Generative AI technologies demand new practical and critical competencies, which call on design to respond to and foster these. We present an exploratory study guided by Research-through-Design, in which we partnered with a primary school to develop a constructionist curriculum centered on students interacting with a generative AI technology. We provide a detailed account of the design of and outputs from the curriculum and learning materials, finding centrally that the reflexive and prolonged ‘hands-on’ approach led to a co-development of students’ practical and critical competencies. From the study, we contribute guidance for designing constructionist approaches to generative AI technology education; further arguing to do so with ‘critical responsivity.’ We then discuss how HCI researchers may leverage constructionist strategies in designing interactions with generative AI technologies; and suggest that Research-through-Design can play an important role as a ‘rapid response methodology’ capable of reacting to fast-evolving, disruptive technologies such as generative AI.