Unless otherwise noted, information contained in each edition of the Kansas School Naturalist reflects the knowledge of the subject as of the original date of publication.
The Metric System: An Introduction to Linear Units and Conversion
by Dr. George Downing
ABOUT THIS ISSUE
Published by Emporia Kansas State College
Prepared and issued by The Division of Biology
Editor: Robert J. Boles
Editorial Committee: James S. Wilson, Gilbert A. Leisman, Thomas Eddy, Robert F. Clarke, John Ransom
Online format by: Terri Weast
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THE AUTHOR
The subject chosen for this issue of The Kansas School Naturalist is by popular request. The author, Dr. George Downing, is an Associate Professor of Mathematics at ESU. His course on the Metric System has received high acclaim by both students and teachers. This issue is the first of a two part series. The second, dealing with the metric units as they apply to weights, volumes, and areas, will be printed as the October issue next fall.
The Metric System: An Introduction to Linear Units and Conversion
by Dr. George Downing
In the late 18th century, the French Academy of Sciences decided to throw on the scrap-heap the hodge-podge of measurement units commonly called the English system (the one we use with its inches, pounds, quarts etc.). In its place was substituted a newly devised system called the Metric System.
In this system, the standard unit of length was defined to be one ten-millionth of the distance from the North Pole to the Equator, measured along a meridian of longitude. This unit was named the "meter." Thus, the standard unit for length was defined in terms of a natural phenomenon, the earth's circumference, rather than in terms of some English King's outstretched arm-length or foot length as was the customary way of defining English units of length.
An outstanding feature built into this system was that each measurement unit, whether length, mass or liquid capacity, was defined to be exactly 10 times larger than the next smaller unit. Thus, conversion from one metric unit to another one could be done quickly and easily since all conversion factors were powers of
10 -- that is 10, 100, 1,000 and so on.
This new measurement system slowly gained acceptance among the nations of the world. However, the United States, from its colonial days to the present, exhibited considerable reluctance towards adopting the new system. The launching of Sputnik in 1957 began to change this attitude. This incident was the beginning of a period of renewed interest in science and mathematics in this country. The federal government poured money into these areas, and students flocked towards them in ever increasing numbers. With this movement came an increasing use of and familiarity with the Metric System -- the measurement language of science.
More recently, it became obvious that the United States, by clinging to the obsolete English System, was at a considerable disadvantage on the international scene of world trade. For instance, if an American manufacturing firm wants to sell its products abroad, generally, it is expected by the foreign nations that the measurement specifications on the products will be in metric units. This requires the American manufacturer to list two sets of specifications for its products -- one in English units for American purchasers and one in metric units for the consumers in other countries. Providing dual specifications on
manufactured items requires more time and expense than most manufacturers are willing to expend. Thus, we see more and more of them changing to an exclusive use of the Metric System.
In December, 1975, President Gerald Ford signed the Metric Conversion Act of 1975. This act called for a voluntary conversion from the English System to the Metric System. It also established that a Metric Board be instituted to guide the nation in the conversion. Although the act did not specify a time limit for the changeover, it was generally conceded that a decade was about what would be needed. However, United States teachers must not let this 10-year period lull them into complacency, for it is generally agreed that educators and their students should be in the forefront of the changeover movement. Thus, the primary purpose of this issue is to aid you in your understanding and teaching of the Metric System.
The Metric Prefixes
The following table includes the six most common metric prefixes together with their abbreviations and their meanings. It is of particular importance that these prefixes and their meanings be memorized. The
reason for this will become apparent as you continue your study of this bulletin.
THE METRIC PREFIXES | ||
Prefix | Abbreviation | Meaning |
kilo | k | 1000 |
hecto | h | 100 |
deka | da | 10 |
deci | d | 0.1 |
centi | c | 0.01 |
milli | m | 0.001 |
(Spellings and abbreviations are those recognized by the U.S. Bureau of Standards). |
In the event that you or your students have trouble memorizing the meanings of the metric prefixes, certain memory clues are listed below. They will help.
kilo | -kilowatt, thousand watts | -thousand |
hecto | -starts with h as does hundred | -hundred |
deka | -sounds like " decade" -- ten years | -ten |
deci | -first part of "decimal" -- one-tenth | -tenths |
centi | -cent is a hundredth of a dollar | -hundredths |
milli | -mill is a thousandth of a dollar | -thousandths |
In the next section, we will see how useful these prefixes can be as we examine the metric units of length.
