Jacobs Physics has moved to a new location!

Hi, all!  The Jacobs Physics blog, archives and all, have migrated to jacobsphysics.org.  I'm escaping Google, for reasons.  Come visit!  I'm still rearranging furniture in the new home, but you should be able to access all the archives there.

All future posts will go to jacobsphysics.org.  You can contact me there with a comment; on bluesky where I'm @jacobsphysics; and via email at my woodberry.org address.  

The 5 Steps books should publish Oct. 15 2024.  

My next possible project is a daily Live Physics Time audio show; still ironing out details, but it would begin mid-August 2024. 

See you on the (new) site!

-- GCJ

Conceptual Physics: teaching the "false calculation"

 In conceptual physics, students are offered three possible ways to justify an answer:

1. Fact: Write a fact verbatim from our fact sheet, then write connecting prose to figure out the answer;

2. Calculation: Write a relevant equation, make a table of known variables (including units) and do math to figure out the answer;

3. Equation: Write a relevant equation, indicate which variable is unchanging and how you know, then draw arrows indicating how the remaining variables increase or decrease.

Education professors might label these approaches as  verbal, quantitative, and semi-quantitative reasoning.  

I've always started the year with fact-based reasoning with reflection, refraction, and lens/mirror ray diagrams.  (A properly drawn ray diagram counts as a "fact".)  Then as we study waves, we do calculations with v=(lambda)f, v=(lambda)/T, and f=1/T.  

A major emphasis of a first year physics class, though, is to get at semi-quantitative reasoning.  "A wave speeds up as it moves from shallow to deep water.  Does its wavelength increase, decrease, or stay the same?"  First, use the fact that says "when a wave travels from one material to another, its frequency stays the same."  Next, in the equation v=(lambda)f, f doesn't change.  Since v and lambda are directly related - mathematically, when one increases, so does the other - an increased speed means an increased wavelength.

Thing is, no matter how hard I tried, 9th grade first-time physics students didn't understand this direct relationship.  They randomly guessed at how v and lambda changed.  Worse, they didn't at all see what the equation meant.  I have them draw up-arrows to show that v and lambda increase, and a line over the f to indicate that frequency didn't change.  But these arrows were clearly meaningless to a large portion of the class.  Even when I drew every possible permutation of the equations and possible relationships on the board, it didn't help.

What was missing was the connection between the equation-in-variables and the underlying arithmetic.  So I made that connection explicit.

Instead of teaching the equation-with-arrows approach to semi-quantitative reasoning, I taught the "false calculation".  What's that?  I'll show you with the above example.

"A wave speeds up as it moves from shallow to deep water.  Does its wavelength increase, decrease, or stay the same?"

Using the relationship v=(lambda)f, make two different charts, one for the wave in shallow water, one in deep water.  The question doesn't require a precise numerical answer for a wavelength - just how the wavelength changes.  So MAKE UP NUMBERS that are easy to work with!

We know that the frequency is the same for each, because when a wave changes material, its frequency doesn't change - so call it 1 Hz for each.  We know the speed is faster in deep water than shallow water - so let's call the speed 1 m/s in shallow water, and 2 m/s in deep water.  (Any speeds/frequencies will work!  The point is to make calculation simple.)

SHALLOW                DEEP                    Using v = (lambda)f

v = 1 m/s                    v = 2 m/s

lambda = ?                 lambda = ?

f = 1 Hz                     f = 1 Hz

******************************

(1) = (lambda)(1)          (2) = (lambda)(1)

lambda = 1 m               lambda = 2 m

Thus, the wavelength is greater in deep water.

Evaluating the false calculation  I expect to see each of the following elements when a student uses a false calculation to answer a question involving semi-quantitative reasoning.  You can "grade" a response that includes a false calculation by awarding one point for each of these items:

1. The relevant equation is written clearly and used.
2. The variable that does NOT change is indicated, with evidence as to why that variable is unchanged.  This evidence usually includes either a fact from the sheet, or direct language in the problem statement.  "The frequency doesn't change because it's constant" is not sufficient.
3. Two charts, like the ones above, filled out with correct units on all values and a question mark indicating the unknown variable.
4. The values in the chart plugged into the relevant equation, and a conclusion drawn.

That fourth point often is awarded even if the calculation is executed incorrectly.  That is, if the student screws up the 5th grade math and gets lambda = 0.5 m and thus says the wavelength is greater in shallow water, that student will get full or nearly-full credit.  The whole purpose of this methodology is to give beginning students a scaffolding to make predictions in a rigorous way rather than using guesswork.  

Of course, my students are often using false calculations to make predictions in the laboratory.  When they do the experiment and find out that the wavelength is in fact greater in deep water, the context is exactly right to show them that they did the math incorrectly - and they've advanced their understanding, which is the whole point of the exercise!

5 Steps to a 5 AP Physics 1 2025 edition RELEASE DATE! And link to request review copy.

Aha!  We have a date... McGraw-Hill says that both the Physics 1 book and the Physics C book will publish on October 15, 2024.  Here is a link to preorder (none in stock yet as of June 2024), and to request a review copy.  

The 5 Steps series has migrated from McGraw-Hill's professional side to their education side.  Which means they are treating the 5 Steps prep book like a textbook for the purpose of marketing.  Awesome!  You can now request a review copy for yourself.  And, you can request to purchase a class set at a discount!  

Fluids for AP Physics 1: buoyant force demonstration and/or lab

I have an aluminum cylinder here.  I hang the cylinder from a string, and attach the top end of the string to a force probe.*  The probe reads 1.1 N.  

*Or a spring scale.  This particular experiment can be done with 1960s equipment.

Next, I am planning to keep the cylinder attached to the force probe, but submerge the cylinder completely in a beaker of water, without the cylinder touching the bottom of the beaker.  What will be the reading in the force probe when the cylinder is submerged?

This is a force problem.  Even though I might be doing this demonstration during the new AP Physics 1 fluids unit, it's still a force problem.  And thus the starting point is a free body diagram, regardless of the exact question being asked.

The free body for the cylinder includes an upward tension T, an upward buoyant force Fb, and a downward force of the earth mg.  The cylinder will be hanging in equilibrium, so up and down forces balance: T + Fb = mg.  I'm looking for the reading in the force probe, which is the tension in the string.  Solving for tension gives T = mg - Fb.  

In this case, we already know mg, the weight of the cylinder, because of the initial force probe reading before we submerged the cylinder: mg = 1.1 N.

