# Monthly Archives: July 2012

## Light Reflection, Refraction, & Diffraction

I hope it’s not getting old, but I would like to say that for those that have never attended a modeling workshop, although I hope this series of blogs peaks your interest, in no way should they be viewed as a replacement. Compared to the first set of workshop, I will not be posting links to the materials used as ASU and the AMTA have chosen to keep them under lock and key. If you are offended by that, I’m sorry, but they have chosen to restrict some materials, and I am in no place to challenge them. The basic thought process is that there research has shown that one best learns through doing, not through being told.  Therefore, the best way to learn how to implement modeling is to experience modeling for yourself, not to read about it.  In short, if you just read how to model, try to use it and find it ineffective, it could very well be you implemented it incorrectly.  Go to the workshop, you’ll thank me.  For those readers that have attended the Mechanics workshop, this may help you sort through the materials to which you have access, but again, you might want to try to get to one of these if you can.

2-D Wave Motion

We begin this portion of the unit by putting together ripple tanks.  Although these contraptions clearly show the behavior of water, I think there is a much cheaper solution.  By no means do I want to go full scale \$2 whiteboard here (I mean no disrespect by that comment, I think that blogpost might be one of the greatest I’ve ever read), but I don’t see why you need that much complexity.  Why not just use a cafeteria tray, your finger and/or a ruler as a wave generator, and some simple objects (or even the edges of the tray) to show boundary interactions?  The time it takes to setup just doesn’t seem to justify the fact that you can see everything with a simple tray you’ve taken borrowed from your school cafeteria.

Anyway, mini-rant over, in using the ripple tank we are to look at how the waves react when a boundary is placed in the water.  Lo and behold, the waves reflect just like we saw with light!  We then try to look at refraction.  This part was very problematic.  The teacher notes say to put a thin piece of plastic in the water to change the water level.  For us, we did see a change in wavelength, however, we were never able to create a scenario in which the direction changed.  I think that side of things needs a little more refining (it has to be the procedure, it couldn’t be your humble blogger).

After a circle meeting to share the behaviors we saw, we worked on the second worksheet which brings in a new way of showing the behavior of waves: wave-front motion maps.  In some sense, just imagine a ray diagram with lines added to show the wave fronts (I’ll try to add pictures at some point, but the server here is way to slow).

One thing that we noticed in discussing this worksheet is that we needed to build the relationship between index of refraction “$n$,”  the speed of light in a vacuum, and the speed of light in a given medium.  I’m not sure if that jump to light is needed here, rather I think this might be better served taught separately as just mechanical waves (more on this to come in a later blogpost).

From there, we started a lab to show the diffraction of light.  The group set up a flashlight with a screen (open 3-ring binder) casting a shadow on a wall.  They then asked us, if a laser is placed under the flashlight and just touches the edge of the binder, where will the laser dot hit the wall in relation to the edge of the shadow?  We said that it should hit right at the edge of the shadow.  Once they set it up, we saw that the laser hit a few inches away, on the light side of the shadow’s edge.  I think care needs to be taken when setting this up to make sure that the students see that the laser is directly underneath the flashlight, and that the flashlight is small.  Several “students” (myself included) think that big of a shift seems unlikely to be due just to diffraction.

From there we move into a new behavior of light, that being diffraction.  By placing two boundaries in the water, we could look at what happens as the wave front passes between a small opening.  We were encouraged by the “teachers” to look at how the size of the opening effected the resulting diffraction pattern as well as how adjusting the frequency of the source on the diffraction pattern.

We finished up by working on worksheet 3: Diffraction to apply some of the concepts just developed.  One thing I really like was at the end of the worksheet.  The students are asked to predict what should happen if there are two openings.  After a good discussion, I left thinking, whatever the kids think here, as long as they can defend it using the wave model, is ok.  We now have a natural tie-in to a behavior of light not yet studied.  I’m ok leaving the question without a definite solution, as they’ll get it in the very next step of the unit, but that’s for the next blogpost.