Metric Units of Length
The meter is the standard unit of length in the Metric System. It is a unit that is just slightly longer than the familiar unit -- the yard. As we think how we might teach youngsters to become familiar with this new unit (and others to be defined shortly), it is tempting to consider having them study it in terms of an already familiar unit. Thus, we find textbooks which state that the meter is 39.37 inches long or that it is 1.094 yards long. Many times, such texts will follow up with problems such as:
5 meters is how many feet?
How many meters is 78 inches?
Billy is 1 2/3 yards tall. How tall is he in meters?
The major drawback to this approach is that the student is not motivated to think in terms of the new unit -- the meter. He is asked to think with and use English units, and then, for some seemingly arbitrary reason, translate them into metric units. The slogan, "Think Metric" is really the fundamental clue as to how to teach youngsters the new system. Encourage them to use the new system as an independent one. Do not require them to make conversions between the English System and the Metric System even though your text may still include an abundance of such exercises.
A list of teacher designed exercises could include the following types of problems which emphasize that the new system does not depend on the old one.
1. Estimate the width of the classroom in meters. Measure it with a meter stick or a metric tape to the nearest meter to check your estimate.
2. Estimate the length of the hallway in meters. Measure it with a metric tape or meter stick or trundle wheel to check your estimate.
3. Estimate the length of the teacher's desk in decimeters. Measure it to check your estimate.
4. Estimate the heights of several students in centimeters. Measure each one to check and refine the estimates.
5. Estimate the thickness of your pencil in millimeters. Measure its thickness to check your estimate.
Such problems make it necessary that the student begin to picture these units in his mind, and they require that he estimate how many such units would be needed to equal the length of an actual physical object. In other words, the learner is, indeed, required to think in terms of the new measurement unit instead of basing an answer on some English System unit and some hasty arithmetic conversion. This is the approach that is recommended as the best one to help students begin to learn and to use the metric units of measure.
Sometimes the meter is referred to, not only as the standard unit, but as the base unit. This simply means that it is the base upon which its fractional parts and its multiples are built. This is where the metric prefixes come in as you can see in the following table. A lower case "m" is used to abbreviate "meter."
METRIC UNITS OF LENGTH | ||
Unit of Length | Abbreviation | Meaning |
kilometer | km | 1,000 m |
hectometer | hm | 100 m |
dekameter | dam | 10 m |
meter | m | 1 m |
decimeter | dm | 0.1 m |
centimeter | cm | 0.01 m |
millimeter | mm | 0.001 m |
Now, let us take a closer look at the meter and its fractional parts. We will try to visulaize these new units in relation to rather common objects having these approximate lengths. Remember, as we help our students become familiar with these new units of measure, it is important that we do so without unduly appealing to their previous knowledge of the units in the English System. As much as possible, we want them to begin to "Think Metric."
The meter. As has been mentioned, the meter is the standard unit of length in the Metric System. This is, perhaps, the easiest unit to visualize because its length is so very close to that of the familiar yard. But,
since we want to try to see it in our mind's eye in relation to other objects, let's mention some different
ways we might use to visualize this unit. The meter is close to the width of a twin bed mattress, and it is about the same as the height that most door knobs are placed up from the floor. Most kitchen counters are about one meter high. The hoods of most large cars are about one meter above the roadway. Most professional basketball players are about 2 meters tall.
A two-meter tall basketball player showing a |
The meter is the unit which we will use in the future for measuring distances such as lengths and widths
of rooms, dimensions of city lots, dimensions of game fields and distances associated with field and track events. In fact, most of the measurements which are now stated in feet or yards will be stated in terms of this new unit -- the meter -- after the changeover is completed.
The meter is the unit which we will use in the future for measuring distances such as lengths and widths of rooms, dimensions of city lots, dimensions of game fields and distances associated with field and track events. In fact, most of the measurements which are now stated in feet or yards will be stated in terms of this new unit -- the meter -- after the changeover is completed.
