Embrace Chaos: science teaching and New England's deflated footballs

Which football is Belichickian?

In the runup to Super Bowl XVXIVXIVXIVXIVIXVIX, the NFL is investigating the New England Patriots for, perhaps, systematically underinflating the footballs they use on offense.  The sports media has gone crazy wearing out the "deflated ball" meme with puns and giggles well below the maturity level of the 9th grade boys I teach.  

In the true spirit of American anti-intellectualism, those who live outside New England condemn the popular and successful Patriots for cheating without waiting for any evidence better than "Cheater, Cheater, Pumpkin Eater."  Meanwhile, those who live *in* New England reflexively condemn the haters who dare to impugn the saintly Pats, even though not even his staunchest supporters would deny that head coach Bill Belichick would trip his own aged grandmother in a race if doing so raised the probability of victory.  

[For those of you who do not follow American football, the above paragraph must sound made up.  Trust me -- the students, faculty, and staff of my school have talked about little else for days.]

Our department has been asked to articulate "best practices of science teaching," things that we do that might be foreign to teachers in other disciplines.  Paul the chemistry teacher's response: 

Embrace chaos. What I mean by that is that, while organization and having a plan for where a lesson is going are important, it's equally important to leave room for serendipity. The "what would happen if we do this?" question that I'm not expecting is one that, as often as possible, I try to answer with "let's try it and see." Those questions are little clues as to what about the topic is going to be the hook of interest that keeps the student going through the difficult parts. Those are also authentic science experiences, in the sense that it's the way science really works---someone has a question and tries to find out the answer, and that investigation doesn't always go in the expected direction.

Paul embraced the chaos of the sports media's obsession with inflation pressure of footballs.  He asked the football team equipment manager to provide him with two new footballs, one inflated to 12 psi, one to 10 psi -- this was roughly the originally reported difference between legal footballs and Patriots footballs.*

*subsequently, it was found that reporters or someone had exaggerated -- the Patriots footballs were only 1 psi short of legal, not 2 psi.  That's going to be important to the next calculation.

First he had his students thrown the footballs around a bit.  While they all suspected that at least one was illegally deflated due to their overexposure to the sports media's "deflategate" meme, to the students both felt like normal footballs.

Next, Paul revealed that sure enough, one football was legally inflated, and one was underinflated.  He asked the class to handle the footballs and guess which was which.  The results showed that, even upon close examination, noticing the difference was pretty much a random proposition.  Paul is not entirely sure of his notes, but either his students today identified the correct ball by a 19-13 margin; or, they were dead split, 16-16.  I made my own guess, which happened to be right, but I was in no way sure of myself.

Finally, Paul introduced the ideal gas law to his students by way of a football inflation calculation.  We checked that a football was properly inflated to 12 psi in our 20 degree Celsius office.  Imagine now that we take the football outside on a 50 degree Fahrenheit day, like the day of the most recent Patriots game.  That would drop the Kelvin temperature by about 4 percent.  By the ideal gas equation, that would likewise drop the absolute pressure in the football by 4 percent.  Absolute pressure in the football would be the 12 psi gauge pressure plus the 14.7 psi atmospheric pressure.  Reduce that by 4% and then subtract the atmospheric pressure again and you find that the gauge pressure would drop to... 11 psi.  Down by 1 psi from the legal standard as measured by the officials before the game.  And exactly what recent reports indicate was measured by the NFL.  And that's a controversy, apparently.  

We haven't had the time yet to do the experiment -- we should leave the ball outside overnight to see if the pressure reading does in fact change by 1 psi or more.  That's next on the list.

Now, if Paul wanted to make this lesson truly interdisciplinary, he might discuss how the NFL conveniently leaked word of their investigation, knowing that the two-week media vacuum leading up to the Super Bowl would thus be dominated by ball inflation questions rather than pointed queries about the NFL's coverup of multiple instances of domestic abuse by their players this season.  Or Paul would discuss Mike Tanier's investigative report that found the Patriots footballs to be primarily filled with nitrogen.*  But Paul says he'll stick to chemistry.

* as well as the wonderful responses from the humor- and science- impaired.

Mail Time: How do you convince a student that motion is not always in the direction of the larger force?

Two identical blocks of mass m are connected by a string over a pulley.  Block A is on a horizontal, frictionless surface; Block B hangs from the string. Consider now that, having previously been given a brief initial shove, block A is sliding to the left across the smooth tabletop.

•Is the tension in the rope greater than, less than, or equal to mg?

One of a reader's students asked, "Why is the tension less than mg if the block is sliding left?"

She continued... "How can it slide left and the tension not be greater than mg if the block is pulled up even if it is slowing down?"

