A Practicum I Can Believe In

I've had some difficulty coming up with a good end-of-term practicum for the physics class for a while. This year, we put motion up front (CVPM and CAPM, all graphical analysis) and then went into oscillations (this used to be our first topic).  In oscillations, I've traditionally looked at period/frequency and amplitude, oscillation graphs, using proportional reasoning to solve problems, and qualitative restoring/driving/damping forces.  Proportional reasoning is something that this crew needs to work on in practice, so much of the year is topics that yield to it fairly well.

There have been several benefits to changing the order, though:
- the oscillation graph analysis seems to come just after they cover it in Pre-Calculus, so that saves me a lot of headache
- I can add motion analysis (where's the acceleration the highest?, find the max v from the position graph, etc.) that I couldn't do before
- the reasoning goes down better after they've done a lot of it in CVPM and CAPM, even though it should've been better to start with the easier reasoning here. It probably has something to do with their familiarity with speed vs. their unfamiliarity with period

Another benefit is that I can put together a robust practicum that uses both CAPM and OPM:

• On the day, I will set up a ramp of unknown length.  There will be a pendulum at the end, oscillating perpendicular to the track.
• I will let them observe the pendulum in motion.
• I will demonstrate for them, three times, the cart starting at rest at the top and traveling freely to the bottom.  I will tell them the length of the track.
• I will assign each group a number of oscillations - the pendulum must complete this number of oscillations between the time the cart is released and the time that the cart gets to the end, and the pendulum must collide with the cart as it reaches the end
• They need to have a procedure ready to determine how far up the track the cart needs to be released in order for these things to happen.

I let them work for a couple of days in groups, with a pendulum and a 1.2 m cart track. They need to develop and test their method so that it can work in any situation that I give them. 

I give them a packet with several pages: one for outlining a plan of attack (which they need to revise, if that plan changes), and several pages for completing each sub-task. Identifying that they need to determine how long the cart will have to travel, and that they need to measure the period of the pendulum and use the given number of cycles to find that time, is one example of a sub-task here. 

Students tend to be bad at laying out an abstract 'path' through a problem, especially if there's unknown information there. It's a tricky issue to tackle, but requiring these kinds of tasks of the students is certainly part of the equation. It's basically the same thing that I'm trying to address with the chains of reasoning exercises.

I laid out that structure on the first day, and students jumped into the problem at different spots, and most figured out a couple of sub-tasks at least.  There was a lot of average velocity vs. final velocity confusion, as is typical for these students.  On the second day, I had them start by writing out a list of the sub-tasks that they had identified - this is the "flow" of problem-solving that I'm trying to help them with. Most were good at this point, even though most groups hadn't figured out how to accomplish all of the sub-tasks yet. Here are the summary boards: interestingly, the first section was able to parse the task very well, but the second section had a great deal of difficulty understanding what the task was, which numbers were measurements and which were calculations, which variables explicitly affect their calculations (and should be measured, like the amount of time for the cart to go down the track) and which implicitly affected it (like the angle of the ramp, which affects the acceleration, but which doesn't appear in their calculations).

For the practicum itself, I'm using my 5 meter (!) air track :)

Chains of Reasoning: Standing Waves and Tension

I'm clearing out a few "meant to" posts from the year.  Here's a chain of reasoning problem I had kids do about a slinky hanging from the ceiling. I asked this as the 'advanced' question on their big waves assessment, and we came back to it in groups the next day. Students were asked about what would happen to the frequency, wavespeed, and wavelength of the waves from the top of the slinky to the bottom, and then to draw a standing wave diagram to reflect that.  I prefaced it with a question that elicited from almost everybody that the tension in the slinky was greatest at the top (almost 100% success even though they never studied forces - you just have to ask it specifically for them to realize it).

Since most had trouble with the question, I wanted them to work through it, rather than just forget it and move on.

The whiteboards are below. I was very particular with them writing down all of their reasoning - mostly "how do you know that's true?" and "what did you observe or assume to get to that?".  If I stayed on them, they did well and their answers were all correct! It's a year-long process to get them to internalize that process. It's not that they don't have the ability, but being a true self-critic is much more difficult than giving up.

