A. Introducing the Metris System
The teacher should make an effort to have the students familiarize themselves with the metric system. For this any elementary book on math should suffice. The U.S. is lagging behind the rest of the world in converting to the metric system. This is an unreasonable position. There is a great beauty and convenience in working with this system. The American system, on the other hand, is one filled unnecessarily with difficulty. It is not so easy to remember that a yard is divided into three feet, the foot into twelve inches and the inch into sixteenths. It would be a lot easier working with the metric system which uses only multiples of tens. There is beauty and simplicity here. Instead of saying 12 inches in a foot and 5280 feet in a mile we would say 1000 millimeters to the meter and 1000 meters to the kilometer. These measurements are good not only for distances but for weights and volumes as well. Instead of using ounces and pounds (lbs.) one would use grams and kilos. Instead of pints, quarts and gallons we would use the liter. The American system seems one arbitrarily arrived at making it difficult to remember.
B. The Powers of Ten
In working with numbers there is a particular notation, called the scientific notation, that is useful and lends a certain beauty in working with very large or very small numbers. The system works briefly as follows:
10 multiplied by itself a certain number of times to reach an intended number, say 100 is 10x10 = 10
2
; 10x10x10 = 10
3
or 1000. Multiplying a number by itself produces a power of that number; 10
3
is read as “ten to the third power”. It is much easier and clearer to write or say 10
14
than 100,000,000,000 or one hundred trillion. We even run out of names when it gets that high. The small number written above is called an exponent. These numbers10-2 (powers) can also be written as a negative. We can have 10
-2
or 10
-3
etc. Instead of a number multiplying itself a certain number of times, this shows that the number will divide itself a certain number of times.
(figure available in print form)
or 0.001, etc.
You can multiply one power of ten with another simply by adding their exponents:
10
6
x10
3
= 10
9
Subtracting the exponents is equivalent to division:
07 Ö 10
5
Ð 10
2
All numbers, not just numbers that are exact powers of ten, like 100 or 1000, can be written with the help of exponential notation. The number 4000 is 4x10
3
; 186,000 is 1.86x10
5
.
There is a delightful movie called, The Powers of Ten.
This video is available in New Haven( see bibliography). The teacher should show this in class for a good demonstration of the powers of exponents. The students will also get a concrete idea on size, from the unimaginably big to the unimaginably small. The movie is also out in book form.
Following are a few examples of size comparison to help make the concept of size more concrete and palatable for the student. The teacher can, with the participation of the class, arrive at other interesting examples.
C. Illuminating comparisons
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1. The ant and the elephant
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____
If an ant were 3mm long and an elephant were 3m and if the ant became as big as the elephant how big would the elephant become proportionally?
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ant is 3mm
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elephant is 3m
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3mmx 1000 (elephant
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so,
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-
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is 1000 times longer
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3mx1000 (same
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than ant since 1m is 1000
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number you multiply
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times more than 1mm)
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ant by)
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= 3000m or
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____
____
roughly 2 miles
____
This elephant, then, would be about the size of a small city.
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2. If a drop of water were to grow as large as a basketball and the basketball grew proportionally how big would the basketball grow to be?
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drop of water is
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Basketball is about
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about 1mm
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300mm
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1mmx300
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300mmx300
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=300mm
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=90000mm or
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____
____
= 9000 m or about 100 feet
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3. If one million one dollar bills were stacked up one on top of the other how high would the pile reach?
____
1 dollar is about 0.1mm
(figure available in print form)
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4. If the Earth were the size of a basketball and the sun were proportionally small, how big would the Sun be? The diameter of the Earth is about 8,000 miles. The Sun’s diameter is about 800,000 miles. That makes the Sun’s a basketball
-
____
This is a very graphic demonstration of how the Earth’s size compares with the Earth.
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5. How many atoms are there on the head of a pin?
(figure available in print form)
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____
across. This is only in one direction! In order to find the area we would have to multiply this number by itself;, 1,000,000 x 1,000,000 =1,000,000,000,000 or 1 trillion atoms. This is only the surface area! In order to find the volume, which would be the head of the pin, then we would have to multiply it again by itself or 1,000,000 x 1,000,000 x 1,000,000 or, 1,000,000,000,000,000,000. A staggering number! Enough to make the mind reel.
