Exploring the locus of envelopes: an initial approach with GeoGebra

Introduction: The teaching of Ordinary Differential Equations (ODEs) presents significant challenges, particularly due to the predominance of algebraic methods that often limit the exploration of qualitative behaviors. The mathematical concept of an envelope, understood as the geometric locus formed by a family of curves, offers a visual and structural approach that may enrich the teaching of ODEs in a more intuitive and exploratory way. Objective: This study aims to develop a structured teaching proposal focused on integrating the concept of envelopes into the context of ODEs. It seeks to examine how this concept, combined with the dynamic representations provided by the GeoGebra software, can support and enhance the teaching of the qualitative aspects of differential equations. Methods: The adopted methodology is Didactical Engineering, with emphasis on the first two phases: Preliminary Analysis and Conception with A Priori Analysis. The study includes a review of theoretical foundations, an analysis of the instructional context, and the design of a didactic sequence. GeoGebra is employed as a digital tool for constructing dynamic graphical representations that assist in the conceptual development of the topic. Conclusion: The expected result is the development of a theoretically grounded teaching sequence that integrates conceptual rigor with visual strategies. Although implementation has not yet been carried out, the proposal is intended to serve as a foundation for future classroom applications.

Abstract licence: CC BY

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