A hands-on activity that introduces at an elementary level a theme of 20-century mathematics—the exploration of algebraic structures less regular than a group—is presented. Students play with an algebraic structure that generates codes in Mullin’s encoding of the plane graphs, and thus generates sequences of craft actions that produce a genus zero (handle-less) basket while working in a closed loop. Abstract algebra is an advanced topic, but this hands-on activity requires only facility in making algebraic substitutions. Starting from the algebraic structure’s identity element, 1, the young algebraist derives an original basket by making a sequence of permitted algebraic substitutions, ad libitum. Upon declaring her derivation complete, she fabricates her basket out of folded paper strips via an easy unit weaving technique—a task made easier by the fact that the variables’ letter shapes are mnemonic for the craft actions they represent. Play in this microworld reveals that all such baskets are related by a connected network of algebraic substitutions: in mathematical terms, they constitute the equivalence class of 1.