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 under-emphasized. 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 pre-algebra 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 real-life 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

1012

Billion

1,000,000,000

Giga

109

Million

1,000,000

Mega

106

Thousand

1,000

Kilo

103

*******

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*** = 1012. "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 Giga-dollars

$ 278x1012

$ 2.78x1014

52,700 Meters

52.7 Km

52.7x103 m

5.27x104 m

4,500,000,000 Years

4.5 Gyr

4.5x109 yrs

4.5x109 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:

  1. Write the number in metric, scientific notation, and engineering notation. (note: At the beginning, don't obsess over mixed systems eg: accept answers like mega-people, kilodollars, and micro-inches, this type of difficulty can be worked out with time.)
  2. 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 on-line and power points were created to compare magnitudes.
 

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