{The Wave Model, brought to you by FIU, CHEPRO and the National Science Foundation}

## 1D Mechanical Waves

Once again, I would like to say that for those that have never attended a modeling workshop, although I hope this series of blogs peaks your interest, in no way should they be viewed as a replacement. Compared to the first set of workshop, I will not be posting links to the materials used as ASU and the AMTA have chosen to keep them under lock and key. If you are offended by that, I’m sorry, but they have chosen to restrict some materials, and I am in no place to challenge them. The basic thought process is that there research has shown that one best learns through doing, not through being told.  Therefore, the best way to learn how to implement modeling is to experience modeling for yourself, not to read about it.  In short, if you just read how to model, try to use it and find it ineffective, it could very well be you implemented it incorrectly.  Go to the workshop, you’ll thank me.  For those readers that have attended the Mechanics workshop, this may help you sort through the materials to which you have access, but again, you might want to try to get to one of these if you can.

As we ended the particle model unit, we had a good discussion about the overall storyline of this collection of units.  I think I’ll let this play out a little more, but I can’t help but think that this storyline is making it more difficult than it needs to be.  For those that don’t know, there is modeling material for just mechanical waves.  I haven’t had a chance to really go through them, but there are units for the oscillating particle model, 1D wave model, sound, and then 2D waves.  To me, that would flow very nicely out of the central force particle model.  With that in mind, I’m thinking you could just go straight into the wave model of light.

I’m fully aware that the current progression of starting with the particle model of light is true to an historical approach, but the major aspects of reflection and refraction could just as easily be first introduced with a wave model.  As you then move in to the photoelectric effect and the beginnings of the Bohr Model, you could then bring in the particle nature of light (yes, the photon model is distinct from a particle model, but what is the end product the kids NEED to know.)

Anyway, just some food for thought.  From what I can tell, this collection of three units is set up to be taught without teaching the mechanical wave units, thus this post will look at introducing mechanical waves to students.  To me, if you do include those units, you can skip this content and move directly into building the wave model for the behavior of light.

We begin this unit by asking the question, “What would you have to do to make a wave?”  You then provide some Super Slinkies or “Snakies” (or both) and have the kids go play.  By time they are done, your should be able to discuss the characteristic variable (frequency “$f$,” wavelength “$\lambda$,” speed “$v$,” and amplitude “$A$“), what you need to do to increase the speed (effects of tension, amplitude, and distance on wave speed), and how non-conservative energy loss (due to friction) effects wave propagation.  If you want, you can pose these as questions to answer, or just let them play (and if need to send them back out to play a second time).  How much guidance would depend on the maturity of your kids.  One other important concept that you need to get across is that with waves, energy moves, even though there is not net displacement of the medium.

By connecting the Slinky to the Snaky, you can also begin to show that the speed of the wave depends on the medium.  If the students are really careful, by sending two quick pulses, they can feel that the frequency is independent of the medium.

The second major activity/lab is to get the students to see that the speed of the wave is a product of frequency and wavelength.  The modeling materials say to do this through the introduction of standing waves. To me, that might be too big of a jump.  Instead, I think you could have the kids measure the speed of a pulse  by measuring distance and time (possibly include various tensions) first, then bring in standing waves (and find frequency times wavelength) and compare to speeds found already.  Another option would be to video tape the waves and use video analysis to determine the speed of the wave.  In the end, with standing waves you need to build in the terms node and antinode.

During the standing wave lab, we had “major” issues trying to measure the variables using the snaky.  We later saw that the teacher notes say to use a slinky since it has less tension.  The other option is to use a wave generator which can better oscillate at a controlled rate.

At some point during this process you need to talk about reflection, interference, and superposition.  Student should see that both the wave model and the particle model can account for reflection (As you get to 2D waves, you can further show the behaviors of reflection discussed in the particle model).

From there, we completed worksheet #1 and later whiteboarded our results. If the kids didn’t notice it already, you can discuss transverse and longitudinal waves, along with terms such as crest, trough, compression, and rarefaction.

As mentioned at beginning, to me, this could just as easily be its own unit (most likely leading into another unit on 2D waves in water.

