29 August 2012

Every possible geometric optics fact from Hewitt

I'll speak more later about the Phillips Style of teaching ninth grade conceptual physics.  For now, know that we spend part of each class highlighting in the Hewitt text; most days start with fill-in-the-blank quizzes based on the facts we highlighted the previous day.  The key is, students are the first time allowed to use any personally handwritten notes for the quiz (meaning that they had BETTER get a perfect score); then the next day, we give the same quiz, verbatim, with no notes allowed.  My colleague Curtis Phillips developed this approach last year, and it worked wonders.  I'm going to use it myself, starting in two weeks when school begins.

To prepare to teach in the Phillips Style, I've got to decide exactly what's important enough in the text to highlight.  I've typed out the facts that I want my students to know.  Take a look here at the google doc I'm using as a guide.  Most of these facts are word-for-word from the Hewitt text, with the page number listed in parentheses. When I didn't like Hewitt's phraseology, I wrote my own version; anything that's different from Hewitt will be written on the board and copied by students as notes.

Now, don't think I've gone all soft in my old age.  I'm still going to expect my students to use these facts to REASON about physics.  The Phillips Style is so wonderful because it avoids the whole "duh, what's a 'normal,' again?"  When I subbed a couple of times for Curtis I was so impressed at the depth of the physics that I could cover -- and it was because I never had to remind students of the basic, basic facts in the middle of a demonstration.

More soon, hopefully.  Posts might be sporadic for a while, because I'm moving into a new house AND getting ready for the school year.


20 August 2012

Preseason camp (guh), and managing the quick students in lab

Katherine Leonard, of Henrico County, Virginia, writes in:

On Wednesday my AP Physics B students will be coming to school for a little AP Jumpstart.  Last year I focused on collecting data, linearizing graphs and extrapolating data.  I was wondering, if you could have your kids for 3 hours before the beginning of the school year what would you try to instill in them?  

Yuck.  I'm not into the whole "preseason camp" business for school.  Will you have your whole class there?  I heard from a teacher in New Jersey who has to run a whole week of preseason classes for AP, but his students don't *have* to show up.  Disaster.

Anyway, I can only suggest making it interesting and active.  You're competing with other AP classes for attention.  You might be fighting some resentfulness from your students at having to be there, but maybe not; they might be excited to see their friends, and if you can make physics somewhat fun, in contrast to sitting for three hours listening to an English or history teacher, they might have a good time.

In your situation I might do my first lab, the one with the cart on the incline, measuring the tension in a scale holding the cart on the incline as a function of the angle of the incline.  They get to use their cell phones to measure the incline angle (woo!).  They're collecting data actively, but you can establish the standards for data collection during a time when they aren't necessarily hoping to leave quickly to do other homework.  When they try to be "done" after collecting three data points, you can say with a smile, "What, you've got something else to do?  Come on, man, you're here until noon with all of us, take the time to do it right."

Then three hours should be enough time for everyone to make graphs, linearize the graphs, and take the slope of the best-fit line. 

Students who finish early in lab at first resent being told to stick around -- "But I'm finished, I should be able to leave.  Why should I stick around just because everyone else is slow?"  If you tell them they have to stay because of school rules, they get even more antsy, because they feel like they are being treated like elementary school children who must be supervised at all times.  The real reason no one can leave early from lab is because if they COULD leave early, they'd all race to finish rather than take the time to do it right.  By setting the rules such that there's little reward for speed, they all relax and have a good time.  Make it clear that no one leaves early because YOU say so, and because there's nothing more interesting or important right now than physics lab.

Nevertheless, you will have groups working at wildly different paces.  Obviously make sure that the students who work quickly work accurately as well.  Nitpick until data is complete, until graphs are perfect.  When a group is truly finished with the graphs, ask them to start working on the lab sheet to turn in.  Most of my class will start the lab sheet during lab; what they don't finish in lab is done for homework.  This way, students don't mind finishing early and being "stuck" in class, because they're doing work that otherwise would have to be done at home.  You only have to insist the first day over protests that no, you can't leave and do the lab sheet later, you have to do it right now.

And if someone finishes the lab sheet?  Have that first problem set available, so anyone who finishes early has something productive to work on.  Once again, you limit resentment at having to stay if students can see clearly that they're saving themselves time that would have been spent at home, anyway.

