Teaching Units of Measurement
Dan Flath and Danny Kaplan
Macalester College
Spring 2011
What Students Know
1. That a liter is a volume, a meter and kilometer a length,
and a kilo is a weight? as is a gram. Some know that a
hectare is an area.
2. They know the meaning of “kilo”, “milli”, and
(sometimes) “micro”.
3. If they are US students, they know that there are 2 cups
in a pint, 2 pints in a quart, and 4 quarts in a gallon.
They know that an acre is an area and a square mile is a
different area.
4. If they have had some high-school physics or chemistry,
they know some numbers, e.g., 9.8
meters/second-squared and 32 feet per second per
second.
None of this has any relationship to what they studied in
high-school algebra.
? Yes,
we know that it’s not.
Units are a Hodge-Podge!
Many people regard units as a odd collection of archaic names:
barrels and bushels and tablespoons and liters and gallons,
pascal and psi and atmospheres,
I acres and square-feet and square-miles, tons and tonnes,
I seconds and minutes and hours and months and years and
decades
Many believe that it’s important to adopt a single, sensible
standard of units, like the metric system: grams for weight? ,
meters for length, cc for volume, seconds for time.
This (if it ever happened) would avoid the need to teach the
traditional names, but there is still a reason to teach about
units:
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To show the relationships among the different physical
quantities, e.g., length, volume, area, force, energy, pressure,
etc.
What do Units have to do with Calculus?
The fundamental operations that we teach in calculus —
integration and differentiation — are also fundamental in
relating different physical quantities to one another. Examples:
The surface area of a sphere is the derivative of volume
with respect to radius.
I Velocity is the derivative of distance with respect to
time. Acceleration is the derivative of velocity with
respect to time.
I Energy is an integral of force over distance, while
momentum is an integral of force over time.
I Obscurely to most people ... Temperature is one over the
partial derivative of entropy with respect to energy.
It’s not just that energy has units with funny names
(kilowatt-hour or British Thermal Units), but that energy has
a fundamental relationship to force and distance.
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Isn’t that Physics and Chemistry and Engineering?
Yes, but ...
I Let’s not forget that physics and engineering is a
fundamental motivation for calculus, both in terms of
student requirements and the historical development of
calculus.
I In Physical Chemistry, students see unfamiliar
relationships among quantities like entropy, temperature,
energy, pressure, volume, quantity. They need calculus to
understand these, and the ACS requires Calc III for a
Chem major.
I The same relationships between different “physical”
quantities show up in biology and economics, and involve
units that the physicists and engineers don’t discuss.
We can use units to illuminate mathematical topics, enhance
student ability to model, and to motivate the fundamental
operations of calculus in terms of real-world topics.
Introductory In-Class Activity
Group the following in a natural way:
foot, hour, mile, radian, dollar, liter, yen, second, day, euro,
gallon, acre, hectare, cubic inch, lightyear, teaspoon,
acre-foot, meters per second, newton, mile-per-gallon, miles
per hour, meters per second square, kilowatt, kilowatt-hour,
atmosphere, pound, kilogram, body-mass index,
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Students ALWAYS do this easily.
The groups are called dimensions
The group members are units.
Students know the concepts but not the words “unit” and
“dimension”; they need the words.
Class names additional units for given dimensions.
Ask class for additional dimensions.
Units and Dimensions
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Length : foot, mile, lightyear
Area : acre, hectare
Volume : liter, gallon, cubic inch, teaspoon, acre-foot
Time : hour, second, day
Mass : kilogram? , gram
Angle: radian, degree
Money : dollar, yen, euro
Velocity : miles per hour, meters per second
Acceleration : meters per second-square,
Force : pound, newton
... and so on. Next analyze dimensions.
? Finally,
we get there.
Fundamental Dimensions
The abstraction that underlies units is dimension.
The fundamental dimensions that will be recognized by all
students:
I Length L
I Mass? M
I Time T
I Temperature Θ (fundamental for our course)
... and, we might as well add ...
I Money $
There are additional fundamental dimensions that appear
mainly in specialized contexts: electric charge, luminance
? Although
they think this is the same as weight.
