Students often get stuck in problem solving, ignoring their physical intuition in favor of plug-and-chug or pattern-matching approaches. We suggest that examining the extreme cases is a useful way of moving students towards more expert-like problem solving. Based on a case study, we show that novice students can quickly learn to use extreme cases productively in problem-solving. In reasoning about extreme cases, students blend conceptual and mathematical cognitive resources. At the same time, they can generate new and creative uses for extreme-case reasoning, here recasting it from a tool for evaluating answers to one for generating them. Extreme case reasoning may prove a valuable instructional goal at the introductory level.