ACTIVITY pubs.acs.org/jchemeduc
Using Role-Playing Game Dice To Teach the Concepts of Symmetry Anthony K. Grafton* Division of Science and Mathematics, Lyon College, Batesville, Arkansas 72503, United States
bS Supporting Information ABSTRACT: Finding and describing the location of symmetry elements in complex objects is often a difficult skill to learn. Introducing the concepts of symmetry using high-symmetry game dice is one way of helping students overcome this difficulty in introductory physical chemistry classes. The dice are inexpensive, reusable, and come in a variety of shapes and can be painted to reduce their symmetry. Because the dice have index numbers on sides or vertices, they can provide a uniform frame of reference for students and instructors. The hands-on activities using the dice can be done in class, as a dry lab, or as an out-of-class assignment. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Hands-On Learning/Manipulatives, Molecular Properties/ Structure, Student-Centered Learning
C
hemistry students often struggle with the ability to process spatial information even though it is a critical skill.1 5 In particular, determining the point group symmetry of a molecule or ion—an important step in applying symmetry in areas such as molecular orbital theory, spectroscopy, and crystallography— requires a student to “think in 3D” and mentally manipulate a complex shape to locate symmetry elements and test potential symmetry operations. In addition to simple 2D drawings and physical ball-and-stick models, other, more active techniques such as paper constructions,6,7 collections of everyday objects and molecular models,8 and numerous computer applications9 18 have been developed to help familiarize students with locating symmetry elements to determine overall symmetry. Of the computer-based activities, many have predetermined operations that students can simply select to watch an animation of the operations being carried out without ever being forced to find the associated symmetry elements themselves. The developers of at least one relatively recent computer program have recognized and attempted to rectify this shortcoming.9 Presented here is an alternative way to introduce students to the skills necessary to find and describe symmetry operations in complex three-dimensional objects using high-symmetry dice (Figure 1). Such dice come in several shapes, and they have certain properties19 that make them a simple, convenient, reusable, and inexpensive way to explore symmetry in and out of the classroom. Whereas the concept of using dice in this way has not been published before, dice are not totally unfamiliar to chemists. For instance, they have recently been used to represent complex packing in condensed phases20 and, as reported in a comment on a Web page describing a symmetry scavenger hunt, at least some students have recognized how high-symmetry dice relate to chemistry.21
Figure 1. Dice are readily available in a variety of shapes. Consistent side and vertex numbering makes it easy to direct a student’s view to the proper orientation. High-symmetry dice can be painted or marked to provide lower-symmetry objects.
12-, and 20-sided dice are readily available. The dice are inexpensive, with a bag of approximately 100 assorted dice manufactured by Chessex22 available from distributors such as Amazon.com23 for about $20. Such a bag will generally contain at least several complete sets of the six shapes, so an instructor with a class of 20 students working in pairs may want to purchase two such bags. Barring loss, a set of these dice should last for many years. Rather than having pips in each side to denote values similar to most common cubical dice, high-symmetry dice normally have Arabic numerals imprinted into the surface of each side. Dice are constructed so that, when possible, the values on opposite sides add to one more than the total number of sides on the dice. (This general rule does not apply to 4-sided dice because they do not have parallel opposite sides or to 10-sided dice, where opposite sides actually add to one less than 10 owing to the inclusion of
’ GEOMETRY AND PREPARATION OF DICE The dice used in some role-playing games have several interesting shapes. Besides the common cubical 6-sided dice, 4-, 8-, 10-, Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
Published: June 30, 2011 1281
dx.doi.org/10.1021/ed101023k | J. Chem. Educ. 2011, 88, 1281–1282
Journal of Chemical Education zero.) Except for color, any two dice with the same number of sides will have the same relative placement of numbers, although because there are many potential manufacturers of dice, it is certainly recommended that instructors verify this in a given set before attempting to use them as examples. Because dice have numbers that are recessed into the surface, sides or vertices can be readily painted or marked without covering up the numerical values. It is therefore easy to reduce the symmetry of a given die without sacrificing the common frame of reference provided by the numbers. A 4-sided die, for instance, which has Td symmetry, can quickly be changed to C3v by coloring one vertex or to C2v by coloring two vertices. The coloring can, of course, be done with paint, permanent marker, or even something as simple as correction fluid. Instructors may wish to create their own reduced-symmetry dice or they may allow students to color their own to achieve a given symmetry.
