Cheap-arse demonstrations: Goldfish bowl as refraction tank

Yeah, please feel free to pay $50-$200 for fancy refraction tanks from science supply companies.  I've gone that route myself -- twice -- but I discovered  that the flimsy pieces of junk lasted only a year or two performing as advertised.  You can get the same results with a goldfish bowl from the pet store.

I introduce the concept of refraction using the "4-wheel-drive vehicle* in the mud" analogy.  If one drives a pickup truck at an angle from a paved surface into mud, the truck's path bends simply because it slows down in the mud.  Similarly, light bends when traveling from one material to the next because its speed changes.  While light doesn't have tires, the muddin' analogy is sufficient to predict the direction of refraction, i.e. toward or away from normal.

* I teach at a boys' school in the South.  It's pronounced "vee-HICK-uhl."  And it has floodlights and a gun rack.

Next I introduce the index of refraction and Snell's law.  We imagine light incident from air to water at a 40 degree angle.  We calculate from Snell that the transmitted ray will be 30 degrees from the normal -- good, since the muddin' analogy suggests that since light slows down from air to water, the light should bend toward the normal, and thus the angle of transmission should be less than 40 degrees.

Thing is, all we've done so far is imagine and calculate.  I want to see this light bend, and measure 30 degrees from normal.  Here comes the fish bowl.  

The picture above shows a fish bowl half filled with water.  Key point:  A dry-erase marker writes and erases on glass just as well as on a dry-erase board.  

I have a student mark the water surface with the dry-erase marker.  Then he uses the protractor to mark the normal to the surface with a dotted line,* and a line at a 40 degree angle to the normal representing the incident ray.

*Watching otherwise brilliant AP physics students attempt to construct a perpendicular line segment with a protractor can be an amusing distraction.  You'd think I'd asked them to use nothing but a compass.  P.S. Do *you* remember how to use a compass to construct a perpendicular?  And how much is added to my nerdity score because I do remember?

The chalk erasers to the left of the fishbowl are clapped together so as to visualize the laser in air.  (A drop of coffee creamer allows the laser to be seen in the water.)  I hold the laser steadily so that in air the laser travels along the 40 degree line as drawn.  The student now traces the path of the ray in the water; the result can be seen in the picture.

Finally, a student measures the angle of the refracted ray with the protractor.  We always get within a few degrees of the predicted 30 degree angle.  Physics works.

The fish bowl works exactly like the expensive refraction tank.  And the fish bowl allows us to do the same experiment with the light originating in the water:  Since the tank is curved, I can just shine the laser in from the bottom, and I get light traveling from water to air.  

GCJ

Waves on a wave machine: quantitative iphone picture demo

Photo Credit Sam Russell, using my iphone in class.

I've got this ancient "wave machine," which consists of a set of iron* bars connected to one another atop a wooden base.  You can send a wave across the machine by pumping one of the bars up and down.  These waves are visually exciting, to the extent that students can never resist playing with the machine before and after class.  Good.

* Yes, iron, not some other metal:  the rust due to storage in my leaky classroom for decades is a bit of a clue.

I use the machine to perform a quantitative demonstration with v=lf.  

Step 1:  I ask the class how I might measure the speed of waves on the machine.  Someone will manage to suggest measuring the distance across the machine with a meterstick, and dividing that distance by the time measured on a stopwatch for a wave pulse to travel that distance.  We discuss how that method assumes constant  wave speed -- that's a good assumption, because wave speed depends on the material through which the wave moves, and that material is the same throughout the wave machine.*  

* I do not use the term "homogeneous," but y'all reading know that's what I mean.

Okay, so we do the experiment:  A student measures the wave machine to be 90 cm long.  We let a wave pulse go back and forth, so it moves 180 cm; they measure about 4.2 s for the wave to travel the 180 cm.  That gives a measured wave speed of 43 cm/s.  

Step 2:  I ask what I should do in order to put a wave with frequency 1 Hz onto the machine.  It takes a moment of thought, but generally the class figure out that this means I push the end up and down once every second.  Seems silly, but not everyone has internalized the meaning of a frequency.  Here's an opportunity to translate "1 Hz" into "one wave per second."

Step 3:  I ask the class to predict the wavelength of the 1 Hz wave on this machine.   The first answer that someone always spews:  "43."  That student looks sheepish for a moment as he realizes he didn't put units on.  He quickly computes the units:  in his mind, the frequency was "1", so 43 cm/s divided by 1 = 43 cm/s.  

Um, so a wavelength has units of cm/s now?  

[Don't think I'm being silly or hyperbolic here.  This happened in all three sections yesterday.]