{The Wave Model, brought to you by FIU, CHEPRO and the National Science Foundation}

## Modeling and the Next Generation Science Standards

One of the traditional parts of a second modeling workshop is to design your own unit.  Since there is so much changing with physics with the Next Generation Science Standards rough draft and the much rumored AP B redesign, I wanted to take a look at how the current curriculum materials matched.  By no means do I confider myself an expert, but I figured, if I was going to build a unit, might as well have something I can possibly use in the future.  With that in mind, I made a very crude list of how the modeling units matched the NGSS.

At the bottom of this post, I list the units as per the ASU Modeling website (can’t get access? Go to a Modeling workshop, it’s the only way to really use the mateyails anyway).  Skipped standards, to me, fit other classes than physics.

NGCC-Modeling topics

• HS.PS-FM.a            BFPM &UFPM
• HS.PS-FM.b            MTM
• HS.PS-FM.c            MTM
• HS.PS-FM.d            MTM
• HS.PS-FM.e            MTM
• HS.PS-FM.f             MTM (w/relativity)
• HS.PS-IF.a               BFPM, UFPM, E&M-#1&4
• HS.PS-IF.c               E&M
• HS.PS-IF.e              E&M (extention of curriculum needed)
• HS.PS-E.a                ETM
• HS.PS-E.b               ETM, 2nd Law of Thermo
• HS.PS-E.c                ETM, Entropy
• HS.PS-E.d               ETM, Entropy
• HS.PS-E.f                ETM
• HS.PS-E.h               ETM, Machines
• HS.PS-FE.a          Energy & Fields (g, E, B)
• HS.PS-FE. b         Energy storage in fields
• HS.PS-FE.c           systems of Energy storage due to fields
• HS.PS-W.a           Mech Waves #1
• HS.PS-W.b          Mech Waves #2
• HS.PS-W.c           Mech Waves #2, 3, &4
• HS.PS-ER.a          Light #1&2 (&3?)
• HS.PS-ER.b          review lit about speed of light
• HS.PS-ER.c          review lit about effect of various wavelengths of matter
• HS.PS-ER.f           review lit about digital vs analog communication

(yes, I love Kelly O’Shea’s Naming system)

Mechanics Units

1. Scientific Thinking
2. Constant Velocity Particle Model (CVPM)
3. Constant Acceleration Particle Model (CAPM)
4. Balanced Force Particle Model (BFPM)
5. Unbalanced Force Particle Model (UFPM)
6. Projectile Motion Particle Model (PMPM)
7. Energy Transfer Model (ETM)
8. Central Force Particle Model (CFPM)
9. Momentum Transfer Model (MTM)

Mechanical Waves Units

1. Oscillating Particle Model
2. 1-D Mechanical Waves Model
3. Sound Wave Model
4. 2-D Mechanical Wave Model

Models Of Light

1. Light Particle Model
2. Light Wave Model
3. Light Photon Model

Electricity and Magnetism

1. Charge and Field
2. Potential
3. Circuits
4. Magnetism

## Particle Model of Light

First off, I would like to say that for those that have never attended a modeling workshop, although I hope this series of blogs peaks your interest, in no way should they be viewed as a replacement. Compared to the first set of workshop, I will not be posting links to the materials used as ASU and the AMTA have chosen to keep them under lock and key. If you are offended by that, I’m sorry, but they have chosen to restrict some materials, and I am in no place to challenge them. The basic thought process is that there research has shown that one best learns through doing, not through being told.  Therefore, the best way to learn how to implement modeling is to experience modeling for yourself, not to read about it.  In short, if you just read how to model, try to use it and find it ineffective, it could very well be you implemented it incorrectly.  Go to the workshop, you’ll thank me.  For those readers that have attended the Mechanics workshop, this may help you sort through the materials to which you have access, but again, you might want to try to get to one of these if you can.

Day 1:

We began today with a simple circle meeting (what one of our fearless workshop leaders calls the circle of truth) in which we introduced ourselves and shared a few war stories from the previous year.

