Corrections to the AP Physics Sunken Ocean Liner Problem

Picture of the Andrea Doria from the Associated Press

The 2004 AP Physics B exam includes a clever problem combining kinematics and static fluids.  That's problem 2, linked here.  I assigned this problem on my trimester exam, and got reasonably good results. 

Anytime you're teaching a class of 37 students, though, you're going to have frustrations grading their exams.  SOMEONE is going to make the silly mistake you warned them about a million times.  Someone, somewhere, is going to lose five points out of fifteen because they didn't think properly under pressure, even though that same person will later slap his head and say "d'oh!"

I make my students correct their mistakes on all tests and exams.  The corrections process is not only pedagogically sound, it is cathartic to the physics teacher's soul, as well.  Some of the questions I ask on corrections are to point out the most common misconceptions, so that students can eliminate said misconceptions from their brains.  Other questions are designed less as a teaching tool than as a deterrent to future silliness.  When I grade the exams, and I scream to an empty room, "How can you fail to put units on an answer after three months of AP Physics?!!!!!" I find that I feel much better by designing a "deterrent" style test correction.  Here's what I mean.

Part (a) of 2004 B2 asked students to find the gauge pressure at the bottom of the ocean, given an absolute pressure of 413 atmospheres.  Those who missed this question usually couldn't recall the meaning of "gauge pressure."  So, the correction is pretty simple:

(a) Don't solve, just tell me briefly what is the difference between gauge and absolute pressure.

Part (b) required students to use the absolute pressure and the equation for pressure in a static fluid column to find the depth of the sunken ocean liner.  The most common mistake was to fail to convert properly from atmospheres to pascals.  That's not a huge deal to me.  Of greater consequence are the numerous students who get an answer of 41 m for the depth of the ocean, and don't recognize why that's so silly.  (The ocean liner itself is probably longer than that!)  So all I ask is:

(b) Explain in no more than two sentences why the answer “41 m” is physically unreasonable.

 

But the other major problem with parts (a) and (b), and with ALL parts of this question, is the students who write down 8 significant digits, or who fail to put units on their answers.  My students certainly know how to use proper sig figs, and they have been called out numerous times for lack of units.  On an exam, units and signifcant figures are a matter of focus, not of true understanding.  So, here's the correction I use.  Imagine reading this correction with an evil cackle:

If you lost points for units or significant figures here, or anywhere in this problem, write out in your own handwriting:  “From now on I will be sure to write units on every numerical answer, and I will limit my answer to two or three significant figures unless the problem requires more significant figures.  And, the value “41,205,100 Pa” has six significant figures, which is too many.”

Ah, I feel better now.  And it works -- I've heard students saying to one another, "Hey, don't worry, I had to write that after our earlier test, and I made sure to put units on every answer this time."

The only other additional question I ask students in correcting this problem is in the last part, which asks for the time for the ocean liner to fall to the bottom.  Some students set up a single kinematics chart, with an initial velocity of 0 m/s, a final velocity of 10 m/s, and a distance equal to the depth of the ocean.  Problem is, they were told that 10 m/s is the terminal velocity of the ocean liner, which was reached after falling for 30 s.  So it's a two-step problem -- find how far the liner falls in the first 30 s while it's accelerating, then use constant-speed kinematics to find the time to fall the rest of the way.  I ask students to re-solve this problem, but first, they answer:

To find the time of sinking, why can’t you just make a kinematics chart with v_o = 0, a = 0.3 m/s², and Dx = 4200 m?

The goal, of course, is for students who got a part wrong to figure out their mistakes.  Just asking students to redo the same questions without acknowledging their misconceptions or articulating in words their mistakes is, while better than NOT doing corrections, insufficient.  These kinds of additional questions make corrections engaging and effective.

Multiple Choice questions may have more value than you think

It is common for teachers in other disciplines to view multiple choice questions as the lazy teacher's way of avoiding grading.  In physics, that could hardly be farther from the truth.

Even physics teachers often think of multiple choice questions merely as a useful way of evaluating student understanding broadly and quickly -- after all, it takes a student only about 1-2 minutes per question to respond, and a teacher 1-2 hundredths of a second to grade by machine.  A multiple choice question can be even more valuable.  Some ways to use multiple choice questions creatively:

* I've detailed many times the "test correction," in which students earn back half credit on a multiple choice item they miss by explaining the answer thoroughly.

* I've also explained my typical "clicker exercise," in which teams of two students each have a chance to respond to a multiple choice item on the classroom response system.  The ensuing discussions of each questions can be more valuable than the best-designed homework question.

