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Re: [Phys-L] Conceptual Physics Course



Easy... start with (1/x) + (1/y) ? (x+y) Really, use the ?. Then multiply both sides by x. Make sure they handle the distribution properly. If they don't do that right, you've got more to teach than this relationship.
Now you will have (x/x) + (x/y) ? x(x+y). Ask them what the first term is. If they don't answer 1, you've got more to teach than this relationship.
Now take 1 + (x/y) ? x(x+y) and multiply by y.
They now have y + (xy/y) ? xy(x+y). Hopefully they will see the y/y and comment on it. Reinforce that y/y is 1 and 1 * x is x.
y + x ? yx(x+y). Ask them if it's okay to commute the left side. If they don't know, tell them it's always okay to reverse the terms of addition (but not subtraction or division). I don't know whether to lie and say mult. can be commuted or say it usually can be.

Now they have x + y ? yx(x+y). Ask, Are these identical? No, okay, let's keep going. (This whole time you might have to force the student to stay with it if they are weak in algebra. The main point is NOT to disabuse this relationship. The point is to get them to figure it ou whether the ? is and = (like JD has said, "firgure it out"). You can guide them to it.
Algebra lets us divide both sides by the same quantity (if the quantity isn't zero) so let's divide by x + y. Now we have 1 ? yx. Is yx always 1? No. Is it sometimes? Yes. When? x = y = 1 or x = y = -1 (and two complex solutions). But not always. So, ? is only = in very special circumstances. In general, you can't write =. So don't do it.

Make the explanation a teaching moment of algebra of reinforcing concepts.

Seems to me that this particular problem is illustrative of poor arithmetic and not poor algebra.. They've forgotten about common denominators.




________________________________________
From: phys-l-bounces@mail.phys-l.org [phys-l-bounces@mail.phys-l.org] On Behalf Of Paul Lulai [plulai@stanthony.k12.mn.us]
Sent: Thursday, May 17, 2012 7:43 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Conceptual Physics Course

I (perhaps mistakenly) assumed the question was asking one thing and implying another.
There was a conversation about students being uncomfortable using variables and preferring numbers. I read some of the posts, but not all. I apologize if I missed something. From what I read, I got the impression that a theme of something similar to: students need to practice using variables so they can see general principles, understand the nature of what is going on etc...
This question asked how to prove (1/x + 1/y) =/= x+y
Is there a clever (or not clever), simple way to show a kid with weak algebra skills the above? We all default to plug numbers in. The kids are probably a bit weaker than we are at algebra. They default to plugging numbers in much more quickly to a wider range of problems. If we go to plugging numbers in to prove something does or does not work, why shouldn't we expect them to do the same?

Have a good one.
Paul.

________________________________________
From: phys-l-bounces@mail.phys-l.org [phys-l-bounces@mail.phys-l.org] on behalf of John Denker [jsd@av8n.com]
Sent: Thursday, May 17, 2012 3:09 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.
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_______________________________________________
Forum for Physics Educators
Phys-l@mail.phys-l.org
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