Estimating the size of a molecule

The elementary kids have been learning about molecules, and they
learned about area and volume earlier in the year, so I thought it
would be great to use area and volume to estimate the size of a
molecule.  This is an appropriate challenge for 4-6 graders, but
beyond the 1-3 graders, so I did a different molecule-related
experiment with them (see next post).

I introduced the basic idea with a bunch of blocks (which happened to
be Post-it notepads, but could have been anything).  When the blocks
are bunched up together as close as possible (a 2x2 wide times 4-high
stack in this case), the bunch has a certain volume which does not
change when the blocks are rearranged. If I spread them out only
1-high, they cover a 4x4 area on the table, so their "footprint" has
quadrupled.  But the height of the arrangement now has to be 1/4
of its former height so that the total volume ("footprint" times height)
is the same.  So we can infer, or indirectly measure, the relative heights
of the two arrangements after we directly measure the relative areas of the footprints.

In equations, it looks like this:

volume before rearranging = volume after

but volume = footprint times height so that

volume before = footprint after times height after

so, dividing both sides by the footprint after rearranging, we get

volume before divided by footprint after = height after

If we manage to spread something out so that it's only one molecule
thick, the "height after" is just the thickness of the molecule, which
is exactly what we wanted to estimate.  Oil spreads out very thinly on
water, so that's what our experiment will involve.  We can't see
directly that it spreads out only one molecule thick, so the estimate
we get this way will be an upper limit on the size of the molecule;
each molecule could be smaller if the thin layer actually consists of
several molecules.  But these kinds of limits are very important in
science.  Often scientists can't measure some small effect but can put
an upper limit on it which is useful for reasoning about the causes
and consequences of that effect.  And the upper limit we will get from
this method is impressively small for a tabletop experiment.  (Maven
alert: oil is hydrophobic so the molecules actually stand on end like
blades of grass.  Thus, our upper limit will actually be on the length
of the oil molecule.)

Before getting the kids all messy with water and oil, I wanted to run
through a practice calculation with a collection of macroscopic
objects, which I thought would clarify the idea for them and make the
oil calculation easier when the time came.  I brought 30+ tennis balls
in a bag and had them determine the volume of the bag, then I poured
out the tennis balls one-deep on a table (with trays to bound the
area) and had them measure the area after spreading.  The ratio of
"volume before" to "area after" should equal the height of the tennis
ball, which we can measure directly at the end to confirm the
soundness of the method.

The problem was, the kids were overwhelmed with the math.  I think
they were rusty on area and volume, and also not used to thinking
about so many variables.  There's a "before" and "after" version of
any variable you care to mention: volume, area, height, length, width.
Although these are related in simple ways which allow us to solve the
problem by thinking only about the "volume before" and "area after",
this does not mean that it was easy for kids to restrict their
thinking to these two variables, nor that I made it as clear as I
could have.  We spent most of our time trying to work through this
problem.  Calculators might have helped, as kids were in cognitive
overload trying to do the arithmetic while simultaneously struggling
to understand what it meant.  So I would hear students proudly come up
with answers like "10" and I'd have to say "do you really think a tennis
ball is 10 inches high?"  In retrospect, I should have asked
Teacher Chris to conduct a review of area and volume on Thursday so
the kids would not be rusty on those topics.

Another idea, in retrospect, would be to get rid of the tennis balls and
have them estimate the thickness of a post-it note, given just a pad
of post-it notes and a ruler.  This seems like a better analogy,
because you can't directly measure the thickness of a post-it note,
and the strategy of dividing the thickness of the pad by the number of
sheets is pretty clear.  To make the most direct analogy, I would
implement the counting strategy by having the kids stick the notes in
an array on a tabletop, which is less boring than just counting them,
and then multiplying out the length of the array by the width of the
array.

In any case, only two groups got to spread oil on water, and only one
really got numbers out of it, but at the end we looked at the result
all together.  A small drop of oil was estimated to be 3 mm
across, for a volume of roughly 27 cubic mm, and when spread out that
drop covered an area of 100x130 mm, or 13,000 sq. mm.  The ratio of
27/13,000 then gives 0.002 mm or 0.00008 inches or 0.000002 meters.
That's a bit large because I think the drop size was overestimated,
but not too far off, and an impressive accomplishment for fifth
graders. 

Summary: this was too much for a self-contained 45-minute session.  I
should have coordinated with the teacher to have the kids brush up on
the math, and I should have offloaded the cognitive burden of
calculating by either supplying calculators or requiring rounding
(9x9x5 inches should be rounded to 10x10x5 inches for the bag of
tennis balls).  And I could have used post-it notes instead of tennis
balls as an analogy.  But next time, with these things in place, it's
going to be a fantastic activity for the 4-6 graders.

Link: here is a nicely illustrated page on this topic.