Equilibrium: quantitative demonstrations (or, how I teach vector math without teaching vector math)

That's a 2 N weight hanging by two strings.  The left-hand
rope passes over a pulley to a 1 N weight; the diagonal rope
is attached to a digital scale at the top of the picture.

In Honors or AP Physics, I begin the year with equilibrium, not with motion.  In a few days, the class gets comfortable with free body diagrams, forces, and two-dimensional vector analysis. 

Next comes motion in one dimension (graphs first, then algebra), followed by projectile motion.  Finally, we cover Newton's second law.  Since we did equlilibrium already, students are comfortable with free body diagrams and writing a vector sum of forces; since we did motion already, they already have some idea of what acceleration is.  Rather than a tough *new* topic, the second law becomes a way to review and solidify the first two topics.

The picture shows my third or fourth demonstration of the school year.  We begin by equating horizontal tensions in ropes pulling on a stationary block.  Next, I hang a 200 g mass vertically to find the tension in the supporting rope.  Easy stuff, so far.

And then, I attach a horizontal rope over a pulley attached to a 1 N weight. (In the picture, the hanging weight on the left is below the table, and out of the frame.)  When we make the free body diagram, everyone's comfortable with the weight of the 2 N weight acting down, and the horizontal rope pulling leftwards.  We draw the arrow representing the diagornal rope's force at an angle, of course.  Then what?

The class has already been taught that in equilibrium, up forces = down forces and left forces = right forces.  I ask, which is this diagonal rope, an up force or a left force?  Someone always comes up with a reasonable answer:  "both."  I redraw the free body diagram, with the tension in the diagonal rope replaced by two arrows, one up, and one to the side.

The class is totally comfortable with the upward component being equal to the mass's weight of 2 N; and with the leftward component being equal to the weight hanging over the pully of 1 N.  (They're also totally comfortable with me using the term "component" without preamble.)  It only takes a suggestive diagram to get someone to suggest that the resultant tension in the rope itself will be not 2 N + 1 N, but the pythagorean sum of 2.2 N.

The clincher comes when  I call a student to the front to read the digital scale attached to the rope.  It reads... 2.2 N.  Physics works.  I can even predict (and then measure) the angle made by the diagonal rope. 

The key to this whole process is that I'm *not* just telling the class how to solve abstract vector addition problems.  I'm not telling them anything at all, really; I'm drawing diagrams and asking questions, getting someone in the class to suggest the next step wherever possible.  I model the correct problem solving method for equilibrium problems on the board, of course, but everything I do flows naturally from the fundamental principles of equilibrium: up = down, left = right.  I don't use any words like "vector" or "reference frame" or "coordinate system." The only technical term I'm introducing is "component," which was introduced organically.

The final demonstration with this setup involves adding a "mystery weight" to the stuff hanging over the pulley.  I measure the new angle that the diagonal rope makes, so I have a chance to suggest how to use sines and cosines to "break an angled force into components."  We predict the reading in the spring scale, and the amount of mystery weight that I added.  Once again, physics works. 

Forget spring scales, look what I found!

At this year's AP reading, I was pointed toward dealextreme.com, a Big Lots style online retailer.  You can't rely on them to have exactly what you want, but if they do have what you want, it will be cheap.

When I browsed the site last month I found some "portable electronic scales," pictured to the right.  Huh... I use crappy spring scales that are easy to break.  My set is in need of replacement. 

This electronic hook scale doesn't measure in newtons, but does in kilograms (or ounces or pounds or "Jin," whatever those are).  The range is up to 20 kg -- way more than I ever need -- and has a 10 g resolution.

Dealextreme sells these for 6.90 apiece, with free shipping.  That's much less than I paid for crappy spring scales from a major science supplier last year.

The reader who recommended dealextreme.com warned me to order one of anything before I buy a class set.  He said that sometimes the products are defective or just lousy.  But because they're so cheap, it's worth the occasional lemon.  I bought two of these portable spring scales, and both work fine.  I'll set up my opening demonstration next week using these rather than spring scales; I ordered eight more so I'll have a class set for our first laboratory exercise.

Oh, and if these do die in a year, they are no worse than what I ordered from Fischer or Flynn or Sargent-Welch. 

Mail Time: What if everyone had physics last year, too?