Katherine is in a tough spot with this "preseason" gig.  Me, I would be tempted to be done in less than three hours -- that's a long time to be working on any one subject, and it's still summertime, after all.  But at the same time I don't want to set a precedent that we leave early from classes, because once school starts I use every available second.  If you're not careful, then every lab during the year will start with "Ms. Leonard, are we gonna use the whole period today, because I really need to blah blah blah."  

Perhaps I'd be transparent at the beginning of the session.  "Folks, during the school year, we use every available minute.  But this is preseason... we're going to do a lab today, and we're going to leave at 11:30, not noon.  But everyone will be working on physics until 11:30.  If you finish one step in the laboratory process, tell me, and I'll get you started on the next step."

Good luck...


16 August 2012

Lab Idea from Jaime Skiba: Volume of a Rising Air Bubble

rising bubbles from alaska-in-pictures.com
New York physics teacher Jaime Skiba posed a novel (to me) idea for a thermodynamics laboratory exercise.  She teaches chemistry as well as physics, so she and her colleagues are even more heavily invested in teaching gas laws than we physicists.  Her colleague, she says, long ago took excellent picture s of a gas bubble rising in a long column of liquid.  The diameter of the bubble increases as the bubble rises, because the pressure decreases while the temperature remains essentially constant.

Jaime proposes having students create some rising bubbles in a long, transparent graduated cylinders just by plunging an air hose to the bottom.  Smartphones can created video of the rising bubble; then (if you have the knowing of the software) the frames can be analyzed on a computer.  The diameter of the bubble in a frame can be measured by knowing the diameter of the cylinder itself, and scaling proportionally.

Students would make a graph of bubble diameter d on the vertical axis vs. depth h on the horizontal axis.  A straight line plot can be created by plotting the diameter cubed vs. the reciprocal of the depth.  Why?

Start with the ideal gas law, PV = nRT.  Pressure in the column is ρgh, where ρ is the density of the water.   The volume is the volume of a sphere.  By solving for the cube of diameter, I get 

Using this equation and the slope of the d-cubed vs. 1/h graph, you could solve for the number of moles in the gas bubble.  

13 August 2012

Waves on a string, and cost-benefit analyses for AP physics

An interesting and detailed email from Virginia physics teacher Drew Austen, who attended my Richmond workshop back in June, asked:

...You mention [in my Just the Facts: Waves post] that students should know that the speed of a wave depends on the material through which it travels. This is easy enough for waves on a string, since students should understand both tension and linear density, as in Question #50 on the 2009 AP exam. However, the speed of a sound wave depends on bulk modulus and density, and we don’t cover anything about bulk modulus in AP Physics... Any suggestions?

Bulk modulus is not necessary.  In general, it's simply necessary to know that the wave speed is determined by the properties of the material through which the wave moves, without the details about that dependence.  For example, a student should understand the deep water waves can move at a different speed than shallow water waves; however, the student will NOT be expected to know offhand which move faster, or how much faster.

For years I've considered the precise equation relating wave speed to tension and linear density to be unnecessary.  Most wave questions on the AP exam that I recall don't require recall of the exact form of the equation.  For example, problem 8 on the 1998 exam showed a standing wave on a string, then asked how to change the mass hanging from the string's end to get "more loops" in the string.  But the problem was explicit that a bigger tension meant bigger wave speed -- the tricks were to recognize that more mass meant more tension, that "more loops" meant a smaller wavelength, and that the frequency provided by the tuning fork would be unchanged.  

Drew paid closer attention than I to 2009 multiple choice question 50.  It described a wave moving on a string, then asked "which of the following will cause the wave speed to increase?"  One of the options was to use a string with identical length and tension, but a smaller linear density.  Oops -- knowing in general that changing the string's properties somehow changes the wave speed wasn't good enough.  You had to know, either conceptually or mathematically, that a "lighter" string gives faster waves speed. 

Yet, I'm still not going to teach the equation 

Why not?  AP Physics B students have to memorize loads and loads of equations already.  I've finally, after fifteen years of teaching AP, been pointed to a question that required use of this equation.  A student who had not been taught the specific relationship between wave speed, tension, and linear density would still have had a good shot at the correct answer, either using intuition about strings, or by eliminating the two answer choices that were obviously incorrect.  