Derived Dimensions
We take the set L, M, T, Θ, and $ as the fundamental
dimensions. Each is the abstraction of a physical idea.
Other dimensions are derived from the fundamental
dimensions. Examples:
I Area L2 — a length times a length
I Volume L3 — an area times a length, or a length cubed
I Velocity L/T — length divided by time
I Acceleration L/T2 — velocity divided by time, or length
divided by time squared.
I Force M L / T2 — acceleration times mass: that’s what
F = M A is about!
I Wage $ / T — money per unit time.
Arithmetic and Dimensions
Arithmetic operations are allowed only when they produce
results with a meaningful dimension.
Only integer powers of the basic dimensions are allowed: only
Lk T l M m Θn $p for integer k, l, m, n, p.
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Addition and Subtraction: Both quantities must have
identical dimensions.
Multiplication and Division: Anything goes!
Exponentiation:
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The exponent must be dimensionless.
If the exponent would result in a non-integer power of
some dimension, the base must also be dimensionless.
Transcendental Functions: The argument must be
dimensionless.
Consequences of Arithmetic and Dimensions
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Choice of parameterization of functions: make sure to
have a parameter that can render the argument
dimensionless.
Example: sin(2πt/P ) or ekx but not sin(t) or ex .
Coefficients of polynomials have dimensions.
Example: In the polynomial a + bx + cx2 , the dimension
of b must “undo” the dimension of x and give the
dimension of a.
Parameterizing functions in this way gives a physical context.
Connecting Units with Calculus
Two basic operations of calculus: Integration and
Differentiation.
Each of these operations produces a result with a dimension:
I df /dx gives a rate: dimension is dim(f )/dim(x)
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f (x)dx gives an “area”: dimension is dim(f )× dim(x)
Of course, it’s not really an area, it’s a quantity whose
dimension differs from that of f (x) or x itself.
Interpreting derivatives and integrals
Knowing the units helps
The time, L (in hours), that a drug stays in a person’s system
is a function of the quantity administered, q, in mg, so
L = f (q).
1. Interpret the statement f (10) = 6. Give units for the
numbers 10 and 6.
2. If f 0 (10) = 0.5, what are the units of the 0.5?
3. Interpret the statement f 0 (10) = 0.5 in terms of dose and
duration.
Integral example
Pollution is removed from a lake on day t at a rate of
f (t) kg/day.
1. Explain the meaning of the statement f (12) = 500.
R 15
2. In the integral 5 f (t) dt = 4000, give the units of the 5,
the 15, and the 4000.
R 15
3. Give the meaning of 5 f (t) dt = 4000.
Example: Dimension and the Chain Rule
Calculus students learn ...
I Rules for derivs, e. g.,
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d t
t
dt e = e
d
dt sin(t) =
cos(t).
The chain rule.
But where are the units in
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d t
e
dt
= et ?
Chain Rule and et
If the variable t has dimension T , then et is not a meaningful
quantity.
We’ve been parameterizing it as ekt , where k has dimension
T −1
Typically, we are referring to the amount of something —
population, a radioactive quantity, money — so it’s Aekt ,
where A has dimension of the quantity being described.
The derivative d/dt is a rate by time, so it must have
dimension dim(A)/T . Thus, kAekt , since k has dimension
T −1 .
Physics-Motivated Examples I
Differentiation with respect to time
Position → Velocity → Acceleration.
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Time has dimension T
Position has dimension L.
Velocity has dimension L/T — position / time.
Acceleration has dimension L/T 2 — velocity / time
Physics-Motivated Examples II
Newton’s Second Law: F = ma
Dimension of force is M × L/T 2
Energy is Force times Length
Dimension of energy is M L/T 2 × L = M L2 /T 2
Momentum is Force times Time
Dimension of Momentum is M L/T 2 × T = M L/T
But what’s really happening is an accumulation: an Integral.
For Our Next M-Cast ...
1.
2.
3.
4.
Unit conversions
Simple dimensional analysis
Units and scale in economics, biology, etc.
Description of Homework Exercises