Research studies support caregivers' involvement in children's mathematical journeys, as foundational to their cognitive development and academic success as mathematical learners. The purpose of this intrinsic case study was to understand how a caregiver initiated and/or continually engaged their child in spontaneous mathematical moments during a family engineering project. Through the analysis of approximately 13.5 hours of video data, we identified three ways in which a caregiver guided, supported, and challenged their child through a shared endeavor of designing a remote-controlled delivery robot--discussing mathematical concepts and engaging in mathematical practices; encouraging use of physical objects, tools, and materials for mathematical application and visualization; questioning that invited mathematical exploration, contribution of ideas, and reasonableness of responses. These findings highlight caregivers' potential to engage their children in mathematics through guided participation in an out-of-school learning context not designed to elicit mathematical moments. Implications for how early childhood practitioners can develop opportunities and supports for caregivers and children to engage in mathematics in a variety of learning environments is included in the discussion.