* Multiple choice questions can be expanded into free response-style homework question with the addition of three words: "Justify Your Answer."  Just today I decided that my class had had enough AP free response review homework.  So I took three of the tougher magnetism questions off of the recently released 2009 AP multiple choice exam, printed them out on a page, and assigned the justifications for homework.

* Even after a question has been assigned and justified, you can develop a further quiz based on the situation presented.  For example, consider a typical multiple choice question in which two railroad carts bounce of each other.  Originally, students may have had to find the amount of mechanical energy dissipated in the collision.  For some reason, that calculation frequencly causes trouble.  So, after I've demanded a thorough justification, I give a quiz -- same question, only this time the carts stick together after collision.  If the student truly understood the concept and calculation on the original problem, the new one should be no trouble.

Condider the multiple choice question below:

A car collides with a mosquito.  Which experiences more acceleration in the collision?

(A) The car, by a factor of about 10⁶

(B) The mosquito, by a factor of about 10⁶

(C) The car, by a factor of about 10¹

(D) The mosquito, by a factor of about 10¹

(E) Both experience the same acceleration.

When correcting this problem, some students will obediently go through the Fnet=ma calculation, estimate the mass of the car to be a million or so times the mass of the mosquito, and correctly answer B.  But not everyone will truly recognize the underlying principle here: This reasoning depends on Newton's Third Law, which demands that the forces experienced by each object in the collision must be the same.

So I'll ask this follow up question on a quiz:

A car collides with a mosquito. The mosquito sticks to the car after the collision.

(a) Which experiences more acceleration during the collision, the mosquito or the car?

(b) Which experiences more impulse during the collision, the mosquito or the car?

(c) Which experiences more force during the collision, the mosquito or the car?

GCJ

Department of the Obvious: Test Corrections Work

Don't know whether you saw the December 2009 edition of The Physics Teacher. This generally excellent magazine has given me untold helpful hints, lab ideas, and physics concepts to think about in the context of teaching high school (and low undergraduate) physics. In fact, I have a co-written article being published in next month's edition, about the USAYPT, the organization that made the mistake of appointing me President. (Check us out at http://www.usaypt.org/!)

This month in TPT, Charles Henderson and Kathleen Harper explain how they use "Assessment Corrections" as a teaching tool. Great idea, obviously. What bugs me about this article is not that I think they "stole" the idea of corrections from me. Of course they didn't. In fact, I "stole" the idea from Haverford professor Lyle Roelofs -- pictured above --  who inflicted test corrections on us in Advanced Quantum Physics in 1994.  He offered half credit back on the test if we corrected out mistakes.  A classmate astutely commented, "Lyle, you know we're going to do the corrections, because without them our grades are lousy, but with them the grades are good.  So even though corrections aren't required, you're insidiously getting us to do them."  Lyle just smiled.

Anyway.  What bugged me about the Henderson and Harper article was the conceit that they were determining, through the use of a scientifically valid theory, that assessment corrections are useful, and that corrections help students learn.  The article is full of phrases like "formative assessment"  and "metacognition."  AARRGH!  Look, readers, I don't care what your "theoretical basis for assessment corrections" is, or whether you even have one.  Does anyone ever ask Roger Federer for the "theoretical basis" for his forehand?  Does anyone ask Albert Pujols for the "theoretical basis" of his swing?  No, these folks just do what works.  They're probably happy to share what they know about what works for them, but what works for them may or may not work for another professional. 

Physics may be a peer reviewed science, but physics teaching is far, far closer to art than science.   Good artists may do things in a similar manner, but they don't need peer-reviewed, buzzword-filled evidence to know they're doing something right.  All anyone -- INCLUDING ME -- can tell you about a physics teaching method is, "it worked for me, it worked for lots of other people, here's how I do it, now try it if you'd like."

Test corrections work for me.  Test corrections apparently work for Mr. Henderson and Ms. Harper, too -- you can read the article for useful examples of how other teachers have made the corrections assignment.  Corrections worked for Lyle Roelofs.  They have worked for a number of attendees at my summer institutes.  They will probably work for you.

Follow-up to multiple choice test corrections

Those of you who have attended my workshops know that, in Jacobs Physics, test corrections are one of the two most important components of the course. Sometimes, though, even the test corrections need correction.

Instead of assigning another round of “correction corrections,” I tend to just give the whole class a quiz when I find consistent misunderstandings. For example, consider the two multiple choice questions below. These were originally AAPT Physics Bowl questions, I believe…

1. A 2 kg object initially moving with a constant velocity is subjected to a force of magnitude F in the direction of motion. A graph of F as a function of time t is shown. What is the increase, if any, in the velocity of the object during the time the force is applied?

