Re: [Phys-L] Conceptual Physics Course

[John Denker <jsd@av8n.com>]

On 05/17/2012 08:22 AM, Jeffrey Schnick wrote:
> On a related note, what do you do to convince a person in  a lasting
> manner that the reciprocal of (1/x + 1/y) is not, in general, x+y?

Well, when I was in kindergarten, the emphasized the idea of
"Check Your Work".

In this case, even the simplest check would tell the tale.
I'm pretty sure the reciprocal of (1/x + 1/y) is not *ever* 
equal to x+y, not for any real-valued x and y, except in
the trivial case where they are both zero.

Apparently the authors of certain widely-used introductory physics 
texts have never heard of "Check Your Work".  I say that because
of the astonishing amount of bogus physics in those books ... stuff
that would not withstand even a moment's scrutiny.

Still, we can hold students to a higher standard.  We can remind
them to "Check Your Work".


=================

Here's another suggestion, more focused on this particular
case:  Tell them to choose y=1 or some other positive constant, 
and then plot z = 1/(1/x + 1/y) as a function of x.  Also show 
z = x and z = x+y on the same plot.

It should be clear that x+y is always larger than x, while
1/(1/x + 1/y) is always smaller (for positive real-valued 
x and y).  This is something well worth knowing, in the
context of resistors, which presumably have positive 
real-valued resistances.

====

As a related theoretical exercise:  Define the mathematical
operator "∥" (pronounced "parallel") according to

               1
  x∥y := ------------
           1/x + 1/y

Then ask students:  prove or disprove:
  a) is ∥ commutative?  
  b) is ∥ associative?
  c) does multiplication distribute over ∥?
  d) does ∥ distribute over addition?
  e) does addition distribute over ∥?
  f) based on the above, would you say that ∥ has the same 
   operator structure as addition, or multiplication, or 
   subtraction, or what?

In real-world electrical engineering, the concept of x∥y is
well known and quite useful.

Also this is an example of the spiral approach to learning.
We are spiraling back to the axioms of arithmetic, reinforcing
them and extending them ... making *connections* between old
ideas and new.