After completion of the circle meeting, we returned to our lab tables (and told to begin “student mode”) and were asked to answer the question, “What is light” as a group.  Some of the groups were better than others at staying in “student mode” (yes my group was one of the bad groups), but overall we tried to reflect what we thought I students would give as an answer to that question.  We shared our whiteboards from our tables (there are only four tables, so we’re already in somewhat of a circle).  Overall, to me this question provides great insight into seeing what one’s students already know about the topic.  You can see how much they remember from physical science and/or chemistry (depending obviously on the sequence of classes at a given school).  One of the interesting side questions that came up was where would these three models fit into the overall flow for the year.  We agreed we’d come back to that later, but the question stands, should you teach mechanics -> light -> E&M, or mechanics -> E&M -> light.  One other side note that I thought was interesting was that LCD Projectors project Red and Blue light in one polarization and Green in the other.  If you’re able, you can show kids this by using a big enough filter for your class setup (assuming you have an LCD projector).

After the discussion came to a close, the next major task (and I do mean major) was the Light Concept Inventory.  By no means do I mean to disparage whoever put in the time to make this, but it was much more of a bear to get through than the FCI.  They gave us an hour and forty-five minutes and most of us weren’t done.  I’m not sure if that is more about us, or just how long that test is.

From there we got started with some of the introductory hands-on stuff.  Each table was given a bucket of equipment with a small maglite flashlight, a candle, a torch lighter, laser pointer, and a small fluorescent light bulb (vertical coil not spiral).  We were each given a small whiteboard (about the size of a piece of computer paper, and asked to individually write down what we noticed about light as we messed around with the contents.

On a side note, our fearless leaders told us that having individual WB’s allows each student to have their say, then let the group talk, and either combine onto one big whiteboard, or just share individually with the class.  Since my school is going 1:1 with iPads, I’m thinking I’ll just find a whiteboard like app and do this step using the iPads.

During the subsequent discussion, we developed some working vocabulary (through socratically led discussion) such as luminous objects vs illuminous objects, a working definition of a shadow (which in the next activity we modified to distinguish between umbra and penumbra shadows), and transparent, translucent, and opaque objects.  We also agreed on how to best represent how light travels -> rays. One of the cool highlights was spraying something (I didn’t catch what it was, but it looked like a can of hairspray or spray deodorant) in the air along the length of the path of a laser pointer, think classic laser scenes from Mission Impossible type movies.  As I said, not sure what it was, but the particles stayed in the air for a very long time, making a very cool effect.

After that, we were told to take the “shroud” off of the maglite and use it as a base, in so doing creating a nifty point source of light.  Again, we were told to play around and on our miniWB’s write down what we observe with respect to the formation of shadows.  After some playing in the dark, they brought a few extra maglites to each group so you could make shadows with more than one source of light.  We again shared our findings, during which we made the distinction between umbra and penumbra.  Some highlights of the discussion were me making the classic barking dog shadow puppet on the wall and using a string to reinforce the linear path of light by stretching it from the light source to the shadow on the wall, just touching the “students” finger along the way.

From there, we made two different types of pinhole viewers.  The first was just a Styrofoam cup, a piece of wax paper, and a rubber band to attach the two.  Pull the wax paper across the opening and secure with the rubber band.  Then punch a small hole with a push pin.  The second was a little more advanced, using two pieces of PVC pipe (one of which just slides inside the other).  Put aluminum foil across one end of the larger PVC pipe and wax paper around the narrower one.  Attach both with a rubber band, and poke a hole in the middle of the aluminum foil.  Point either at a light bulb with an arrow (cut I’m guessing with a Xacto Knife out of electrical tape) and the leftover piece of tape with the missing arrow) and observe.  (I have a picture, but I’m having trouble getting it to upload, so I’ll try to add it soon)  Again, we wrote down our individual thoughts on the miniWB, shared with our lab group, and later with the entire class.

At the end of the first day, we recapped what we know as to the basic characteristics of light and discussed what might be able to explain them (a stream of particles).  They let us know that starting tomorrow, two of us would be the “teacher” for each of the coming sections in the rest of the unit.  They posted on the board what part (whiteboarding worksheets or leading the next couple of investigations) we were to get ready to lead.