The buoyant force on a submerged object is equal to the weight of the displaced fluid.  This is written mathematically by the equation Fb = (density of fluid)(volume submerged)(g).  I write words rather than variables here because it's so easy to get the wrong density, or the wrong volume.  Generally, density times volume gives mass, and mg gives weight.  The mass of the displaced fluid is the density of the fluid times the volume of the displaced fluid.  

Well, we know the density of water: 1000 kg/m^3.  (See the previous post for a brief digression about "as much as you can hug.")

But how can we figure out the volume of this cylinder?  I ask the class for ideas.  There's no one right answer; and this creative experimental brainstorming is exactly the kind of practice that can help students approach AP Physics lab questions.

Idea 1: It's a cylinder, which has volume equal to the area of the base times the height.  So take a ruler and measure the diameter (and thus the radius) of the base; measure the height.  The volume is pi*r^2*h.  Excellent.  Any other thoughts?

Idea 2: Use water displacement.  Pour water into a narrow graduated cylinder.  Look at the initial volume reading when the cylinder isn't submerged; look at the final volume reading when the cylinder is fully submerged.  Subtract those volumes to get the volume of the cylinder.  Great.  Any further thoughts?  Anyone?  

Idea 3: You said it was an aluminum cylinder.  We can look up the density of aluminum; we know the mass by dividing the cylinder's weight by g, meaning the mass is 110 g.  The cylinder's volume is its mass divided by its density.  Fantastic!  Any OTHER ideas?  No?  

My idea: I look at the cylinder, or at least the part of the cylinder that was facing away from the audience.  "It says right here on the cylinder, written clearly in permanent marker, "42 mL".  Okay, okay, it's a trick.  Ha ha ha.  The point has been not to get the right answer, but to have this particular conversation.  I've got the accurate volume pre-measured (I pre-measured using the water displacement approach), the same way a cooking show will have a soufflé pre-baked so the audience doesn't have to sit through the, um, less exciting parts of the cooking process.

I use google to find that 42 mL is equal to 4.2x10^-5 m^3.

And now we can do the buoyant force calculation.  The buoyant force is (1000 kg/m^3)(4.2 x 10^-5)(10 N/kg) = 0.42 N.  The reading on the scale will be (1.1 N - 0.42 N) which should be about 0.7 N.  

Sure enough, I press collect on the app connected to my force probe, submerge the cylinder... and the reading drops from 1.1 N to 0.7 N.  

Physics works.

Extensions:  So many great questions you can ask.  Does submerging halfway down make the buoyant force greater than, less than, or equal to 0.21 N?  

How would the scale reading change if the cylinder touched the bottom of the beaker?

What if you put the beaker on a platform scale and submerged the cylinder, without allowing it to touch the bottom of the beaker?  What would the platform scale read?  This one's rather complex.  I analyze this question here.

Fluids for AP Physics 1: pressure in a static column, demonstration and/or lab

Fluid mechanics is trading places.  Since 2015, fluids has been part of the P2 curriculum, for the 25,000 or so students who take that exam.  But next year, fluids moves over to P1 and its 150,000 students.  So now's probably a good time to share some thoughts about teaching fluids.  

I do recommend coming to an AP Summer Institute, where you can see me do this and other fluids demonstrations live!  You can see my APSI schedule in the left sidebar.

Pressure in a static column

When a tank of fluid is not moving, the pressure anywhere in the fluid is given by P =P0 + ρgy. Here P0 represents the pressure at the surface* of the fluid, ρ is the fluid density, and y is the depth below the surface at the position where you're measuring pressure.  

*Yes, this is *usually* atmospheric pressure... but not always.  Consider one fluid on top of another.  The pressure in the bottom layer is the pressure at *its surface* plus ρgy.

I have one of them giant graduated cylinders filled nearly to the top with water.  My pressure sensor is connected to my ipad under the document camera; I set the Vernier Graphical Analysis app to produce a graph of pressure vs. time.

I attach a long tube to the pressure sensor.  First, I read the pressure when the tube is NOT submerged - this is P0.  The sensor generally reads in kPa; we want a reading in Pa, where 1 Pa is equal to 1 newton per square meter.

Next, I announce that I will predict the pressure sensor reading when the tube is submerged to the very bottom of the container.  The relevant equation is P =P0 + ρgy.  What additional information do we need?

We have the surface pressure - usually around 102,000 Pa, but that varies by weather and especially altitude.  We can measure the depth y with a meterstick - this is usually about 18 cm, i.e. 0.18 m.  We know the gravitational field g to be 10 N/kg.  

What about the density of water?  

AP Physics 1 does not require unit conversions!  The number of times that a student is even asked to give a numerical answer to anything is minimal.  So for this in-class problem where conversions are necessary, I do the conversions in my head and state the result; or I use google ("convert 102 kPa to Pa").  De-emphasize number crunching, and you'll un-de-emphasize concepts.  :-)

That said, thought experiments are well within the scope of AP Physics 1.  Students don't hesitate to volunteer that the density of water is "one."  One what?  The standard units of density are kilograms per cubic meter.  So one kilogram per cubic meter for water, right?

How big is a cubic meter, anyway?  That question provokes several seconds of pure puzzlement, followed by students waving their hands about a meter apart.  "It's as much as you can possibly hug," I say.  Close enough.

So, consider a tank filled with a cubic meter of water - as big a tank as you can possibly hug.  Can you lift that tank?  Oh.  No way.  Well, you can lift a kilogram without trouble - here, catch this 2 pound thing.  That huggable tank doesn't weigh 2 pounds.  No, it weighs a ton.  The density of water is a THOUSAND kilograms per cubic meter.

Back to the calculation.

The gauge pressure at the bottom of the cylinder - gauge pressure means just the ρgy bit, ignoring P0 - is (1000 kg/m^3)(10 N/kg)(0.18 m).  That's 1800 Pa, or 1.8 kPa.  

Thus, the pressure sensor reading should increase from about 102 kPa to 104 kPa.  The actual sensor has a precision in the neighborhood of +/- 0.2 kPa, so this difference will be pretty much dead on when you send the tubing to the bottom.

Follow-up questions include "what's the gauge pressure halfway down?   Greater than, less than, or equal to 0.9 kPa?  Answer?  Equal to: because in ρgy, y is in the numerator and neither squared nor square-rooted, halving y halves the gauge pressure, too.  Sure enough, submerging the tube halfway verifies this prediction.

