In an Inquiry-oriented math course, students are primarily engaged in mathematical activity, and math content is developed from students' activity.
Classroom resources for inquiry-based courses in college mathematics
Open-access curricula for inquiry-based courses in college mathematics
In an interactive lecture, content is introduced by the instructor through a lecture, similar to “traditional” courses. However, rather than devote the entire class time to lecture, interactive lectures devote a portion of time to engaging students in activity and sense making, and instructors are responsive to students' activity (e.g., they adjust their lecture based on the outcomes of the activity).