The soup can and pocket comb have lengths of approximately one decimeter. |
The decimeter. The decimeter is 0.1 meter. It is about as long as a pocket comb or a Campbell's soup
can. The cover on an electrical wall switch is about 1 decimeter long. The diameter of the dial ring of your
telephone is close to 1 decimeter. The standard size square bathroom tile has a length of about 1 decimeter on a side.
The decimeter will be a good unit to use in the future for lengths of objects which we commonly use the foot for now. This is not to say that it is approximately a foot long. In fact, it is about as long as the width of your palm, but is is an appropriate unit to use for some lengt!1s shorter than the meter. For instance, a table lamp might be 6 decimeters high; a book could be 2 decimeters long; a picture frame might have a width of 4 decimeters; and, a bucket may have a depth of 3 decimeters.
The centimeter. The most common short unit which elementary students will need to know and use is the
centimeter -- 0.01 meter. It is a unit of length which is about as long as the width of your little fingernail. The diameter of a thumbtack head is about 1 centimeter. The width of the switch (just the switch, not the cover plate) on a wall light switch is about 1 centimeter. The distance between the prongs for a wall electrical plug is about 1 centimeter. As you can see, this unit of distance is relatively small.
A nail of the little finger is |
The centimeter will be the unit we will use for measuring distances which we commonly use the inch for now. Heights will usually be measured in centimeters. Johnny may be 120 centimeters tall. A newborn baby might be 50 centimeters long. Any relatively short distance can be measured using this unit. For example, your textbook may be 3 centimeters thick. A pencil might be 12 centimeters long, and a piece of chalk could be 8 centimeters long.
The millimeter. The millimeter is the shortest unit of length that we will have need for in everyday living. It is 0.001 of a meter-about the thickness of a dime. The diameter of the wire used in a paper clip is about 1
millimeter. A grain of salt on a pretzel stick might have a diameter close to 1 millimeter. The pencil lines of a student's drawing could be close to 1 millimeter wide.
A dime, held so that you can see it edge-on, is close to one millimeter thick. The diameter of the wire in a paper clip is also about one millimeter. |
Primary students will have practically no need for such a small unit of measure. Most of their attention should be directed to exercises requiring measurements in meters and in centimeters. Intermediate grade students, junior high students and senior high students will need to use the millimeter unit as their work in Science, Industrial Arts, Home Economics and Mathematics requires more and more of them in terms of accuracy of measurements.
To help your students become more and more familiar with these units of measure, you can use verbal drill exercises with them. Perhaps, on a given day, you may want to emphasize the unit -- the meter. You might ask your students to estimate the length of the chalkboard in meters. Make them write down their guesses -- commit themselves to actually thinking in terms of this unit. Then, have one or two students make the actual measurement to the nearest meter and announce it to the class. Follow this exercise up with a question such as, "How long is this room in meters?" Again, students should estimate this length, then measure it. "How high is this room in meters?" Use the same follow-up.
The tiny spheres are tick eggs. Each is about one-third of a millimeter in diameter. |
Allowing the students to measure after each question helps them to refine their estimating techniques. Consequently, they should become better and better at estimation, and they should become more comfortable in using the new unit. When you are satisfied that your students have a good feeling for a
given unit, then you can use the same sort of verbal drills placing the emphasis on a different unit of measure. Once your students get very proficient with each of the units, you can start devising your questions to mix the units. For example, you might ask for a length in centimeters for a given object and follow it with a question about a length in meters for another object.
Another approach which you might use is thi1 : Say something like, "I see something that is 2 meters long. What is it?" "Something in this room has a diameter of 3 decimeters. What is it?" "I see something in the room which is about 12 ceratimeters long. What is it?" In other words, make up exercises which require the student to match objects with given distances. Of course, you must be prepared to accept many answers for this type of verbal drill, for there will be many objects in your room with measurements close to the one you give.
Now, let us consider some of the units of length which are longer than the meter. These units will be more
difficult for the students to visualize simply because they are longer. It's relatively easy to pick up and handle a meter stick, but grasping something that is a dekameter long is usually not possible. In general, this is why most of us experience difficulty in estimating longer distances, but, let's give it a try!
