How to teach AP students to derive equations? Annotate.

From Anna, a regular reader:

Do you have any tips you offer students for questions that ask them to derive equations?  I am finding my students get into a tissy with this type of question, and every question is so different, I can't quite put a finger on "where to start" for students...

And that - where to start - is a holy grail of physics teaching.  Fact is, the vast majority of AP Physics students are fully capable of carrying out the relevant mathematics for a college-level physics problem if they know where to start.  Sure, they might make silly algebra errors like canceling across a plus sign.  Who cares.  If we're going to successfully teach students how to derive equations as on a qualitative-quantitative translation question, those students must become comfortable figuring out where to start, and then not fretting over algebraic manipulation.

My primary word of advice for derivations is "annotate."  The phrase I use on homework, quiz, and test questions is "Annotate your derivation with a description of your approach."  Then I simply don't accept a response if there aren't clear annotations - no matter how clear, no matter how correct, a derivation without words must be redone from scratch.*

* Whereas I generally accept the math errors and move on.  It's more important to reward physics done nearly right than to use political capital to correct a math error.

Once I convince students that annotations at all are non-negotiable, it's then a long process to explain how annotating physics (like, "writing newton's second law for the top block") is different from annotating math (like, "dividing both sides by g and canceling the m").  

I really want students to tell me two things.  First, what principle of physics are they starting with?  And then, if they make an algebraic substitution, why did they do that?  

The most important annotation is that very first - where did you start?  This annotation is just as important in a numerical problem as in a derivation.  So I demand this sort of annotation even on problems that say "calculate the acceleration of the block".  

Okay, how do students learn where to start?  The good news is, AP Physics 1 doesn't include that many starting points.  Start with 

• A free body diagram and newton's second law
• A kinematics chart
• The impulse-momentum theorem / conservation of momentum
• An energy bar chart

That's about it.  Universal gravitation, circular motion, and two-body problems are just special cases of newton's second law.  Rotation questions can still be categorized exactly as above, just with torque, newton's second law for rotation, angular momentum conservation, etc.  Yeah, I'm probably missing something somewhere; yeah, often two of these approaches must be combined; but this is a really good short list for students to refer to.

Once a student decides which of these approaches to use, they should annotate each equation with where it comes from.  "Writing N2L for the block at the top of the circle using this free-body."  "Acceleration in a circle is v^2/r."  "To get the smallest speed, the normal force must be zero."  I'd love for these words to be alongside the mathematics, perhaps with circles and thought balloons or other graphical means of indicating how the words relate to the mathematics.  This process is the soul and center of the qualitative-quantitative translation: the annotation process could also be described as translating between mathematical and physical reasoning.

I try to integrate derivation into problem sets gradually throughout the year.  Students are always more comfortable at first using numbers rather than variables.  So I set them up for success with calculation problems.  But I still require an annotation process.  When I add in some full-on derivation problems without numbers, it's a straightforward step to replace their numbers with variables.  Not an easy step for most, but a straightforward step.  

By year's end, we're doing full-on QQTs from the exam.  I don't mean at all to suggest that my students think these are easy!  But from the long year of gradually developing skills, they do know how to start and generally what to do.  So they can learn effectively from the mistakes they make.