This paper examines the process of instrumental genesis through which students develop their proficiency in making use of movable points and slider bars--two tools that dynamic mathematics software provides for working with variable coordinates and parameters in the field of functions. The paper analyses students' responses to task sequences designed to support a planned instrumental genesis and then examines how features of these task sequences might be modified accordingly to improve such support. Evidence about student responses was collected during the first iteration of a design-based research study conducted in collaboration with four upper secondary school teachers. Three task sequences were trialed with four 10th grade classes, involving a total of 85 students. Orchestration and examination of the instrumental genesis were guided by the identification of key elements--both conceptual and technical--of the instrumented action schemes involved.