Intergenerational relationships, crucial for emotional support and stability, can be significantly enhanced through technology. While existing research mainly explores long-distance connections, the potential for technology to foster bonding during regular physical meetups between grandparents and grandchildren remains largely untapped. This study addresses the gap in families with regular intergenerational interactions. We conducted participatory card-based interviews with grandparents and children, analyzing the data with mixed methods including sentiment analysis, mapping the influencing factors of bonding, and thematic coding. This informed a framework for designing technology to support intergenerational connections. Two key contributions of the study are a novel mixed-methods approach that analyzes the same interview data to yield diverse results, and an expansion of the current understanding of intergenerational interaction through a layered model, which was validated by five design experts and tested with three additional families.