris mentioned that if/when you have students whiteboard this, to make sure that they display the graphs vertically.
• Worksheet: Speeding up/slowing down - we liked that this allowed us/students to predict what they thought would occur, then later them seeing the results. 
• We liked that there were multiple labs that were short, as you could get more hands on time, but not use multiple days to do different activities.
• We liked seeing the graphical proof of kinematic equations
• We liked the reading from Aron's book, especially the misconceptions he mentioned, and tools to help overcome them.

What didn’t work:

• We said that we would like more insight into the "mechanic" of implementation the modeling cycle
• What does the day to day flow look like
• When do learning objectives come into play (some are at schools that must display the objectives for that day's lesson at the start of class)
• Pacing of course
• Some of us that aren't familiar with the content want more time to complete activities, and we also recognize that we need tools to overcome what we see in the workshop; some people are done with nothing to do, while others are struggling to keep up.
• Some asked, "What to do if we don’t have loggerpro/equipment?"
• One other thing we liked in the first cycle that didn't occur here was the division of Labor/variation of control variables.  {I'm not sure how you would fit that in, but that's what came up in discussion}

 {To those in the workshop (merely reading) that want to see the pacing of a class, one website I found was Mark Schober's Website.  Another great blog that you might find useful is Action-Reaction, one especially nice feature is that he organized his blogroll for different subjects (I'll try to get to that at some point).}

Miscellaneous Questions:

• How often do the students need to do formal lab reports, and how do those "Work?"
• As already mentioned, what are some ideas for extensions of labs for “faster” students

For the first question, Jon referred us to some of the resources at the beginning of the modeling binder (here, here, and here).
For the second question, Jon mentioned that he often splits up the groups that are done and have them help the groups that are going slower.

One other point that came up, was that if you need help keeping everyone engaged, assign each person in the group a roll. {When I need to do this, I use "Leader," "Secretary," "Technician," and "Gofor."  The leader in is charge of making sure the group is on task.  The Secretary is in charge of recording all necessary information/procedures/equipment/etc.. The technician is in charge of running the actual experiment.   The gofor (some call it the Yeoman) is the person in charge of "going for" stuff.  He/she gets the equipment at the beginning, is in charge of cleaning up at the end, and the assistant for all other jobs.}

One other conversation that came up was to make sure that everyone in the group knew one anothers' names.  Jon mentioned that he was surprised how many problems could be avoided if they knew that one simple fact.  He makes it a point to quiz students each others' names at beginning of new lab groups.

When Chris got a chance to address the question about objectives, he said he often uses some of the resources from the modeling curriculum to make review's

Kelly O'Shea has a blog that I love, which focuses a great deal on Standards Based Grading.  (One oversimplification of SBG is you report grades based on learning objectives of the unit. ) (Here are her objectives for Honors Physics, by the way).  When I asked her when she reveals her objectives to her students, she said:

Usually try to hand them out at the start of the unit. I would say few students look at them before they are preparing for an assessment. Some probably don’t look at them until they get the test back and look at their scores.

Jon mentioned that he often uses the provided Unit Objectives sheet to create a review.  Chris said that he often has the 3 ring-binder out when groups are whiteboarding, and often asks questions right out of the teacher notes (post lab discussions especially)

This was a lengthy discussion, but some great ideas came out of it.  From there we began Unit IV:

Jon began the unit by asking us to describe the motion of the following event:
{He tweeked the activity as we do not have student desks, what he said to do in the classroom was to have a student sit at a desk that everyone can see.  Push the desk so that it starts moving at near constant speed, and then stop pushing.}

We were to describe the motion using our 4 tools (written description, motion maps, kinematic equations, and kinematic graphs (x vs t, v vs t, and a vs t).  {We were to make those tools describe the motion from before it started moving until after it stopped.}

After we were done with our individual answers, Chris shortened the process by just asking individuals to share their ideas/draw their graphs on the board.  {I'm guessing that he was trying to make up for lost time due to our lengthy discussion earlier, and would do this through whiteboards, but I could be wrong.}

During this process, students often want to jump to why the objects are doing what they are doing.  Which leads to a discussion on forces.