 Hmmm... I can't get this one rotated - Blogger issue.  The original's fine!

Time to Bounce

I usually begin reflection with an investigation of the full-length mirror.

This year, we prefaced that with the question "How does reflection work?"  They grouped, I gave them a rubber ball, and they went "to the boards!"

I was mostly looking for the law of reflection, and I got that from many:

• This group rolled the ball off of a meterstick and traced the initial, final, and bounce points.  They then used a protractor to determine that the pre-bounce angle equals the post-bounce angle.

 Another couple of groups did the same sort of thing, but bounced light off of a mirror, with the protractor vertical, measuring the incident and reflected angles. 

One group set up so that they measured the horizontal distance of the initial, final, and bounce points - doing a slick end-around on the sine function!

• This group looked at reflection off of a curved surface (a flexible meterstick), and used the ball to simulate it.  They picked up on some good trends, though they communicated more verbally than they left on the board.  We'll get to reflection off of a curved mirror in a few days, but it's great that we're noticing already that it's not really any different than reflection off of a flat mirror (or anything else)!

Almost everyone measures the angle between the "mirror" and the path the first time out, which isn't what we usually do (the angle between the normal and the path is more useful, but that doesn't happen until we get to refraction).  No worries - the concept's there.

• This group did a very good job comparing the differences between the ball model and the light model: mostly, it's about the effects of gravity on the ball.  Under most circumstances, gravity's influence on light isn't noticeable, but there are notable (and awesome) exceptions, of course.  The first group above also noticed a difference in the speed of the ball after the rebound, which they dissociated from light's behavior.

Evaluating the limitations and implications of your model, on the first day in a new topic?  How much more do you want?!

While the law of reflection was nominally what I was "after," a great deal of other discoveries were made and shared with the class.

• This group figured out the law of reflection and noticed some things about the transmission and reflection of light by paper.

•  This group initially thought that the mirror might make light spread out (because the spot of light on the mirror from the flashlight was smaller than the spot on the wall after the reflection), but then realized that it only appeared to do that - a laser beam didn't spread at all.  The reason?  It was right there in the ray diagram the whole time: the flashlight's beam spreads out naturally, and the reflection had caused the path to lengthen, so the mirror hadn't actually done anything!

• This group came up with the law of reflection (with a snazzy diagram) and also accessed some previous information about color, and why different objects are different colors.  Even better, they were able to use the law of reflection and some ray tracing to show why letters can appear reversed in the mirror.  Several groups noticed it, but they were able to put the pieces together to figure out why.

• Finally, this group (which has been doing very well with chains of reasoning) took a tree-type approach to organizing their knowledge about light.  It wasn't what I had in mind, but they were getting good work done, so I let them see where it went.  I'm glad that I did: they were able to jump ahead a day and determine the difference between specular (mirror-like) and diffuse reflection!  There are diagrams of each off to the left side.

Chains of Reasoning: Static Electricity #2

Ahh, Volta's Hail. It's my favorite static electricity demo. It really has it all: conduction, polarization, charge induction, attraction and repulsion, grounding...

If you're not familiar with it, here's the setup (image from www.winsco.com):

The top plate is put in contact with a Van de Graaff generator, and the conductive pith balls are resting on the bottom plate.  After that?

Let's warm up first, just like my kids did last week.  They broke into groups and each tried to come to agreement on one of the conceptual questions from the text (Giancoli) that I had given them for homework.  In the past, I have given these short shrift, but they can be a valuable part of your teaching arsenal, if you let them be.  I do clicker questions and conceptual ranking tasks in class, but I had always shrugged them off for HW before.  Anyway, this warmup worked really well for one section, and really well for two groups in the next section, but the other groups hadn't done their homework, and...  well, you know how well that goes.

The questions (paraphrased) and the whiteboard solutions:

• If a plastic ruler is rubbed with cloth, it can pick up small pieces of paper.  Explain why, and why this doesn't work as well on a humid day.

• What balances the repulsive force between the leaves of a charged electroscope?

