Remember the Anna Karenina principle: Every successful
student understands the material more-or-less the same,
whereas each unsuccessful student is confused in his own
way. Therefore I'm not going to be bold enough to guess
what "the" right answer is. Let me offer a bunch of
comments, some of which might turn out to be helpful to
this-or-that student.
There are at least five ideas on the table here:
1) The Subject: line speaks of ratios. That's even more
basic than scaling. For example, the scaling law for a
pendulum says that the period scales like the square root
of the length. It's not a simple ratio.
If a student can't handle proportional reasoning, in all
likelihood there isn't a problem with the proportionality;
there's a problem with the reasoning. Directly teaching
"ratios" per se is generally a waste of time. For further
discussion, see below.
2) There is also basic dimensional analysis ... such as
checking each expression to see that the RHS and the LHS
have the same dimensions. This can be considered a scaling
argument, because if the dimensions aren't correct, the
expression won't scale.
As discussed below, this is pretty much mechanical, not
requiring much "feel" for the physics, or much creativity.
3) There are also scaling arguments that do involve physics.
Simple examples include asking how the area scales with the
perimeter, or how the cooking time scales with mass, or how
the braking distance scales with speed, et cetera.
4) In addition to dimensional analysis and dimensional
scaling, there is also /nondimensional/ scaling. This
requires quite a bit more feel for the physics.
5) At an even more sophisticated level, scaling arguments
can be used to conjure new laws out of thin air, almost
like magic. This is highly creative.
I will concentrate my remarks on items (2) and (3). There
is a tremendous amount that could be said about the other
items, but I'll leave that aside for today. If anybody
wants to hear about that, please re-ask the question.
As the proverb says: Learning proceeds from the known to the
unknown. So whenever a student is struggling, my immediate
reaction is to backtrack as far as necessary to find
/something/ the student understands. Then we can build on
that.
Let's start with item (2), i.e. asking students to check the
dimensions of any given expression. If they can handle that,
it's something we can build on. It is a poor man's scaling
argument: If the dimensions weren't right, it wouldn't scale.
If students can handle that, the next step is to make it a
firm habit. Check every equation to see that the dimensions
are correct.
If some students can't handle it, then we need to figure out
why. There are two main pedagogical possibilities:
a) If some students aren't checking dimensions because
they never heard of the idea, then they just need to
learn it and get into the habit. This takes time and
effort, but it's straightforward.
b) If students can't do it because of some deeper deficit,
attacking the symptom is pretty much a waste of time.
You have to backtrack until you get a handle on the
deeper deficit. Only then can you start moving forward.
Note that people sometimes don't think of dimension-checking
as being algebra, since it doesn't involve ``solving for
x''. However, IMHO it is algebra. It uses the /language/
and /logic/ of algebra. Dividing velocity by acceleration
is isomorphic to dividing X/Y by X/Y^2. In the latter case
the symbols are variables, whereas in the former case they
are some wacky abstractions, namely "dimensions", but the
logic is the same either way.
So ... the obvious hypothesis is that if a student is having
a hard time with dimension-checking, the problem isn't
scaling or dimensions or physics at all; the problem is an
algebra deficit. I'm not saying this is always the case,
but it is a hypothesis you can check, on a student-by-
student basis.
For some students this is a double-or-nothing proposition,
since the dimensional analysis can serve as a /motivation/
to learn algebra. All too often students never paid
attention to algebra because they couldn't see what it was
good for. The usual approach is to teach algebra in the
abstract and then move on to applications. This works for
some students but not others. For the others, it helps to
teach both together, or to go back and forth in tight
spiral.
====
Dimension-checking is not particularly creative. It can be
done by rote, or even by a mindless computer program,
without having any "feel" for the physics. It requires some
grasp of the language of algebra, but that's about it.
Dimension-checking applies to pretty much everything, even
the simplest numerical plug-and-chug exercises.
Let's now move on to item (3). That is, we consider
exercises where the answer is not merely numerical, but
involves some sort of formula.
Every formula on earth has "some" scaling behavior. I
mention this in order to combat the idea that scaling is
separate from other techniques. It is not something that
you sprinkle on as an afterthought, like powdered sugar on a
donut. It needs to be baked in, like the carrots in carrot
cake.
As always, it pays to start slow. A lot of people are
clueless about the most basic scaling ideas.
*) Let's start with a square. We make every length
in the problem bigger by a factor of 2. The area
goes up by a factor of 4.
*) Now consider a triangle. We make every length in
the problem bigger by a factor of 2. The area goes
up by a factor of _______ ?
It's amazing how many people answer "3". You can see where
this comes from: Inside the big square, they put four baby
squares, one in each of the four corners. Inside the big
triangle, they put three baby triangles, one in each of the
three corners.
At this point I say: I don't mind that you guessed. I don't
even mind that you guessed wrong. I mind that you didn't
check the work. It's a simple rule: Check the work.
I encourage equation-hunting. Yesterday I hunted five
equations before lunch. The point is, after you hunt up an
equation, you have to check that it gives the right answers.
In this case, there are lots of super-easy ways of checking.
*) For starters, you could draw the picture. https://www.av8n.com/physics/scaling.htm#fig-scale-tri-
This is one of the most basic reasoning skills: Draw the
picture. Feynman got a prize for drawing diagrams of
stuff that would otherwise have been nearly impossible to
keep track of.