So finally we all think carefully, and agree that 43 cm/s divided by 1 Hz gives 43 cm for a wavelength -- the units of seconds cancel in the improper fraction, and a wavelength has to be a length, anyway.

Step 4:  What good is a prediction without verification?  We must measure the wavelength.  

For years, and in my first section yesterday, we've had a student use his hands to estimate a peak-to-peak distance between the waves, and then use a meterstick to measure that distance between his hands.  

Yesterday, someone suggested taking a picture of the wave machine with a meterstick nearby.  "Just read the peak-to-peak distance right off of the meterstick in the picture," he said.  Certainly I've heard that suggestion before, and it's most definitely an appropriate method.  But until now, I've never had the technology to get this done in class.

I whipped out the iphone, and... handed it to a student.  What, you think that I can handle newfangled technology better than a whippersnapper?  Bah.  

I created the 1 Hz wave, and the student took the picture.  What he actually did was take a video, then play back the video to capture a still shot when the wave was in a good place.  He emailed the photo to me; I got the email, and put the photo up on the screen in my classroom.  No more than about 3-4 minutes elapsed.  That's still too long, but I'll bet with practice I can bring that time down.

The picture we took is at the top of the post.  Problem:  it's awfully blurry, so much so that we can't see the markers on the meter stick.  The iphone camera is convenient, but not so, um, good.  Nevertheless, we can make a quick estimate -- the peak to peak distance is clearly less than half a meter, but more than a quarter meter.  That's as good a measurement as I used to get with a student holding his hands above the wave.

After class, I imported the picture into Microsoft Paint.  I measured several times, to get about 680 pixels from one end of the meterstick to the other.  I found the peak to peak wave distance (again, using several measurements) to average 270 pixels.  Dividing the scale out, I get a wavelength of... 40 cm.  Difference predicted to measured wavelength: 14%.  Not bad for a blurred meterstick.

Don't "go over" an exam!

We're back from spring break today.  On Woodberry's strange trimester schedule, we took a set of exams in the first week of March, and compiled trimester grades right before we left.  

In our Physics C - mechanics seminar, I'm responsible for testing, while my colleagues are responsible for daily problems and discussions.  I showed everyone the exams, which were quite strong.  I gave some suggestions for where to pilot the course in the next few weeks, so that the hard stuff (rotation) would be covered thoroughly, and so the easy stuff (harmonic motion, linear momentum) would be glossed over for the sake of time.

One colleague asked the reasonable question, "Shouldn't we go over the exam?"  To which my answer was probably a too-strong "NO NO NO!"

Any major exam comes at the end of a significant, pressure-packed period of preparation, causing a "thank the almighty Bob" catharsis when it's over.  Our March exam period is followed by a 2.5 week break.  All continuity has been lost -- I doubt that anyone in my honors class could even recall a single question on the exam right now.

 I don't generally "go over" even regular-season classroom tests, opting to assign corrections instead.  But on the day after a test, I will frequently mention briefly one or two major issues that caused trouble.  How do I know whether to mention something?  The day after an in-class test, I can feel a buzz through the class when there was an interesting problem.  They want to know whether they got it right -- if they got it wrong, they just want to know why their approach was incorrect.*  If I hear that buzz, I throw out a point of discussion.

*Note that on the day after, students don't care about the solution to the problem.  All they care about is whether they did or could get credit.

What's to gain, then, by "going over" the exam?  There's no buzz about any of the long-forgotten questions.  No one really remembers how they approached a problem; to discuss the exam effectively, a student would have to re-solve each problem from scratch.

Hmmm...

Instead of an in-class discussion, I merely give a test corrections packet.  We take two class periods plus a no-other-homework night for students to correct every answer they got wrong initially.  The corrections count as the first test grade of this trimester.

By not "going over" the exam, I'm getting two tests for the price of one.  They took this exam under test conditions.  Now, after so long, the questions have that new-car sheen back; all the students know is which parts were marked wrong.  So the corrections become effectively a brand new take-home test.*  One way or the other, everyone has to approach these problems again very seriously, gaining the benefit of test taking practice one more time.

* Sure, they're allowed to collaborate -- call it a group test.  I've asked enough additional questions about each problem that just copying a friend's answer would do little good.

Over the years I've observed classes in which a teacher discussed a long-ago exam, point by point.  Watch such a class some time.  Look at the students' body language.  See who is truly engaged with what the teacher is saying, and who's staring off into space.  When the teacher shows the question about the relationship between weight and mass, are the two students who missed that question (again) paying attention?  Or are those two students looking at their laps, while the rest of the class sits bored because that question was so obvious?  If you gave the same exam the day after going over it, would the class truly do better?