• Explain why clothes that have just come out of the dryer can sometimes stick to you.

• When a charged plastic ruler picks up small pieces of paper, occasionally one will stick to the ruler and then quickly jump away.  Why?

The honors classes haven't done as many reasoning chains, and the results are certainly mixed here.  There are all the classics: the insufficiently justified, the over-written, the under-written, etc.  Some of these are about communication and learning what's really telling the story, and some represent holes in the conceptual understanding ("But I know the answer!" We all know that having the answer doesn't necessarily mean having the understanding.  They haven't all gotten the message, but we're getting there.)

The chains that we did last class and the conceptual homework have, however, delivered far greater understanding than what I've done in the past.  All of that demonstration, lecture, etc. gave them the sense that they knew what was going on, but...  this year, they're so much stronger with their conceptual understanding.

The difference really became apparent when we went to Volta's hail.  I laid out the scenario: the materials that the apparatus is made from, what I'm going to do, etc., but did not demonstrate and did not tell them what would happen or indulge their questions.  Get in groups, get on that whiteboard, and figure it out.  This part of the cycle I've done before (at least two years).  It's always a colossal bust.  Almost no groups figure out the complex set of things that are going to happen. 

This year, though... all of the groups eventually "got it," only about half went significantly down a blind alley (and they only needed a small prompting question from me to get them back on track), and they really discussed the concepts like folks that knew what they were talking about... ...because they did!

Here are a few of their whiteboards.  It was great seeing them move from their gut reactions (usually just that the pith balls would move up to the top plate) on to making a complete and correct prediction.  It's especially awesome that their understanding and reasoning process was able to overcome their initial guesses without any intervention from me at all!

These aren't bad at all, and I think that most of the omissions here are communication-oriented rather than about gaps in understanding.  Here's a fuller chain:

I will, though, give a video of the demo in action.  If you don't use it in class, I'd consider it, because it's really slick and is one of the few really active electrostatics demos.

Oh, the barbell

Last week, we tackled this excellent "Figuring Physics" problem by Hewitt (as posted in TPT):

I love this problem - it helps students differentiate between translational and rotational motion.

It's also a great example of a chain of reasoning that's not too difficult to lay out, but which has a critical choice to be made right at the beginning. This choice bifurcates the reasoning tree into "the real answer" and "bizarro world".  One's true, the other's the opposite, and... it all relies on getting one little piece correct.  Here are the dueling trees:


Now it's time to make the call: is the bottom of the bar moving forwards or backwards relative to the track?

If it wasn't moving at all - like it wasn't as the dumbbell rolled at the beginning, and like it won't be after the friction sets the rotation "right" again - then the bar would be rolling without slipping, which is a super-important concept for students to have a handle on.

The most amazing thing about rolling without slipping is that the total velocity of the contact point of the wheel is zero!  Here's a great picture of a wheel that's rolling without slipping, taken by Archan Baldev Luhar of Medfield High School, for the AAPT Photo Contest (any Tatnall students interested?!):

 

Great photo!  The v due to the rotation is the same as the translational v, but in different directions at the top and the bottom (and zero in the middle).  Excuse my lame graphic:

This means that the top moves at 2v, the hub at v, and the bottom doesn't move at all!

This is an excellent entryway into how friction works with wheels, both static and kinetic.  The direction of the static friction on a wheel can be hard for students to get, but it's just resisting the potential slippage of the tire (just like static friction always does), so you just need to figure out how the wheel would slip if there weren't any friction, and the static friction force is the opposite direction.  Neat, but a bit beside the point.

Here, we're looking at kinetic friction.  What's happening is that either the bar of the dumbbell is rotating too fast for the translational speed (which means that the bottom is moving backwards, relative to the track) or that the bar is rotating too slowly for the translational speed (which means that the bottom of the bar is moving forwards relative to the track).  It starts with the "correct" rotational speed, but that's the speed for the larger radius of the weight, not for the bar.

This brings us back to the question at hand, the choice that will send us down one path or the other (BTW, our dueling trees did narrow it down to a choice between C and D, so we already know more than we did at the beginning!), the measurement that creates two parallel bits of the multiverse, etc.

