This paper links educational psychology research about curiosity to teacher moves that are effective in an inquiry-based mathematics classroom. Three vignettes will show explicit teacher moves (staging disagreement, intriguing anecdotes, and creating a safe space) for different audiences (math majors, mathematics for liberal arts students, and future elementary school teachers) and tie them to particular studies about curiosity. The goal is to deepen our thinking about inquiry-based teaching by considering curiosity as one of the starting points of inquiry. Educational psychologists have found ample evidence that curiosity improves learning. We claim that curiosity and inquiry are deeply connected in a "curiosity-inquiry cycle" and become most active in a classroom that exhibits a culture of asking questions.