And most importantly, why rehash an old exam when you could be discussing new physics?  Move on, get the class excited about something, get them solving new problems... then make them correct their mistakes on their own time.

GCJ

Which quantity stays the same?

Suppose you repeatedly dip your finger into a sink full of water to make circular waves. What happens to the wavelength of these waves as you move your finger faster?

 The wavelength increases

 The wavelength decreases

 The wavelength remains the same

Justification:

This is one of the first justification questions I ask after spring break in general physics, the night after I've introduced waves and v = lf.  At this point in the year, everyone knows to use either an equation or calculation to justify an answer appropriately.  

Even with such a straightforward equation, my guys will often wrap their minds in a huge knot attempting this kind of justification.  Why?  Well, some folks think that moving the finger "faster" means the wave speed increases.  But even for those who start on the right track, they get confused because they don't recognize what quantity doesn't change.  Their reasoning will sound something like:

"The faster finger means that the frequency increases.  If f goes up then v goes up, by v = lf with v in the numerator.  And since the velocity increases, wavelength must also increase, because l is in the numerator of the equation v = lf."  

The trick to justifying an answer qualitatively with equations is to start with the UNCHANGING quantity.  I train the class to start each of these kinds of questions with asking themselves, Based on facts of physics, what must stay constant?

Then the correct justification leaps out at them:  "Okay, the speed of a wave depends on the material (fact of physics). Since the wave is still traveling through the same water, the speed v doesn't change.  Now, we want to find wavelength, so rearrange to l = v/f.  The numerator doesn't change.  Frequency increases, because the finger produces more waves per second.  With f increasing in the denominator, wavelength must decrease."

Not only does identifying the unchanging quantity avoid confusion, it also guarantees some partial credit.  Even if this student had said frequency decreased, he would have earned something for stating that speed doesn't change, and attempting to use this fact in his justification.

Just the Facts: Geometric Optics

Geometric optics -- meaning, the behavior of light without reference to its wave properties -- can be a fun spring topic for upperclassmen.  After the rigors of Newton's laws, after the abstraction of electric and magnetic fields, geometric optics seems straightforward, both mathematically and experimentally.  And for a freshman physics class, geometric optics provides an easy-to-verify set of behavioral rules that can be  made semi-quantitative.  

But what specific parts of geometric optics should you cover?  And to what depth?

I've heard positive responses to my "Just the Facts" post about circuits.  Teachers have said they appreciated the short list in simple language of exactly what circuit topics to teach at each level of physics. So, here's a similar post about geometric optics.  Look down the list to the approximate level that you are teaching.  

I've written this list from my own notes -- those of you who teach Regents and AP Physics, please post comments or send emails if something here needs correction, addition, or clarification.

Woodberry Conceptual Physics

1. Light as a ray, definition of normal

2. Light reflects at an angle equal to the angle of incidence.

3. The “index of refraction” tells how much slower light moves in a material than in air

4. When light slows down, the ray bends toward the normal; when light speeds up, the ray bends away from the normal.

5. When light would speed up into a new material, but hits at an angle greater than the critical angle, total internal reflection happens.  The value of the critical angle depends on how much the light would speed up.

6. For a concave or convex mirror, parallel rays reflect through (or from) the focal point, and rays through (or toward) the focal point reflect parallel to the principal axis.

7. For a convex lens, rays through the center are unbent, and parallel rays converge to the far focal point.

8. For a concave lens, rays through the center are unbent, and parallel rays diverge from the near focal point.

9. Categorization of images as upright/inverted, real/virtual, enlarged/reduced

[Those who can figure out on their own how to use Snell’s Law and the thin lens / mirror equation will be eligible for advancement into Honors Physics.]

Regents (please correct me if I screwed up here, New York teachers) / Woodberry Honors Physics:

1. Light as a ray, definition of normal

2. Light reflects at an angle equal to the angle of incidence.

3. The “index of refraction” tells how much slower light moves in a material than in air, n = c/v

4. When light slows down, the ray bends toward the normal; when light speeds up, the ray bends away from the normal, by snell’s law n₁sin θ₁ = n₂sin  θ₂.

5. When light would speed up into a new material, but hits at an angle greater than the critical angle, total internal reflection happens.  The critical angle can be calculated by sin  θ_c = n₂/n₁.

[Looks like curved mirrors and lenses came off the Regents exam in what, 2002?  Honors Physics will include more involved questions than Regents, of course.]

