Choosing a Unit of Measurement

Date: 05/06/2002 at 21:05:08
From: Karen A. Olenych (Mrs. O.)
Subject: Proper unit of measurement to use when measuring butterflies

Hi!  I am a second-grade teacher at a DeSoto County, Florida 
elementary school. My students just took the County's math assessment 
and faced this question (worded something like this):

If Sue were to measure the length of all the butterflies in her 
collection, which would be the best unit of measurement for her to 
use?
a-millimeter
b-centimeter
c-kilometer

I observe butterfly and moth biology for a hobby and could not figure 
this one out for myself so went home and consulted some standard 
field guides including Audubon's, Peterson's, Simon and Schuster...  
Some of the guides gave approximate wingspans in centimeters, while 
some did in millimeters. 

I expressed this fact to my County administrators, thinking there 
was a problem with the answer choices to the question. The County 
claimed the test question was valid because in math, generally, the 
unit used to measure an object is the one that is the smallest 
possible unit where the object is not less in length than the unit in 
question....  I think I am getting the wording mixed up here...  For 
example, by this argument, one would measure a car's length in yards, 
not inches, feet, or miles.  The county also said that second graders 
were not responsible for understanding measurement in millimeters.  I 
need to check on that too.  

I am surprised that they did not care to invalidate the question.  
Some of my students chose cm and some chose mm.  Please send me some 
sort of help for this situation.  I am looking for an outside, 
objective opinion. Thanks!

Date: 05/06/2002 at 22:28:26
From: Doctor Douglas
Subject: Re: Proper unit of measurement to use when measuring 
butterflies

Hi Karen, and thanks for writing.

I'm not exactly sure I understand you here. If the object is *not* 
less in length (emphasis mine), then we should always choose the 
smallest unit (e.g. micron, or nanometer, or even smaller) possible.  
I think probably what is meant is that one should choose the *largest* 
unit such that the unit (e.g. yard) fits at least once into the object 
(e.g. car).

Having said this, I don't believe that this rule is the best one to 
use.  In my opinion, one should use the units that give the most 
convenient numbers, or are consistent with what other people use. In 
other words, while it is true that cars might be "best" measured in 
yards (and give nice, small numbers such as 3.18 yards), what is often 
relevant is how that object compares to OTHER objects in various 
contexts: How many (thousands of) feet to the toll booth? Is my car 
too big for my garage? Am I parking too close to the stop sign? Here 
it becomes clear that feet are probably more convenient than yards, 
because of the variety of contexts in which we need to know the 
dimensions of cars. After all, it's rare that we're driving a car on a 
football field.

Now, what would be "most convenient" for the measurement of 
butterflies and moths? Here I would probably lean toward cm, because 
to adequately measure the spread in different sizes, the cm seems to 
be the most natural unit in that most butterflies and moths that I 
know are somewhat bigger than a centimeter. However, if there were a 
multitude of butterflies with wingspans under one cm (or perhaps we 
are interested in their sizes at various stages in their growth), then 
I would probably lean toward using mm.

I think the County question is poorly written, because both cm and 
mm seem to be natural choices. If instead the mm is changed to 
"micron" or something quite a bit smaller than a butterfly, then the 
"correct" answer would be clear.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/16/2002 at 23:42:17
From: Doctor Peterson
Subject: Re: Proper unit of measurement to use when measuring 
butterflies

I agree with Dr. Douglas that millimeter and centimeter are both 
perfectly reasonable units for this case. It's not clear to me 
whether the problem deals with length or wingspan, but in either case 
most butterflies will probably be in the range from a centimeter or 
two to ten or twenty centimeters at the most; and in that range, both 
units will give small numbers that are reasonably easy to handle.

In my mind the "best" unit would be one for which all commonly found 
values will be greater than 1 (to avoid needing small numbers like 
0.43, where the decimal point is easy to miss. But also, in many 
cases, I would like a unit that gives reasonable precision without 
needing a decimal point at all. I might prefer to say 12 mm rather 
than 1.2 cm.

Moreover, for some purposes centimeters are to be avoided. The SI 
metric system recommends keeping to powers of 1000, avoiding "centi-":

    How Many? A Dictionary of Units of Measurement - Russ Rowlett
    http://www.unc.edu/~rowlett/units/prefixes.html 

    The prefixes hecto-, deka-, deci-, and centi- are widely used
    in everyday life but are generally avoided in scientific work.
    Contrary to the belief of some scientists, however, the SI
    does allow use of these prefixes.

So rules besides the "small number" rule may make us choose 
millimeters rather than centimeters, just as Dr. Douglas pointed out 
that we should use whatever units are commonly used by others. In 
particular, if, say, the wingspans of eagles are typically in the 
tens of centimeters, we would not choose to use dekameters, both 
because that is a rarely used unit, and because eagles are likely to 
be compared to, say, sparrows, which would certainly be measured in 
centimeters or millimeters.

If this much can be said about a problem, then it is certainly too 
ambiguous!

-Doctor Peterson
  http://mathforum.org/dr.math/