From there, Chris said that he then does a mini-lecture on contact forces (forces caused by to objects being in contact) and fundamental forces (Gravity, Electro-magnetic, Strong, and Weak). {He said that he doesn't get into Normal and Friction forces at this point, but I might.  I'll have to think on this some more.}

From there, he makes use of a hovertoy (examples here and here, or make your own similar by hot-gluing the top to a "sport-top" water bottle to a CD, and slipping an inflated balloon over the cap. If your school has an airtrack, that would obviously work as well. ).  With the air turned off, push the puck across the table.  Then turn on the air, and push the puck.  Ask the students what is different about the two trials.  Guide the discussion until they realize that, for the whole series of demos, no force acting on the object leads to no change in motion; force applied leads to a change in motion. "No Force, No Change."

{Newton's first law, but like much of modeling, focus on the concept not the name.  Chris mentioned that he steers the conversation away from the term inertia, and instead focuses on the terms "balanced" and "unbalanced" forces.}

From there, Chris introduced Free Body Diagrams (FBD), in which you show the forces acting on the object (or system).  He started with a FBD for the puck resting on the table:



The circle/dot in the center represents the object, the arrows represent the forces acting on the object.  Chris said it was up to you if you wanted a convention such as all arrows point away from dot, or arrows point to show how the force is acting (Push-inwards arrow, Pull-outwards arrow).  The convention used in modeling for the name of the force is the numerator is the acting object, the denominator is the object in question.  So the two forces here are the force of the table on the puck $F_{T/P}$ and the force of Earth on the puck $F_{E/P}$ (AKA the force of gravity/ weight of the puck).  If need be, remind students that the earth is the object that creates gravity, not that gravity is an object itself (Thus $F_{G/P}$ would be incorrect).

{By the way, when I was a wise-a$$, acted like a student, and said that the puck is resting on the table, so the table can't be pushing up, Jon went into the closet, found a bowling ball and rolled it to me.  He told me to lift the ball and hold it shoulder-high at arms length.  Then asked if I'm pushing the ball to keep it at that same height. Touche Jon!}
{By the way, a bowling ball is another cheap prop you can use for the earlier part of the lab.}

He then showed a FBD for the puck at the instant he first pushed it:



The added force is the force of "your" hand on the puck $F_{H/P}$

From this discussion, Chris had us work on Worksheet 1, and then had groups whiteboard answers. {My only concern with this is that many of these problems get into 2D FBD's.  I'm not sure if I want to get to that before I've really had the students do any hands-on with forces}

Chris mentioned that this worksheet does a great job of bringing out student misconceptions about forces.  Most students get stuck, so instead of whiteboarding answers, for this problem, he has them whiteboard their questions about the worksheet.


At this point, we moved on to the paradigm demonstration: Dropping a bowling ball from shoulder height.
We again went through the usual questions, however, Chris added one more to the mix:
What do you notice? What can you measure? What forces are present? What can you manipulate?

We then created the purpose: To determine the graphical and mathematical relationship between the force of the earth on the object and mass.

We were then thrown a curveball for the experiment, we were given a Vernier Dual Range Force Sensor (Jon mentioned that spring scales work just fine) and some standard masses, and guided to plot mass vs Force.  As we saw that the data made a straight line, we could find the slope of that line. 

Each group then whiteboarded their results.  About the time the groups were getting lazy with the presentation (since we all had approximately the same numerical results), Chris threw out a question, "What is the connection between the slope of your line and dropped bowling ball from the start of the lab?"

For the groups that plotted Force in units "N" (which are, as of this point in the process, possibly unknown units) vs mass in kg, we found that the slope was eerily similar to the number we measured when finding the acceleration of object dropped (picket fence and rubber ball over motion detector) in the previous units.  That acceleration describes the acceleration of the dropped ball.  If all the groups used grams (which are the units printed on most standard masses), guide them through questioning to the value of slope with mass in units of kilograms.

Chris also noted, that making the connection between $N/kg$ and $m/s^2$ will payoff when Electric fields come up later in the year. {Obviously, this point will need to be reinforced throughout this unit and others for the students to remember it during E&M.}

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.