Teorema da divisão euclidiana: retomando conceitos na primeira série do ensino médio

Abstract

This study aims to develop a didactic proposal focused on the first year of High School, with an emphasis on the Euclidean Division Theorem as a starting point. This purpose has the goal to consolidate and expand the understanding of this fundamental operation, exploring its implications in topics such as Number Systems, Greatest Common Divisor (GCD), Least Common Multiple (LCM), and Divisibility, which are part of the Elementary School curriculum. Furthermore, it introduces more advanced concepts in an accessible way, such as the Remainder Theorem and the Euclidean Algorithm, which assist in the development of mathematical reasoning and analytical skills that are useful in all areas of life. To achieve these objectives, an analysis of students' prior knowledge on the subject was conducted among high school students at Coronel Olímpio Cunha State School, in Espírito Santo. Additionally, a study of these contents and related topics in the last two collections of Mathematics textbooks approved by the “Programa Nacional do Livro Didático” (PNLD)–National Textbook Program–and used by the teachers and students of this school from 2017 to 2023 was carried out. The “Base Nacional Comum Curricular” (BNCC)–National Common Curriculum Base–was also consulted to assess the inclusion and depth of these topics in the school curriculum. Based on this research, an investigative didactic sequence is proposed, focused on problem-solving and enriched by the use of technological and historical resources, aiming to spark students’ interest in the subject and develop skills such as autonomy, decision-making, information analysis, idea experimentation, argumentation, and conclusion verification. After the implementation of the didactic sequence, students underwent a final examination to compare the results with the diagnostic assessment, demonstrating the progress of the students and the effectiveness of the pedagogical strategies employed in this study.