Physics works.

Make this a laboratory investigation!

Put pressure on the vertical axis, depth on the horizontal.  The slope of the graph should be ρg, and the y-intercept should be atmospheric pressure.  You can use the slope to determine the density of water.  Or, use a different fluid like vegetable oil, and you'll determine the density of the mystery fluid!

Edna meets Mr. Chubbs at the Conceptual Physics Tournament

Will Keay, who teaches in Fairfax, Virginia, came to judge the Woodberry Forest Conceptual Physics Tournament last weekend.  His class pet, Mr. Chubbs, met my pet hippopotamus Edna.  They seemed to like each other.  

We've been doing the tournament as our 9th grade final exam for seven years now.  Feedback from students is universally positive.  They like that they don't sit for yet another two-hour written exam, of course - even though, to a person, they recognize that they work harder to prepare for the tournament than they do for any written exam.  

This year they particularly articulated how much they like the collaborative nature of their preparation.  They get to give presentations and argue about physics with their classmates during their nightly study period.  The process feels social!  It doesn't feel like studying!  Yet, everyone recognizes how much their understanding of their particular physics fight problems - and of physics in general - improves through this social process.

I pointed out that there's nothing stopping these 9th graders from doing similar practice for their history exam as if they were going to have "history fights".  This point met with very confused and skeptical faces.  

The other interesting piece of feedback this year was about the difficulty of the AP-style problems that we posed.  More students than ever said that, once they had dug into the problems for a day or two, they felt easy.  Really!  A couple of students even suggested tougher problems for the following year.  While this peaceful, easy feeling was engineered by design - familiarity over the three-week preparation period breeds comfort - the fact that college-level problems seemed straightforward to general-level 9th graders means the tournament is accomplishing its goal.  We want students to leave physics with a good experience, feeling challenged, but good about meeting the challenge of a difficult subject.  

And, of course, the end-of-year positivity is helped significantly by the invited tournament jurors.  The conceptual physics teachers help students prepare for the tournament, but we don't judge the tournament.  When we give feedback, that feedback is authentic.  "No, you don't *have* to redo your graph, of course not.  But the jurors are more likely to say 'the data points on the graph don't take up anywhere near half a page' than 'oh, I'll bet you spent ten minutes on that graph, you shouldn't have to spend ANOTHER ten minutes making the scale more reasonable.'"  It's always been true that I am the publisher, not the author, of my students' grades.  The large jury, from a pool external to our school, makes my role even more clear.

New online APSI Physics 1 offering - June 24-27, through PWISTA

By popular request, my June 24-27 APSI through PWISTA will now be online.  You can sign up through this link.

Online means you can attend from anywhere, and even bug out for a moment to feed/clean up after the dog/kid.  :-)  Demand was not there for a New York session in person, but we've had a number of requests for another online session!

The PWISTA online session runs synchronously through Zoom from 9am to 3pm each day.  I'm also available for "office hours" from 8:30 each morning, and until 4pm each day.  And I'm happy to arrange for further conversations outside of these times.  As with all institutes, online or in person, you'll get access both to the official College Board materials, AND to my personal files with problem sets, lab activities, quizzes, and tests.  

The AP Physics 1 exam is changing for 2024-25, adding fluid mechanics and a few other minor topics.  And, the exam format will change significantly.  I'll discuss all the changes, but I'll also do all the labs, demonstrations, teaching/culture building tips, etc. that are the hallmark of a Jacobs Physics institute.

Join us in June - you won't be disappointed!  Please feel free to contact me via email with questions.

GCJ

Upperclass AP Physics students in the spring

In the spring, our headmaster reminds us how this is a tremendously emotional time for seniors as they approach a significant ending of one life chapter / beginning of another.  I overheard some parents (from another school) the other day rightly noting how much difficulty their kids are having holding themselves together.  These kids are making a good faith effort to complete so, so many year-end capstone requirements from their classes and their school, on top of life events that matter like proms and sports and social events.  Even seniors who try to be good citizens are being pushed to their mental limits in May.  

What's my reaction to this (unfortunate, I think) feature of 2020s schooling as an AP physics teacher?  I try to make my class one of the most fun parts of the students' day, full of camaraderie.  I'm trying to cash in the culture building I've done for four years with these seniors at our school.  For students who do want to leave the school on a positive note, those who want to work in good faith in each class, they are incredibly happy to be in a comfortable place where they can learn together.  My P2 students are having fond memories of freshman year, when they prepared for AP physics 1 in a similar atmosphere; but their confidence is through the roof now in physics 2, and they know their classmates as true brothers* who've bonded over four years in classes with one another.  

*I'm at a boys' school. You're likely gonna want to use the word "siblings".  :-)

I know, teaching upperclasspeople is not all Care Bears and Smurfs.  Teenagers trying to manage social and parental pressure plus their own boiling crock of hormones can behave in frustrating and nasty ways.  I have to remind myself not to let my seething anger at a few students color my relationships with their classmates who haven't been nasty.  

This year more than ever before, I'm getting the sense that most of my seniors are appreciative of their teachers, their classmates, our school.  This is the class who entered my boarding school in the fall of 2020... right after they had been locked down for months.  They were being denied the social contact that is so critical to all of us, but especially to young teenagers who are developing their sense of themselves and the world.  This senior class seems more grateful than any class I've been around... because, perhaps, they've seen that teachers who care about them are in fact a force for good, not merely a set of jailers.

So in AP Physics 2, we do the "quizzes" that I've been posting each day, but not for grades.  We do experiments where I join in as a regular lab partner.  We sometimes just talk about things other than physics.  This group has learned physics well; they are intrinsically motivated to be as ready as they can be for the exam.  I, and they, can relax and enjoy our time together.

AP Physics 2 - fundamentals review #3

With three weeks to go before the AP physics exams, it's worth remembering that our students don't need MORE practice problems; rather, they need to pay careful attention to the practice problems they do.  This is my application of a John Dewey principle, that we don't learn by doing; we learn by paying attention to what we do.  

My first-year students in physics 1 are in a cycle of AP problem / quiz based on AP problem / corrections to AP problem if their quiz or problem shows they didn't get it the first time.  In AP Physics 2, which is a second-year course, students have already internalized that they must pay attention to what they do.  And, P2 students have a level of earned confidence in their skills that my 9th grade P1 folks lack.  So truly all we're doing is these daily quizzes, in-class experimental and problem solving work, and each week one take-home 25 minute "quiz" with 5 multiple choice and one free response.  Less is more when dealing with upperclasspeople in the spring.

