Using a lookup table for a conceptual physics lab

In conceptual physics, I want to do an experiment with a 60 Hz frequency generator and waves on a string.  The setup is shown in the picture to the right: the hanging mass is varied, varying the tension in the string and thus the wave speed and the wavelength.  We move the generator left and right until the standing waves are clear; then we measure the wavelength with a ruler.

I want to plot wave speed vs. wavelength, so that the slope of the straight-line graph will be the 60 Hz frequency.  

Problem is, I don't have an instrument to measure wave speed on the string.  In AP physics, I'd just show the students the equation 

and let them figure out the wave speed for themselves.

Well, this is 9th grade conceptual physics. Most of my students either have not completed algebra 1; most wouldn't know a square root if it bit them on the arse.*  I can not expect my class to be able to plug into this formula.  But I still need them to be able to graph a wave speed, knowing only the mass of the hanging mass.

*That happened to me once.

One thought I had was to create a quick app to make the calculation:  On an iphone or ipad, it could ask "What's the hanging mass?"  Then, using the linear mass density value I measured for the string before class, I could program the app* to spit out "the wave speed is 3000 cm/s."  Yes, I know I could do something like this in excel or on wolfram alpha, perhaps, but anything beyond a mass input in grams followed by a speed output in cm/s is too complicated for me.

*That is, if I knew how to program ios apps.  Hey, now, if I had access to a 1985 version of applesoft basic, I'd pwn all of ya in a programming contest.  And I'd have that "app" ready in five minutes.

Without the ability to make the program I want, I realized that I could go all 1940s and just create a lookup table.  Excel will do the calculation... in fact, I learned how to get excel to round the speeds to two significant figures.  So I put mass values from 5 g to 300 g in one column.  I made excel use the equation above to calculate the wave speed in units of cm/s.  

Then I just printed the two columns.  I'll hand this out to each lab group.  I think it's totally reasonable to expect freshmen to use this table to relate the hanging mass to the wave speed... then to graph wave speed on the vertical, and the measured wavelength on the horizontal.

GCJ

Graph paper link, and setting up a graph for conceptual physics

Above picture from
free online graph paper

Wednesday is our first day of class.  New students arrived here on Sunday, so we're in the midst of two days of orientation and partying.*  I've spent the past week making final preparations for teaching 9th grade conceptual physics -- no wonder I haven't posted much.

*Unfortunately, the skee-ball machine is delayed, so probably won't be at my house for my advisee group's get-together tonight.  I'll have to use it for physics classes, instead.

In the first class session, we will set up a mirror and ray box.  Students will take turns using a protractor to measure angles of incidence and reflection.  Each student will make a graph of angle of incidence on the horizontal, and angle of reflection on the vertical.  The point here is to establish context for reflection and refraction, and to be sure we all know how to use a protractor before the first true laboratory exercise on refraction.

In a 12th grade course, I'd have students figure out how to scale the graph.  Dealing with a range of 0-90 degrees, when the graph paper has seven major ticks and 70 minor ticks on the horizontal axis, is a serious skill that I try to teach, and which takes significant patience and practice.  However, for 9th graders I'm happy if they can plot points accurately at all.

So I've gone to this link for free online graph paper.  This page is particularly nice because you can choose from common templates (e.g. 1 inch major with 1/10 inch minor gridlines), OR you can customize the weight of the line, number of lines per inch, etc.

For that first class session, I can bring the class to understand that we need to scale each axis from 0-90 degrees.  Making that scale will be easy as pi, because I've printed out customized paper that has nine major gridlines across the page, and five minor gridlines per major gridline.  The class can see quickly that each major line will represent 10 degrees, so each minor line will represent two degrees.  And the graph will take up the whole page.

GCJ

Conceptual physics lab: refraction without using Snell's law

We're starting the year in conceptual physics with geometric optics.  On the very first day, we'll set up a mirror and a ray box.  The class together will measure a bunch of angles of incidence and reflection, and we will together graph θr vs. θi.  We'll set up both axes from zero to ninety degrees, and show that we get a straight line.  We'll show from the graph that, say, doubling the incident angle also doubles the reflected angle.  No slopes, no calculations, just graphical experimental evidence for the law of reflection.

The next week, we'll use the ray box to shoot light into a plastic block.  We'll set up the same axes, and this time graph refracted vs. incident angle for light refracting in the block.

Now, in AP physics, we'd make a new graph of sin θi vs. sin θr; the slope of that line would be the index of refraction of the plastic.  But this year I'm not teaching AP physics.  This is ninth grade conceptual physics.  So what are we doing?  And why are we doing it?

Well, I will certainly spend about five minutes asking students to predict what the θr vs. θi graph might look like.  I'd be happy if anyone recognizes that the data won't ever approach 90 degrees on the refracted angle axis.  But prediction is not really the essential issue.  My goals for this experiment are:

(1) Establish norms for data collection.  This is a simple experiment.  I showed my nine-year-old how to do it just once, and he collected data without my help.  So there's no reason we shouldn't be able to acquire data quickly, accurately, without asking a bazillion questions.  I will go around the class and hurry them along, teaching them not to second-guess data, not to be overly precise... and importantly, to collect data rather than argue with the partner.  It's also a good experiment with which to establish the rule that since no one is leaving early or doing work for another class, we might as well take as many data points as humanly possible.

(2) Practice graphing data as it is collected.  I recognize that many freshmen will struggle with a skill as simple as graphing data by hand.  So I use this straightforward experiment with axes scaled identically on the vertical and horizontal to start practicing.  Each partnership will graph data as it is acquired, one graph per group; then I'll show them where they need to collect more data in order to define the shape.  When they're done collecting, I'll give them a homework sheet, part of which will ask them to regraph their data on axes that I've prepared.  The homework is one graph per person.  This way, a partnership can split duties efficiently during lab, but nevertheless everyone will have to physically make a graph.

(3) Make a first stab at interpreting graphs.  A novice physics student might think it "stands to reason" that since the angle of incidence vs. reflection was a straight line, the angle of incidence vs. refraction should also be straight.  That's not the case.  One of the homework questions will ask whether doubling the angle of incidence doubles the angle of refraction as well... just looking at the graph shows that the angle of refraction does not quite double in this case* -- no use of Snell's Law is required at all.

*Okay, for small enough angles, doubling θi does in fact double θr as well.  If someone is actually astute enough to point out the small-large angle difference, I'll be pleased, but I'm not expecting that level of analysis.

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...

GCJ

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.  

Experiment: density of mystery fluid, and the audience for a lab writeup

Materials for the "density of unknown liquid" experiment

Fluid mechanics first entered the AP physics B course description in 2002.  That year, the laboratory question (#6, I don't have a legit link, but it's easy enough to look up) asked students to determine the density of an unknown liquid by submerging a mass on a spring into that liquid.  

My own classes did a version of that problem on their trimester exam.  The ones who got it wrong -- usually by conflating the density of the liquid with the density of the submerged mass -- did an exam correction on which they described the procedure correctly.  

And then this week, I had everyone actually, honest to goodness, do the experiment for themselves.  I've often suggested that the AP exams since 1996 provide a wonderful laboratory guide, if you can improvise a bit.  Pick a lab-based question, set it up with whatever equipment you have lying around, and there you have a college-level physics laboratory exercise.  I took my own advice and tried out this new experiment.

I gave each group a beaker of fluid and a mass.  (You can see the cubical masses in the picture -- they all are made of different materials.  I just dug them up in an old storeroom.  I have no idea where they came from.)  The groups were encouraged to pick the spring of their choice.  They could use any other equipment they wanted, including a balance scale; the only action I forbade was to directly measure the mass of the fluid in the beaker.

Now, most of my experiments, and many AP lab questions, call for a linear graph, a best-fit line to copious data on the graph, and interpretation of the physical meaning of the line's slope and intercept.  This particular experiment doesn't lend itself well to a graph, at least not the way it's presented on the 2002 exam.  So I came up with an alternative approach to the writeup.

I used a true mystery liquid rather than water.  I won't reveal what I used (because my students occasionally read this, and the writeup isn't due 'til the end of the week), but I don't even know its density right now.  We're going honestly double-blind here.

I'm asking each partnership to write up one typed page describing their results.  I give no specific instruction other than to imagine that they've been hired to determine this mystery density, and that their financial well-being depends on the quality and accuracy of their work.  They get one shot to impress their potential client with their writeup.

That "client" is Peter, the captain of our USIYPT research physics tournament team, and the rest of his class.  I will collect the typed pages with no names, and hand them to Peter and the research students.  They will rank the papers from best to not-so-best.*  I'll assign grades based on the rankings, and give a prize to the winners.

