1.5 hr workshop on Measurement
Designed by Miranda Elliott Rader for the programs Thinking Through Craft: Wood and
Thinking Through Craft: Metal
October 13, 2011
Goals:
Students will appreciate the history of different units of measurement and their
relationships to one another.
Students will be able to differentiate between the concepts of precision and accuracy.
Students will be able to discuss the importance of precision in measurement.
Students will be able to predict how the precision of a measurement will affect scaling
that measurement up or down.
Outline:
- 5min: Warm-up activity: measuring ourselves
- 15min: A brief history of systems of measurement in modern American society
- 15min: Your turn: why do we measure, and how do we choose what to use to measure?
- 15min: A scientists view of precision and accuracy
- 30min: Activity on precision
- 10min: Wrap-up
Materials:
Powerpoint presentation on history of measurement
Ruler for each student with both inches and centimeters
Graph paper for each student (to make it easier to compare area)
Rectangles of cardboard or tag board cut to uneven sizes (not an even number of inches
or centimeters). Trimmed index cards work well.
-Warm-up activity: Measuring ourselves. Stand up, shake it out. Write on board after
saying: How many fingerwidths from your elbow to the tip of your finger? How many
fingerwidths tall is your head? How many fingerwidths long is your hand? How many
hands tall are you? Compare your findings with your neighbors.
- History of systems of measurement in modern American society: brief powerpoint
presentation:
All over the world, many people use units of measurement based on the human body. In
many circumstances, they’re the most practical and useful forms of measurement. Think
of a tailor-made suit, or a chair or table that’s the right height for you. Ancient Romans
used paces, the yup’ik of Alaska use the yagneq, the length from fingertip to fingertip,
and subdivisions based on other parts of the body. The “cubit”, the distance that you
measured between your elbow and the tip of your finger is a classic unit used historically
all over northern Africa and the middle east.
Story of the Smoot: Even in the modern era, body-based measurements can turn into
commonly accepted, useful measurements. In 1958, an MIT fraternity used the height of
one of their new members, Oliver Smoot, to measure the Harvard bridge (a bridge across
the river between Boston and Cambridge) in Smoots – about 5’7”. The entire bridge was
found to be 364.4 Smoots long The “smoot markings” are refreshed in paint every year.
Eventually, they became commonly used – police used the smoot markings to locate
crashes, and since no other measurement of the bridge has been undertaken, the cement
company that made new sidewalk slabs made them in 5’7” slabs instead of the usual 6’.
Interestingly, Oliver Smoot later became president of the International Standards
Organization.
Many large civilizations made standardized measurements for length and weight(mass) to
facilitate trade and taxation (chinese empire, Japan, northern india, 16th century songhai
empire in western africa). They made “systems of measurement” – specific, defined,
agreed upon basic quantities – units – for things that needed to be measured: length, area,
volume, weight/mass, time, angle, etc.
The current systems of measurement that we use most in the US are the US Customary
system, which is basically the same as the English or Imperial system (feet and miles,
pounds and ounces, pints and gallons) and the metric or international standard system (SI
units). However, there are many different particular units, or different definitions of
units, that apply to only specific substances (hardware, board feet, cord of wood,
women’s clothing sizes, and beyond)
The English system is based on the measurements that evolved in English society,
primarily by royal decree. These units began as body-measured units, whether the greek
proportion-based body units or Roman units. As English units began to be standardized
some were based on body part lengths of the king, etc. Edward the 1st (13th century
England) standardized the yard, foot, and inch (the inch was later standardized as the
length of three barleycorns laid end to end) and their proportions, and measures were
further standardized and defined by successive monarchs. The pound is the base unit of
weight in the English system and divided into 16 ounces.
The metric system began in the late 1700’s in France just before the French revolution.
