O conjunto dos números irracionais: fundamentos teóricos e estratégias de ensino na educação básica.

Abstract

The teaching of irrational numbers represents a recurring challenge in Basic Education, both due to their abstract nature and the difficulties students face in understanding their practical applications. This dissertation aims to investigate the theoretical foundations of number sets, with emphasis on the set of irrational numbers, and to propose didactic sequences that foster a more meaningful and contextualized learning of this content. The work is organized into three chapters. The first presents a conceptual overview of number sets — natural, integer, rational, and real. The second deepens the discussion on irrational numbers, addressing different demonstrations of irrationality and its historical relevance. The third focuses on the development of didactic sequences aimed at Middle and High School students, covering: (i) the derivation of irrationals through the Pythagorean Theorem, (ii) the experimental approximation of Pi using everyday objects, and (iii) the intuitive introduction of the number e through compound interest situations. Part of the activities was implemented in the classroom, enabling reflections on their relevance, pedago￾gical effectiveness, and possibilities for replication in different educational contexts.