Day 2

Our homework for the previous night was to complete the first three worksheets of the particle model packet.  As we began setting up our whiteboards, we had a little bit of a side discussion.  So of the big things that came up were: 1) To keep in mind that although modeling is unstructured, it is still very goal oriented.  Make sure, as the teacher, you have looked over the materials and especially the teacher notes, to make sure that, no matter the path your class takes, you get them to the correct destination. 2) Make sure you are continually helping the students to extract the core concepts, one way to do that is to make sure that you take a few minutes at the start of one day/beginning of the next to summarize the concepts or important observations that are needs moving forward within a given unit.  One other thing I offered was Kelly O’Shea’s awesome Mistake Game as a way to improve discourse during whiteboarding. (Kelly just made an updated post to this awesome way of whiteboarding.  Definitely check it out.

A couple of the key ideas that are beginning to emerge for this unit are after our introductory interactions with light and having completed the first three worksheets:

1. Light can be modeled as particles that move very fast in straight lines.
2. Light can bounce off of objects and take a new straight path.
3. There are too many rays emitted by a source to count.
4. Some objects create light, some block/reflect light, some allow “all light” through, some only partially allow light through.

The readings talk about a pinhole making a reproduction, not an image, but so far, we’ve been just using the term image.  I think the official packet is trying to make the distinction since at “all locations” the reproduction is visible, while an image is only visible at fixed locations due to the geometry of the optical system.

After whiteboarding the first three workshops, we than began the paradigm lab for this first unit on light.  From here, 2 of the participants are now running the show, so we get to lead the group of misfit “students.”  (By the way, in my opinion, the worst students are teachers.  We tend to break down and do all the things we find so annoying too quickly.)  The leaders for this section began walking through the tradition modeling introduction to a lab: what do you notice/see, what can you measure/change/control, what can you manipulate.  If you’ve read this far, I’m sure you’re somewhat familiar with the process, so I’ll keep this somewhat simple.

The equipment we were eventually given was a meter stick with support stands, a square piece of paper and clips to attach it to the meter stick, a square piece of paper with about 400 dots punched out (approximately have the diameter of traditional 3 hole punch dots), a mini maglite, and a ring stand w/clamp to hold the maglite.  At some point, I’ll get a few pictures imported, but so far I’m having trouble doing so with the FIU guest server.

The overall goal was to compare the number of dots seen within the square screen as a function of distance to the source (as a group, we didn’t make a good distinction about distance to source or to the dot paper, the groups that measured to the source had much more reliable results).  Just about all the groups acquired data that looked like an inverse relationship.  As we then linearized the data, some groups had data that then looked parabolic and linearized again.  During the whiteboard discussion we compared results and procedures to try to flush out the correct relationship.

The original plan was to then whiteboard the fourth worksheet and have a board meeting, however the workshop was interupted due to the, at the time, upcoming announcement of the Higgs Particle.  FIU has a few particle physicists involved in the CMS project at CERN, so one of them gave a pretty substantial preview of the coming announcement from the other side of the pond.  As a side note, we are planning to have a Higgs party with them at 3 am!

Update: Higgs party was awesome in a nerdly way.  Hearing room full of scientists gasp at a little bump on a curve and then give a huge applause at the mention of 5$\sigma$, was quite an interesting moment.

Day 3

We began our third day with whiteboarding worksheet 3 from the modeling packet.  Some interesting thoughts that came out of the discussions were as follows.  One misconception that were battling over was whether or not students would think that decrease in light intensity as you move away from the source would be due to light spreading out or due to absorption (meaning a loss of energy do to some non-conserved force).  One of the problems led to a great discussion of the concept of a point source of light emitting light in all directions vs light in a beam (laser light).    One of the groups did an awesome job of explaining the solution to one of the problems by explicity starting from the relationship derived in the first lab ($I\propto \frac{1}{d^2}$).  During that discussion, we also developed the idea that intensity of light is like a “density” of light rays.  A few other tricks of the trade that were discussed was the idea that we need to reinforce the idea of mathematical symbols.  Although we got a little side track nitpicking on the math teachers, the important aspect of the discussion was that we can cause confusion by calling any independent variable x part of  the time and using x for only position other times.  We need to try to stress that the axis are horizontal and vertical, not x and y.  We also need to continually push our students to try to figure out the meaning of the slope of a graph, not just jot the numerical value down.

