
Large Numbers
Metric and Engineering Notation
Purpose: Students are typically taught large numbers,
metric, scientific notation, and engineering notation in different
units, in different classes, in completely different years, with
little or no effort made to link the ideas. These lessons are based
on linking the ideas. Also, in many schools measurement tends to be
underemphasized. These lessons are designed to develop measurement
concepts, and get students used to speaking about the parts of a
number.
Background: These lessons were used in a 7th grade
prealgebra setting.

Reformatted: January 2010 

Overview: In most modern applications large numbers, and
extremely small number, are represented with either metric,
engineering notation, or scientific notation. All numbers used in
reallife applications have units (meters, degrees, dollars, etc) and
the units should be understood as part of the number. Also, all
measurements contain uncertainty, and this may also be considered
part of the number. The three modern notation types break the number
into three parts:
 the basic part of the number which contains an estimate of
certainty
 magnitude  rough size of the number (eg: how many
digits does it have?)
 units of measurement (does 3.2 mean 3.2 Km or 3.2 mm?  Its a
big difference.)
The three modern systems for large numbers:
 Metric  uses a prefix (to the units) which corresponds to the
traditional name, or number of commas
 Scientific Notation  uses a power of 10 to show the magnitude
of the number
 Engineering Notation  a compromise between the first two 
the exponent always corresponds to the metric prefix, that is, the
exponent is always a multiple of 3.
Old Nomenclature

Traditional Notation

Metric Prefix 
Engineering and Scientific Notation

Trillion

1,000,000,000,000

Tera

10^{12} 
Billion

1,000,000,000

Giga

10^{9} 
Million

1,000,000

Mega

10^{6} 
Thousand

1,000

Kilo

10^{3} 
*******




hundredth ***

0.01

centi

10^{2 }* not part of Eng. Notation

thousandth

0.001

milli

10^{3} 
millionth

0.000001

micro

10^{6} 
billionth

0.000000001

nano

10^{9} 
Students need to practice the meaning before rushing into
practicing conversions. Students should know that billion = Giga*** =
10^{12}. "What is meant by 57 Kilometers?" is more important
than "How many centimeters make 57 Kilometers?"
Examples:
Traditional

Metric

Engineering

Scientific

$278,000,000,000

278 Gigadollars

$ 278x10^{12}

$ 2.78x10^{14}

52,700 Meters

52.7 Km

52.7x10^{3} m

5.27x10^{4} m

4,500,000,000 Years

4.5 Gyr

4.5x10^{9} yrs

4.5x10^{9} yrs

**note: many math text books still present the metric
prefixes: deci, Deka, Hecto, etc. The prefixes are no longer in use
in the international scientific community, and need not be taught.
Similarly, some older text books use mixed notation, such as: 2
kilometers and 320 hectometers. The proper notation for this is 2.32
Km. The mixed forms are incorrect.

Related Pages at this site
Worksheet


Reinforcing:
Step [1]: Have student collect numbers (>1000 or
<0.01) from meaningful sources: newspapers, sciences texts, social
studies lessons.
Have them report on the following:
 Write the number in metric, scientific notation, and
engineering notation. (note: At the beginning, don't obsess over
mixed systems eg: accept answers like megapeople, kilodollars,
and microinches, this type of difficulty can be worked out with
time.)
 Compare the number to another meaningful number: for example
the number of people afflicted with AIDS might be compared to the
number of people in your city, or the number of people with TB.
The federal deficit might be compared to the federal budget, or
the number of taxpayers. (Again, initially students do not need to
know the value of the number being used for comparison, only that
such a number exists, what its units would be, what the comparison
will tell you, and what operations can be used for comparison.)
Step [2]: Have students organize the numbers they have
collected by units, one pile for meters, one for dollars, one for
people, etc. This is now a good time to talk about conversions. The
inches pile should be put together with the meters pile, but only
after inches are converted to meters to make them consistent.
Step [3]: Make a magnitude chart (each step up represents a
power of 10.) for each units pile. This will help reinforce
comparisons eg: "Do more people have STDs or cancer?" "How does that
show on the chart?"
Step [4]: Once the chart is made, it can be used for
further research. Are there blank spaces that could be filled in?
Should meaningful numbers for comparison be looked up?
Comments:
 Modifications to this lesson would be great for thematic
planning: Math & Health, Math & Social Studies, Math &
Science.
 The Chart part of this lesson can be a great introduction to
logarithmic graphing. (An
example.)
 Students could make wall charts or data bases for use by
future students.
 Save this year's charts to motivate next year's project.
 Notice, from the chart, how easy it is to convert between
engineering notation and metric. For some it might be easier to go
from traditional to metric to engineering then to scientific.

This lesson was done at one school as math homework. Articles were cut out of newspapers and the numbers were discussed. At another school it was done as a technology project where numbers were researched online and power points were created to compare magnitudes. 