A utilização de problemas matemáticos em aberto no ensino médio

Abstract

In math an open problem is an unresolved question. This research aims to show out the importance of using open problems in teaching mathematics to high school students, breaking the conception of students of this school level that all mathematical problems have a solution. The use of open problems encourages the use of investigative activities that are fundamental for the construction of knowledge, but it is absent in the school environment, causing losses in the training of students. Thus, we performed a research of open mathematical problems accessible to the high school student with the development of contents necessary to understand the problems researched, including Number Theory with the Principle of Mathematical Induction and several results on prime numbers, Combinatorial Analysis with topics on minimal superpermutations, magic squares, diagrams and games, and Geometry with problems of finding points with rational distances to the vertices of a regular polygon, the Fagnano’s problem and packaging problems. The present study presents three suggestions of activities that relate mathematic contents of high school with open problems for teachers to use in the classroom. The first activity proposes to insert the Principle of Mathematical Induction in high school through the search for an expression for the number of diagonals of a convex polygon and problems related to the Fibonacci sequence, the second activity works the properties of arithmetic progressions through magic squares and the third activity applies geometry concepts to a packaging problem. The research presents a selection of questions that cite open mathematical problems that can serve as a source of inquiry for teachers or as a list of exercises for the student who wants to delve into math.