France had previously had a completely non-standardized system of measures of which a
main one was the “king’s foot”. The spirit of pro-rationality and pro-science among the
elite, the intense need for a standardized measuring system, and the anti-king spirit of the
revolution all combined to make France adopt the metric system. A council of French
mathematicians and scientists decided to use 1/10,000,000 (one ten-millionth) of a
quarter of the earth’s circumference as the meter; they surveyed a longitude line through
France and used this to calculate. Unfortunately, the Earth is not actually perfectly
spherical, so their measurement was inaccurate (we’ll talk more about what “accuracy”
means exactly later). Nonetheless this measure was adopted as the meter. A standard
meter-long metal bar was constructed in 1889, and since 1983 the meter has been defined
as the distance that light travels in approximately 1/300,000,000th of a second (one 300
millionth). The kilogram is defined based on a prototype (a metal kilogram) in a vault in
France. It was originally conceived as the mass of a cube of water whose sides are 10
centimeters long. A gram is one-one thousandth of a kilogram, or the mass of one cubic
centimeter of water. Other useful facts about grams: a nickel weighs five grams and 1
ounce is about 28.35 grams.
- Your turn: why do we measure, and how do we choose what to use to measure? Make a
group of three with people sitting near you. For about 5 minutes, discuss these two
questions: Why do we measure? How do we choose what to use to measure?
(discussions happen). Let’s come back together. Who would like to share what their
group came up with? Record ideas. Get out: To find a value or an amount. To be able to
compare. To make things work. Allow things to interface with other objects from the
same system of measurement.
- A scientist’s view of precision and accuracy: I heard people mention precision, being
exact. I’m going to give you some scientific definitions of the terms precision and
accuracy. Scientists mean something very particular by precision. Write on board:
Precision: How reproducible a measurement is – with high precision, you measure
multiple times and get the same result. Also the degree of “granularity” or “detail” to a
measurement, how “fine” the measurement is – what is the smallest subdivision that you
can measure? Example: two scales at the co-op. Analog scale has markings every ¼
pound, and the pointer is fat: you can know the weight within ¼ of a pound. Digital scale
has readouts to within one 1/100th of a pound: it is more precise.
Accuracy: closeness of a measured value to the “real” or “actual” value. Example: the
scale might read 2 and ¼ pounds but the scale is off by one pound, and really your fruit
only weighs 1 and ¼ pounds: the measurement is inaccurate. Another example: one time
the QuaSR Center made rulers. But they photocopied them, which inadvertently shrunk
them slightly. So while the new rulers are just as precise (they also measure to within
one-eighth of an inch) they are inaccurate because they report inches which aren’t inches.
Measurements can be more and less precise, but no measurement is exact. Every
measurement has some amount of inherent uncertainty. Even the definition of the meter
is know only within 2 ten-millionths of a meter. Therefore, we have to be careful and
observant as we measure in order to not introduce unnecessary uncertainties – warping,
making big marks, etc – and lose some of the precision that our tool had. For scientists,
the most important thing is to keep track of how precise your measurements are, and not
report results that are more precise than the original measurements. Especially when you
do calculations!
- Activity on precision: Now we’ll get to try out measuring to different levels of
precision, and see how it affects scaling. What happens to the precision of our
measurements when we calculate with them? Get into groups of three again. Each
person gets a piece of graph paper and a ruler, each group gets a rectangle of cardboard.
Choose one person to be A, one to be B, and one to be C. Each person is going to
measure the rectangle and then draw it scaled up by a factor of two (twice as large). But
– each person is going to measure to a different level of precision. Person A measure to
the closest inch (round to the nearest inch) before you scale the measurement. Person B
round to the nearest half-inch before you scale. Person C measure to the nearest
millimeter before you scale up. (if there are some groups of four, have someone measure
to the nearest centimeter). Before you get started, have a little conversation so you are all
on the same page – what does it mean to scale up by a factor of two? Is that the same as
making the rectangle twice as large? When you’re done, compare your rectangles in your
group. I’ll write up questions: How are your rectangles different? Are they different in
proportion (the ratio of the sides)? Are they different in area? How? After your group has
had time to discuss, find another group and compare your conclusions with their
conclusions. Decide together: what happens to your error as you scale up?
-Wrap-up. Report back on answers to questions above. What else did you see? What
did you find out? What happened that you didn’t expect? It’s especially important to
have different groups compare and try to generalize about their conclusions. One group
may come up with an idea that is contradicted by another group!