Several highly motivating lessons particularly pertinent to the study of mathematics in high schools and the first years of college are presented. These lessons fall into four categories: (1) problems that present a challenge and which are interesting for their own sake; (2) problems based on real situations that can be understood by the student; (3) puzzles and mathematical games; and (4) project-oriented mathematics. Included are notions from graph theory, geometry, number theory, logic, probability, statistics, consumer mathematics, and set theory. (MP)