*Donald Trump would say "worst."  Why is it that teachers tend to get in trouble for such language, while Mr. Trump is lauded for his bluntness?

Sean measures the extension of the spring

Sure, this is a nice cutesy little game.  But there's a real message here.  Too often when students are asked to describe the results of an experiment, they use stilted, overly-formal language that stifles meaning in favor of big-arse words.*  I want them instead to write informally for an audience at their own level of physics.  A major obstacle to improved writing is the faceless audience.  High school students have never published anything; their understanding of a paper's audience is poor even in English class, let alone in a subject where they struggle both with the content *and* the writing skills.

*  "In this experiment, the experimentors carefully and consistently used a decimal-labeled wooden shaft to record precisely the extent to which the PASCO brand spring was extensively extended.  We ensured safety by wearing goggles and grounded electrical outlets." 

So, I put a face to the audience:  Peter.  Everyone knows Peter.  They talk to him in normal language.  They are not in fearful awe of him, but they are all quite clear that he and the research team have no use for incorrect physics.  (I wonder where they got that from...)

I've never done this particular exercise; but I have had students write with a named fellow student as their audience.  Their writing doesn't become perfect, but some of the filler gets filtered out.  And so we focus on the physics... which is what I want, anyway.

Describing a procedure concisely

From Homestarrunner.com.  Don't get the reference?  Check out
"English Paper" at the homestarrunner wiki.

Much of the physics teaching world, and I, have leapt away from multi-page, badly written "formal lab reports."  Instead, it's typical now for students to be asked several directed questions about an experiment.  Formal lab reports are only useful if the instructor takes the time and energy to truly teach the scientific writing process -- that means grading drafts, participating in writing conferences, paying as much attention to style and language as to results.  I decided long ago that the benefits of the formal lab were in no way worth the costs.

That said, I still do teach a few writing skills.  Particularly, my class learns how to describe an experimental procedure while defining relevant variables.  AP exams, as well as my Honors Physics exam and even the Regents exam, occasionally ask for a description of an experiment.  Your class needs some minimal tutelage so that these questions become easy rather than time- and stress- consuming.

Describe the procedure you used to measure T, q, and any other relevant parameters.

This is a question I ask after our first experiment, phrased identically to the style of an AP question.  I explain to the class that they should use no more than three sentences, telling me in prose what they measured and how they measured it.  I expect an answer such as:

We kept a cart in place on an inclined track using a string held parallel to the track.  The tension in the string, T, was measured with a spring scale tied to the string.  The angle of the incline from the horizontal, q, was measured with the "clinometer" iphone app.

Note how everything, including definitions of variables, is included as part of the prose.  Some folks want to just give a list of variables* ; they lose considerable points in my attempt to get them to write sentences.  

* "T:  measured with a scale"

But consider what I often see at the AP reading, or after my class's first experiment:

We came into lab today in order to find the tension in a string when we hold a cart at many different angles.  First we found are partners; I worked with Joe.  Then we gathered our materials:  A 250 g cart, some string, scissors, an aluminum track, and a spring scale.  Goggles should be worn, as always in the laboratory.  We tied the string to the cart, being careful that the string could be held parallel to the track.  Joe tied a sailor's knot so that we could attach the spring scale to the cart.  I downloaded an app from the itunes library that shows an angle.  Now, with all materials in place, we could begin the experiment.  Joe carefully read the scale to the nearest 0.2 N, and he wrote down both readings in his notebook.  Finally, I graphed the data on a graph with a pencil, which I forgot to include in my materials list above.

[Is it considered satire if it's true?]

My purpose here is not* to make fun of a student who would write the above passage.  The question is, how do we get this student to write a proper, brief, clear, 3-sentence procedure?

*primarily

I've had some success reminding the class of their audience.  They are not writing a "how to" manual, they're not writing for their English teacher who is ignorant of laboratory methods; the audience for a laboratory procedure is OTHER SCIENTISTS.  I make the audience even more concrete by identifying a recent alumnus: "Peter Chen, whom youall know took my class last year, should be able to figure out what you measured, and how you measured it."  They see with minimal prodding that Peter doesn't need to be told to gather materials.  Peter doesn't need a silly safety lecture.  Peter doesn't need to be told to record data carefully -- we take all these basic "skills" for granted, because we are scientists.  

The other useful reminder is about time.  A procedural lab question like this might be worth three points on an AP exam.  At the going rate of one point per minute, the test expects a few minutes of work -- no more. If you spend a lifetime writing multiple paragraphs, even if such paragraphs are pulitzer-prize worthy, you will earn... three points.  And you will NOT earn the other seven points available in this problem, because the bell has rung, the sun has set, and the exam is over.

It's important to model the correct writing style for the class at some point.  This is a different kind of writing than their English teacher has required -- after all, I used no drama, no interesting transitions, nothing about my feelings.  They have to see for themselves that it is okay to be short and "boring."  

And finally, perfect practice makes perfect.  Some folks will ignore all your advice until you take off points; then their next writeup will be perfect.  Others will need a figurative bashing over the head.  But with patient, persistent work, you can get your students comfortable with describing experimental procedures.

GCJ

Hanging Mass on an Incline lab

Mr. Johnson shows off his hanging mass setup

In my general physics labs, we follow a familiar formula each week.:

(1) Students collect data, constructing a linear graph as they go.

(2) They take the slope of a best-fit line using far-separated points on the line that are not data points, including units on the slope.

(3) They use a relevant equation to relate the physical meaning of this slope to a measurable physical quantity.

That's it.  I keep things as simple as possible so as to work on just these three skills.  The trick is, of course, finding straightforward yet interesting experiments which lend themselves to this approach.  Oh, and these experiments must stick within the topic areas in general- (approximately Regents-) level physics.

The procedure is essentially the same each lab day.  I demonstrate a method of data collection.  I write on the board the graph that students are to make.  Students collect and graph their own data in groups of two.  Once I've approved a group's graph, then I hand them a two-page homework assignment for them to work on either the rest of the lab period, or that night if they need more time -- no one leaves early.

Ideally, I'm designing an experiment such that I know what the value of each group's slope should be, but the students do not.  This is a bastardized version of the "recurrent lab" as described by Mikhail Agrest.  Students earn credit either for predicting something using their slope, or for matching their slope to an independent measurement.

So, Greg, give us an example of such a laboratory activity.  Sure -- see the picture.  I have students set up an inclined track, on which they place a Pasco cart.  The cart is connected over a pulley to a hanging mass.    The hanging mass is adjusted until the system hangs in equilibrium.  The angle of the incline is what they are eventually going to predict; I surreptitiously come around during the period to measure each incline with my iPad clinometer app.

We graph the mass of the hanging stuff on the vertical, with the mass of the Pasco cart on the horizontal.  Some mass is added to the Pasco cart, the hanging mass is adjusted to equilibrium, and another data point goes on the graph.  Rinse and repeat for an easily obtained straight-line graph in about 30-60 minutes (including setup and cleanup).

The homework assignment based on this lab activity is available here.  In sum, students take the slope, and then are guided to identify the slope as the sine of the incline's angle.  Most groups easily match my measurement.  

This is one of my better lab exercises, because (a) it fits the formula we've been using all year with no deviation, (b) it allows for accurate prediction of a measurable but initially unknown quantity, and (c) it reinforces the content in the problem solving portion of the course.  Try it... post comments or questions.

GCJ

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.

What if my force vs. length graph for a spring is weird for small displacements?

Tim and Andy measuring the force applied by a spring

I think every physics class in the known universe does the F vs. x experiment for a spring:  The force on a spring is measured with a spring scale or hanging masses, and is plotted on the vertical axis of a graph.  The length of the spring (or the displacement from the resting position) is measured with a meterstick and plotted on the horizontal axis.  Because F = kx, the slope of this linear graph is the spring constant k.  

(As an aside, I've written up a detailed approach to this experiment for the College Board -- take a look here.)

This experiment is beautiful because the data are easy to take, and because even the worst experimenters get something resembling a line.  However, occasionally you'll see something weird -- the graph will be a line most of the way, but very small displacements will give a significantly steeper slope.  See the graph to the right (and click on it to enlarge if you can't quite see).  

What's going on?

First of all, quash the inevitable misconception:  "Oh, that makes sense because the more the spring stretched, the more force we had to use."  Well, of course -- that's what F = kx means.  We should need more force to stretch the spring for larger displacements.  

The slope of this graph represents the spring constant k, which indicates the stiffness of the spring. What's happening here is that the spring is significantly stiffer under about 3 cm of stretch.  Does that make any physical sense, though?

Well, in this case, yes.  If you get this sort of data, take a careful look at the spring you're using:

See how many of the coils are touching each other?  I asked the class to be very quiet... and then I began to stretch the spring a couple of centimeters.  We could all hear the "poing!" sounds of the individual coils unsticking from each other.  All the coils were fully separated when I had stretched the spring... about 3 cm.