One absolutely fabulous moment, in this bloggers humble opinion, was during a discussion of how long it takes light to get from distant stars to Earth, and is the light from “now” or from the past.  To keep it short, just watch this:

From there, we moved into another lab, images from a flat mirror.  The group began by showing a flat mirror and a laser, and asked what we noticed the laser was moved around while being pointed at the mirror.  Keeping the story short, we set out to find the relationship between the incoming angle and the outgoing angle for reflection in a flat mirror.  To help show how awesomely powerful Kelly’s Mistake game (I don’t know if she invented it or not, but she introduced me to it and has given numerous blog posts explaining it).

I convinced my group to measure all of the angles relative to the mirror on the side of the mirror on the incident beam (Meaning a beam coming in very close to the mirror is $10^o$ and the reflected beam is $170^o$)  At first the other groups were baffled that we didn’t measure from the angles from the normal line, but it lead to a great discussion of how to justify the convention to measure from the normal line.  Two great lines of questions that eventually emerged, do the ingoing (we haven’t really stressed the term incident so far) rays and out going rays look symmetrical?  If you agree that they do, should we have convention of describing the angles that stress how symmetrical they look?  Which was then followed up by “Where do you normally stand when you look at yourself in the mirror?”  If that’s where you normally stand, isn’t that a great reference point?  To me, that might be the best reasoning to call it the “Normal Line” I’ve ever heard.

After that, we started building towards developing a way to find the image in a mirror.  They gave us this awesome red pseudo-mirror (You can get one here).  Draw something, most of us just drew a letter, then trace over what you see on the paper/mini-whiteboard behind the red pseudo-mirror.

Another great discussion then ensued in which we tried to figure out how to convince students that the image was behind the mirror, not inside/on the surface of the mirror.  One great point was to get something with small writing.  Stand at a distance where it can be read, and where twice the distance it becomes illegible.  Then stand at that distance from the mirror, and try to read the writing.  One other suggestion was to get a manual focus SLR type of camera (most likely using the manual setting, since all of them come with auto functions).  Measure how far away you are from the mirror, then look at the distance the camera is reporting when your image is in focus.

A second part of this side discussion was over trying to explain why images “flip” or “reverse” side to side but not up and down (seen with the pinhole viewers).  What Jeff suggested was to write on wax paper.  Hold it so that it looks normal and then look at what you see in the mirror.  (If you really need to convince someone, then turn and face them and ask them what they see {text is backwards for them but normal for you [yes I love parenthetical comments]}).

Once we got back on track, we began whiteboarding the next worksheet (#4A).   In the process, we summarized two laws for reflection:

1. The normal line, incident ray, and reflected ray are always in the same plane.
2. The incident angle is equal to the reflected angle.

We also began to distinguish between reflection off a “rough” surface (which we eventually defined as diffuse reflection) versus reflection off a “smooth” surface (specular reflection).  We also began to wrestle with the terms real versus virtual images.

One great quick take home assignment/quiz that one of the workshop leaders uses is to have the kids go home and look at their face while being very close to a mirror. Using either a whiteboard marker (lipstick, tape, or any other removable form of making a couple of marks), make a line at the top and bottom of where your head appears to be on the mirror.  Then answer the question, what happens to the position of the top and bottom of your head relative to those marks as you walk backwards (away from the mirror).

The last part of today’s class was spent with a paradigm lab to study curved mirrors.  This was the part in which a fellow workshoper and I became the “teachers” for the class.  We began by setting up a candle in front of three mirrors at slight angles to each other (similar to what you see in dressing rooms).  We then directed a laser, from in front of the set up, at one of the mirrors such that it hit the back of the candle, and asked the class how does the laser get to the back of the mirror.  We then asked is there another place we can direct the laser such that it will still hit the back of the candle two times (answer move to the side so the laser hits a different mirror).

The reason being that we wanted the class to recognize that, since the mirrors were not coplanar, they had normal lines that were not parallel to each other.  Thus, the “same” path of light behaved differently due to different normal lines.

We then asked, how would the setup change if we had several smaller mirrors, each at different angles around the candle (set up in an arch).  To which, they said, there would be more ways of directing the light to the back of the mirror.  From there we showed them a concave mirror, and ask, “So then what should this mirror do?”