The reader explained that block A's acceleration and net force are to the right, since the system will slow down after the initial push. If tension is greater than mg then block B would have an upward acceleration, which would mean that the block would speed up while moving upward -- that doesn't happen.  And finally, the FBD  shows the block on top with only one force, that being to the right -- rightward net force on Block A requires a downward net force on Block B.

She wasn't satisfied with these explanations. Is there a better way of putting it?

I wouldn't say I have any better ways of explaining this issue; I use all of the above explanations.  This student is still conflating force and motion.  Any object -- not just these blocks -- can move opposite the direction of the net force acting on it.  That just means the object is slowing down.

Ask her and the class for examples of objects that move in one direction while experiencing a net force in the other direction.  A ball moving up in free-fall is the canonical example.  

I'd then set up this described system* in class, using a force probe or a spring scale to measure the tension in the string.  It's fun to watch the spring scale reading dip below the weight of the hanging mass as soon as you let go.  If your class is too large for all to watch the spring scale dial, use the Vernier force probe and project its reading on the screen.

* It's often called the "modified atwood" when two block are connected by a string over a pulley, but one of the blocks is on a horizontal surface.  See AP Physics B 2012 exam problem 1.

I ain't saying this experiment will solve all your student's misconceptions, but it should at least stop her from arguing.  That's what I love about physics: my students can argue, sure, until a smiley but facetious "Bet you $100 that the experiment works the way I say it will?" shuts them up a treat.  :-)

Adapting an AP Physics 1 question: motion graphs of a student in an elevator

The picture to the right is from the first problem on the 1993 AP Physics B exam.  That problem

asked for calculations and numerically correct graphs of position-, velocity-, and acceleration-time graphs given the force vs. time graph shown.  

When I adapted this problem for my AP Physics 1 class, I took into account two major considerations:

(1) The AP Physics 1 exam is not likely to require twelve(!) sets of kinematics and Newton's Law calculations.  So I need to find and ask about the conceptual essence of the problem.

(2) Short answer questions on the AP Physics 1 exam are only 7 points.  The original AP Physics B problem was graded on a 30(!) point scale*, looking at the results and methodology of each calculation and graph segment separately.  The revised question must be doable in 15 minutes -- that generally means only three lettered parts to the problem -- and scored with "fatter" points.

* The 30-point score was divided in half to get a standard 15-point problem.  This is the only AP Physics B problem in recorded history with such a nonstandard rubric.

The point of this post is not to show you a finished product, ready for the College Board to pick up for a future exam.  Note that I also am not correlating this question to any standards or learning objectives.  No, I'm just trying to respond to the numerous questions I've received about how to write test questions for AP Physics 1, while we don't have much in the way of officially published resources.  This question ain't perfect, but I hope I'm revealing some of my own thought process in writing problems; and then I hope you'll take my thoughts and make them your own.

Here's the revised AP Physics 1 style problem.  The rubric is below, too.  

(a) Describe the motion of the elevator.  In each of the five-second segments, be clear about the direction of motion, and whether the elevator is speeding up or slowing down.  Justify your answer.  [Comment: This question takes a good bit of writing to answer.  But it really rewards students who understand the physical process represented by the original graph.  No one can skate by, or even get partial credit, with just memorized equations.]

(b) On the axes below, sketch a graph of position vs. time for the 20 s shown in in the graph above. 

(c) On the axes below, sketch a graph of velocity vs. time for the 20 s shown in in the graph above.  [Comment:  Parts (b) and (c) are subsets of what the original problem asked, just with no calculational element, nor a justification.  When we do test corrections, I ask for justification with respect to facts about position-time and velocity-time graphs.  But since I asked such a verbally intense part (a), I don't think students would have time to justify these parts as well.]

The rubric I used to grade this problem:

(a)        3 points

1 pt for using N2L to correctly justify that acceleration or net force is upward from 5-10 s, zero from 10-15 s, and downward from 15-20 s

1 pt for describing upward motion the entire time from 5-20 s

1 pt for describing speeding up from 5-10 s, constant speed from 10-15 s, and slowing down from 15-20 s

(b)        2 points

1 pt for curved graphs of any sort from 5-10 s and 15-20 s, coupled with a straight graph of any sort from 10-15 s

1 pt for completely correct graph

(c)        2 points

1 pt for straight segments throughout

1 pt for completely correct graph

Remember, this rubric hasn't been vetted by anyone else; it seemed to work okay when I graded my one class's work one time.  At the real AP Physics 1 exam, we'll be grading three orders of magnitude more student responses than I graded.  I've no doubt that this rubric would have to be amended somewhere, somehow.

GCJ