(A) 0 m/s

(B) 2.0 m/s

(C) 3.0 m/s

(D) 4.0 m/s

(E) 6.0 m/s

2. A deliveryman moves 10 cartons from the sidewalk, along a 10-meter ramp to a loading dock, which is 1.5 meters above the sidewalk. If each carton has a mass of 25 kg, what is the total work done by the deliveryman on the cartons to move them to the loading dock?

(A) 2500 J

(B) 3750 J

(C) 10 000 J

(D) 25 000 J

(E) 37 500 J

Many students showed an iffy grasp of these two questions on their test corrections. So, I posted to our class folder early last night. I noted that we would take a follow-up quiz today on these problems. I wrote the quiz to address specifically the mistakes that I had repeatedly seen on the first attempt at corrections. Here’s the quiz:

1. (a) What’s wrong with the statement “Work is done both up and to the right in order to move the boxes up the incline?”

(b) What is the direction of the force necessary to carry one box up the incline at constant speed? Justify your answer. Your justification should include a free body diagram.

2. (a) Explain why the average force during the time interval t = 1 s to t = 5 s is NOT 1.0 N.

(b) How do you get impulse from this graph WITHOUT trying to find an average force?

Test - Correction - Recall Quiz

There's only one assignment you can give on which you can be sure of the class's full attention:

A test.

Thus, it's useful to sqeeze every last bit of usefulness you can out of a test. The cycle I try to use on every test all year is test - correction - recall quiz.

Consider the exam I gave last week. One of the questions was #3 from the 2007 AP Physics B exam. I can't post the question itself here, but you can get it at the College Board's archive of AP physics B exam questions. The problem shows a simple circuit and asks for a ranking of resistors in order of their current and voltage. Then, a capacitor replaces a resistor; students are asked to calculate current through the resistors and voltage across the capacitor.

The Test
On the test itself, most of my students get the ranking task and the calculation. The difficulty comes when the capacitor is added. Still, my class averaged 10.9 out of 15 points. That's 73%, or well into the range for an AP score of 5. (That's also a full standard deviation above the national average of 6.0 points. Statistics for many exam questions can be found at the archive page linked above.)

The Correction

In a test correction, students earn back half the points they originally missed by redoing the lettered parts they got wrong. Of course, I sometimes ask an additional question, or change the given values, so that they can't just parrot, they have to think about what they did wrong.

The most common mistake on the voltage ranking task is to assume that the first resistor takes the most voltage simply because it's first in line; or, to assume that the first resistor takes all 12 volts of the battery. Thus, for part (b) I ask the additional question:

First, draw me an example of a circuit in which the FIRST resistor does NOT experience the largest voltage across it. Explain your answer.

Now, justify the ranking you gave to the three voltages.

For the calculations in part (c) and (d), I simply change the R value from 100 ohms to 200 ohms so that they'll have to pay attention when they redo the circuit problem.
The most extensive part of the correction is for part (e), calculating the charge on the capacitor -- because that's the part that most people missed. I use the original question, but I add the following parts:
i. Diagram the new circuit.

ii. What does a capacitor do in a circuit?

iii. Explain how to figure out the voltage across the capacitor.

iv. Now calculate the charge on the capacitor.


The Recall Quiz
You might well be familiar with my "fundamentals quizzes," which test the basic facts that must be memorized before higher-level physics problem solving can happen. The third step in my testing process is to remind my class of the fundamental physcis facts behind each question. The questions on these recall quizzes are a bit too involved to ask on my regular fundamentals quizzes, because I assume familarity with the problem. The recall quizzes usually only ask about the most-missed issues on the original test -- there's no use beating a dead horse. Thus, the the recall quiz dealing with AP Physics B 2007 #3 only discusses the capacitor:

A capacitor is in parallel with R_C in the circuit shown above.

i. Check one and explain briefly: After a long time, the voltage across the capacitor is

☐ Greater than e
☐ Less than e
☐ Equal to e

ii. Rank the current through the three items R_A, R_C, and C from greatest to least, with number 1 being greatest. If two items have the same current, give them the same ranking. Justify your ranking briefly.
____ R_A ____ R_C _____ C

iii. Rank the voltage across the three items R_A, R_C, and C from greatest to least, with number 1 being greatest. If two items have the same voltage, give them the same ranking. Justify your ranking briefly.
____ R_A ____ R_C _____ C

GCJ

Rainwater in a cart – why does the cart slow down?

A multiple choice question on, I think, the 1998 AP B exam, asks about rainwater falling into a moving cart. If the rain falls vertically, does the cart speed up, slow down, or maintain constant speed? And is that because of conservation of momentum or energy? (Note that we're not considering a donkey-pulled cart, just a freely-rolling cart.)