GOOD GRAPHS: a sequel to BAD GRAPHS

I do have a couple more BAD GRAPHS.  These are utterly obvious, so I won't post pictures:

(BAD GRAPH #9) Failure to draw a best-fit at all means the slope cannot be taken properly
(BAD GRAPH #10) Failure to label the axes of the graph and to include units means the graph is worthless.

Now that we've washed our hands of those, it's time for some GOOD GRAPHS. 

GOOD GRAPH #1: y-intercept is clear

The y-intercept may have physical significance.  Often it's useful to be sure that the y-intercept can be recognized by inspection.  However, this is not the only GOOD GRAPH.

GOOD GRAPH #2:  You don't HAVE to start scaling from the origin

This graph is just dandy.  In fact, there has been at least one AP question (2005 problem 6) on which the scale could not have begun at the origin in order to scale the data to at least half a page.  Students will attempt to demand a hard-and-fast rule about scaling graphs from the origin, but such a rule does not exist.  The scaling of a graph depends on the circumstances of the data.

One warning, though, while we wrap up today's feel-good episode of GOOD GRAPHS:

GOOD GRAPH #3:  If you don't scale from the origin, be careful about the y-intercept.

This graph is quite fine.  Proper labels, scale, points, and best-fit.  However, gotta be careful... the circled point looks to be the y-intercept.  But no!  The horizontal scaling starts from .01 kg.  The actual y-intercept has to be extrapolated.

BAD GRAPHS:  Summation
I've created this series of posts on request from several teachers.  Our students come to us with essentially zero experience making useful graphs of experimental data.  We have to bust all sorts of misconceptions. 

Ideally, we bring our class to an understanding of the purpose of an experimental graph.  A graph communicates not just the result of the experiment, but also the data acquired, the calculational methodology behind that result, the precision of the result.  A scientist who says merely "From my data, I conclude that the density of this oil is 0.9 g/ml" must be taken at his word.  It is so much more transparent to say, "The density of this oil is 0.9 g/ml, as determined by the reciprocal of the slope of this graph here."  Of course, a BAD GRAPH undermines this point.

It's great if you can get your class to see why they should not make BAD GRAPHS.  But the other usefulness of this series of posts is more functional.  When your student tries to argue that his graph is okay, and when he's not listening to or believing your rationale, you can point him here:  "Johnny, look at BAD GRAPH #5.  That's why you're going to redo the graph you submitted."

Bad Graphs part II: don't force the best-fit through the origin

In today's episode of Bad Graphs, we begin with another poor scale.

BAD GRAPH #4:  Scaled to less than one-quarter page

As you can see -- or maybe you can't, it's so small -- these data points are plotted correctly, but in a teeny weeny portion of the page provided.  A proper graph takes up well over half the available room on the page.  The standard for credit on this particular AP problem (2010 B2) was for the graph to take up more than 1/4 page.  Scaling across a whole page is a skill that must be taught -- it does not come naturally out of math classes.

BAD GRAPH #5:  Can't see the data points without a magnifying glass

If you want to get extra-technical, the size of the data points on the graph should reflect the experimental uncertainty in each quantity measured.  That is, if you could measure to the nearest milliliter, than the data points should be as big as half of a box in the vertical direction.  (And if large uncertainty would make the points ridiculously big, then you're supposed to use error bars.)

For the purposes of AP exam questions or labs within my course, all I ask is that the data points be clearly visible, as in all of the other BAD GRAPHS shown in this post.  The graph above, though, shows itty bitty dots and a nice best-fit.  Sure, the best-fit will yield a reasonable slope, but without easily seen evidence of where that slope came from.

BAD GRAPH #6:  best-fit line forced through the origin

A best-fit line should reasonably indicate the trend of the data.  There is no one "best" best-fit, but rather a range of allowable best-fits.  I've occasionally had my class draw the steepest possible best-fit, then the shallowest, and note that the value of the slope is somewhere between these two extremes. 

The problem with the graph above is much more subtle than with some of the other BAD GRAPHs.  This student has drawn the best-fit line by starting at the origin of coordinates, and only then trying to approximate the trend of the graph.  Problem is, for one thing, the origin is not a special spot on the graph.  The point (0 kg, 0 m³) is no more important than the point (.04 kg, .000054 m³).  Even in the case where (0,0) is a data point, it's a data point like any other.  Would you insist that the best-fit line always go through the third data point?