The dekameter. The dekameter -- 10 meters -- is a unit of length which will not be used extensively in the future. We can get along without it. For example, although we could speak of a 4 dekameter distance, we will not. We will simply refer to it as 40 meters. Although this unit will not be used, teachers should, at least, be aware of it and have some feeling for it. For one thing, it is essential to consider it, if we are to understand that the Metric System is an orderly one built systematically on powers of 10, because this unit is intermediate between the meter and the hectometer. The dekameter is about the length of half of a tennis court -- from the base line to the net. Most city streets are close to one dekameter in width. As you walk, take 10 rather exaggerated steps and you will travel a distance close to 1 dekameter.
Half the length of a tennis court is approximately one dekameter. |
The hectometer. The hectometer -- 100 meters -- is another unit which will not be used for every day purposes. Most of us who watched the summer Olympic games recall that the 100-meter dashes were called just that, rather than the hectometer dashes. However, as teachers, we should have some knowledge of this unit. We need to know that in the orderly arrangement of metric units of distance, the
hectometer is the unit which is 10 times larger than the dekameter and one-tenth as large as the kilometer. This unit of distance is about the distance from home plate to the fence along the foul lines of a big league baseball park. In fact, in Royals Stadium in Kansas City, the distance down either foul line from the plate to the wall is almost exactly 1 hectometer.
Royals Stadium -- the distance from home |
The kilometer. The kilometer is the long unit which will be used in the future to measure distances for which we now use the mile. The kilometer -- 1,000 -- is approximately 6 city blocks long. It is a little more than half of a mile -- about 0.6 of a mile to be more precise. Elementary students enjoy laying out a course of 1 kilometer and then walking it and timing the walk. Your students should be able to walk a distance of 1 kilometer in about 13 or 14 minutes.
One way for you to investigate the relationship between the kilometer and the mile is to layout a walking course of exactly one mile, but in the course, also denote a position which is exactly 1 kilometer from the starting point. Then walk the course, pointing out when the kilometer distance is completed. Be sure to time both parts of the walk, the kilometer portion and the total walk, for later comparisons. Also, have someone keep a count of the number of blocks walked in the kilometer part of the walk as well as the total number of blocks walked so that the two units -- the kilometer and the mile -- can be compared in terms of blocks.
Exercises such as these point out that the kilometer is considerably shorter than the mile. With this understanding, we can more readily accept the notion that the number which describes a distance to a familiar place in kilometers should be larger than the number which describes the same distance in miles.
CONVERSION WITHIN THE METRIC SYSTEM
Do you remember working problems like the following ones when you were a youngster? How many inches are there in 17 miles? (We thought that this was a ridiculous problem, but we worked it anyway). How many rods are there in 594 inches? These were relatively difficult problems. In the first place, we had to remember rather unrelated and arbitrary conversion factors. Then, we had to perform some rather messy arithmethic, and the chances of making mistakes were considerable. For example, to work the first problem above, you needed to multiply 17 by 5,280 to get the number of feet in 17 miles. Then, this result had to be multiplied by 12 to convert the answer to inches. In the second problem, you must divide 594 by 12 to convert inches to feet, and then, divide this result by 16.5 to convert the answer to rods. Try these two problems on your own. If you do, you will appreciate how nicely conversion in the metric system can be accomplished. Problems such as these two will provide excellent motivation for your students as you
begin to teach them conversion of metric units within the Metric System.
In the English System, the conversion factors are not" nice." In the Metric System, they are. Let's see how conversion works in this system. How many meters are in 17 kilometers? The prefix "kilo" immediately should convey to you a conversion factor of 1,000. Since the kilometer is a longer unit than the meter, there will be more meters than there were kilometers in your answer. Thus, we multiply 17 by 1,000 and get 17,000 meters in 17 kilometers. Every conversion in the Metric System is done in exactly this manner. First: let the prefix convey to you the conversion factor -- 10,100 or 1,000. Second: You should think, "Will there be more or less of the new unit?" If more, multiply -- if less, divide.
Here is an important additional hint. As you teach conversion this way, you must keep in mind that the metric prefixes are relevant only to the base unit -- the meter. Hence, your problem must always involve
converting to the base unit or from the base unit. Here are some more examples to help you understand this point.
Converting To Base Unit From Larger Units. How many meters are there in 16 kilometers? Remember that we should multiply by 1,000. The answer is 16,000 meters.
