As artificial intelligence (AI) increasingly enters K-12 classrooms, what do teachers and students see as the roles of human versus AI instruction, and how might educational AI (AIED) systems best be designed to support these complementary roles? We explore these questions through participatory design and needs validation studies with K12 teachers and students. Using human-centered design methods rarely employed in AIED research, this work builds on prior findings to contribute: (1) an analysis of teacher and student feedback on 24 design concepts for systems that integrate human and AI instruction; and (2) participatory speed dating (PSD): a new variant of the speed dating design method, involving iterative concept generation and evaluation with multiple stakeholders. Using PSD, we found that teachers desire greater real-time support from AI tutors in identifying when students need human help, in evaluating the impacts of their own help-giving, and in managing student motivation. Meanwhile, students desire better mechanisms to signal help-need during class without losing face to peers, to receive emotional support from human rather than AI tutors, and to have greater agency over how their personal analytics are used. This work provides tools and insights to guide the design of more effective human-AI partnerships for K-12 education. [This paper was published in: "Proceedings of the 20th International Conference on Artificial Intelligence and Education" (pp. 1-14). Chicago, IL.]