RUME 28
2026
Alexandria, Virginia
A Learning Trajectory for Area Under a Rate Graph
Page: 642
We made a short sequence of interactive tasks in which a student must estimate the total amount of water in a pool, given the water rate (gals/min) at a handful of moments. Through working on these tasks in a small teaching experiment, an undergraduate who had not seen calculus learned to conceptualize the area under a linearly increasing water rate graph as representing the total accumulated volume of water. From this we developed a learning trajectory, highlighting requisite and developing concepts, actions, and teacher-researcher interactions that were key to his learning. His learning trajectory, which entailed imagery of lower and upper Riemann sums, reveals ways to learn key calculus concepts while maintaining the relationship between rate and amount quantities as a central, rather than an easily jettisoned, part of students’ understanding of ‘area under a curve’.