P2 Fundamentals Review #3

21. A battery is connected to two resistors in series.  The resistors each take 20 V of voltage across them.  What is the voltage of the battery?

22. Write the first law of thermodynamics, which is an expression for the change in internal energy.

23. Two waves are initially in phase with one another.  One wave has traveled a small extra distance than the other.  Under what conditions does constructive interference occur?

24. Define the period of a wave.

25. What is the equation relating the image and object distances for a convex mirror?

26. Two light waves undergo constructive interference.  What physical effect will be observed?

27. Name two items that can produce a magnetic field.

28. A battery is connected to two resistors in series.  The resistors each take 20 mA of current through them.  What is the current provided by the battery?

29. A gas consists of molecules moving around.  What feature of these molecules’ behavior causes the macroscopic effect called pressure?

30. What is the physical quantity that means energy produced in one second?

AP Physics 2 - fundamentals review #2

My AP Physics 2 class is an ungraded honors course.  There's not even a contract.  There is a careful selection process - students are selected not based on their previous grades, but holistically based on their demonstrated authentic interest in the subject.  Basically, if a student passed P1, wasn't a jerk, and put forth reasonably consistent effort, we take them into the P2 class.  

Even without published grades, the ground state of our class is to begin with a 4-5 minute fundamentals quiz.  We grade the quiz; students tell me their scores.  The quiz grades have no extrinsic meaning, won't be seen by parents or counselors or universities.  So what!  Motivated students still care about getting things right.  But, with no published grades, the students are insulated from the shame or world-ending dread of receiving a published grade that is not an A.  If someone gets 3/10 - which happens somewhat regularly! - they don't have to fear that someone is waiting with a (hopefully figurative) cane for their poor performance.  They just should try to do better next time.

P2 Fundamentals Quiz #2

11. Light is incident on a thin film.  Under what conditions does the light change phase?

12. An ammeter measures ____.  It is connected in _____ with a resistor.

13. Write the equation for induced emf.  

14. And electric field points →.  A positron moves ← in the electric field.  Sketch the path of the positron, and describe briefly how it moves.

15. An electric field points from position A to position B.  Which position is at higher electric potential?

16. A gas expands adiabatically.  What is the sign of the work done on the gas during this process?

17. Write the units of electric field.

18. Kirchoff’s loop rule is a statement of conservation of _____.

19. A concave mirror has radius 50 cm.  What is the focal length of this mirror?

20. For a convex (diverging) mirror, how does a ray parallel to the principal axis reflect?

(Solutions will be in the comments in a few days!)

Fundamentals checks for AP Physics *2*

Last year I wrote a bunch of fundamentals checks in preparation for the AP physics 1 or C-mechanics exams.  This year, I'm teaching a physics 2 section.  So, I'm writing daily quizzes for P2 now!  I'll post the quizzes here, then follow up a couple of days later with the answers in the comments.

Quiz 1:

1.       A substance has index of refraction 1.5.  What is (or how do I figure out) the speed of light in this substance?

2.       Write the equation relating focal length, image distance, and object distance for a mirror.

3.       Write the equation for the energy carried by a photon.

4.       Under what conditions is the image distance d_i positive, and under what conditions is it negative?

5.       The diagram shows light hitting a mirror.  What is the angle of incidence as the light hits the mirror?

6.       A resistor is connected to a battery, and the current in the circuit is measured.  The voltage of the battery is increased, and the current is measured again.  Which Ohm’s Law variable(s) remain unchanged after the voltage is increased?

7.       Under what conditions is the electric field given by the equation E = kQ/d²?

8.       One mole of monotomic ideal gas begins at pressure P₁ and volume V_1­.  Next, the pressure is increased to P₂ without changing the volume.  Write an expression (including a + or – sign) for the work done on the gas.

 9.       Write the equation for the capacitance of a parallel plate capacitor.

 10.   A photon initially has wavelength 500 nm.  The photon collides with a free electron which was initially at rest, giving the electron 3.0 eV of kinetic energy.  The photon bounces back opposite the direction it was initially moving.  What is the speed of the reflected photon?

Energy of various systems in an inelastic collision

 

I was asked about the situation above, in which two carts of different masses are released from rest and roll down frictionless ramps.  The carts collide and stick together on the flat surface.

Let's treat this as a goal-less problem:

Edna, Bertha, and Anthony by @Aldescary

Which way do the carts move after they collide?  

Well, Anthony (whom Edna calls a mean hippopotamus) says that the carts don't move after collision - they both have the same energy but in opposite directions, which cancel.

Oy.  So many things wrong with Anthony's answer.  Let's start with the fact that energy is a scalar quantity - energy can't have direction at all, let alone "cancel" other forms of energy.  And whenever we see a collision, energy should never be the first port of call - momentum should be.

It is true that the gravitational energy of each earth-cart system is converted to kinetic energy at the bottom.  And since the gravitational energy 2mgd is the same for both earth-cart systems, each cart will have the same kinetic energy before collision.

However, the momentums of each cart will be different.  I like to use the shortcut equation K = p^2 / 2m in this case to see that with the same kinetic energy, the cart with a greater mass will also have greater momentum.  You could also convert gravitational energy to kinetic energy to show that the speed at the bottom will root 2gh for both carts with mass canceling; then by p = mv twice as much mass with only 1.4 times as much speed means bigger momentum.

So the cart moving left has larger momentum than the cart moving right, meaning the two-cart system has a leftward momentum.  System momentum must be the same after collision as before, so the momentum is still leftward after collision... and that's the way the stuck-together carts will move.

Is mechanical energy of the both-carts-and-earth system conserved from release to just BEFORE the collision?

Anthony says mechanical energy is potential plus kinetic energy, and is always conserved, so yes.

Well, even a blind squirrel, or hippopotamus, finds a nut once in a while.  Anthony is pretty much correct.  Mechanical energy is conserved when no work is done by external forces and when no internal energy conversion occurs.  Here, the only external force acting on the carts-earth system is the normal forces of the surfaces on the carts.  These forces are perpendicular to the carts' motion, and so do no work.  Mechanical energy is, in fact, conserved here!

Is mechanical energy of the both-carts-and-earth system conserved from release to just AFTER the collision?

Anthony is perturbed... he already answered this question!  Mechanical energy is conserved, period, full stop, end of sentence.  Hemph.

