Choosing the Most Appropriate Unit

We’ve been looking at measuring and drawing devices (compass, ruler, protractor); let’s move on to units of measurement. A fairly common question for students learning about measurement is, “What is the most appropriate unit for measuring ___?” The answer is not always clear, as we’ll see.

General principles

Here is a question from 2003:

Choosing Appropriate Units

I don't get how to choose appropriate units for measurements. For example, what units would you use to describe the size of a garbage can?

Some other typical questions of this sort that we have seen are:

Which is the best unit for the length of (a) a paper clip; (b) a pencil; (c) a railroad route; (d) scissors? You may use meters, centimeters, decimeters, or kilometers.
What is the best unit of measure for finding the surface area of a rectangular prism measured in inches?

... for finding the distance around a table measured in cm?

... for finding the space inside a carton measured in feet?
The BEST unit to measure the weight of a person would be:
a) ounces  b) grams  c) milligrams   d) kilograms

The amount of blood circulating through a person's body at any one time would best be measured in:
a) kiloliters  b) liter  c) milliliters  d) gallons
In which case would it be most appropriate to use miles as a unit of measurement?

A. length of a soccer field
B. distance from Washington to New Orleans
C. length of a crayon
D. diameter of a penny
What is the more reasonable unit of measurement?
A. length of an arm   m  cm  mm
B. length of an automobile   dkm   m   dm
C. distance from NY to LA   km   hm   dam
D. weather satellites orbit earth at the altitude of    km   hm  m

These questions are meant to develop a student’s sense of how large each unit is; and a large part of the thinking involved will be visualization. But there can be more to it than that.

Doctor Ian answered by making up three such questions of his own:

Suppose someone asks you

1. The distance from Los Angeles to New York.

2. The distance from your house to the one across the street.

These are all examples of length, right?  And there are lots of different units of length: millimeters, centimeters, meters, inches, feet, yards, kilometers, miles, light years, city blocks, and so on.

In the present question, we haven’t been told whether to use metric or other units; that is probably an omission by the student. Most such questions we see seem to be about metric units, which in American schools would be emphasized because they are less familiar.

How do you know which units to choose?  Well, suppose you tried to tell someone the distance from LA to NY in inches.  That would seem silly, right?  Why?  Because the number would be enormous!  And people have difficulty dealing with numbers that are really large.

On the other hand, saying 'about 3000 miles' is something you can deal with.

So the main issue is the size of the numbers involved (180 million inches, in this case).

On another occasion (unarchived), Doctor Ian said, “In general, a unit is ‘appropriate’ if it gives you a number that is ‘mind-sized’… a number that you can visualize because it is within your daily experience. Usually this means no smaller than 0.01, and no larger than 1000.”

What about the distance from your elbow to your wrist?  To express that in miles would give you a really, really small number!  And people have difficulty dealing with numbers that are really small.

But saying 'about 10 inches' is something you can deal with.

This would be about 0.00016 miles.

What about the distance from your house to the one across the street? Miles will give you too small a number; inches will give you too large a number.  Feet might work, or yards.

If it’s 200 feet, we’d have to say 2400 inches, or 0.038 miles. Or we could say “about 70 yards”. There’s some choice there, isn’t there?

The point is, people strongly prefer numbers that aren't really big or really small.  So this is why we _have_ so many different kinds of units!  They let us keep the numbers manageable, regardless of whether we're talking about the distance between two atoms (which might be a dozen angstroms) or the distance between two stars (which might be a dozen light years).

One benefit of the metric system is that all the units of a given type (say, length) are related in a simple way, so whether we choose centimeters or meters, all we’re doing is moving the decimal point. It makes a bigger difference in American traditional units.

A teacher wrote us in 2002, wondering if a test question on this subject was inappropriate — and bringing some real-life insight into the question:

Choosing a Unit of Measurement

Hi!  I am a second-grade teacher at a 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!

Hmm … if millimeters are not in the vocabulary of a second grader, why are they in the question? (I’m guessing the question was remembered wrong, inserting millimeters because that’s what some books use.) But more important, scientists (who should know) don’t agree on which unit to use! There’s a lot going on here.

Doctor Douglas was the first to respond:

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).

Yes, Karen got the wording mixed up a little, as she suspected.  The claim is that the best unit to use is the largest possible unit less than the length of the object.

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.

So we might choose feet rather than yards for a car because we will be comparing that length to a variety of other things, and it is customary to use feet for other items within a fairly broad range of sizes. Using the same units makes it easier to compare them. (It would be interesting to study where yards are used, and figure out what special feature calls for that choice.)

So we have at least two considerations: small numbers greater than 1 in our measurements, and few different units used for a range of sizes. (The yard is relatively rare perhaps because it competes for “space” with the foot, which differs from it by only a factor of 3. The metric system is spared this difficulty because everything is in tens.)

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’ll have another possible issue here below.

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.

In real life, there is often more than one reasonable choice; and there is no “official rule” about this that makes one definitely correct.

Then I joined in, with an additional perspective:

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.

Since 1 centimeter is 10 millimeters, and 20 centimeters is 200  millimeters, neither gives unreasonable numbers.

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.

If our preference is to avoid decimals, then we will be inclined to choose smaller units. This, I think, is a matter of taste.

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.

My suggestion here is that scientists stick to the power-of-3 units (kilo, mega, milli, micro, …) in order to have fewer different units in play, which each cover a broad span of sizes in order to make comparison easy. This could be why some of the field guides use millimeters: They are following up-to-date guidelines for scientific usage. This recommendation seems to be missed at the elementary level – at least in part because authors there are largely dealing with everyday use of units, not scientific usage.

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

So, yes, the problem should have been changed.

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