The authors asked fifth-grade and eighth-grade students to pose stories for number sentences involving the addition and subtraction of integers. In this article, the authors look at eight stories from students. Which of these stories works for the given number sentence? What do they reveal about student thinking? When the authors examined these stories, they noticed that the students' work reflected different ways of thinking about integers. They call these mathematical uses of integers "conceptual models". The authors describe five models that emerged from their research: bookkeeping, counterbalance, relativity, translation, and rule.