How many meters are there in 53.75 kilometers? Again, we multiply by 1,000 to find an answer of 53,750.
Remember, multiplying by 1,000 requires only that we move the decimal point 3 places to the right.
How many meters are there in 48 hectometers? The prefix "hecto" brings to mind a conversion factor of 100, so we multiply 48 by 100 to get our answer -- 4,800 meters.
How many meters are in 23.58 dekameters? Multiply by 10 since "deka" brings to mind the conversion factor of 10. The answer is 235.8 meters.
Converting From Base Unit To Smaller Units. How many decimeters are in 537 meters? The prefix "deci" brings to mind the conversion factor of 10. Since we are converting to a smaller unit, we should multiply by this factor. The answer is 5,370 decimeters.
How many centimeters are in 12.506 meters? Here, the prefix "centi" conveys to us a conversion factor of 100. Again, we should multiply since there should be more centimeters in our final answer than there were meters in the original problem. Multiply 12.506 by 100, and our answer is 1,250.6 centimeters.
How many millimeters are there in 8 meters? The prefix "milli" causes us to think 1,000, and we multiply 8 by 1,000. Our answer is 8,000 millimeters.
All of the problems above have one thing in common. You should have noticed that each one required you to convert from a larger unit to a smaller one. Such conversions always require the operation of multiplication rather than division, and most teachers concede that multiplication is easier to teach and learn than division. Thus, it is recommended that students first learn to do conversion problems which require the operation of multiplication rather than division.
After they become proficient at converting from larger units to smaller units using multiplication, then they can try their hand at converting from the smaller units to larger ones. Conversions of this type will require that they know how to divide by such conversion factors as 10, 100 and 1,000. The following sample problems illustrate the technique.
Converting To Base Unit From Smaller Units. How many meters are there in 47,000 millimeters? The prefix "milli" brings to mind the conversion factor of 1,000. However, this time, since we are asked to convert from a smaller unit to a larger one, there should be fewer of the larger units. Thus, we shall divide 47,000 by 1,000 and obtain an answer of 47 meters. Again, the key idea is to let the metric prefix be your clue as to the correct conversion factor. Then, we divide by this number if we are converting to larger units.
How many meters are there in 11 centimeters? Let "centi" bring to mind the number 100. Divide 11 by 100
since you are converting to a larger unit of length. Move the decimal point two places left and the answer is 0.11 meters.
If a desk is 15 decimeters long, how many meters long is it? Here, the prefix "deci" is your clue to think of
10 as the conversion factor. But, again, since we are converting to a larger unit, we shall divide the original measurement by 10. The answer is 1.5 meters.
Converting From Base Unit To Larger Units. How many kilometers are there in 200,000 meters? The prefix "kilo" should cause you to think of the number 1,000. You should divide 200,000 by this number since you are converting to a larger unit. The answer is 200 kilometers.
How many dekameters are there in 75 meters? The conversion factor is 10 since "deka" stands for 10. Divide 75 by 10 and the answer is 7.5 dekameters.
A certain city block is 225 meters long. How many hectometers is this? The prefix "hecto" should bring to
mind the conversion number 100. We divide 225 by 100, and our answer is 2.25 hectometers.
As you can see, conversion within the Metric System is very easy. In the first place, all conversions can be
done by applying conversion factors of 10, 100, or 1,000. Second, the arithmetic associated with multiplaction or division by these numbers is very easy to perform and can usually be done in your head.
Two-Step Conversions. After your students have had a good deal of practice in carrying out conversions
on problems such as those mentioned in the preceding sections, you may want them to become proficient at working two-step conversion problems. For example, you might require them to convert 318 kilometers to centimeters. (Seldom would such a problem be encountered in everyday life, but it is a good problem for checking to see if a student really understands conversion). Notice that in this problem, conversion to the base unit or from the base unit is not explicitly required. However, the approach that is recommended is that the student should convert to the base unit first, and then, convert to the required unit. In other words, he should convert 318 kilometers to 318,000 meters by multiplying 318 by 1,000. Of course, he does this because the prefix "kilo" reminds him of the converstion factor of 1,000. Next, he should multiply
318,000 by 100 because "centi" brings to mind 100. The final answer is that 318 kilometers is the same as 31,800,000 centimeters.