Oh, Anthony... when carts collide and stick together, they undergo an inelastic collision by definition.  Mechanical energy may never be conserved in an inelastic collision - rather, some mechanical energy must be converted to internal energy.  

Is mechanical energy of the both-carts-only system conserved from release to just AFTER the collision?

Anthony says he's done answering these tricky questions involving systems.  He's gonna put his head down on his desk and listen to Edna for a change.

Edna thinks this one is pretty simple... because an object by itself can only possess kinetic energy!  (Potential energy can only exist when a spring or the earth is incl
uded in the relevant system.)  So the mechanical energy of the carts is just their kinetic energy.  On release from rest, the carts have no speed and therefore no kinetic energy.  After collision, the carts are moving, so they have kinetic energy.  The KE has gone from zero to not-zero, and therefore has increased.

Call for jurors: 2024 Conceptual Physics Tournament, on May 19. We pay $100.

In my school's conceptual physics program, we give cumulative written tests after the first and second trimesters.  In lieu of a final exam*, we are once again running the Woodberry Forest Conceptual Physics Tournament!  This is a competition for 9th graders, to be held at 1:00 on Sunday May 19 2024.  We've done this before, including the last two years after a pandemic-enforced break.  We're happy to be back to annual.

*No, to be clear to all, we're not giving an A to the winner and an F to the person in last place.  That's silly.  We're just having a fun, competitive tournament, to determine a winner.  Jurors engage in discussion and conversation with participants about their problems.  Jurors then award scores and write comments for the participants; jurors aren't assigning grades!

How does this tournament work?

On May 1, I will reveal a slate of three problems to the 64 participants.  These problems will be in the style of AP Physics 1 "paragraph response" questions.  Except, rather than just answer in a paragraph, the students will spend the month of May setting up experiments to provide evidence for their answers. By tournament time, each student will be expected to be prepared to discuss the solution to two of the three problems, with both theoretical and experimental support.

At the tournament, each student will participate in two "physics fights."  Think of these physics fights like a miniature version of a graduate thesis defense.  Students will have a strict limit of two minutes to present their solution to a group of two or three jurors, who then will engage each student in conversation about the problem for five minutes.  The students are evaluated by the jurors not only on the quality of their solution, but also on their ability to discuss the solution, to confidently hold a conversation with the jury.

Importantly, jurors are explicitly instructed on their primary role - to find out how much the students DO know, not merely to expose what they don't know.  

How do the students prepare?

Starting on May 1, all conceptual physics classes the rest of the year will be devoted to tournament preparation.  They'll work together to set up experiments in class, they'll be assigned to write up their solution as homework, they'll practice presenting.  They'll get intense instruction and guidance from the conceptual physics teachers, from their peers in the AP classes, from those who've been through this tournament before.

We need jurors.

The key, I think, to any class project is external assessment.  I and the other conceptual physics teachers will play the role of coach and advocate, always encouraging and helping the students to deepen their understanding of the problems and to improve their presentations.  Our relationship will be purely supportive, enthusiastic, positive.  

We can't then turn around and grill these same students as examiners!  That'd be like our football team's coaching staff refereeing the state finals.  Even -- especially -- if their officiating were fair, the coach-student relationship, both in before and after the game, would be irrevocably compromised.

So we need jurors.  We can pay.

Would you like to come to Woodberry on May 19 to be a juror?  We'd ask you to arrive at 11:30.  We'll have a meeting of all jurors in our beautiful dining hall over lunch.  

Then we'll ask you to be a juror for a couple of hours' worth of physics fights.  You'll be partnered with several other examiners over the course of the afternoon, getting to know a diverse set of fun folks from all over.  When all students have presented their two physics fights (to two separate juries), we'll gather the jurors for conversation, coffee, snacks, and their paycheck.

In any case, our goal is to be done by 3:30, or possibly 4:00 if there are logistical issues.  No later -- our students will be attending the final seated meal with their advisors that night followed by study hall, so we can't run late.

We will pay you $100 plus lunch for your time.  (If you're coming from more than a few hours away, we can put you up on campus on Saturday or Sunday night - please let me know if you're interested in this option!) I think you'd find that the camaraderie among the jurors and the engagement with the students will make the trip worthwhile.

Who's eligible as an juror?

Anyone who has passed a college-level physics class.  This includes alumni of your advanced physics class, even if they're still juniors or seniors in high school - we've had several teachers bring a caravan of students, and they've had an awesome time.  We've had local college or graduate students on juries, we've had parents, alumni, colleagues who teach other subjects, grandparents, friends... Anyone willing to engage in conversation about physics at the high school level, as long as you can recognize good and bad physics, we'd love to have you.  We are looking for a diverse juror pool, which especially includes diversity in age - truly, we want folks in their teens as well as folks in their 70s, and everywhere in between. When I ran the USIYPT, I found the mixture of undergraduate / graduate / professor / high school teacher / industrial physicist / retired physicist on the juror panel allowed some amazing relationships to develop.  I'd love to create a similar vibe here.

How can I sign up?

Send me an email via greg dot jacobs at woodberry dot org.  I'll send you more information, including the three problems, and our current draft of the scoring rubric.

We would like to get 45 jurors - pretty much the first 45 who sign up.  I can't wait to see some blog readers!  I'll even introduce you and your students to my pet hippopotamus, Edna.  :-)

April 8 2024: Free AP Physics 1 exam prep live show! (Archive link available)

McGraw-Hill, publisher of the 5 Steps to a 5 AP prep book series, is sponsoring a free live physics show for students and teachers.  I'll be presenting from my lab on Monday April 8, from 3:30-5:00pm eastern time.

All are welcome!  The way to join is, teachers (preferably) should "register" for free at this link.  They won't ask for you to make a username and password, nor to receive marketing emails.  Just give your last name and email.  Then, you'll get a message with the link to join - a link you can share with all your students, colleagues, friends, whoever.*

*McGraw-Hill would prefer not to be in the business of collecting student emails.  So they ask for a teacher's email, and ask the teacher to share the link.  They're trying to be entirely above board here!  There's no sneaky agenda to get your personal info!   The only agenda is, they want your students to buy the 5 Steps book.  :-)

I haven't planned the show yet, other than the general vibe, which will be similar to my 2020 live shows or my AP classroom videos (for Physics 1, units 2 and 3).  My pet hippopotamus Edna is excited for the event, and no doubt will make an appearance.

Do you have any requests?  My initial brainstorms are perhaps to set up experiments based on the 2023 P1 exam questions 1 and 2 - about harmonic motion, and a cart rolling down a ramp.  And I'll definitely leave plenty of time for improv, in which Bob the master of ceremonies will read the chat, and relay questions or requests to me.  However, I'm open to all sorts of suggestions now.  Post in the comments, or contact me via email or Bluesky!  I can do a lot in a 90 minute live show.  Tell me what you and/or your students want to see, and I'll try to make it happen!

And spread the word.  Last year, this Physics 1 live show was the best-attended of all of the subjects they offered... by a factor of about 20.  Let's keep that momentum going!  See you on April 8.

Update April 9 2024: The archive link is https://mcgrawhill.info/43yhAkc

AP Summer Institutes 2024 - will the new exam content and structure be discussed?

Of course it will!  

I'm doing several online and in-person P1 institutes in summer 2024 - see the sidebar to the left for details.  Please sign up!  The institute will certainly discuss content and structure changes for the 2025 exams.

What, specifically, will we do?

• We will do a number of fluids demonstrations and lab activities, including all three major fluids topics.
• We will do demonstrations with the three minor content changes in P1: parallel axis theorem, quantitative understanding of elliptical orbits, and center of mass location.
• We will discuss the new exam format - though I will emphasize that in P1, preparing students for the 2025 exam format looks exactly like preparing students for the 2015-2024 exam format.

Of course, the institute will still do all the AP Summer Institute things that you expect.  I'll give an overview of the AP program, the course audit, and AP classroom.  Much more interestingly and importantly, I'll discuss give you access to my course files, problem sets, laboratory activities, quizzes, tests, and a day-by-day planner.  Teachers new to AP Physics 1 can use these verbatim to get themselves started; veterans can use the materials to supplement and inform what they already do well.

The highlight of the in-person institutes is the "studio time" in lab on the final day, in which we all work together to set up and develop laboratory exercises based on released AP questions.  You'll come away with a dozen or more pre-tested new lab ideas!  In the online institutes, which are broadcast live from my actual classroom, we'll instead do "improv time" - challenge me to set up an experiment, or to show how I use demonstrations to teach any topic on the exam.

As of the 2025 AP exam revision, are Physics C mechanics and Physics C E&M two separate year long courses now? (No.)

On February 29 2024, the College Board released the course and exam descriptions for the revised version of all four AP Physics courses. You can find all the information and links at this page.  

For the AP Physics C exams, the course content will not change.  However, all AP physics exams - P1, P2, C-mechanics, and C-E&M - will be in an identical format as of the 2025 exams.  The format is, 80 minutes for 40 multiple choice questions; and 100 minutes for 4 free response questions.  That means an entire exam takes three hours.

But wait!  For decades, the two physics C courses have each had 90 minute exams, not three-hour exams, because the physics C courses have each represented a single-semester college course.  Has that changed?  

It has not.  Each of the FOUR courses now represents what is taught at the college level in a single semester.  AP Physics 1 represents the first semester of a college-level algebra-based introductory course.  AP C-E&M represents the second semester of a college-level calculus-based introductory course. And so on.

Thing is, we are teaching high school classes on a high school schedule.  The vast, vast majority of high school students taking physics for the first time should do a full year of mechanics.  This full year can be AP Physics 1; this can also be AP Physics C-mechanics for advanced students who are taking calculus.  Both cover substantially identical concepts.

For those taking a SECOND year of high level physics, well, AP Physics C-mechanics probably isn't challenging enough.  It's absolutely normal, acceptable, reasonable, typical for a student to take AP Physics 1 one year, then the combination of C-mechanics and C-E&M in their second year.  This post gives a recommended course sequence for such students.  

Me, I like to teach AP Physics 2 as my second-year high school course.  It's rich in diverse content, meaning that physics veterans won't ever say "oh, geez, not another cart on a ramp".  It's also particularly well adapted to seniors who need the course front-loaded - start with the hardest stuff like electricity and magnetism, and end with the simpler and more concrete topics like optics and thermodynamics.

But in any case, in any way you adapt the courses to your particular school ecosystem, the three-hour AP Physics C exams don't mean anything about how long you spend teaching the material.  Rather, the longer exams are a response to the fact that the pre-2025 APC exams were quite "speeded."  They were as much a test of how fast a student could do physics as how well a student could do physics.  And that's not what anyone wants to test.

Experimental procedures in AP physics, the redesigned free response section, and Wally the Astronaut

Wally the Astronaut, from The Physics Aviary

Above is a screenshot from the "Work to KE" simulated laboratory exercise on The Physics Aviary. In the exercise, you press start, and a fire extinguisher causes Wally the Astronaut to speed up.  You press stop, and the fire extinguisher ceases to apply a force.  Wally coasts, then passes through two photogates separated by 10 meters.  The time for Wally to cross the photogates is displayed.

You can do a thousand different sorts of classroom exercises with this single simulation.  I like to give this quiz here, go over the quiz, then have students go through this laboratory exercise. But the simulation here is so, so rich, you could do many different things.  Propose your favorite in the comments!

I was asked how I would describe* an experimental procedure on the AP physics 1 exam.  Some teachers ask their students to write a step-by-step instruction manual, including safety procedures and calculation instructions, for an in-class laboratory exercise.  Is that what the AP exam demands?  Should a procedure include calculational instructions?

*The "task verbs" on the AP physics exams will be in boldface as of the 2025 administration.

Historically, the AP readers expect students to communicate what they measured, and how they measured it.  If the experiment could in fact be done in a reasonable high school laboratory, the procedure is legit.  

The prompt on the AP exam - especially the redesigned 2025 AP exams - will be more targeted than what I often see in classroom lab handouts.  For example, the exam might write:

(a) Students are asked to take measurements to create a graph that could be used to determine the mass of Wally.  Describe an experimental procedure that the students could use to collect the data needed to determine Wally's mass. Include any steps necessary to reduce experimental uncertainty.

My response might be, "Measure the force F exerted by the fire extinguisher with a scale.  Then in each of many trials, turn off the fire extinguisher after Wally has traveled a distance d, a different distance in each trial.  Measure d; and divide the 10 m photogate distance by the time output of the photogate to find Wally's speed v."

The analysis - that is, how to do the necessary calculations - is usually in a separate lettered part of the question.  It's fine generally to write the analysis part in the same section as the procedure!  But the procedure will earn points independent of the analysis.  One being wrong or incomplete doesn't affect how the other will be scored.