AP Physics B / Woodberry 11^th grade general physics / (1990s era Regents):

1. Light as a ray, definition of normal

2. Light reflects at an angle equal to the angle of incidence.

3. The “index of refraction” tells how much slower light moves in a material than in air, n = c/v

4. When light slows down, the ray bends toward the normal; when light speeds up, the ray bends away from the normal, by snell’s law n₁sin  θ₁ = n₂sin  θ₂.

5. When light would speed up into a new material, but hits at an angle greater than the critical angle, total internal reflection happens.  The critical angle can be calculated by sin θ_c = n₂/n₁.

6a. The focal length of a concave or convex mirror is half the mirror’s radius.

6. For a concave or convex mirror, parallel rays reflect through (or from) the focal point, and rays through (or toward) the focal point reflect parallel to the principal axis.

7. For a convex lens, rays through the center are unbent, and parallel rays converge to the far focal point.

8. For a concave lens, rays through the center are unbent, and parallel rays diverge from the near focal point.

9. Categorization of images as upright/inverted, real/virtual, enlarged/reduced

10. For all converging and diverging lenses, the image distance, object distance, and focal length are related by 1/f = 1/d_i + 1/d_o.

11. f is positive for converging lenses and mirrors, negative for diverging.

12. d_i is positive for real images, negative for virtual

13. In all situations we will deal with, d_o will be positive

[For #6-13, Regents exams before 2002 provide some wonderful if straightforward sample questions.  The AP exam asks more involved questions, and often combines geometric optics with wave optics on the free response.]

I'm teaching ninth grade conceptual physics next year.

We used the Hewitt book
in 2012. No longer!
I recommend using no text at all.

I've spent the last few years getting my school's upper level physics classes in shape.  I now have a skeleton of problem sets, quizzes, labs, and tests that my colleagues can use to teach 11th-12th grade general physics, honors physics, and AP physics B or C.  The general physics course is based loosely on the New York Regents curriculum; the honors physics course on my version of what a future AP physics 1 test might look like.

So now it's time to tackle the only level of physics that I've never taught (cue ominous music): Ninth Grade Conceptual Physics.

(As an aside, I *have* taught ninth grade.  Once.  My first year of teaching.  "Integrated Science."  At a fluffy school.  Team teaching with an idiot who denied the results of experiment, who undermined me to students and administrators at every opportunity.  The scars still give a wee twinge on rainy days.)

The juniors and seniors I've worked with for years enter the school year as reasonably mature students; it's been my job to provide them with a challenging course to which they can apply their well-developed study skills.  Upper level physics can almost be thought of as a master class -- here's where all of the math, organization, relationships with classmates, writing, all of everything you've ever done as a student must be used in combination to conquer a difficult but manageable subject.

I am well aware that ninth grade is a different boat of gravy altogether.  I've begun talking to some of my school's best ninth grade teachers, listening to their thoughts and ideas of how they structure their course, how they develop relationships with 14 year olds, what different types of issues I can expect once I start teaching an entirely new species.  

One overriding goal over the next two or three years is to develop my own version of a ninth grade conceptual physics course, complete with a course structure, problem sets, laboratory activities, quizzes, tests, etc.  In terms of the level of physics, I want to aim at a low-arithmetic adaption of Regents-style questions, as I explain in this post.  Some of the course structure ideas that I know I'm going to implement:

Pace of the course:  I need to invert my usual approach.  With seniors, I've got to shoehorn in as much material as possible in the first half of the year.  That's when they're still motivated by their grade, that's when they are still afraid that any slackage might lead to the world ending and having to go to (gulp!) a different college than their first choice.  In the spring, I get some work out of seniors by demanding less.  They feel like I'm legitimizing their senior slide, so they actually do the minimum amount of work that I ask of them without complaint.  Thus, I'm always pushing the pace in the fall, and tapering through the spring.

With freshmen, I recognize that the fall is NOT the time to push hard in physics.  Adjusting to high school, and in my case to boarding school, is a difficult process for an adolescent.  Sure, a few students are ready for serious academics from day one -- these folks will be siphoned into Honors Physics within a few weeks.  Most need a gentle introduction to high school.  Then, in February or so (just as the seniors start to slack), freshmen are ready to move fast.

Sequence of coverage:  It's been argued that a physics class can seem friendlier by starting with more straightforward topics like ray optics.  At the AP level, I completely disagree -- the last thing I want to do is to give the immediate impression that memorizing facts alone will lead to physics success.  I want to start tough and get easier.

