Symmetry Lab · For Teachers
Teacher Guide
Group theory through movement: what to watch for, and where it leads.
📥 Printables
About this lab
Symmetry Lab is a hands-on encounter with D₃, the symmetry group of the triangle, the standard first example in every abstract algebra course, made playable. Students discover closure (any two moves combine into another move), identity, inverses, and, the headline,non-commutativity: flip-then-rotate ≠ rotate-then-flip.
This is the perfect third act after Algebra's Hidden Structure (whose color tables ARE commutative) and Pattern Hunter (whose Lesson 4 names the "same pattern in disguise" idea).
Facilitation notes
- Physical warm-up first: cut a paper triangle, label corners A/B/C on both sides. Every screen move can be done with paper, let hands lead.
- The order-matters moment: have students predict before testing flip∘rotate vs. rotate∘flip. The surprise is the lesson: their first non-commutative operation ever.
- Table Builder: filling 36 cells sounds long, pairs finish in ~15 minutes with the composition calculator. Look for the "every row contains all six!" observation (each row is a shuffle, a Latin square).
- The subgroup hunt: the three rotations {identity, R120, R240} close among themselves, a group inside the group. It behaves exactly like a 3-color ⊕ table from the algebra course (an isomorphism!).
Discussion prompts: Why does every row of the table contain each move exactly once? · Where does your body use "undo moves"? · A square has 8 symmetries: can you list them?