From the observation of regularities to the construction of algebraic generalizations

a report of mathematical investigation

Abstract

This article reports a mathematical investigation of an autonomous and reflective nature, conducted by a student in the final years of Basic Education in 2025 under the academic supervision of the other authors. The study aimed to investigate numerical patterns related to parity, numerical sequences, and the relationships among addition, multiplication, and exponentiation, seeking to attribute meaning to the observed regularities. Methodologically, the investigation was developed through the formulation of hypotheses, numerical testing, identification of regularities, and the construction of algebraic generalizations. The study includes the development of representations for even and odd numbers, the algebraic formulation of the nth odd number, and the analysis of the relationship between the sum of consecutive odd numbers and square numbers. The results indicate that the investigative exploration of numerical patterns can foster the construction of mathematical meaning, contributing to the development of algebraic thinking and conceptual understanding of Mathematics in Basic Education.