Joe Konieczny, who teaches in Georgia, writes in with a problem some folks would love to have, but a problem nonetheless:

This is my third year teaching AP Physics B and I'm at a new school with students who have already taken a year of Honors Physics (just switched from public to private school).  The past two years my students have all taken the AP class as a first year physics course and so this year, three weeks in, I've yet to actually teach them something they don't know.  I'm struggling to keep them challenged.  How do you approach your AP students who have already seen a year of physics and keep them challenged? 

Joe says he's "three weeks in" because in Georgia, summer vacation seems shorter than the lifetime of a muon. 

Well, Joe, to be honest, I've never taught an AP B course in which ALL of the incoming students had taken physics before. And even then, the vast majority of those who have taken physics did so as freshmen. I just went at it as if it were a first year course, and everyone seemed happy.

The first step for you would probably be to evaluate to your satisfaction the scope and rigor of their previous physics course. Did all students take the same first-year course? Was it with the same teacher the whole way? Was the instructor competent? What topics did they cover, and how well do your students understand those topics? The first place to start, if possible, is to have a conversation with the students' previous teacher.

Next, right now in your current class, try getting into a topic they *haven't* seen before. Then you can get a sense of the students' true ability. Furthermore, depending on the topic, you might be able to see whether the students really understand what they learned last year. For example, maybe they didn't cover fluids last year. Dive in. (Hah!) See how they approach an Archimedes problem that includes equilibrium or Newton's third law. See how they handle a lab.

If your class truly has mastered the fundamentals of solving problems with Newton's Laws, energy conservation, and momentum conservation, then you can think of AP Physics B as applying those fundamentals in a variety of different topic areas, spending more time on the areas left uncovered last year. If, on the other hand, most of them keep saying things like "Wait, what do you mean, work done by friction?" then maybe the start-at-the-beginning approach would be better.

Regardless, by insisting on outstanding problem presentation with thorough explanations, it's likely that you can keep your strong students engaged, even if they think they have seen the material before. Setting sky-high expectations for problem sets is likely critical... if the students are acting unchallenged, then there's no reason they shouldn't be doing picture-perfect, suitable for framing in a museum, jobs on their homework. Perhaps you might think of last year's course as a wonderful head start, which could allow you to actually finish teaching all topics thoroughly in AP Physics B. This would make you practically unique amongst your peers.

Good luck. Let me know how things go, or if you have specific topics areas in which you want some feedback.  It is *tough* to teach second-year physics when you weren't the first-year teacher.

Classic Posts: First Day of School

Now might be a good time to refer readers to the classic post exhorting the masses to DO PHYSICS on the first day of school.  In AP/honors, I start with forces, free body diagrams, and equilibrium; in general physics, I begin with motion graphs.  In all cases I have live, quantitative demonstrations, which start within 15 minutes of the opening bell.

The "First Day of School: Do Physics" post is linked here for your reference.  Enjoy! 

GCJ

Rewriting Problem Sets for Honors / AP Physics

Above is an example of the layout style I'm applying to
problem sets.  Every problem will require both verbal
AND mathematical response.  See this link for a
google docs example:
projectile problem on google docs

Traditionally in my algebra-based AP-level physics course, I've assigned about two homework problems per night.  When I began teaching AP, I selected these problems from the textbook.  All textbooks seem to label their end-of-chapter problems by difficulty:  level I or * means easy, level III or *** means hard.  In every major textbook, the problems at the middle level of difficulty tended to be approximately on target for AP.

Now, there's much more to learning college-level physics than solving end-of-chapter problems.  Although textbooks are trying to improve, still their problems are heavily calculational.  Conceptual questions requiring verbal responses are shunted off into another section, rather than integrated into every problem.  That's not how an AP exam is structured. 

An AP free response question will have 3-5 lettered parts.  Some of these parts will likely require calculation; some parts will require explanation.  It is rare nowadays that a single free response item does not include BOTH verbal and mathematical sections.  I want to mimic this style in my own nightly assignments -- partially as a tool to prepare for an AP-style exam, but primarily because I think it good pedagogy to integrate verbal and mathematical questions.

When I began teaching AP, I scoured my textbook for good, level II, end-of-chapter problems.  The assignment would be stated as, "Do chapter 10 problems 29 and 64."  After I had taught the course for a few years, I began to add my own additional parts to the textbook problems, such as "... for problem 64, also describe as you would to a non-physicist the size of the boat."  And in the past few years, I've re-written most problems entirely to phrase them the way I want.