But in ninth grade, we will start with ray optics.  Refraction, total internal reflection, lenses, and mirrors all can be taught well diagrammatically and conceptually, with absolutely no mathematics.  But, I can use Snell's Law and the thin lens equation as an "application" for the honors course -- students who *can* handle quantitative predictions with these equations can be moved out, while the remaining students learned some serious physics without feeling bowled over by mathematics.  

I haven't decided on a precise sequence of coverage, but I do know that we want to gradually add arithmetic and basic algebra as the year progresses.  By the end we will certainly have covered the "Big Three" skills of reasoning with equations, interpreting graphs, and understanding the meaning of numbers.  It's just that we'll get to these skills gradually, after we start with a topic that allows straightforward conceptual prediction and straightforward experimental verification.

Types of test questions: I talked extensively to Bruce Oldaker, who at one time was in charge of helping the physics department at West Point streamline their testing at all levels.  He put into words a point about test construction that I have always done by feel:

First consider the portion of a test that is essentially recall, asking students to state facts or solve simple problems in situations they've seen before.  Then consider the portion of the test that asks students to synthesize multiple concepts, to extend problem solving techniques to new situations.  By sorting test items into bins of "recall" and "synthesis," it's possible to control the perceived difficulty of the test while adjusting the rigor of the evaluation of the students' physics knowledge and ability.

Now, as a long-time AP teacher, I have always advocated (though I didn't say it this way) keeping the "recall" and "synthesis" portion of a test consistent throughout the year, and in similar proportion to what students will see on the cumulative national exam.    Sure, that makes the first test of the year seem difficult; but soon enough students are old pros at physics tests involving considerable synthesis.  The shock of a test that doesn't just present homework problems with the numbers changed is going to happen sometime in the year; so, I say, deal with the shock right away when the class has plenty of time to recover.

Freshmen, though, need to build up to "synthesis" testing gradually.  Bruce and some of my colleagues point out that the same students who can't do anything but spit back facts at the beginning of the year will often develop their reasoning skills so that they can handle difficult questions by June.  Freshmen are growing that much physically and intellectually.  So, Bruce suggests that early tests be as much as 80%-90% recall... and that by the end of the year the recall percentage can be reduced to 40%-60%.  Without wasting too much time on meta-analysis, I'm going to be conscious of starting simple, and adding complexity to my tests throughout the year.

Got any ideas?  I'd love to hear 'em.  I've got an enormous amount of work to do to develop the freshman course to my liking.  It will take several years, and it will take plenty of failed attempts, too.  Maybe in a couple of summers I can hold a "Conceptual Physics Summer Institute" where we all get together to talk about teaching freshmen...

Update Feb 2021: I did hold that Conceptual Physics Summer Institute last year - it was well received, see the reviews in the comment section of the linked post!  I will be doing another CPSI in 2021, likely the weekend of July 24-25 and July 31-Aug 1. I'll post details soon.  

Newton's Third Law -- Test Question

A short-answer item, worth three points out of 100 on a 2 hour test:

A block of wood sits at rest on a tabletop.  A free body diagram for the situation is shown above.  State TWO action-reaction pairs of forces, in accordance with Newton’s Third Law, which involve the block of wood.

What fraction of your students do you expect to get this right?


Last year, I would have branded this a difficult question, expecting correct answers from about 1/3 to 1/2 of my honors-slash-AP class.  You may recall my post asking for help teaching the third law.  Sure enough, a few folks chimed in.  The theme of their comments: if you're gonna teach the third law properly, you've gotta establish the language of force pairs from the very beginning.


In particular, this year I piggybacked on John Burke's suggestion that every force, from day 1, be described in clear Newton's Third Law Language.  Starting with the very first free body diagram I drew, I added a list of all the forces in the problem, indicating the object creating the force and the object experiencing the force.  I drilled my students that they must always include such a list.


As an example, the free body diagram above would be incomplete in my class, now.  Students are trained to add, "Normal force:  Force of the table on the wood.  Weight: Force of the earth on the wood."  Sure, they squawked a bit because it took so long to write out all those words.  They stopped squawking when they realized that the points they lost for laziness were never coming back, no matter how much they argued.


Then, when I finally got around to introducing the Third Law, most of the class understood quickly:  to get the   Third Law force pair, just switch the objects.  Force of the wood on the table.  Force of the wood on the earth.  The process was easy, but only because everyone had long ago come to grips with the idea that, for example, weight is the force of the EARTH, not some nebulous concept like "gravity," on an object.

And now I see the enormous payoff.  The test question above seemed so easy to most of the class, that they wondered why I would even ask such an obvious thing.  Only two students out of 42 wrote an incorrect force pair.  Thanks for the ideas, folks.  


GCJ