Is the bar rotating too fast or too slowly for the translational speed?

How fast does something rotate if it's rolling without slipping?  The rotational speed must be (you might have to click through to the post to read the LaTeX if you're on a reader):

We can easily determine the rotational speeds that the barbell already has and that it'll need to roll without slipping when the bar is on the track:

Here, r is the bar's radius and R is the larger radius of the weights.

It's easy to compare them with my favorite tool, the ratio:

If the needed angular velocity of the bar is greater than it currently has as it rolls on the weights,  we can say that the bar is rotating too slowly when it hits the track.  This means that the bottom of the bar is moving forwards (the forwards translational velocity is not canceled by the "rotational part" of the velocity), which puts us on tree #2!

As von Braun tells us, one experiment is worth a thousand expert opinions, so... (click through for video)

 The reason that I make a big deal about experiment here is that we all talked our way through the long chains of reasoning, and were smugly satisfied that the first chain was correct.  Sometimes, it really does pay to write a little something down.

In particular, it really helps to write things down when there's a symmetrical comparison to be made - it's really easy to get things backwards!

The reason that I bring up the experiment being worth so much is that the kids (bless their hearts) recognized that they hadn't quite wrapped their heads around the situation, and leapt to "let's do it!"  A quick field trip to the fitness room while others set up the track, and we were off.

The explanation came quickly, and the understanding was much more solid (and correct!).

Doing is learning.

Oh yeah - it works backwards, too: now the bar speeds up and the rotation slows! (click through for video):

Estimation Nation

So, it's the day before spring break.  As if that weren't motivation enough (for the students) to not move on with content, there were lots of kids missing and this was an orphaned 'A day' (we're on an A/B schedule, and I have one section of physics and one of honors physics on each of the days, so I have to be careful about missed days in order to keep them together).

So... let's do something fun! (as usual), but non-content oriented (not as usual).  Let's do...

Fermi problems!

Here are the ground rules:

1. No calculators, long division, or long multiplication: stick to 1 or 2 sig figs (maybe the only time I say those dreaded words all year!) and concentrate on getting the right power of ten for the answer
2. No research: you know more than you think you do, and you can figure things out from distantly related facts (remember all of those chains of reasoning that we've been doing?!).  Anything else?  Intelligent estimation.

I presented a big set of questions, and students, in groups, chose a question and dove in.  In the past, I have given everybody one question or assigned each group one question, but I thought that this was more in line with the WCYDWT? spirit that we've been going with this year.

Here's the list of questions (many creditable to Maryland's list), along with student solutions.  Many of the reasoning chains are pretty well-communicated, which is great.  Some haven't gotten there yet, but we're working on it.

• How many piano tuners are there in NYC?
• How many hairs are there on your head?
• How many pencils would it take to circle the equator?

• What's "your share" of the land area of the Earth?
• How much (per hour) will you pay for classes in college?

• How many drops of water are there in the Great Lakes?
• How many blades of grass are there in a typical lawn?
• If you remove all of the string from all of the tennis rackets in the US, how far could it stretch?
• How heavy is the rain that falls on the school's roof during a big storm?
• How many hours would you have to work (at minimum wage) to pay "your share" of the national debt?

• If the whole US were a pool, how deep would it be when filled with all of the milk consumed in the US in a year?
• How many flat tires are there right now in the US?
• How long would it take you to reshelve every book in the library?

• Could we build a pipe organ in the classroom that covers the whole range of human hearing?

I thought that it was interesting to look at which ones were popular and which weren't.  Mostly, students stuck to things close to their experience and/or in their personal interest (pencils, tuition cost).  These are also very concrete ideas - easy to get a handle on for students.  At least we have a consensus that it'd take some number in the low hundreds of millions of pencils to circle the equator!  You can rest well during spring break now, secure in that knowledge.

The next time we do Fermi problems, I'll force some engagement with some of these that just seem impossible to them (until they dig in), like the drops of water in the Great Lakes, the number of piano tuners, or (my personal favorite) the amount of water flowing through the Mississippi River in a year.