Chronology | Current Month | Current Thread | Current Date |
[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
-----Original Message-----
From: phys-l-bounces@mail.phys-l.org [mailto:phys-l-bounces@mail.phys-
l.org] On Behalf Of John Denker
Sent: Thursday, May 17, 2012 4:09 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Conceptual Physics Course
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.
_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l