In the increasingly data-driven digital era, bridging the gap between ideal mathematical models and noisy real-world data is a critical competence. In the Japanese curriculum, Grade 8 students often face cognitive “transitional gaps” when extending “direct proportion” (y=ax) to “linear functions” (y=ax+b). Specifically, they struggle with data approximation and a persistent “proportionality bias” regarding the y-intercept. Adopting a spiral mathematical modelling framework based on Borromeo Ferri (2018), this study investigates how physical robot manipulatives can scaffold students in navigating these gaps. A qualitative case study involving two Grade 8 students was conducted through three experiential phases: modelling sensor noise, establishing the physical necessity of the intercept, and representing dynamic piecewise motions. Analysis of discourse and worksheets revealed that modelling "noise" helped students transition from viewing data variation as "error" to using it for predictive models. Furthermore, the robot-based tasks facilitated a structural discovery of the intercept, transforming students' understanding of functions from static formulas into dynamic models for explaining reality. These experiential activities serve as an effective bridge to future STEAM education.