In this particular experiment from the 2010 AP exam, the y-intercept of the graph was explicitly non-zero.  (In fact, the last part of the question demanded students to figure out that the y-intercept represented the volume of fluid displaced by the floating cup alone, without any additional mass.)  Forcing the best-fit through the origin not only artifically steepens the graph's slope, but it obscures the physically meaningful y-intercept.

Of course, forcing best-fits through the origin isn't always as subtle.  Trust me.  When we graded this problem, we saw the not-totally-unreasonable version above, but also we saw plenty of these:

BAD GRAPH #7:  Curved to get to the origin

 Yuk.  But this one takes the cake...

BAD GRAPH #8:  Forced through the origin that isn't even the origin

It's perfectly acceptable, and sometimes desirable, not to begin an axis at zero.  However, you gotta recognize that what looks like the origin isn't necessarily the actual origin, in that case.  This grapher would have been fine, except for forcing that line through the origin that, after all, isn't the origin.  Boux.

One more set of BAD GRAPHs tomorrow.  But I promise, I'll include a couple of GOOD GRAPHs as well.

 



Bad Graphs -- Common mistakes on data-graphing test questions part I: horrid best-fits

In the previous post, I discussed the rubric for an AP Physics question that required graphing data.  A number of folks requested that I show and discuss the most common mistakes on this type of question.  I should emphasize that while I am speaking in the context of grading the AP Physics exam, the graphing issues here are germane to experimental physics at any level.  Even in the most basic conceptual physics course, even in our professional level Research Physics course, appropriate graphing skills should be developed.

Don't let your students' graphs look like these.  You may laugh at some -- just the mere fact that you're reading this blog implies that your students would be less likely to make most of these mistakes.  But understand that every one of these mistakes is made FREQUENTLY on the AP exam. 

BAD GRAPH #1:  Non-linear axes

Aarrgh!  This is the most horrid of bad graphs, suggesting that this student has never graphed data in his life.  The only time I've seen it in my own class was the first year I taught, in the first lab I assigned to my regular 9th grade class.  That was an eye opener -- we stepped back and had a new lesson the next day.  On one hand, I used to think that AP students generally wouldn't make this mistake; however, having graded graphs on the actual exam, I'd now bet that one exam in twenty does this.

BAD GRAPH #2:  Dot-to-dot

At least this student has graphed data before.  Connecting data like this implies that we have theoretical or experimental support that the slope of the graph is or should be different in each region.  Since the slope of this particular graph is related to the fluid density, the implication is that the fluid density changes depending on what mass we float on the water.  Really?

BAD GRAPH #3:  Curve fudged to go through each data point

This is for the folks who have been told never to connect dot-to-dot, but who are still uncomfortable with the idea that data points indicate a trend -- they are not delivered unto us on stone tablets by the Almighty.  Some students do even more obvious fudging, making sure their curves go through the center of every point.  They are implying theoretical justification for a 6th order function modeling the data.  I remember the eye-opening I experienced when someone pointed out that if you make excel use a high enough order polynomial, you can produce a curve that will seem to fit ANY data set perfectly.  I counter this misconception not only by fiat (minus one million points for drawing a baloney curve), but also by insisting on an enormous amount of data in every experiment.  It's hard even for first-year students to justify fudging a fit through 20 data points.

It's not hard, folks -- when there is theoretical support for a linear graph, and/or the data look linear, just place the danged ruler down on the paper, align it approximately with the trend of the points, and draw.  When done right, a proper best-fit line takes much less time than any of the baloney above.

So that this post doesn't go on for pages, I'll stop here.  Tune in tomorrow for the "scaling issues" edition of BAD GRAPHS.

Graphs in laboratory -- a rubric

The 2010 AP Physics B exam, question 2, provides a typical lab-based question involving graphical analysis of data.  Students were asked to graph a small set of volume-vs.-mass data on the axes provided; the density of the oil used in the experiment was then determined by the inverse of the graph's slope.

It's instructive to look at the portion of the rubric (look at pages 5 and 6) relating just to the graph.  Graphical analysis is an important skill, one evaluated in our classes and tested on the AP exam.  But an equally necessary skill is that of creating and presenting a graph in the first place.  You might think that merely making a graph is child's play compared to understanding the graph's meaning, but even strong students don't usually do a good job presenting graphs until they've practiced many times.

