Re: [Phys-L] playing for keeps

["John Clement" <clement@hal-pc.org>]

The solution is to have them actually measure it.  I used a very nice WS
from Minds on Physics which had students calculate working for a father
pushing a merry go round with a constant force for 2 revolutions.  The
diagram showed the distance across.  So many students write 4 F D for once
around.  I would ask them to draw that path or visualize 4D, and of course
it is a square.  Then I would ask them to use a scale drawing to figure out
the distance around.  It is easy if you roll up a piece of paper to measure
the distance around.  I carefully avoid the word circumference.  Eventually
they come up with a number a little bigger than 3 to multiply by D (which I
never call diameter).  Then I ask them if there is a number from math which
might be the exact figure and they immediately know it is pi.  They only
know the equation with the memorized trigger words diameter and
circumference.

Similarly you can have them put 4 squares on top of a circle with the
corners at the center.  Then they can easily estimate the area which is a
number which looks like 3.xx r x r.  Once they visualize this, the equation
can be invoked.

They have never done the exploration necessary to make connections to the
formulae.  Actually the exploration should come first before the formula,
because once given the formula they stop thinking.

Part of the problem is that the majority of students are not proportional
reasoners, so these formulae do not seem logical and right.  So a better
method would be to promote proportional reasoning in middle school, then
explorationw with the formulae introduced later.

John M. Clement
Houston, TX 

> Good engineering and science students at good universities 
> have memorized the formulas C = 2*pi*R and A = pi*R^2 but 
> often do not know which applies when to what, nor that there 
> is any connection to things they really do know. It would be 
> awfully useful somewhere to start with the circumference 
> being about 4*(2R) = 8R (the circumference of the bounding 
> square) and then see that this is only a bit bigger than 
> 2*pi*R, and to start with the area being about (2R)^2 = 4R^2 
> (the area of the bounding square) and then see that this is 
> only a bit bigger than pi*R^2.
>
> An example of confusion is the frequently seen A = 2pi*R^2.
>
> Bruce
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