Laboratory quiz question: pressure in a static column

A primary laboratory skill, one that is frequently tested on the AP exams, is determination of the physical meaning of the slope and intercept of a linear graph.  My own approach to such a question is to solve the relevant equation for the vertical axis of the graph, then to identify the variable representing the horizontal axis.  Anything multiplying this variable is the slope of the best-fit line; anything added to this term is the y-intercept.  We religiously go through this process of identifying the slope and intercept of a straight-line graph in every laboratory activity.

However, just doing laboratory work isn't enough to develop this skill.  In a 90 minute lab period or a lab report, a large subset of students will parrot their friends' answers or my suggestions without sufficient understanding.

So then, how do I check for "sufficient" understanding?  I give quiz and test questions that ask directly about the physical meaning of graphs that the class hasn't seen before.  For example, a recent "justify your answer" question showed a graph of weight on the vertical axis, and mass on the horizontal; what is the physical meaning of the slope of that graph?

It was instructive to read the justifications.  Most folks got that the slope is g, the gravitational field.  The stronger students recognized the relevant equation weight = mg; since weight is on the vertical and mass on the horizontal, whatever is multiplying m must be the slope.

The weaker students, though, got the correct answer reasoning from the units of the axes.  The vertical axis, they said, "was" newtons.  The horizontal axis "was" kilograms.  Since we've shown that g has units N/kg, the slope must be g.

I've got to force these weaker students to get away from the crutch of using units to determine a slope's meaning.  While such an approach is better than nothing, often the units of the slope won't obviously match any known quantity; or, a factor of 1/2 or 2π will be missed.  It's not like the method I'm proposing (of first writing the relevant equation) is too difficult for anyone.*

* The correct method does require remembering or looking up the correct equation, though, which is sometimes an obstacle; but that's a separate issue.

Below is a quiz that will help practice the skill of identifying the physical meaning of a slope.  Note that, by this point in the year, if we just graphed GAUGE pressure vs. depth, most of my class would have little trouble seeing that the slope is ρg.  The addition of the Po term in the equation for pressure in a static column causes difficulty.

1.    In the laboratory, you are given a tall graduated cylinder full of fluid, along with a pressure probe which reads absolute pressure.  You submerge the pressure probe in the fluid and record the reading in the probe P at various depths d below the surface.  The pressure at the surface is 1.01 x 10⁵ Pa.

A graph is made of P on the vertical axis and d on the horizontal axis.

(a)    Is the graph linear, or curved? 

o  Linear

o  Curved

(b)   If the graph is linear, explain how the density of the fluid r could be determined from the best-fit line.  If the graph is curved, explain what quantities could be graphed in order to produce a linear graph from which the fluid density r could be determined.

Multiple Choice quiz: two-body problem in an elevator

Diagram for today's problem, modified from
something in (I think) Serway & Vuille

A couple of nights ago, I assigned a two-body problem in an elevator, from (I think) Serway & Vuille.  Two blocks were hanging from an elevator as shown in the picture; the acceleration in the original problem was upward.  On the homework, I asked (among other things):

• Draw a free body diagram for each object.
• Is the tension in the lower rope greater than, less than, or equal to 35 N?
• Calculate the tension in each rope.
• The ropes have a breaking tension of 85 N.  Calculate the maximum acceleration that will cause a rope to break.
• When a rope is observed to break, explain how the elevator was moving.

This problem is one of the best at separating those who are following an appropriate physics problem solving procedure from those who are just trying to plug numbers into some random equation.  The students who used the free body diagrams to write (up forces) - (down forces) = ma got the right answers, and got them quickly. 

On the other hand, the students who didn't carefully write the equations were confused for most of an hour, got the final answers correct because they asked friends for help, but usually earned little credit -- if after collaboration they just wrote "T = ma + 35 N, so T = 40 N" I marked the answer wrong.  Why?  Because I saw no evidence of how they got to that equation, other than listening to a friend without understanding.  Would an English teacher give credit for a one-sentence essay, even if the one sentence is spot-on in its conclusion?  Of course not.  So why on homework should I reward the correct numerical answer when it was essentially derived through magic?

