Abstract:

This work presents a didactic proposal focused on the exploration of sequences and regularities in basic education. The study of sequences provides a particularly rich context for developing mathematical reasoning, identifying patterns, and formulating generalisations, while also playing an important role in the early development of algebraic thinking. Building on this idea, the proposal connects the discovery of the Fibonacci sequence with the exploration of other growth sequences, encouraging students to identify rules, formulate conjectures, and communicate mathematical reasoning.

The proposal is aligned with the Essential Learnings for Mathematics for the second cycle of basic education and adopts an investigative perspective that requires students’ active participation in the construction of knowledge. The tasks were designed to promote the identification of regularities, the comparison of different sequences, the explicit formulation of rules that generate their respective terms, and the establishment of connections between mathematics, nature, and different cultural contexts.

This didactic intervention is organised into three sequential lessons. In the first two, aimed at 5th grade students, learners are invited to investigate different numerical sequences, identify growth patterns, and formulate explanations for the rules that generate their terms, with particular emphasis on the Fibonacci sequence and its associated regularities. At this stage, special attention is also given to identifying relationships between Fibonacci numbers and geometry, namely through the exploration of constructions such as the golden rectangle, the Fibonacci spiral, and natural forms that exhibit proportions close to the golden ratio. In the third lesson, designed for 6th grade students, these ideas are further developed through the creation and exploration of new sequences, also integrating the use of digital tools. In this context, Scratch is used to implement simple algorithms capable of generating sequence terms, fostering the articulation between mathematical thinking and computational thinking.

Throughout the activities, students are encouraged to observe, predict, test hypotheses, justify their thinking, and communicate conclusions, promoting the development of reasoning, argumentation, and generalisation skills. The use of exploratory tasks, the emphasis on classroom investigation, and the integration of digital tools constitute key elements of the proposal, contributing to a more dynamic and meaningful approach to the study of sequences.

This work highlights the potential of the Fibonacci sequence as a privileged context for exploring mathematical regularities and for developing fundamental competencies in mathematics education. By combining mathematical investigation, historical contextualisation, and the use of digital resources, the proposal seeks to contribute to diversifying pedagogical practices and promoting more meaningful mathematical learning in the study of sequences and regularities.

Keywords:

Sequences and Regularities, Fibonacci Sequence, Mathematical Reasoning, Computational Thinking.