We then showed them how, but putting a triangular piece of tape on the “window” of a flashlight we could project an image of that triangle onto a surface.  We then did a quick hand waving of the modeling series of pre-lab questions to get the class to go about looking for a relationship between the distance from a candle to the mirror and the distance from the image to the mirror.

As the groups were working, many quickly saw that the data had an “inverse” look.  However, when they tried to linearize by taking the inverse of one data set, many were unsatisfied with their result.  As per the notes in the teacher info, we help most of the groups  by telling them to take the inverse of both data sets.

At this point we had two cool ways of finding the focal point.  The first was that we went outside with a big classroom demo size concave mirror and a piece of balsa wood.  (I’ll eventually upload a video, but I’m not going to bother trying to do so with the limited bandwidth I have here at FIU).

We also used an overhead projector to create a beam of light.  After turning off all the lights in the room, we put one of the small concave mirrors used in lab in the path of the beam.  We then got the special effect spray again and could very clearly see the cones on each side of the focal point.  After quickly measuring that point, we were ready for the whiteboard/circle meeting.  Most of the groups seemed ok with having the graph of $\frac{1}{d_{image}} vs. \frac{1}{d_{object}}$  Just about all the groups agreed that the slope was “-1.”  We then asked what do you notice about the vertical or horizontal intercepts.  Most quickly noticed that they had the same value.  After a little prodding, they realized that they were each the inverse of the focal distance we just measured.  So I then showed them (if we had more time I would have had them do it) the equation of a standard line: $y = mx + b$ and then began replacing the variables with what we had in our graphs.  Since the vertical axis is “$\frac{1}{d_i}$,” that replaces the “y.”  Since the slope is “-1,” that replaces “m.”  The “x” gets replaced with “$\frac{1}{d_o}$,” and the “b” with “$\frac{1}{f}$.” Thus:

$\frac{1}{d_i}=-\frac{1}{d_o}+\frac{1}{f}$

Which if you so choose can be rearranged into the more standard form:

$\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$

Day 4

We began the day whiteboarding worksheet #4 from the particle model packet.  Some important concepts from the were the 3 principle rays (which we said would need to be discussed before the students would be able to do any of the worksheet.  One significant flaw we saw was that the first two drawings on the worksheet did not have the mirrors drawn accurately (all others looked fine).  Due to this, if students tried to complete the worksheet using protractors and the (2nd) law of reflection, they would get a completely different result than if they used the principle rays.  Thus, if and when I use this worksheet, I’ll probably cut and paste from somewhere else in this worksheet.

We also said that somewhere during the curved mirror lab, you need to relate the focal length to the center of curvature (a few of the problems provide one but not the other, which might cause problems).

One great rattle mantra that one of the workshop leaders says he uses is, “If it diverges on one side (of the mirror/lens), it converges on the other.”  I thought that was such a great statement, since I always have to battle the confusion that  “diverging rays don’t intersect.”  Meaning, I often have students that get diverging rays and say that they don’t intersect.  If you can get that saying into their head, they will know to look on the other side of the mirror/lens.

From there, we were given two paper cups with magnets glued to the bottom (doesn’t have to be magnets, just something somewhat flat).  We were told to set them up so we could just see the magnet’s edge, then begin to fill one with water, and write down our observations (on a mini-whiteboard).  We were also given two clear cups with a pencil in each.  Again, we were told to fill one with water, and write down our observations.  After which, we discussed what we noticed, in so doing, beginning the notion of refraction (I’m not sure if this is necessary if you’ve used the mechanical wave set of units as they also build the idea of refraction).  One cool demo they did was using a laser pointer, a thin rectangular water tank (with a few drops of coffee creamer in the water), and the laser fog spray mentioned earlier.  They eventually had the tank hang over the edge of the table so you could get the laser entering the top and leaving the bottom of the tank.

From there, we were shown a small semicircular water dish, and led to the objective of the lab: to determine the graphical and mathematical relationship that exists between the ingoing and exiting angles from the water dish.

