Um estudo teórico sobre as estruturas aditivas e multiplicativas de números racionais em sua representação francionária

Abstract

The theoretical study presented has as its theoretical approach the Conceptual Fields Theory (CFT) of Gérard Vergnaud. As our goal, we are going to identify elements of the CFT such as Situations, Invariants and Representations upon the re-reading of international articles dealing with aspects of the teaching and the learning of additive and multiplicative structures of rational numbers on their fractional representation. In order to proceed with the research, we searched the CAPES website for online journals that presented activities involving additive, subtraction, multiplication and division operations of rational numbers on their fractional representation. For the additive structures, problem situations concerning the composition measures were identified. The main theorem mobilized by the students considering the addition operator was "adding a numerator with a numerator and a denominator with a denominator". For the multiplicative structures, problem situations on the class of the isomorphism of measures were identified, and the main invariants been detected were "he dividend is greater than the divider" and "the quotient is smaller than the dividend". The representations used in both the additive and the multiplicative structures were the natural written and pictorial language. The identification of these elements of the CFT at the conceptual fields of additive and multiplicative structures may assist teachers in choosing activities for the training and development of the concepts involved in these rational structures on the fractional representation.