So I'm making a cost-benefit analysis.  By teaching the equation above in addition to everything I teach already, I've given the students the information necessary to garner 3/5 of a point on one AP test over the past 15 years.  That's the benefit.

What's the cost?  Requiring students to learn yet another equation.  Putting this equation on fundamentals quizzes.  Writing homework and quiz problems to use this equation repeatedly.  (Remember, the benefit of teaching the equation only accrues if students actually remember what you taught.  Just writing the equation on the board once is worse than useless.)  

Is the benefit worth the cost?  I say no.  I show in a demonstration that by tightening a string, I produce a standing wave form that requires the wave speed to have increased.  I also show in a demonstration with a guitar that the light strings produce a higher pitch than the heavy strings, for the same string length -- this requires that light strings produce higher wave speeds.  

I don't repeat these facts again, nor do I make the students "responsible" for that facts on quizzes or homework, nor do I ever write the equation relating speed, tension, and linear density.  [I don't even use the term "linear density, usually, unless someone asks.]  I'm willing to sacrifice 3/5 of a point for less confusion.

Nevertheless, I would bet that my students didn't perform any worse than the national average on this question.  Point is, before you reflexively throw everything on the AP "learning objectives" into the gumbo of your course, consider whether simple matzo ball soup from a box might be easier to cook, and taste nearly as yummy.  Apologies in advance to all residents of Louisiana, but I'm *not* a fan of gumbo.


07 August 2012

Never trust a student with a negative sign

Yesterday's post detailed how I deal carefully with negative signs in six topics where they can't be avoided.  No doubt you are wondering, "how do you avoid negative signs elsewhere?"  Okay, I'll run through some topics in which negative signs *can* be used, but in which I manage to avoid them.

Equilibrium of forces:  In the first topic of the year, I teach to write "up forces = down forces" and "left forces = right forces."  Negative signs never come up:  a force has a direction "to the left" or "up and right, 30 degrees above the horizontal," but there's no "negative" direction.

Newton's second law:  The rule I teach is to write, for example, "up forces - down forces = ma."  But sometimes a problem requires "down forces - up forces = ma."  How do my students know the difference?  I tell them always to start in the direction of acceleration.  Then all force values are plugged in without ever including a negative sign, just as I taught in the equilibrium unit.

I know many folks teach ΣF = ma, assign a positive and negative direction, then just add algebraically all forces with attention to signs.  Go for it, as numerous good physics teachers have had success with this approach.  But I personally have found that causes more confusion than it eliminates.  Understanding intuitively that "negative" in this context* simply means "to the left" is a sophisticated piece of abstract thinking.  I know that you understand, 'cause you're reading this blog and so are presumably a well-practiced physics teacher.  But consider your audience of first year students with no experience whatsoever of applying mathematical methods to the physical world.

*But not in every context, of course -- scalars can be negative, opening up a different can of worms

Sure, I could say the same thing about kinematic acceleration and momentum, two of the topics that I teach with negative signs (as I discussed yesterday).  A number of introductory-level problems in kinematics and momentum cannot be solved without including minus signs in the calculation; and so I use negatives freely and appropriately in these topics.

However, Newton's law problems never require algebraic methods with negative signs; the finessed approach detailed above will always work.

Conservation of mechanical energy:  There's no getting around the sign of mechanical work done by a force.  But try teaching that the lowest position in a problem is automatically the y=0 position for gravitational potential energy -- then no minus signs in mgh.  Elastic potential energy cannot be negative, because (1/2)kx2 will always be positive.  (No need to define x as positive or negative at all -- x is just the distance from the unstretched string length.)  Calculus-based courses must include negative signs in order to relate potential energy functions to conservative force functions, true.  Now you see one reason that I don't teach calculus-based physics as a first year course.  

Spring force:  I know books write F = -kx.  In calculus-based physics, it's necessary to use the negative sign when deriving the period of a mass on a spring using a differential equation.  However, for all calculations in algebra-based introductory physics, the negative sign is an unnecessary distraction.  As with all forces, define the magnitude of the force as kx, and the direction is always toward the equilibrium point.  Done.

(If you allow students to read or use F = -kx, I will bet most of my life savings that someone, sometime in the year will state a spring constant as a negative value.  Send winnings to my school address, please.)

