Student engagement is a central concept for educational practitioners, researchers, and evaluators, especially working with learners historically minoritised in science developing their self-efficacy. Design-oriented projects in invention education show potential for promoting equitable engagement, partly by building upon learners' sociocultural backgrounds and experiences. However, the relationship between more social and more individual conceptions of student engagement is not yet well understood. We took a cultural psychology approach to design-based research for planning, implementing, and analysing a five-day camp in the Northeast US, wherein grades 6-8 students invented an electronic door and a free-choice invention. Our mixed-methods case study for convergence revealed some statistically significant changes in engagement and self-efficacy for inventing, which qualitative analyses suggest were related to campers' cognitive self-efficacy for ability with technology, campers' perceived agency for inter- and trans-disciplinary approaches to inventing, and the camp's social infrastructure for student participation. Further, we found evidence differentiating individual and social levels of affective/emotional, behavioural, and cognitive engagement, supporting a six-part model over previous three- and four-part models. We conclude with conjectures about the camp's enactment, learning processes, and outcomes, providing an educational model that could be useful in designing similar environments for user- or activity-centred design projects.