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?