In a course on chemical applications of symmetry and group theory, students learn to use several useful tools (like character tables, projection operators, and correlation tables), but in the process of learning the mathematical details, they often miss the conceptual big picture about "why" and "how" symmetry leads to the quite dramatic consequences that it does. This pedagogical gap is addressed in this paper by using one of the simplest chemical model systems, the particle in a box, along with a simple symmetry operator, parity, to get a clear understanding of the consequences of symmetry. The analysis of the particle-in-a-box model is extended by analogy to molecules, and connections are made to chemically important concepts like symmetry labels of molecular states, spectroscopic selection rules, and symmetry adapted linear combinations of orbitals.