# Collaborative Team Competition for Identification of Symmetry Operations on (Paper) Models

## In-Class Activity

Submitted by Sophia E. Hayes, Washington UniversityStudents work individually, then compete in teams, to identify symmetry elements and operations present in a high-symmetry structure, such as an octahedron or tetrahedron (without showing the character table until the end of the activity). Students often visualize symmetry elements differently from one another. Creating teams, allows them to work collaboratively, and the competition adds an incentive for finding the most elements. Since some students are better at seeing some symmetry elements (and operations) than others, it allows for them to work in small groups to both teach and learn from one another.

In participating in this activity, a student will:

- Identify symmetry elements/operations on a 3-dimensional model
- Distinguish between a symmetry element and symmetry operation
- Determine equivalent symmetry operations
- Understand how the top row character table is populated with symmetry operations
- Identify classes of operations—why specific operations are grouped together in character tables
- Work with peers collaboratively, to identify these symmetry operations

Paper models, scissors, tape.

Toothpicks (5-10 per group) -- or better, bamboo barbeque skewers (long enough to pierce 2 sides of the model.

Labels (square/rectangular for mirror planes, round for rotations)—possibly different colors.

Paper models of platonic solids (octahedra, tetrahedra) can be found on the web at sites such as this one

http://www.korthalsaltes.com/pdf/platonic-solids.pdf

(there is a copyright mark on the pages that you can print from here)

Students are given materials to construct simple paper models of a high-symmetry solid, such as an octahedron or tetrahedron. (Paper, scissors, tape.)

First: students work for a few minutes individually to identify all the symmetry elements and symmetry operations on, say, the tetrahedron. They write down the # of unique symmetry operations they can identify.

Next: students then get into groups to compare answers and to help one another identify additional symmetry elements. This is especially good because of the usual variation in natural ability for some students to “see” the operation and others not. By tutoring/coaching one another, they can help develop deeper understanding paced to each individual’s needs. Making this into a competition for the group with the most (the most correct) unique symmetry operations instills some incentive into this activity. The subsequent “team number” can be then written down on the same piece of paper, with a list of all symmetry operations.

After groups get started, it’s useful to remind them about equivalent symmetry operations. (For example, a *C*_{2} is equivalent to an improper rotation done twice *S*_{4}^{2}).

Wrap-up: at the end of the activity, the character table listing all symmetry opeerations (not merely grouped into classes--i.e., when it says there are 2 C_{4} elements, it means it's a *C*_{4} and and *C*_{4}^{-1}). This is also a good time to demonstrate unique axes, showing that (for example) a rotation axis going through one face and out the opposite side only counts as one axis, not two.

In particular, this lets students learn actively: how symmetry operations are grouped into classes, how many unique symmetry elements are present (and which are redundant), and find strategies to avoid "over-counting" symmetry operations.

A good endpoint to the lecture is to show the Otterbein Symmetry Visualization site

(http://symmetry.otterbein.edu/gallery/index.html)--to get help "seeing" operations when the rest of your team can't help you!