’ USING DICE TO TEACH CONCEPTS OF SYMMETRY To apply the concepts and power of symmetry to problems in chemistry, students must first be able to assign the correct point group designation to a given molecule. This process is usually learned as a flowchart where each decision is based on locating, or failing to locate, a particular type of symmetry element. Finding these symmetry elements can be a challenge for students in introductory physical chemistry, and using dice may help students master this ability more quickly. Rather than seeing a 3D molecular model on a 2D computer screen or trying to follow an instructor’s movements using a hand-held model, dice can be held in students’ hands, providing a tactile element to the learning process. And because the dice have sides (or vertices in the case of the 4-sided dice) with consistent numbering patterns, it becomes very easy for the instructor to refer students to the correct viewing angle to identify a particular symmetry element or observe a given symmetry operation. When using dice to introduce the concepts of symmetry, initial discussions with students should begin by pointing out particular symmetry elements in a given die. For instance, students could be instructed to take an 8-sided die and place it on the desktop with the side marked 1 up. The instructor could then note the existence of a three-fold rotation axis that is perpendicular to the 1 side and passes through its center. Looking down on the die, the students can see not only the top of the die, but also the sides labeled 3, 4, and 7. This makes it easy to lead them through performing the C13 and C23 clockwise operations, which interchange the 3 and 7 or the 3 and 4, respectively. This demonstration also provides the opportunity to see the equivalence of the C3 1 and C23 operations. Note that it is important at this stage that symmetry elements, which are points, lines, or planes, are distinguished from symmetry operations, which are actions performed about these elements that leave the object in an indistinguishable orientation (discounting arbitrary index marks such as the numbers on the sides of the die or assigned index numbers of atoms of the same element in a molecule). Ultimately, the students should be able to take a die of a given symmetry and, using precise language, describe the location of every symmetry element. Once students are comfortable seeing and describing the location of symmetry elements using the dice, they can then move on to directly determine the point group symmetries of the dice using a flowchart process. Maintaining the same expectation of rigorous language when locating symmetry elements in molecules is important to help students clarify their own thought
ACTIVITY
processes about complex shapes. Examples of possible questions and detailed answers are available in the Supporting Information. Instructors will want to tailor the application of these exercises to fit their preferences. Students could work through examples with the dice entirely during a lecture period, they could be led through the first few examples and left to do the others as homework, or they could be asked to complete the entire assignment as homework. Some instructors may prefer to conduct the entire exercise as a dry laboratory.
’ ASSOCIATED CONTENT
bS
Supporting Information A set of example questions and answers (instructors who wish to assign the entire set as an out-of-class project should consider providing students answers to one or two similar problems as examples of the kind of precise language required). This material is available via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The author would like to thank Lyon College for its support. ’ REFERENCES (1) Tuckey, H.; Selvaratnam, M.; Bradley, J. J. Chem. Educ. 1991, 68, 460–464. (2) Rozzelle, A. A.; Rosenfeld, S. M. J. Chem. Educ. 1985, 62, 1084–1085. (3) Baker, S. R.; Talley, L. H. J. Res. Sci. Teach. 1974, 11, 95–97. (4) Pribyl, J. R.; Bodner, G. M. J. Res. Sci. Teach. 1987, 24, 229–240. (5) Seddon, G. M.; Tariq, R. H. Eur. J. Sci. Educ. 1982, 4, 409–420. (6) Hansen, R. M. Molecular Origami: Precision Scale Models from Paper; University Science Books: Sausalito, CA, 1995. (7) Sein, L. T. J. Chem. Educ. 2010, 87, 827–828. (8) Flint, E. B. J. Chem. Educ. 2011, 88, 907–908. (9) Meyer, D. E.; Sargent, A. L. J. Chem. Educ. 2007, 84, 1551–1552. (10) Cass, M. E.; Rzepa, H. S.; Rzepa, D. R.; Williams, C. K. J. Chem. Educ. 2005, 82, 1736–1740. (11) Cass, M. E.; Rzepa, H. S.; Rzepa, D. R.; Williams, C. K. J. Chem. Educ. 2005, 82, 1742–1743. (12) Vining, W. J.; Grosso, R. P. J. Chem. Educ. 2003, 80, 110. (13) Kastner, M. E.; Leary, P.; Grieves, J.; DiMarco, K.; Braun, J. J. Chem. Educ. 2000, 77, 1246–1247. (14) Lee, A. W. M.; Chan, C. L.; Leung, K. M.; Daniel, W. J. J. Chem. Educ. 1996, 73, 924–925. (15) Vazquez-Vidal, L. J. Chem. Educ. 1996, 73, 321–322. (16) Potillo, L. A.; Kantardjieff, K. A. J. Chem. Educ. 1995, 72, 399–400. (17) Charistos, N. D.; Tsipis, C. A.; Sigalas, M. P. J. Chem. Educ. 2005, 82, 1741–1742. (18) Johnston, D. H. Symmetry @ Otterbein. http://symmetry. otterbein.edu (accessed Jun 2011). (19) Diaconis, P. D.; Keller, J. B. Amer. Math. Monthly 1989, 96, 337–339. (20) Haji-Akbari, A.; Engel, M.; Keys, A. S.; Zheng, X.; Petschek, R. G.; Palffy-Muhoray, P.; Glotzer, S. C. Nature 2009, 462, 773–777. (21) Wile, B. M. Symmetry Scavenger Hunt. https://www.ionicviper. org/class-activity/symmetry-scavenger-hunt (accessed Jun 2011). (22) Chessex Home Page. http://www.chessex.com/ (accessed Jun 2011). (23) Amazon Online Shopping Home Page. http://www.amazon.com/ (accessed Jun 2011).
1282
dx.doi.org/10.1021/ed101023k |J. Chem. Educ. 2011, 88, 1281–1282
