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.