The answer is that the cart slows down due to conservation of momentum. Mechanical energy is not conserved in this situation because much of the kinetic energy of the water dissipates as thermal energy upon splashing in the cart. And since momentum is conserved, the additional mass added by the rainwater causes the cart’s speed to drop in order to maintain the overall mass×velocity.

My students often miss the question on first pass. They’re not really sure what’s conserved and why. “Conservation of energy” is so ingrained in their consciousness – both from a physics and an environmental standpoint – that they nearly automatically choose an answer with the magic phrase.

I asked this question on my first trimester exam back in November. Anyone who missed the question had to write a test correction. Problem was, a lot of folks still made poor arguments. Some thought that energy was conserved. Some thought the cart would speed up because of the water’s initial speed – they didn’t separate vertical from horizontal momentum. I knew many of these folks were ripe for making the same mistake again.

I’m not averse to hammering an idea over and over. That’s the main idea of the “Less is More” teaching philosophy – you don’t have to assign that much work, but you must hold students thoroughly accountable for understanding everything that is assigned. So, I told the class to expect a quiz based on this question. Below is the quiz I assigned...



1. An open cart on a level surface is rolling without frictional loss through a vertical downpour of rain. As the cart rolls, an appreciable amount of rainwater accumulates in the cart. Thus, the cart and water can be treated as if they are colliding.

(a) Which of the following is conserved in this collision? Circle all that apply.

Kinetic energy

momentum

velocity

acceleration


(b) What is the horizontal velocity of the rainwater before it lands in the cart?


(c) What will happen to the speed of the cart? Explain in one or two sentences.

GCJ

(Top photo from donchesnut.com.)

Preparing for the Trimester Exam

It’s nearly trimester exam time! In AP physics, my 2-hour trimester exam will consist of 23 multiple choice questions in 30 minutes, followed by a full-length 90-minute free response section consisting of authentic AP exam questions. The general physics exam is an eight question free response test designed to be 2 hours long (but I allow three hours for everyone). The exams will, of course, cover everything we have discussed all year.

(Students always wonder if the exam will be cumulative… why wouldn’t it be? Why did I bother to teach back in October if you’re just allowed to forget what I taught you? Are you saying that the material I taught isn’t worth remembering? And in AP physics, the May AP exam is cumulative, so I would be doing you a disservice if every test were not cumulative. Now quit asking silly questions.)

With the exam upcoming, prepare for a series of posts about exams. Today, I discuss exam “review.”

I refuse to enter into conversations about what specifically will be on the exam, or to run a “review session.” If I’ve been doing my job, and if students have been paying attention, then exam preparation should be nothing special. Daily homework, quizzes, and discussion are exam review.

Woodberry holds a “consultation day” before each exam period, during which teachers hold court in their classrooms, and students can visit whomever they want to ask questions. I don’t want consultation day to degenerate into “so, want to tell me what’s on the exam?” But I want to encourage my students to stop by. I dangle bait for my class in the form of extra credit.

Yesterday I distributed a sheet with 20-30 multiple choice questions (a different sheet to AP physics and general physics, of course). The extra credit assignment is to do these questions if they were a test – no books, notes, or collaboration. Then, on Sunday’s consultation day, they can come to my classroom to scan their answer sheet. Showing up on Sunday is the first requirement for extra credit.

The second requirement is corrections. For each question they missed, they must explain how to get the correct answer. Corrections must be done on the sheet displayed to the right. (I hope the quality is good enough to read... if you want a ms word version of the sheet, email me.) The standard of evidence for the correction is high – if they don’t thoroughly convince me that they understand the problem, they don’t get credit. The corrections are not due until next Thursday’s exam; but, since collaboration is allowed and encouraged on the corrections, most students stick around on consultation day because so many folks are around to discuss the problems and to help each other out.

Students appreciate my approach, and not just because they get extra credit. Think about how your students will prepare for a physics exam. OCD-style students might think that they must study for hours… and those hours are too often unproductive. Lazy students might not normally prepare at all. But the extra credit multiple choice assignment helps both of these student phenotypes. The exercise helps the OCD folks focus their studying, so that either (a) the questions they missed inform them about what topics need special attention, or (b) they feel like they’ve studied, and so they don’t waste any more time preparing for the exam. As for the lazy folks, the extra credit might well lure them into doing something when they might otherwise have done nothing.

If nothing else, just getting my students physically in my and each others’ presence is a productive exercise, because conversations invariable turn to physics. Extra credit can be an amazing attractor. (I note that food often works as well. First trimester consultation day is well known as Nacho Day in the physics classroom. I sometimes wonder whether the ample supply of official Nacho Man Nachos or the extra credit does a better job of bringing in the sheaves.)

GCJ

(Picture at the top courtesy of falconsscience.wordpress.com.)