Exploring a new geometric-mechanical artefact for Calculus

We introduce a geometric-mechanical artefact designed for laboratory activities related to Calculus topics (3D models and construction instructions are freely available online). With new capabilities and a new design, this instrument adopts some mechanisms historically introduced to solve inverse tangent problems (that analytically correspond to solving differential equations). By such an instrument, besides materially revealing the tangent to a curve (tangent mode), it is possible to trace the graph of exponential functions and parabolas starting from the geometrical properties of their tangent (curvigraph mode). Furthermore, one can perform transformations as derivatives and integrals (transformation mode). Our research project aims to study the use of this artefact mainly for secondary school students. In this paper, we present the analysis of its semiotic potential, referring to the instrumental approach and the Theory of Semiotic Mediation. We also focus on a secondary school teacher manipulating the artefact to identify exploration processes and gestures of usage. The analysis supports the choice of starting the exploration in the tangent mode and suggests that the artefact fosters the emergence of the idea of the tangent line.