Electric fields and forces:  The "bible equation" defining an electric field is F = qE.  My classes practice for nearly a week: never plug in a negative sign into this equation, even if the charge experiencing the electric field is negative.  Instead, memorize the facts.  Positive charges are forced in the direction of E, negative charges are forced opposite E.  I take off mongo points if I see a negative sign included here.  My students are not sophisticated enough to use signs as indicators of direction for electric fields, then to turn around the next week to use signs properly in electric potential, where the signs have an entirely different meaning.  Why not avoid as much confusion as I can?  Signs are universally unnecessary when dealing with anything relating to a force.  

Signs on electric potential are necessary because a potential could possibly be "below zero," and so a positive charge would be forced from -20 V to -40 V and not the other way.  And a negative charge will produce a "below zero" potential.  

Application of above paragraph to problem solving is far more confusing than English homophone rules, and your students certainly always use there possessive pronouns correctly, eh?  So why confuse them more by including even MORE meanings of the negative sign in electrostatics?

05 August 2012

When I *HAVE* to deal with negative signs...

I can think offhand of about six situations in algebra-based introductory physics where negative signs are essential:

(1) algebraic kinematics, where displacement, velocity, and acceleration must include a direction
(2) momentum conservation, in which velocity must include a direction
(3) work done by a force
(4) all variables in the first law of thermodynamics (heat added to a gas, work done on a gas, change in internal energy)
(5) electric potential, where a negative potential means "below zero", and where a negative charge produces a negative potential
(6) thin lenses and mirrors, where negative image distance indicates a virtual image; and negative focal length indicates a diverging lens or mirror

For (1) and (2), I teach students to make a chart with all of the possibly relevant variables.  On top of that chart they are supposed to indicate which direction they've chosen as positive.  The chart helps de-emphasize the algebra.  

If I ask for substitution directly into equations, students get overwhelmed.  They try to substitute and solve all at the same time, losing track of what is known and unknown; a negative sign easily gets overlooked.  Furthermore, the meaning of the negative sign gets completely lost.  "Oh, the acceleration is negative.  Okay, now the math works," is what I might hear.  It's not clear why the acceleration should be negative, just that it gives the right answer.  And conceptually, "negative acceleration" takes on an incorrect connotation, as if the acceleration were somehow bad, undesirable, or naughty.

The process of making and filling out a chart clearly differentiates problem solving into a physics step, and a math step.  When a kinematics chart includes three of the five variables, the physics is done; everything else is mathematics.  In practicing problem solving, students will certainly forget a negative sign occasionally.  But because that math step is so obviously separate from the physics, I can train them to "debug" by checking signs with physical reality -- e.g, "the object was slowing down, so acceleration and velocity are in opposite directions.  Whoops, acceleration must be negative here, then."  The algebra is usually redone from scratch, yielding a correct answer with physical understanding included.

For (6), I teach the assignation of signs before I even introduce the thin lens equation.  We practice ray diagrams using templates until everyone is really good at drawing.  But a key part of each diagram is to measure the image distance, object distance, and focal length, recording each value in a chart.  I'm clear that all entries in the chart require a sign -- if I see a value without a + or - sign, the value is not assumed to be positive -- it's assumed to be wrong.  

Since I teach optics in March, my class has no trouble actually plugging into the relevant equation, and they're used to the chart approach.  All I have to do is prod them to fill in the chart properly.

[In fact, I think that next time I teach the junior-senior course I will use this "required sign" approach even in kinematics and momentum: if I don't see a + or - sign, it's wrong.]

Numbers (3), (4), and (5) are much more difficult.  Electric potential, work, heat, and internal energy are scalars, so the negative sign does not indicate a direction.  I don't generally discuss the "meaning" of the negative sign in these cases; rather, I repeatedly emphasize how to figure out the sign:

* Work done by a force is positive when the force and the displacement are parallel.  Work done by a force is negative when the force and displacement are antiparallel.

* Electric potential due to a positive charge is positive, and due to a negative charge is negative.  Positive charges are forced high-to-low potential, and negative charges are forces low-to-high potential.  A negative potential is just potential "below zero."

* In the first law of thermodynamics, ΔU is positive when temperature increases, negative when temperature decreases; Q is positive when heat is added, negative when heat is removed; and W is positive when volume decreases, negative when volume increases.

You might use negative signs in a bunch of other situations, too.  I know.  I don't.  In a forthcoming post, I will explain how I avoid dealing with signs elsewhere, because I simply don't trust first year physics students with a negative sign.