This summer, I've revised my assignments again for the express purpose of integrating verbal and mathematical responses into every problem.  I've typeset the assignments in MS Word, so that each night's assignment takes up a single page, front-and-back, with room for the answers.  You can check out this projectile problem on google docs as an example of the format and style of an assignment.

Why the room for answers after each part?  For a long time now I've observed that students will tend to use whatever space you provide for problem solving -- no more, no less.  If they use their own lined notebook paper, problems are crammed into as few lines as possible.  When I've provided full sheets of blank paper, they use most of the full sheet -- great, but sometimes I get an essay when I wanted a two-sentence response.  My hope is that I will get my class in the habit of giving just the right depth of response by subtly showing them the space that should be filled.

For those of you who are in their first few years of physics teaching, I would suggest you file this post away for future reference.  It takes enough time at first to figure out how to solve and explain the problems; don't worry about whether the problems are perfectly phrased, or well-typeset.  Just get the students in the habit of communicating their solutions.

But if you've been teaching a while, and if you're wondering how to make your assignments shorter yet more effective, I think this style is worth a try.  Make students write verbal responses on every problem, so they see that physics is about so much more than getting the right number. 

Assigning multiple choice exercises as homework

I'm in the process of writing nightly problem sets for my new Honors Physics I course.  These are essentially the same problems that I used for AP Physics B; however, I'm rewriting the problems to include more verbal explanations, and so that they're typeset on the front and back of a single page. 

Most of my homework assignments are simply rewrites of textbook end-of-chapter problems to make them AP style, and to explicitly demand verbal responses.  Occasionally, though, I want students to work through a set of multiple choice questions.  I *could* give these as a quiz in class; but to make the quiz worthwhile, I'd have to go over the quiz in detail.  I usually want to use class time for other purposes.  How can I usefully assign multiple choice problems for homework?

The issues are probably obvious... It's too easy for students merely to copy the (presumably) correct answer from friends without thinking through the answer thoroughly.  The simplest response is to require students to justify every answer with verbal reasoning.  I do this occasionally... but I don't want to assign more than three or four multiple choice per night in this manner.  How can I get folks to work through a longer set?

Once in a while, I'll pass out a 20-question-or-so multiple choice exercise and a scantron form.  I require each student to answer each question on the scantron by himself, without collaboration.  (You can enforce this either with an honor pledge, if you can trust it, or by using 20 minutes of class time.)

Next, I require each student to check his answers with classmates.  Everyone's final answers go on the reverse side of my two-sided scantrons.  I only grade the final answers.  The trick is, if a student changes his answer based on collaboration, he must write a verbal justification.

Grading is easy, since I can scan the scantrons, and spot-check the justifications.  The assignment is not excessively long, because students only have to write justifications for the ones they missed initially, and since the discussions with classmates when they are finding out which ones they missed will make justifications quick.

Sometimes I'll ratchet up the grade incentive for useful collaboration.  I'll take off one point for the first wrong answer, but two MORE points for the second wrong answer, and three more for the third... someone who gets 16 of 20 right would thus earn a 50%.  This grading system has led to wonderful physics arguments within the class, which of course is the whole point of any homework assignment. 

Rules for Turning In Daily Work: The secret to effective collaboration

Solitude... the counterintuitive secret to effective collaboration.

Ever have a student miss every part of a homework problem because his free body diagram was incorrect?

Ever have a student ask, "How were we supposed to do the problem when you didn't give us the mass?"

The relevant question for these students is, "With what other student(s) did you discuss the homework problem?"

So many folks knowledgable about learning physics will emphasize the benefits of collaboration.  I have my own story about how the only A I earned in an undergraduate physics course -- I worked on the weekly assignments myself on Sunday night, then I had four separate scheduled meetings with four different other students during the week.  By Thursday night, I was so familiar with the problems that I was providing cogent explanations to the procrastinators.  These folks thought me to be really talented at advanced quantum physics; Thomas and Jen, who worked with me on Monday and Tuesday, knew better. 

I'm sure you have your own story about how you or an acquaintance discovered the usefulness of regular collaboration in learning physics.  Our challenge, as physics teachers, is to push our students toward their own epiphany sooner rather than later.

