Communication pubs.acs.org/jchemeduc
Displacement between Orders of Magnitude Method for SI Unit Conversion Ericka N. J. Ford*,† and Yvette V. Gilbert‡ †
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0295, United States Miller Grove High School, 2645 Dekalb Medical Parkway, Lithonia, Georgia 30058, United States
‡
ABSTRACT: The displacement between orders of magnitude (DBOM) method was introduced to help high school chemistry and physical science students perform conversions between units of the international system (SI). Students were taught how to numerically solve for the total number of placements between a set of units. The integer value, representing the DBOM, determines the multiples of 10 by which the coefficient should be reduced or increased. Further, the DBOM method gives a scaling factor between dissimilar SI units. This communication defines and gives instruction on how to teach the DBOM method within the classroom. KEYWORDS: High School/Introductory Chemistry, Misconceptions/Discrepant Events, Demonstrations, Mathematics/Symbolic Mathematics, Nomenclature/Units/Symbols
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of their numerical values or order of magnitude. A logarithmic scale was chosen to simplify the representation of SI units. The basic SI unit has an order of magnitude of zero, prefixes having values larger than one are represented by positive integers, and those smaller than one are represented by negative integers. Other logarithmic representations of measurement constitute values of the pH, Richter scale, and decibel units. According to the DBOM method, students must solve for the displacement between unit orders of magnitude
he ability to convert between units of measurement is a fundamental skill needed by science students; this instruction is covered in introductory science courses. Students are taught the factor label method as an approach to conversion between different units of the international system (SI) and units between different systems of measurement; in either case, the factor label method provides a method of linking dissimilar units.1−4 Base units, derived units, and those accepted for use with SI units are examples of basic units of measurement. When applying the factor label method, basic units such as meters, grams, liters, and so forth provide the link between prefixes of SI units. As a result of conversion, the decimal place within the coefficient of the SI unit is moved by one or more placements to the right or left (Figure 1). Introductory science students may also be exposed to mnemonic techniques for moving the decimal place for SI unit conversion.
O( )Initial − O( )Final = O( )Displacement
(1)
where O is the symbol for order of magnitude. The integer value should be placed between the parentheses of the symbol for order of magnitude. The difference between the initial and final order of magnitude is represented by the term O( )Displacement. The DBOM method (eq 1) enables students to shift the decimal in the appropriate direction and by the correct number of placements.
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DISCUSSION OF TEACHING METHOD
Introduction of SI Units
Students were introduced to the customary table of SI unit prefixes, as shown in introductory science textbooks, but with an additional column entitled “orders of magnitude”, as shown in Table 1. The additional learning of orders of magnitude prepares the students for the application of the DBOM method. In addition to memorizing the prefixes and values of SI units, students were encouraged to look for numerical patterns between the columns of Table 1. For instance, students were instructed to observe how each unit’s order of magnitude was
Figure 1. Decimal placement in SI unit conversion: the arrow shows the possible movement of the decimal point.
Our high school physical science and chemistry students are taught a simple mathematical approach for SI unit conversion, in addition to the factor label method. This communication will introduce that approach, which is called the displacement between orders of magnitude (DBOM) method. We also suggest steps for teaching it within a classroom. According to the DBOM method, SI units are represented by the logarithm © 2012 American Chemical Society and Division of Chemical Education, Inc.
Published: October 29, 2012 134
dx.doi.org/10.1021/ed300006e | J. Chem. Educ. 2013, 90, 134−136
Journal of Chemical Education
Communication
Table 1. Description of SI Unit Prefixes Numerical Value 1,000,000,000,000 1,000,000,000 1,000,000 1,000 100 10 1 1/10 1/100 1/1,000 1/1,000,000 1/1,000,000,000 1/1,000,000,000,000 1/1,000,000,000,000,000
0.1 0.01 0.001 0.000001 0.000000001 0.000000000001 0.000000000000001
Prefix
Abbreviation
Tera T Giga G Mega M Kilo k Hecto h Deca da Basic Unit (e.g., meter, gram, liter) Deci d Centi c Milli m Micro μ Nano n Pico p Femto f
Scientific Notation
Orders of Magnitude
1012 109 106 103 102 101 100 = 1
O(12) O(9) O(6) O(3) O(2) O(1) O(0)
10−1 10−2 10−3 10−6 10−9 10−12 10−15
O(−1) O(−2) O(−3) O(−6) O(−9) O(−12) O(−15)
to introduce students to the concept of scaling. For instance, the numerical values of the (given and final) SI units are defined in relation to the basic unit, and O( )Displacement represents the relative size of the given unit in comparison to that of the final unit. A centigram is 2 orders of magnitude (100 times) smaller than a gram (see Figure 2). The value of O( )Displacement, in Figure 2B, means a decagram is 3 orders of magnitude or 1,000 times larger than a centigram. Upon teaching this method, instructors may want to compare the relative sizes of objects measured with different scales of SI units. For example, students could be asked to describe the relative diameter of human hair (measured in micrometers) to that of a carbon nanotube. On the basis of the units alone, students should determine human hair to be at least 1,000 times larger than a carbon nanotube by calculating O(3)Displacement.
equal to the exponent in scientific notation, the number of zeros (n) in whole numbers, and −n − 1 zeros for decimal values. The relationship between positive orders of magnitude and numerical values greater than zero should also be made. Likewise, students should understand negative orders of magnitude to be indicative of values less than zero. Explaining the DBOM Method
As shown in the examples of this method (Figure 2), the orders of magnitude for the given and final units were entered into the
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CONCLUSIONS In Canagaratna’s discussion of problem solving techniques, the method of equations, which requires students to convey knowledge of definitions and relationships, may prepare students for advanced problem solving.4 Similar to the method of equations,4 the DBOM method requires students to define and relate physical quantities to each other. We also believe the DBOM method to be a more direct approach to metric conversion, which could then help students to focus on the mechanics of more advanced problems. Furthermore, the DBOM method has the potential to develop the problemsolving skills of students, as long as the meanings of displaced units and the derived physical quantities are understood by students. Because of the simplicity of this method, both the instructor and student should discourage rote memorization to use this method, as for other methods of conversion.4 Therefore, the ultimate goal of this communication is to offer the DBOM method as an additional tool for SI unit conversion and as a technique for pedagical research. Once the DBOM method has been taught, instructors can assign existing problem sets, pertaining to the conversion of SI units, to their students. Further, students could be asked to apply the factor label and the DBOM method to problems requiring SI unit conversion. The DBOM method is a mathematical determination that some students may intuitively perform. Nevertheless, we have observed that most students were able to correctly apply the DBOM method.
Figure 2. DBOM method for SI conversion from (A) a small to larger unit and (B) a large to smaller unit (i) without and (ii) using scientific notation.
parentheses of O( )Initial and O( )Final, respectively. The difference between both unit orders equals the displacement between orders, O( )Displacement. The numerical value of O( )Displacement also describes the number of placements by which the decimal point should move. As shown in Figure 2Ai, for conversion from centigrams to decagrams, a negative value for displacement indicates leftward movement of the decimal point, thereby decreasing the value of the given coefficient. The reverse was true for positive orders of magnitude (as shown in Figure 2Bi). Figure 2, parts Aii and Bii, gives examples of how to apply the DBOM method to numbers given in scientific notation. The parameters of the DBOM method (see eq 1) give relationships between units of measurement, which can be used 135
dx.doi.org/10.1021/ed300006e | J. Chem. Educ. 2013, 90, 134−136
Journal of Chemical Education
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Communication
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected],
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We would like to acknowledge NSF Grant 0338261 for supporting the Student and Teacher Enhancement Partnership (STEP) program between Georgia Institute of Technology and Miller Grover High School (in Georgia), which led to the development and practice of this method within introductory chemistry courses. We also thank the STEP program facilitators, Donna Llewellyn and Marion Usselman, for their support of this partnership. Kevin Ashley and Anton Puvirajah provided helpful feedback on how to help science instructors implement this method within their classrooms.
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(1) (2) (3) (4)
REFERENCES DeLorenzo, R. J. Chem. Educ. 1980, 57, 302. DeLorenzo, R. J. Chem. Educ. 1976, 53, 633. McClure, J. R. J. Chem. Educ. 1995, 72, 1093. Canagaratna, S. G. J. Chem. Educ. 1993, 70, 40.
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dx.doi.org/10.1021/ed300006e | J. Chem. Educ. 2013, 90, 134−136
