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This six page document contains: one page of instructor notes, a single page main activity, two pages of follow up activities, and a two page answer key. In the primary activity the students explore the properties of commutativity, associativity, and closure, as well as identities and inverses. The exploration centers on a 5-element set where a novel operator is defined by a 5x5 matrix. Activity two focuses on traditional definitions of inverses and identities, while the third activity allows students to explore the properties in a more open-ended exploration environment, and can be used as a take-home extension.