"Principles and Standards for School Mathematics" (NCTM 2000) proposes that mathematics instruction provide opportunities for students to engage in mathematical inquiry and in meaning-making through discourse. Mathematics teachers are encouraged to build on student discoveries in designing subsequent instruction. In this article, the author describes the experience of implementing inquiry- and discussion-based mathematics instruction in one classroom to illustrate how students' unexpected interpretations of one problem created an opportunity to extend their mathematical investigations. The author also shares the particular demands that building on student ideas placed on classroom curriculum and instruction. (Contains 12 tables and 4 figures.)