Part (b) might ask about the analysis:

(b) Describe how the data collected in part (a) could be plotted to create a linear graph and how that graph would be analyzed to determine Wally's mass m.

And I'd say, "Wally's kinetic energy is equal to the work done by the fire extinguisher, .  [That first sentence is probably not necessary for credit!  But I write it so it's clear where my analysis comes from.]  Plot the work done by the fire extinguisher (Fd) on the vertical axis; plot (1/2)v^2 on the horizontal axis.  The slope will be Wally's mass."

Full credit would be earned for a more bare-bones "Plot Fd on the vertical axis, and v^2 on the horizontal.  The slope is (1/2)m."

A daily quiz based on 2023 AP Physics 1 question 1 - Did you *understand* how to do the homework problem?

It's getting toward the back half of the school year in AP Physics 1.  I've made a first pass at all the major content units; we've done laboratory activities out the wazoo.  We're gearing toward one more half-length practice AP exam before spring break, and then a final half-length practice in mid April.

My students need practice doing cumulative, AP-like problems which require synthesis of multiple concepts; or which require students to choose from the entire year's menu of possible approaches.  Later on, in April and May, I'll have students do authentic AP free response questions in class practically every day, without a safety net.  We're not quite ready to take the safety net away.

No, right now, I'm assigning AP-style free response questions as collaborative out-of-class work.  Everyone is encouraged to collaborate, to seek help when they're stuck.  As long as they get to the correct answers eventually, I'm happy that they're making progress.

You have questions about this approach.  "Even the most honest, diligent students will often just do what their smart friend told them to do, Greg.  Getting done with the assignment is more important than getting it done right.  Even with the five-foot rule religiously followed, at least some students are parroting, not learning, not progressing."  

Unless there's disincentive for pure parroting.  And I don't mean grade disincentive.

The approach I use - which is absolutely not the only effective approach! - this time of year is the daily quiz based on the AP-style problem.  When students come to class, I collect their assignment.  But the first four minutes of class are basic questions about the problem they did for homework.  We trade and grade the quiz, then I collect the quiz.  

Someone who understood the problem, even if they had to be nudged hard in the right direction, can do the quiz just fine.  Someone who truly parroted the smart kid cannot do the quiz.

Yet!  Even the student who parroted and then flunked the quiz has made progress!  The point of the quiz isn't to play gotcha, it's to review the problem in a context in which the students will listen.  If I say "Imma go over last night's homework," no one cares.  But if I say, "here's the answer to question 1 on the quiz and how I know, now mark your classmate's paper right or wrong," I get rapt attention.

My class is contract graded, which means there's no shame for poor performance, no cookie for being perfect.  What's the incentive, then, to take the assignment and quiz seriously?  If someone does particularly poorly on the quiz or problem set, I bring them in for a consultation to redo the quiz.  I just had a student in while I was writing this post.  It took him a relatively short time to redo the problem perfectly, with clear justifications for each part (including the parts that didn't initially require justification).  He didn't get this problem at first, but the combination of attempting it for homework, trying the quiz, and grading someone else's quiz meant that he gained a serious understanding of this problem.

Your ideas are intriguing to me, and I wish to subscribe to your newsletter.  Okay, here's issue 1: a quiz based on the 2023 AP Physics 1 exam problem 1.   Notice how the quiz gets to the essence of the solution without just asking "what was the answer".  This quiz brought forth excellent questions from the class about the physics behind the original question.  It made them think!

A cart oscillates, as shown above and on the problem set last night.

1. Point A on the graph is labeled in red.  On figure 1, draw and label where the cart is located at position A.

2. Point B on the graph is labeled in blue.  On figure 1, draw and label where the cart is located at position B.

3. How is frequency related to period?

4. What is the equation for the period of an object on a spring?

5. When a block is dropped on the cart, does the frequency of oscillation increase, decrease, or stay the same?

6. When a block is dropped on the cart, does the amplitude of oscillation increase, decrease, or stay the same?

7. When a block is dropped on the cart, does the maximum potential energy of the cart-block-spring system increase, decrease, or stay the same?

8. When a block is dropped on the cart, does the maximum kinetic energy of the cart-block-spring system increase, decrease, or stay the same?

9. When a block is dropped on the cart, does the maximum speed of the cart-block-spring system increase, decrease, or stay the same?

Mail Time: how do I have students describe normal and friction forces?

Vanessa asks:

How do you have students list the normal force and friction force on an object experiencing friction? Would both Fn and Ff be described as "the force of the surface on the object"?

Or do you have them specify "the normal force of the surface acting on the object" and "the friction force of the surface acting on the object"?

Just "force of track on cart" or "force of the ground on the cart" or similar, like you said.

I work so hard to get students to avoid excess language (like "the downward force of the earth pulling down on the upward moving cart") that I'd undo that work if I insisted on other language.  The simplicity helps substantially with Newton's 3rd law, for which we just switch the objects experiencing and applying the force.  

The 3rd law force pair to the friction force?  Well, friction is the force of the track on the cart, so the 3rd law pair is the force of the cart on the track.  That easy - but only if the friction force is originally written with this concise language.

Mail Time: In the day-by-day plan, what are "four minute drill" and "dang fool questions"?

In my workshop materials for both conceptual physics and AP physics 1, I provide a day-by-day plan for an entire year-long physics course.  It's not that I expect teachers to follow it word for word, of course!  See, the most common questions I get at workshops are "In what order do you teach these topics?"  "What problems and labs do you assign during the momentum unit?"  "How do you review for the exam?"  and, especially, "How do you pace your course?"  Seeing the detailed list of activities and assignments that I actually used over a full school year can help teachers plan for their own classes, usually by adapting the general framework they see to their actual situation on the ground.

But, Marah wants to know: On that plan, I mention the "four minute drill".  What's that?

Here's the post about the 4-minute drill.  In 2012, this was a technique for getting my AP Physics B class to recall equations.  Nowadays, I riff off of each fact on the fact sheet.  "How do you find speed from a position time graph?"  "How do you find displacement from a position-time graph?"  And so on.  Very, very effective and fun.  I use this in all courses I teach, both conceptual and all forms of AP.

Marah's next question: What is this "Dang Fool Questions" class?  

