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(1) Is there still an important role for log-log paper, or logarithmic
graphs?
I want my students to have appropriate tools to examine and interpret data
from experiments in which they have an idea of how things should turn out,
as well as situations in which they don't know what sort of relationship
to expect. One of the things I was taught to do was to throw data onto
log-log paper and see if it makes a line. If so, find slope.
My students ask me: Why do this, when I can just test out a curve fit of
form y=Ax^B?
(2) How is linearizing data different or more useful from curve-fitting it?
I notice that the AP test is big on linearizing data. I only mention this
technique in class so that students won't be surprised by this if it comes
up on the test. Am I doing wrong by my kids?
Suppose we divided Bob's students into groups and asked them to look at
the sand crater experiment he describes. Here's what they're going to do:
a. graph diameter^4 against height.
b. graph log(d) vs. log(h).
c. graph d vs. h and do a fit to Bd^A and ask for best values of B and A.
Will they come to different conclusions about crater-formation? Will they
learn different lessons about data analysis?
(3) Is there a significant advantage (or disadvantage) to designing an
experiment so that the value of interest may be extracted from a contrived
linear relationship?
example:
I wanted to teach a lesson on uncertainty last week. I walked into class
and said: "This is a speed lab. The school has been moved to another
planet where rents are lower. In order for physics research to continue,
we need to know the value of g in our new home. The Provisional Planetary
Governor wants an answer in ten minutes." No other instructions.
I set out atwood machine parts, meter sticks, stopwatches. Ten minutes
gave them about enough time to get set up and drop weights maybe eight or
ten times. Later we discussed why not everyone's results agreed, and what
level of accuracy the Provisional Planetry Council should ascribe to these
values.
Here's the question:
Most students measured the distance a weight dropped from rest, the time
for the fall, and the two masses. In this "speed lab" setting, everyone
decided to compute g based on the data for each run, then take the
average, excluding values they "didn't like."
What if instead we had tinkered with the mass values in the lab, then made
a graph of (acceleration) vs. (m-m/m+m) and looked at the slope? Is this
value for g somehow a better way of aggregating the data? worse? the same?
I suspect that this linear regression makes the extreme values more
significant than the other values...
(4) What IS an appropriate sequence to suggest for students faced with
data when they have no initial hypothesis about how to model it?
Left to their own devices, my students will go through all the suggested
curve fits in LoggerPro looking for the one with the lowest RMSE. I tell
them this is silly - that they should have some other criteria for what's
a good model.
-Does model predict behavior at extremes? (eg does your model tell you
that a cart has zero acceleration on a flat ramp and a=g vertically?)