This same two-step approach might, also, be used when converting smaller units to larger ones. For example: How many kilometers are there in 53,000 millimeters? First, we divide 53,000 by 1,000 to change millimeters to meters. The result of this operation is 53. Then, we divide 53 by 1,000 to change meters to kilometers. The result of applying this step is 0.053. Thus, we have converted 53,000 millimeters to 0.053 kilometers.
The value of this two-step approach to conversion problems in which you are not required to convert explicitly to or from the base unit is two-fold. First, the student can make the metric prefixes do the work for him using the techniques previously learned. Second, the student need not ever use conversion factors larger than 1,000 using this method.
BASE 10 NUMERATION AND THE METRIC SYSTEM
There is another approach to conversion within the Metric System which is particularly helpful to elementary and secondary teachers because it emphasizes the similarities which exist between our base 10 numeration system and the Metric System.
As we strive to teach youngsters the fine points of our numeration system, we continually point out that
it is a place value system. By this, we mean that the position a single numeral holds in the name of a
number is extremely important in that it names so many units, so many tens, hundreds, thousands and so on. For example, let us take the numeral 5,217, and place its individual symbols in our place value table as shown:
Thousands | Hundreds | Tens | Units | Tenths | Hundredths | Thousandths |
5 | 2 | 1 | 7 |
We should see that the number named here is 5,217 units, 521.7 tens, 52.17 hundreds, or 5.217 thousands. If you desired, you could express it as 52,170 tenths, 521,700 hundredths, or 5,217,000 thousandths by filling in the zeros in the right columns.
So now, what does all this have to do with the Metric System? Suppose we write down our column headings
again. However, this time, instead of units, we will write meters (since the unit is the meter when we speak of metric units of linear measure). Instead of tens, we will write dekameters; instead of hundreds, we will write hectometers and so on, remembering that the metric prefixes have been assigned meanings which make them perfectly analogous to the headings used in our base 10 place value table above. In the table below, the abbreviations of the metric units have been used to name the individual columns.
Suppose we are concerned with a measurement of 5,217 meters. Place the individual symbols of this numeral under the proper column headings as follows:
km | hm | dam | m | dm | cm | mm |
5 | 2 | 1 | 7 |
Looking at this table, we can see that the numeral can be interpreted as standing for 5,217 meters. Do you see that it also can be read as 521.7 dekameters? How many hectometers are in 5,217 meters? The answer is 52.17 hectometers. How many kilometers? Answer -- 5.217 kilometers. Undoubtedly, you notice the similarities between this example involving 5,217 meters and the example which we considered earlier involving 5,217 units. The problems are identical! Only the labels were changed. You should also see that, by filling in with zeros, 5,217 meters is 52,170 decimeters, 521,700 centimeters and 5,217,000 millimeters.
You can see that what we are doing is investigating another technique which can help us to understand conversion within the Metric System. The technique, as illustrated in this problem, is not as quick as the one explained earlier which makes use of the meanings of the metric prefixes. In fact, it should not be emphasized as the way to convert in this system. Its primary value is for you, the teacher, in that it points out the high degree of similarity between the Metric System and the base 10 place value system -- the one we continually use and teach. We, as teachers of this new system of measurement, can appreciate it more and come to understand it more thoroughly by investigating it in the light of our base 10 place value system.
The Metric System. It is a beautiful measurement system! Let's learn to use it!
REFERENCES
1. National Council of Teachers of Mathematics. A Metric Handbook for Teachers. 1976. For sale by the National Council of Teachers of Mathematics, 1906 Association Drive, Reston, Virginia 22091.
2. United States Department of Commerce, Brief History of Measurement Systems. Special Publication 304 A, September, 1974. For sale by the Superintendent of Documents, U.S. Government Printing Office,
Washington, D.C. 20402.
3. United States Department of Commerce, Report To The Congress-A Metric America. 1971. For sale by the Superintendent of Documents. U.S. Government Printing Office. Washington, D.C. 20402.
4. United States Department of Commerce, U.S. Metric Study Interim Report: Education. 1971. For sale by the Superintendent of Documents. U.S. Government Printing Office, Washington, D.C. 20402.
Can you give the measurements, in millimeters, of these objects: the pencil head,
the head of the map pin, the snail, the writing, the fish?
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