I invited in for extra help the students who didn't follow the correct method.  They now feel much more confident about two-body problems, because they see that all they have to do is write the correct Newton's Second Law equations from the free body diagrams.  But it's still worth a follow up quiz -- either I build significant confidence, or I discover further misconceptions.

Below is today's three-question quiz that I'll give at the opening of class.  (It refers to the diagram above, in which the acceleration is DOWNWARD.  Yeah, I switched the direction of acceleration for the quiz.)  The "distractor" answers in the second question quote some students verbatim.  

Two 3.5 kg blocks hang from ropes in an elevator, as shown above.  The acceleration of the elevator is 1.6 m/s², downward.  While the elevator has this acceleration, the tension in the bottom rope is 29 N.

19. Which of the following best describes how the elevator’s speed is changing?

(A) The elevator is speeding up.

(B)  The elevator is slowing down.

(C)  The elevator is moving at constant speed.

(D) Whether the elevator is speeding up or slowing down cannot be determined.

  

20. Which of the following describes the meaning of an acceleration of 1.60 m/s²?

(A) The elevator gains or loses 1.6 meters per second of speed each second

(B)  The elevator gains or loses 1.6 meters each second

(C)  The elevator travels 1.6 more or fewer meters each second

(D) The elevator travels 1.6 m/s² more or less each second

(E)  The elevator is either speeding up or slowing down by 1.6 meters for every second squared.

  

21.  Now the magnitude of the elevator’s acceleration is doubled to 3.2 m/s², still directed downward.  What is the tension in the bottom rope now?

(A) 41 N

(B)  35 N

(C)  32 N

(D) 24 N

(E)  0 N (i.e. the rope goes slack)

It is just fine to give a quiz based on the homework that's due today

Last month, my colleagues who teach 9th grade conceptual physics started using nearly-daily quizzes in their classes.  Great move.  The students initially rebelled a bit, but they started preparing better for class each day when they saw they would be held accountable for the course material.  (Daily quizzes are best established at the beginning of the year so as to avoid the resistance to change.  But these work any time.)

The general approach was to have an in-class discussion, demonstration, or presentation; assign some problems for homework; and take a brief multiple-choice quiz the next day about the previous day's class and the homework.

A major student complaint sounded something like "We did the homework, but then we had the quiz before you graded the homework and gave it back.  How were we supposed to know whether we were doing the homework right?  It's not fair that we have to take the quiz before we get the homework back."

Well, one of the teachers accepted that argument, and postponed his quizzes until at least a day after he had returned the homework.  I ask, will the quiz scores be substantially improved by this postponement?  I say, "no."

There is nothing wrong with giving a quiz based on homework problems that are due the same day. 

A principal battle that I fight about homework is to establish a "correctness" rather than "completion" mindset.  A decade of schooling has convenced my students that homework is a mindless chore that must be done for the sake of doing it.  I think of homework problems as practice, as ways to remember and reinforce problem solving methods learned in class.  Without active participation in the problem solving process, homework is useless.  Whether the answer to a homework problem is right or wrong is immaterial -- what matters is the student's engagement.

A quiz can check that engagement.  Did the student figure out the most important step in the problem?  Can the student solve the problem again with the numbers changed?  Can the student explain why the answer makes sense, or doesn't make sense?  If not, then the homework problem didn't have the desired effect.

This follow-up quiz must be given right away.  By waiting, you're telling the class that the homework isn't really that important... it's okay to give a half-arsed effort because I'll just tell you the answer if you wait a day.  And, if the student didn't check his answers with friends or ask the teacher about a difficult idea on the night he was supposed to do the problem, what is the likelihood that he will follow up after the problems are graded?  What are you doing every day in class in the meantime, while you wait one night to grade the problems, and another night for students (in principle) to look at their graded work and study?  You can move along to new or more complex material immediately if you insist that students pay attention to their problems the very first time they're assigned.  If there's substantial misunderstanding, the quiz can provoke a good class discussion about the problem in a way that "Anyone have any questions?" cannot.

Homework is not merely busywork.  Regular grading of the homework, along with quizzes and other creative accountability methods, promote the appropriate attention to detail on daily work.  Without such attention, you might as well not even assign problems for all the good they'll do.  :-)