One significant flaw flaw with this step is the fact that, with this equipment, every group got data that would be likely misinterpreted as linear.  The teacher notes say to lead the students (either before or during the lab) to compare the semi-chords (relative to the normal line) to lead them to the sine relationship.  We all agreed this seemed very forced.  A few other things we discussed was that if you use error bars in the graph, you very easily see that the line doesn’t fit the data.  However, most of us do not use error bars, believing it’s a level of sophistication above (and not necessary for) the introductory physics students.  In the end, we all thought that if we sufficiently laid the groundwork earlier in the year, most of our students would see that the data can’t be linearized with any of the tools they’ve developed so far (squaring or inverting one set of data).

To me, this is a similar stumbling block as what occurred during the curved mirror lab (having the students invert both sets of data).  What pops in my head for each of these cases is the religious “poem” “Footprints in the Sand.” I think this is where the idea of physics coach/guide (not teacher) becomes a powerful image.  I think this is an example of knowing where a major stumbling block will prevent the success of our students.  In such cases in which they can’t get there easily on their own, that’s where we step in and offer direct instruction.  As a side note, I would not cite the poem explicitly in class as I don’t want my students to infer that I’m a physics god.

One other great idea that came out of this discussion, that I had never heard before, was the “least time” argument. It was described/introduced using the following “smart dog” problem:

Imagine you are playing fetch with your dog at the beach.  Your dog is a certain distance down the shore.  You throw a ball directly into the water in front of you.  Since the dog can run faster on the sand than swim in the water, what is the fastest path for the dog to take to reach the ball.

Here’s a link to a similar problem.

From there, we began working on worksheet 5A, which we will finish discussing tomorrow.

We began Monday’s class by finishing worksheet 5A on refraction using the particle model.  One interesting idea that came out of it was a way to show the connection between concave mirrors and converging mirrors.  What someone mentioned was to take the piece of paper  with a mirror and fold it along the middle of the lens.  What you can see through the paper is a concave mirror (similar result for diverging lens to convex mirror).

One other discussion we had was over the idea that refraction creates an “image.” If you draw two rays reaching two different “observers,” you can extend those refracted rays to one intersecting point.  We tried to figure out if that refracted “image” location.  After a brief attempt to be like one of the physics blogosphere’s super physicists (Rhett Allain), I made an excel spreadsheet that calculated the apparent position based on geometry, and was able to show that the location is not consistent.  If I ever get the chance, I’ll try to make a post about that. If Rhett beats me to it, I’ll post a link.

One of the important concepts that comes out of this worksheet, is the critical angle.  I’m not sure if I would need to give a little bit of a heads up on this, or just let it play out during the whiteboard meeting.

From there, we moved to worksheet 5b, which now builds on the particle model through quantitative questions.  One of the interesting discussion came up when asked about the difference between an image and a reproduction (pinhole camera/viewer).  Since as a group we used them interchangeably earlier in the unit, there was some major confusion here.  I would definitely recommend stressing reproduction earlier (and why it’s not an image then).  You’ll probably still have some misconception at this point, but probably less.

We also had a great discussion about how to lead students to understand that, when placing a screen over have of a lens, you can still see the entire image.  One slight flaw with the worksheet is that it uses a “large” lightbulb, so a ray of light cannot directly reach the bottom of the lens (without passing through the lens itself).  I might alter this problem to show a candle instead to avoid the confusion.

We finished the unit by discussing the “Essay Questions.”  For this unit (and the other light units) students do not write lab reports, but rather write reflective essays.  These essays ask the students to reflect on what the particle model is, and what evidence he or she has for each aspect of the particle model.  The second essay asks the student what aspects of the particle model seem incomplete or unable to fully explain behaviors of light.  The student is then asked to propose a modification to the model or a chance to begin to create a new model all together.

One great demonstration that came out of the talk was to take a squirter bottle.  You can create drops of water to show a “particle” and slowly squeeze until the drops become a stream.  It is very simple, but yet, a great visual to show one particle becoming many.  One other great visual was to use a can of spray paint or hair spray to show particles leaving a flashlight.  This could also be used to model the intensity of light.  You wouldn’t spray paint on a wall from several meters away, no paint would reach the wall.

Well, that’s it, that’s the particle model.  I’m guessing this might not give an absolutely clear picture of the model, but I don’t want it to.  I want this to be enough to help those that have attended to remember

{The Particle Model, brought to you by FIU, CHEPRO and the National Science Foundation}