Multiple options for a Physics C multiple choice question about capacitors

A parallel-plate capacitor is filled with air. Each plate has area 10 cm². The separation between plates is 2.0 cm. The top place stores +200 nC of charge, while the bottom plate stores -200 nC of charge.

Above is just the stem of the question I'm writing for the upcoming edition of 5 Steps to a 5: AP Physics C.  Consider it a "goal-less" problem for a moment.  What could be figured out?

The first and most obvious answer is to calculate the capacitance of the capacitor using C = εA/d.  And yes, that is a calculation fraught with pitfalls, as the plate area and separation have to be converted to SI units.*  But carrying out this sort of calculation is not the point of introductory physics.  If you really want a correct calculation, don't ask a first-year high school student with a calculator - use Wolfram Alpha.  More importantly, I don't expect even strong students to be able to calculate the right answer in about a minute.

* Or I suppose you could convert epsilon-naught to F/cm.  Whatever floats your boat.

Better, more authentic questions might involve semi-quantitative reasoning.  What happens to the capacitance when I double the plate area? Plate separation?  Charge storage?  Using C = εA/d, you can see quickly that the capacitance doubles, halves, or remains the same because the relevant equation is in the numerator, denominator, and not involved, respectively. 

A further step could involve graphs - what would an experimental graph of capacitance vs. plate area, plate separation, or charge storage look like?  Linear, inverse, and horizontal, respectively.

Or, we could go one step beyond single-equation reasoning to ask how the electric potential across the capacitor changes depending on plate area, plate separation, or charge storage.  This question involves combining two fundamental relationships: C = εA/d for the capacitance, and Q = CV for the voltage.  In this case doubling plate area doubles capacitance, but halves voltage; doubling plate separation halves capacitance and doubles voltage; doubling charge storage doesn't change capacitance but doubles voltage.

(And you could even ask about the electric field inside the capacitor, which is uniform and equal to V/d.  Or the field or potential at several positions inside the capacitor.  Or...)

A fun exercise might be to present your class with the stem above and some whiteboards with, say, 10 minutes remaining in class.  Then tomorrow's daily quiz would include two multiple choice questions based on this stem, with only two minutes to answer.  I'll bet you get (a) good conversation amongst the students, (b) mostly right answers on the quiz, and (c) everyone finishing the quiz with lots of time to spare.

Pivot Interactives - these videos are so worth the money.

Screenshot from Pivot Interactives

As you might have noticed, my all-time favorite internet physics resource Direct Measurement Videos has migrated.

Vernier now is selling access to "Pivot Interactives", for $150 per year or $5 per student, whichever is greater.  On one hand, I wish the National Science Foundation had stepped up with a seven-figure grant for primary DMV creator Peter Bohacek so that he could continue to provide these resources free to physics teachers.  I mean, I can list probably 10² NSF grants that flush my tax money down the toilet. Nevertheless, I'm happy that Vernier has seen the extraordinary value in these exercises to keep them alive.  

The question for the physics teaching community is, then, do we spend the couple hundred beans to get access to Vernier's new site?  For me, the answer is definitely "yes."  

I've made it my official teaching goal this year to replace as many textbook-style homework problems as possible with Pivot exercises.  Since Peter and company have been hard at work adding to their video library, I'm finding this goal easy going.

Take, for example, the car-around-a-traffic-circle problem.  I've always started my circular motion unit there, as it's a situation all my students have experienced.  I ask, how fast can the car go around the curve?  We discover that the maximum speed depends only on the curve's radius, g, and the coefficient of (static) friction. 

Great.  But this is an abstract "imagine if" problem.  I can't take my students to a traffic circle for experimentation - the nearest one is 20 miles away, and is too busy for fooling around, anyway.  All I can do is suggest that the yellow suggested maximum speed signs don't include a mass variable - they say "max 25 mph", not "max speed in mph is 0.025 times the mass of your car in kg."  Interesting... but not experimental.

Well, look at the screenshot at the top of the post.  Peter took his drone to a traffic circle.  He drove the gray car around the circle at a speed that was always on the verge of slipping.  When he imported the video, he included tools to find angles around the circle, the radius of the circle, and a frame-by-frame timer.  I can't do this experiment, but Peter can. And did.

So for a homework problem later in the circular motion unit, I link the class to this Pivot Interactives video.  The site allows me to customize the assignment - the default is quite good, but I can add or eliminate questions and guidance.  For me, I like a clean prompt like "Determine the (maximum) coefficient of static friction between the car's tires and the ground."  The site allows students to input their solution and reasoning directly in the space provided; you can then scroll simply from one response to the net, awarding points if you'd like.  I prefer to have students answer on paper, but that feature seems to work as well (paper not provided by Vernier).

There's so much more that Pivot does.  I prefer the simple open-ended "determine this parameter" exercises.  But Pivot also has some modeling exercises, providing an easy graphing interface that allows students to make and linearize plots.  These multi-layered videos allow you to change multiple parameters.  The prompts guide students through what essentially is a complete lab exercise which you might never be able to do in your own classroom; or, a lab exercise you don't have the time to do in your classroom but can assign for homework.  I know that Vernier has gotten some serious physics teaching experts, including Kelly O'Shea, writing these exercises.  

And Peter is adding videos every time I look.  