In recent years I've taken the bull by the horns, and simply required nightly collaboration.  Students must write down the name of the student(s) with whom they discussed the problems, or at least checked their answers.  No collaboration = not full credit.  Such a requirement is easily workable in a boarding school, where a classmate is guaranteed to live no more than 15 yards away.  It's workable in a day school, too, in the age of email and social media.  Discussion via facebook or twitter is okay by me.

Many readers, at this point, are staring at their computer screens skepticipickly.  "Sure, Greg, let's *encourage* our students to copy each others' answers.  Who cares about academic integrity, anyway!"  Ah, but read on... the secret to encouraging effective collaboration is...

...stringently requiring a brief, written, *individual* effort before beginning collaboration. 

Without guidance, students interpret "collaboration" as, "sit down in a group and let the smart guy tell us how to do the problems."  Just a few minutes of serious, solitary work -- reading and processing the questions, writing down the relevant equation, attempting to answer the first part -- provide significant context for later discussions.  Now the smart guy is going to face questions from his or her peers:  "Okay, I didn't think to try using energy conservation.  How did you figure that out?"  Or, possibly opposition:  "Didn't Mr. Lipshutz tell us that kinematics isn't valid here, 'cause acceleration is not constant?"  Since everyone has had a chance to process the questions, even those students who don't see physics instantly will develop confidence in their abilities. 

It's hard for someone to cheat on homework if he or she has put forth individual effort before collaborating.  The context for the solution was established by the individual work; other students are merely helping to fill in the details.  It is critical that the teacher avoid the appearance of being the cheating police.  Always assume good faith... I screwed up royally one year when I got overly frustrated with the couple of students who worked too closely together.  Sure, they cheated -- but by publicly expressing my anger and disappointment, I deterred all the honest and earnest students from legitamate collaboration for fear of punishment.  When a pair of students are working too closely together, talk to both of them quietly, and patiently help them understand your expectations -- even if you believe that they're willfully cheating.  Only go into punitive mode when the same students have three or four times openly defied you.  The goodwill you buy with the rest of the class is worth the occasional dumbarse who thinks he's getting away with something.

Fair enough, Greg, you say, but how in the *heck* can I enforce a serious individual effort?  Most of my students will not put forth that individual effort at home, and we're back to the smart guy carrying everyone through the course.

An effective day-school approach, described by several summer institute participants over the years, is to give studnets the last five or ten minutes of class to begin that night's problems, with no discussion or questions allowed.  When class is over, each student's "ticket out the door" is to show you his or her written progress.  You're not looking for correctness, and you're not offering suggestions or criticism.  No, you're just looking for some sort of physics-related writing on a page.  Early in the year, you might see simply the diagram redrawn and the problem rewritten -- fine.  As the course progresses, your expectations might also progress to seeing a relevant equation or principle written down.

I draw a red vertical line down the page on which I pose the nightly problem sets.  Individual work is required on the left; collaborative work goes on the right of the line.  I only grade the final answer, so incorrect individual work is not penalized.  Other methods can be developed as well -- let me know if you have a useful way to promote individual and collaborative work on nightly problems.

I've been most pleased over the years at the bonding that goes on among my students.  Since they have to work with each other, odd pairings sometimes emerge, leading to friendships.  Most importantly, though, discussions about problems with me can focus on settling arguments between collaborators, not on re-teaching yesterday's lesson.

GCJ

Website -- how to use a breadboard for electronics labs

pic from techdose.com

I'm at Manhattan College (in the Bronx) for an AP Summer Institute.  We are doing my electronics lab, in which students hook up simple DC circuits.

Students struggle using the breadboard, at least until they get used to it.  I used breadboards for many years in college, so I have no difficulty teaching their use, and understanding how they work.  However, if you haven't used these thingummies before, you will have enough trouble learning their use, let alone explaining their use to a novice student.

Michael Morgante, from New York, asked me whether I have an instruction manual of how to use a breadboard.  I didn't; but I googled, and found this wonderful guide through techdose.com.  Read through the five or so pages, and you'll have a better clue.

Should you xerox this for your students?  I don't think so.  They ain't gonna read through five pages of dense text and diagrams during lab!  But you will find this guide useful for yourself.  Learn how to use the breadboard well enough that you can demonstrate its use to a class of novices, so you can answer simple questions.  Hopefully this site will help.

GCJ