*) Another fundamental skill is to check for consistency
with what you already know. You could make a pair of
triangles by drawing a diagonal across a square. We have
already established that for a square, if you double the
length in every direction, the area goes up by a factor of
4. It's a pretty safe bet that the area of the triangles
we just drew also goes by a factor of 4 also.
*) Or you could just remember that the area of a triangle
is half base times height. If the base doubles and the
height doubles, the area goes up by a factor of 4.
*) Another basic skill for reasoning and problem solving is
to solve an /analogous/ problem. In particular, having
solved a problem, look for a /generalization/. In this
case, we have considered the triangle and the square. So
you could ask what happens if we start with a pentagon
and double it in every direction. Or an octagon. Or a
100-sided polygon. Do you really think the area of a
100-sided polygon is going to increase 100-fold if we
double the distance in both directions? It seems kinda
implausible.
In fact any polygon can be dissected into triangles, so
you know immediately that if we increas all the lengths,
by a factor of N, the area goes up by a factor of N^2.
The same result holds for circles or any plane figure,
because we can get a tight upper and lower bound using
inscribed and circumscribed polygons with a sufficiently
huge number of sides.
My point here is that I tend to stop paying attention
whenever people talk about the issue of "proportional
reasoning". The real issue is never the proportionality.
It's the reasoning. We're talking about basic yet super-
powerful principles and stratagems that apply to everything,
from arithmetic to zoology and everything in between. Check
the work. Draw the diagram. Check for consistency. Look
for analogies and generalizations. Et cetera.
far-field behavior of the dipole E-field
That is a relatively advanced example (at least as far as
basic dimensional scaling is concerned). Do these students
known any calculus? The dipole is a lot less mysterious if
they do. They need to understand that μ•∇ is a directional
derivative. Then the scaling is easy: https://www.av8n.com/physics/scaling.htm#sec-calculus
The dipole field can be explained using just algebra, but
it's more work. It can be justified numerically, but that
requires knowing what you're looking for, and I would expect
students to need some pretty enormous clues before they got
much traction on that.
The rule about drawing the picture is tough to uphold in
this case. Sketching the field lines is a pain in the neck
in two dimensions, and even worse in three dimensions. Even
if you have a computer, it takes some pretty specialized
software to plot the field lines. http://docs.enthought.com/mayavi/mayavi/auto/example_magnetic_field_lines.html#example-magnetic-field-lines http://alumnus.caltech.edu/~muresan/projects/esfields/index.html
One way or another, it has to be done. For the introductory
students you pretty much have to hand them the plot of the
field lines. Then you can ask them to verify that it is
consistent with the equations. You can also ask questions
about various sub-features, such as the field on the axis,
the field on the equator, et cetera.
The same scaling result came up in our recent discussion of
tides. It's not the same physics of course, because there
is no such thing as a mass dipole ... but as Feynman liked
to say at every opportunity: "The same equations have the
same solutions." (Even if the physics is different.) https://www.av8n.com/physics/tides.htm#fig-tide-stress-solar
I took it as a good sign that the students were bothered by
the fact that the sun's gravitational field (at the earth)
is larger than the moon's, yet its contribution to the tidal
stress is less. That tells me that they were /trying/ to
come up with a scaling law. They weren't quite succeeding,
but they get credit for trying, and with a tiny amount of
help they will succeed.
The usual advice is
"Check the scaling behavior,
to see that it makes sense."
Very often the hard part is the second part. That is to
say, it's not "just" a scaling issue. The way I look at it,
scaling suggests some questions that you ought to ask, but
you need to do some physics to get the answers.
For example: Suppose you have a formula that suggests the
resonant frequency increases like the square root of mass
and decreases like the square root of the spring constant.
Is that plausible? Scaling tells you to ask the question,
but you need a feel for the physics in order to obtain the
answer.
Along the same lines: If students can't see that the tidal
stress scales like 1/r^3, it's probably not "just" a scaling
issue. There's a physics issue -- the equivalence principle
-- and without that they will never be able to understand
(much less derive) the scaling law.
=========
Here's another issue: A lot of students want physics to be
exact, like arithmetic is exact. I tell them it's not an
exact science; it is a natural science. Almost everything
anybody does is an approximation to one degree or another,
in physics, in engineering, in business, and in everyday
life.
Scaling laws are usually not exact. For example, the tidal
stress of the moon acting on the earth does not scale
exactly like 1/r^3. OTOH it's close, and an approximate
answer is a whole lot better than no answer ... especially
given that other far-cruder approximations have already been
made (such as assuming the moon sits directly over the
equator).
Students need to stop whining about inexact results. This
is not specifically a scaling issue, but part of a larger
growing-up-and-getting-a-clue issue.
-----------
The elementary type of scaling we are talking about today
is part of a package of techniques for checking the work.
Check the units, check the dimensions, check the scaling
more generally, check the symmetry, check that the vectors
and scalars behave as they should, et cetera.
For example, in the neighborhood of equilibrium, to lowest
order, the force had better be an odd function of x, and the
energy had better be an even function. That is a symmetry
check. It's not exactly a scaling law, but it's in the same
spirit.
So in some sense, introducing scaling is part of the
never-ending story of show the work, check the work, show
the checks. (This is far better than trying to have a
National Scaling Day where everybody suddenly learns how
to do scaling.)
The scaling we are talking about here is not particularly
tricky. If students are having a problem with it, the
problem is almost guaranteed to /not/ be what it looks like.
Some of the other things you can do with scaling -- like
conjuring up new equations out of thin air -- involve
serious industrial-strength wizardry. We can talk about
that some other day.