Part of the students' issue is that they perceive the graph creation process as drudgerous busy work.  "I've got the data my teacher told me to take right here in a table.  Why do I need to bother making this graph?  I'll do it because my teacher is making me, but it's stupid."  And they make the graph as quickly and sloppily as they can.

Well, the creation and presentation of a graph was worth 4 of 15 points on AP Physics B 2010 #2.  Maybe significant credit -- or loss of credit -- can convince students to make graphs properly.  It's instructive to look at how those points were awarded.  We can see and communicate to our classes the elements of a graph that college professors, the AP exam, and we as high school physics teachers are looking for.

Point #1:  axes.  Were the axes of the graph labeled properly, with units?  On this particular problem, the axes were pre-labeled, but the units had to be included.  On a lab in class, I ask the students to use the axes to communicate in words the quantity measured, along with its units.

Point #2:  scale.  The scale must be linear (i.e. the space between gridlines must always represent the same value); the scale should allow the plotted points to take up most of the grid.  On 2010 B2, the standard for credit was that the scale must allow the data to take up more than 1/4 of the grid area.  I'm more stringent in my class, requiring the use of more than 1/2 the grid area.

Point #3:  plot.  The points must be plotted correctly and visibly, such that the measurements could be correctly extracted from the graph.  Earning this point is usually a matter of attention to detail, but part of experimental physics is attention to detail.

Point #4:  best-fit.  A best-fit line must be straight, meaning drawn with a straight-edge.  It must never deliberately connect point-to-point.  It must not be forced through the origin.  (That's the most common mistake here.)  It should reasonably represent the trend in the data. 

However you grade your students' graphs, in lab and on tests, the elements in this rubric can provide a guideline for what's important.  Train your students to check each of these elements before turning in a graph.  Perhaps even make them redo a graph that is substantially missing one of these elements. 

Point is, a scientist would never dream of presenting for publication a graph that doesn't meet each of these four standards.  Your students shouldn't, either.

GCJ

How I "conduct" a laboratory session -- NO HANDOUTS!

But Mr. Lipshutz, you didn't tell us
 which graduated cylinder to use!

At my AP summer institutes, I offer attendees my entire general and AP physics laboratory program.  I describe each experiment briefly in writing; I include the "lab report" evaluative exercise that is assigned for homework; and we even actually conduct three of the labs I use.

Confusion generally reigns, though, until teachers actually try out a couple of my experiments.  The issue?  "What do you give to the students so they'll know what to do?  Do you have the lab handout?"

My answer is, I give the students nothing.  I demonstrate the use of the equipment -- moreso early in the year than later in the year.  I tell the class what to measure and how to measure it.  On the board, I draw the graph (always a graph) that they should make.  That's it.*

But how will the class know what to do? 

First of all, they LISTEN to me.  If they know a handout is coming, why should they listen actively to anything I say?  "I'll just read the handout," they'll think.

Secondly, I don't want students blindly following directions.  Figuring out how to measure something is an experimental physics skill, even at the most basic level of "how are we going to find the volume of this water?"  Recognizing that they need a graduated cylinder rather than a beaker, then selecting an appropriate cylinder from the shelf, then realizing that the one they chose was too big -- all that is a learning experience.  My placing the proper-sized cylinder on the lab table and writing "pour the water into the graduated cylinder and read from the bottom of the meniscus" teaches the class to follow directions, nothing more.

And finally... how many times have you handed out a carefully-prepared lab sheet, then had ten students ask a question to which the response is, "Look at the handout here."?  Students DON'T READ LAB HANDOUTS CAREFULLY.  We all know this.  So instead of wasting time preparing a handout, then getting frustrated, then complaining about how these dang kids today don't read anything... just don't give a handout.  Laboratory isn't the time to be teaching the skill of following written directions, I don't think -- that's for homework, tests, and home ec class.

I know you're skeptical.  After all, your teachers all through high school and college probably never failed to hand out a lab sheet with complete instructions.  It's not easy to let go of this crutch.  But flying by the seat of your students' pants in lab does work beautifully.  Joshua Beck, who attended my workshop at NC State University last summer, says not giving a procedural handout on lab days "has been great, for me and them."  He's right.  Try it.

* Sure, early in the year I make sure everyone is acquainted with my lab requirements, as shown here.  These aren't on a handout, they're just oft-repeated guidelines.