About Mathematics Commentary
Chapter 3. Dimensional Analysis
What this chapter is aboutDespite the high-toned terms in its title, much of this chapter is about some very elementary concepts, but they are
concepts that, in the past, have often been scrupulously avoided in mathematics instruction. The subject is labels.
An unfortunate aspect of label use is that it has long differed significantly between mathematics and science classrooms.
In school math classrooms their use is suppressed until the conclusion of an exercise, when a requirement is announced:
“The correct label is required for full credit.” On the other hand, many science teachers employ labels regularly throughout
their calculations in the way that is described in this chapter. For some students this creates the unfortunate illusion that
the use of numbers in science is somehow different from that in mathematics.
As the text points out (on page 37) the suppression of labels is no longer necessary or justified. Whitney has provided
the mathematical basis for their use within mathematical calculations. They can be factored and cancelled just as are the
associated numerical values. The key to much of this chapter, as described formally on page 38, is: if two measuring units
are equal (example: 100 cm = 1 m) then their quotient = 1 (example: 100 cm/1 m = 1) and thus multiplying or dividing by that
fraction (100 cm/1 m) does not change the value of an expression just as multiplying by one does not change the value of an
expression.
It is important to note that for lengthy calculations incorporating labels would simply clutter up the work so judgement
is necessary. The important points are that: (1) labels can lead the way to correct calculations, and (2) they can point out
processing errors.
Section Notes and Activities:Section 3.1 One way to motivate the content of sections 3.1 and 3.2 is to skip ahead to have students read the story of the Gimli Glider and
the Mars Orbiter on pages 45-46. Those episodes highlight the importance of handling labels in computations.
You’ll want to point out to students that labels carry the same meaning for both singular and plural: that is, both 1 in. and 5 in.
are appropriate and it is reasonable to cancel them while simplifying expressions. (That is: inch and inches are thus treated the same
and can be cancelled in appropriate settings.)
The more examples you can work through with your students the better as they need to see how the choice of what fraction to choose
to multiply by is central to the conversion task. It is one thing to convert miles directly to feet, for example, because the conversion
is obvious, but miles to inches or miles to yards require an additional step (assuming students don’t know the number of yards in a
mile. The key to all these conversions is the choice of the fraction by which to multiply.
Section 3.2 Although most students who have studied school algebra have seen the justification for the rote arithmetic rule: “To divide by a
fraction, invert and multiply,” demonstrated algebraically (as it is done on page 42), we urge you to repeat this development with
your students and to include additional numerical examples as well. In doing so you will be once again (as in Chapter 2) showing
that rote algorithms are really recipes justified by basic mathematical concepts.
Once again, the more examples you can work through with your students the better. For many of them this will be unfamiliar
territory.
In the example at the bottom of page 42, “miles per hour” is translated as mi./hr. This use of the word “per”, and to a lesser
extent “for” as fraction indicators needs to be stressed. Note in this regard that “percent” translates as “/100” in the same way.
A note about assigning exercises: The problem set for this section is extensive and includes a number of results that should
prove interesting to your students. If you simply assign a number of them as homework, their interest will often be subverted by
the work involved in the calculations. Another approach is to go over some of the exercises with your students, then discuss what
is going on in one or two others, possibly asking them to guess the outcomes, before assigning them as homework.
Section 3.3 Now the ideas of sections 3.1 and 3.2 are applied to two and three-dimensional situations. For some students the use of
in2 as equivalent to square inch and in3 to cubic inch will be new and it is appropriate
to stress that in2 is equivalent to in. times in. as well.
As with the exercises in Section 3.2, these exercises should be thoughtfully discussed before assigning any of them.
Section 3.4 There are a number of rules for rounding numbers to reflect measurement precision, but applying them reasonably is still
difficult. There is also the problem of zeros in representation. Does 5000 represent a number between 4500 and 5500 or between
4950 and 5050 or between 4999.5 and 5000.5? We don’t know unless we have additional information.
A famous story you may wish to share with your students illustrates this point. In 1856, under the direction of British Surveyor
General Andrew Waugh, the height of Mount Everest was calculated to be exactly 29,000 feet, but was publicly declared to be 29,002
feet in order to avoid the impression that an exact height of 29,000 feet was nothing more than a rounded estimate.
So, what are we left with? A general rule: You can’t do better than the least accurate of the measures you are dealing with.
And a strong dose of common sense. This section is not designed to teach the rules; rather, it is meant to raise concerns about
calculation with measurements. You might want to point out its relation to the problems raised about calculating with rounded
numbers in section 1.3.