Look, I know it's disappointing to have to pay for what had been available for free.  And I generally don't recommend paying for physics content on the internet.  Nevertheless.  

Vernier has hired the varsity for this project.  Everything I see on the site is something that makes me say "wow."  Pivot cannot replace a physics teacher doing active lab work in the classroom, because nothing can.  Using Pivot gets students as close as I believe it is possible to come to an online laboratory experience.  I highly recommend.

GCJ

(Note that Peter and Vernier have not paid me in any way for this endorsement.)

Umpires, and Training Students to Grade Each Others' Daily Quizzes

The most effective tool I've ever discovered for 9th grade conceptual physics is not the daily quiz - it's the grading of the daily quiz.  Students trade papers, pick up the red pens I provide, and mark right or wrong as I go over each answer.  

The purpose here is pedagogical, not logistical.  I'm perfectly capable of rephrasing these questions as multiple choice for ease of grading; and it's not like I don't have time to grade these simple quizzes, anyway.  No, the reason we trade and grade daily quizzes is that students pay attention to each question threefold:  once when they take the quiz, once to figure out how to grade the quiz in front of them, and once to think about whether they themselves got each question right. 

It's essential, though, to establish an appropriate tone and ground rules in order for trade-and-grade to be successful.  Here are some principles that my class learns that make the daily quizzes work.

(1) We are a team.  Daily quiz performance is just like a football team's conditioning sprints.  Teams have fast and slow players.  Good teammates respect the effort and talent of those who are the fastest; good teammates encourage and respect those who are slower, too.  

The team atmosphere must be established from the first day of school.  The highest form of sin in my class is to denigrate  a classmate for a wrong answer.  Even negative body language directed at another student - a teammate - is unacceptable.  

Because we are a team, we can depersonalize the grading process.  A grader is acting as an umpire for the team.  Umpires don't call people out because they dislike the players, they call safe or out honestly based on their judgment, and on a shared respect for the game.  A good team doesn't want a biased umpire, even if the bias is in their favor - they want to win or lose with integrity.  Those who play sports know that a tainted win is nothing to be proud of. 

(2) Don't look at or talk to the person grading your paper.  You have enough to focus on with the paper you are grading.  Trust your classmate to evaluate your work appropriately.  No one respects the player who continually tells the umpires how to do their job.

(3) Don't talk to the person whose paper you're grading.  Imagine an umpire who, presented with a close pitch, asks the catcher: "Hey, did you mean for that pitch to be over the outside corner?"  The batter would go berserk!  So students should never ask "hey, did you mean this line to be straight?"  If you can't tell, mark it wrong.  The burden is on the student writing the quiz to communicate clearly.  

(In my AP classes, I ask regularly: "Are you allowed to travel with me to Kansas City in order to accompany your exam from reader to reader explaining what you meant?  No?  So don't get in that habit now.")

(4) Ask for help on borderline calls.  The class is instructed to raise their hands and read me any answers they're not sure about.  Not generalities like "What if the student kinda hinted at the answer?"  I ask the student to read, verbatim, what is on the page.  I then make a call one way or the other, and we move on - just like an umpire.  

And if the student who wrote the answer tries to chime in what he meant, I politely say, "Since the answer needed clarification, it must be incomplete.  Please mark it wrong."

(5) Be transparent.  Just as a league doesn't assign the same umpire to the same team too many times, be sure you're mixing up occasionally who gets whose paper.  Have the graders write "graded by" and their name at the top of the page, and make it clear that accurate grading is a skill that you are assessing.  Ask students to place their own pencils and pens on the floor while grading goes on, so that the only writing utensil available is the red pen - that way, it's not possible to change an answer dishonestly.  

If you suspect someone is grading inappropriately to benefit (or hurt) a classmate, don't ignore it, but don't get preachy!  I've had small issues early on - students giving credit when they shouldn't, hoping to curry favor with a classmate.  Those were easily and definitively solved when I asked publicly but earnestly, "Luke, can you please show me why you marked this answer correct?  I think it's clearly wrong, but am I missing something?"  That's all it takes.

(6) Make instant replay available.  We have a standing rule - students may never, ever, argue or discuss with the person who graded their paper.  But they may circle in red a question they think was graded inappropriately, signaling me to take a look.  They can't talk to me, they can't tell me what they meant, they can just circle the question.  And, if the grading was indeed incorrect, I will change the score and let both student and grader know what happened and why.

Early on, usually I get a bunch of students circling their answers, hoping to argue for a better score.  That ends quickly when they can't talk to me to lawyer up.  If someone can't stop arguing and circling obviously wrong answers, I tell that student that I'll take off an additional point for each answer they challenge incorrectly.  (Kind of like how in the NFL a coach loses a time out if he incorrectly challenges a ref's call.)

Thing is, students are more careful graders than I am.  The process is quite fair - even moreso than if I combed through all the quizzes.

The most important point to keep in mind, though, is that the grading process is not even about fairness.  The goals are twofold - to provide an arena in which students are paying attention to physics facts, but also to establish that physics has right or wrong answers.  

Within a week my class has stopped arguing with me about grades.  They know what my answer will be - it is the student's responsibility to communicate correct physics the first time, in writing.  They've seen that the entire class is held to the same standard, that no one can sycophantically beg for points after the fact.