GEOMETRIC CONCEPTS OF LOCAL CHANGES AND CUMULATIVE EFFECTS IN BASIC SECONDARY EDUCATION FOR GRADES 8° AND 9°

Abstract

The study of real functions usually begins in eighth and ninth grade with students aged 14 to 16. Generally, some of their attributes, different representations, equation solving, and various applications are studied, without any priority given to the meaning of local variation and the accumulation of these changes over an interval. This may be the cause of the gap between algebra and future calculus, despite some of the objectives of the Basic Learning Rights (BLR) for these courses. The purpose of this article is to analyze the feasibility and demonstrate the educational value of progressively building notions of these concepts as an early conceptual bridge to the local and cumulative variation of linear and quadratic functions without technical formalisms, using estimates, comparisons, and refinements of approximations; of the slopes of the segments tangent to a set of their points and of the area under the graphs in some interval of their domain; through the usual geometric constructions with pencil and paper, simulators, or software. The methodology corresponds to a theoretical-didactic essay as an exercise in didactic awareness to address variational thinking from a geometric visualization perspective. The central proposal rests on the feasibility of introducing these ideas, given their alignment with the DBA for grades 8 and 9 in Colombia. It is hoped that this early articulation will transform the understanding of these functions from a static to a dynamic view that supports the previous concepts for introducing the study of calculus.