Dang Fool Questions in AP Physics are generally the last day before an exam, or before The Exam.  I go through all the topics on the AP exam (linked is the 2015 AP Physics 1 version, you can still use it for P1 but cut the waves and electricity stuff) as fast as I can, in about 10 minutes of riffing.  Then I ask for questions.  They're "Dang Fool" questions because I say no judgment, ask whatever is on your mind, even if it's the simplest or most obvious thing in the universe, I'll answer patiently and kindly, no worries. Usually the questions I get are things they've encountered in the past but are worried they won't remember how to approach.  

Like, "How would you approach a flying pig-style problem?"  "It looks to me like forces and circular motion... so that means we use the force approach:  free body, components, N2L... where acceleration is v^2/r to the center."  "What are the typical things where you can't use kinematics, again?"  "Anything without constant acceleration.  The canonical situations are, object moving on a spring, object swinging on a string, or cart on a curved track.  In those cases, the free body diagram changes throughout the motion; so you can't use kinematics.  Make an energy bar chart instead."

How do you teach students to use simple scales for graphs?

From a physics teacher message board:

This topic recently came up on a Facebook group: Some students when plotting graphs will choose an axis scale that uses every line-division, but in turn makes the scale difficult to use and read when plotting points and calculating slopes, etc.

I agreed with most responses that say the student deserves full credit. But I also agree that the scaling makes the graph difficult to use when plotting points and calculating slopes. I always tell my students to select a scale that uses at least 50% of each axis AND is easy to use for plotting points and reading values. However, I have struggled to teaching students HOW to do this. I would appreciate any help or suggestions of exercises or teaching strategies that can reliably help students choose good scales that are easy to use and that work for a variety of graph paper types.

These recommended scaling instructions are exactly what I give my students.  Using 50% of the available grid space on each axis earns full credit for every AP physics rubric I've ever seen.  As for getting students to actually use simple and appropriate scales... as with so much of physics teaching, I think we have to let students mess up, then explain the better approach when the context for that advice is exactly right.

It doesn't matter what I say before we start a lab - no one is listening then.  Nor is anyone listening when we have finished an experiment and are ready to move on.  Just as when teaching Newton's Third Law, there's no One Weird Trick for perfect comprehension.  Rather, the best you can do is fight a year-long war of attrition.  Each time just one more student gets the idea of scaling experimental graphs, rejoice; then hope another student comes on board next week.  

I usually have the conversation about scaling at two key moments in the course of an experiment.  And I design the general approach to full-on graphical analysis laboratory exercises so these two moments are likely to happen.

(1) Data collection: All data is required to go on a graph as it is collected, though it's fine to first take one or two data points for the purpose of estimating the overall scale.  And yes, I have to holler and cajole and (figuratively) poke students until they actually follow this requirement.  But as I'm moving from group to group checking their graphs, I'll often see a graph as you describe - with each line-division representing 3.67 m/s, or 0.13 N, or something similarly silly.  Such a group is generally behind others, perhaps a bit frustrated, possibly arguing with one another.  That means they're ready to listen.  

"Hey, can you tell me what x and y axis values this dot (I point at the graph randomly) represents?  Yeah, it's tough to figure out, isn't it.  But what if you rescaled so that instead of the graph going up exactly to 0.82 N at the top, the top line were 1.0 N, or 1.2 N, or 1.25 N or something like that that's easy to deal with?  Take a look here (where I do a simple rescale on a draft piece of paper, and graph a data point easily)."  Generally at least someone in the group says "oh!" and does the rescale.  

"But Greg, only that one student understood - the other group members will make the same mistake again."  Yup, very likely.  My responses are, (1) see above about the war of attrition; and, 

(2) Analysis:  I only ask for one experimental graph per group.  If the fastest grapher (or the only person in the group who is any good at all at graphing) makes this initial graph, that's fine.  But the next part of the experiment generally requires students to linearize their graph, and then to use the slope of a best-fit line to determine an unknown quantity.  And for the linearization and analysis, each student must make their own graph individually.  They all take pictures of their group's data table, but then they graph the data by themselves.  Collaboration according to the five-foot rule is acceptable, which explicitly means students can't just copy other students' work.  They can talk, they can look, but they have to *do* the scaling and the plotting by themselves.

Students come and show me each step in their analysis, including the linearized graph.  When someone shows me a graph with inefficient scaling, the time is right to have a conversation.  Usually, if the scale is awkward, the data isn't quite plotted correctly; and even if the data is technically correct, they can't answer the "what x and y values does this dot represent?" question quickly.  Either way, I explain how an easy-to-read scale avoids these issues, I suggest a simpler scale... and I ask the student to redo the graph.

Here's where my approach differs significantly from what you're probably seeing on Facebook.  You're right that on an AP exam, a graph done correctly but with suboptimal scaling will earn full credit.  Thing is, my laboratory is emphatically NOT an AP exam.  It's a studio, a place for practice, collaboration, and learning.  The point of plotting data points isn't to earn points.  

Generally, a student sighs, goes back and replots their graph, and finds the new graph much easier.  Aha!  I can rejoice over this student's progress.  Sometimes they got help from a friend to do the rescale.  That's fine, too - because they have to physically create the graph by themselves, they also see for themselves the elegance and utility of a simple scale.  

Very occasionally, a student might try to argue that they don't need to rescale.  I say only very occasionally, because I'm not taking off points, I'm not docking a grade, I'm just asking the student to redo the graph, the same way an art teacher might ask their student to mount their painting to a more attractive frame before the big art show.  The only thing this student has lost is some time.  So such a student generally gets a sympathetic smile from me along with a firm, "I'm sorry you have to regraph, I know it's a pain in the butt, but it's gotta be done."  

(And if a student were to become hostile, the problem is beyond one of whether the graph is acceptable or not.  Passive-aggressive or actual-aggressive argument-for-the-sake-of-argument is unacceptable in the classroom, whether we're talking about scaling graphs, using free body diagrams correctly, or starting a justification with a fact of physics.  Culture building from day one of the class means that generally students listen rather than lawyer up.) 

This student who feels like they have "wasted" their time with a badly-scaled graph usually doesn't make that mistake again!  Lost time matters to students in a way that grades do not.  I cannot recall having this conversation about proper graph scaling in the analysis section of a lab twice in a year with the same student.   

Because my students have been graphing large data sets by hand all year, when the AP exam asks them to graph 5 or 6 data points, they practically laugh at the simplicity.  The large amount of in-class time spent on experimental graphs pays off in that moment.