applications anJ analogies
edited by
RONDELORENZO Middle Georgia College Cochan. GA 31014
Dimensional Analysis An Analogy to Help Students Relate the Concept to Problem Solving James R. McClure Southern Illinois University at Edwardsville Edwardsville. IL 62026
Dimensional analysis is one of the most critical skills that must he mastered by beginning chemistry students a t any educational level. Dimensional analysis is a required skill throughout most general chemistry courses, with common a. ~.~ l i c a t i o nincluding s introductorv unit conversions, stoichiometw, and concentration units The crucial natuie of dimensional analysis has led to the introduction of a varietv of technioues to assist students in mastering the concept. Nearly all instructional techniques involve the development of two properties of conversion factors:
Figure 2.Solution of the initial problem. lntroduction of the Analogy between Conversion Factors and Dominoes Two conversion factors are first derived from a unit equivalent and then related to a "unit" domino, as shown in Figure 3. Unit Equivalent: 1 yd = 3 fl Conversion Factor 1
-
(Divide both sides by 1 yd)
(1)two
conversion factors can be derived from a single unit equivalent "Unit" domino
(2) conversion factors are ratios equal to unity.
While most students quickly develop a n understanding of the properties of conversion factors, a significant number have difficulty grasping dimensional analysis a s a problem-solving technique (i.e., linking information given to information sought through conversion factors).
"Unit" domino
information given x conversion factads) = information sought
Figure 3. Derivation of conversion factors and "unit"domino analogy.
The following method of introducing dimensional analysis has led to more rapid student mastery of the problemsolving technique and a smaller frequency of inversion of conversion factors in student calculations.
lntroduction of a Sample Problem Given the following unit equivalents, convert 2.00 kilometers into units of feet:
lntroduction of Linking Using Dominoes
Students are first presented the problem shown in Figure 1:to use three dominoes to link the gap between two isolated dominoes. Most students easily provide the solution shown in Figure 2.
Isolated domino
Isolated domino
Dominoes to form the link between the isolated dominoes
Figure 1. Presentation of the initial problem
"Kilometers" is introduced as the unit of the information given and "feet" as the unit of the information sought. The unit equivalents are converted to "unit" dominoes and the problem presented in "domino" format (Fig. 4). Students are then asked to provide the solution to the "domino" problem a s shown in Figure 5. By rotating the linking "unit" dominoes connter-clockwise, the solution to the "unit" domino problem can be converted to the mathematical equation shown in Figure 6. Figure 7 illustrates the final steps in the solution of the dimensional analysis problem. The numeric values are added to the given unit and to the conversion factors. I t is then demonstrated that the appropriate units cancel leaving only the multiplication and division operations to solve the problem with the remaining unit being carried over to the final answer. Conclusion An analogy has been developed that has led to a n observable decrease in the time required for students to attain a Volume 72 Number 12 December 1995
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Information given
Information sought
"Unit" dominoes to link the information given to the information sought L
Figure 4. Presentation of dimensional analysis problem in "domino" format.
km XInformation given
Information sought
mi krn
x-
Yd mi
x - =ft
Yd
ft
Figure 6. Conversion of domino analogy into a mathematical equation.
Linking "unit" dominoes Figure 5. Solution to "unit"domino problem fundamental proficiency in dimensional analysis. The initial presentation of a domino problem acquaints students with a familiar concept of linking one domino to another through a series of dominoes. Extension of the analogy to "unit" dominoes introduces the conceptual linking of one unit to another through a series of unit equivalents ("unit" dominoes). The analogy concludes with a simple mechanical operation that converts the linked unit equivalents into a mathematical equation consisting of a series of conver-
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Journal of Chemical Education
Figure 7. Solution to dimensional analysis problem. sion factors and provides the solution for a sample dimensional analysis problem:
