Contextualization
The language of Mathematics is universal and permeates various areas of knowledge, its applications are vast and, often, surprising. Among many mathematical concepts, we find Geometric Progressions (G.P), the subject of this project. To make it more understandable, we will identify patterns, observe sequences, and deduce formulas.
Geometric Progression is a sequence of numbers where each term, starting from the second, is equal to the product of the previous term by a constant called the ratio. This is a simple definition, however, throughout the project, we can deepen this understanding, reaching to identify G.P's in discrete domains and associate these sequences with exponential functions. In fact, the interpretation and resolution of problems involving G.P's will lead to the development of new skills and the improvement of mathematical reasoning.
Furthermore, the study of G.P's is not limited to its definition. They are an important tool in various branches of science, such as Physics, Economics, Engineering, and Biology. In Economics, for example, G.P's are used in the calculation of compound interest in investments and loans. In Biology, they help to understand the population growth of species. In Physics, they are useful for the analysis of phenomena involving free fall, thermodynamics, among others.
In the second part of the project, students will be encouraged to carry out a practical activity involving the planning and execution of an experiment that proves the application of G.P in real life. Then, they will have to produce a report with the results, contributing to the development of writing skills, as well as mathematical competencies and skills.
Practical Activity
Activity Title: Hanoi Tower and Geometric Progression
Project Objective
The objective of this activity is to apply the characteristics and knowledge of Geometric Progression (G.P) in solving the famous puzzle, the Hanoi Tower. From this activity, students will be able to visualize the concept of G.P in practice and also develop reasoning and problem-solving skills.
Detailed Project Description
Students, divided into groups of 3 to 5 members, must build a version of the Hanoi Tower with, at least, 5 discs of different sizes and three bases. The mission will be to move all the discs from one base to another, following the game rules that, surprisingly, are related to G.P.
After solving the puzzle, the group must develop a general resolution strategy for any number of discs and realize the relation of this strategy with G.P.
Necessary Materials
- Discs of different sizes (can be made with cardboard, for example);
- Three bases (can be plastic bottles, for example).
Detailed Step-by-Step for Activity Execution
- Each group must build their version of the Hanoi Tower with 5 discs and three bases;
- Understand the game rules: the goal is to move all the discs from one base to another, with never a larger disc on top of a smaller one, and you can only move one disc at a time;
- Solve the puzzle;
- Develop a general resolution strategy for the puzzle for any number of discs and record this strategy;
- Realize the relation of this strategy with G.P and record this relation.
After completing the practical activity, students should write a report to describe the entire process of realization, observations, discoveries, and conclusions of the project, following the proposed structure: Introduction, Development, Conclusions, and Bibliography.
In the Introduction, students should contextualize the theme and the project objective. In Development, students should describe in detail the methodology used, the observations made during the game, and the resolution strategy developed. They should also explain, in detail, the relation of the strategy with G.P, presenting the corresponding theoretical knowledge. In Conclusions, students should synthesize the main findings of the project, the learnings, and skills developed. In Bibliography, they will indicate the sources of information used in the project.
This project will provide students with a playful and collaborative environment to learn in a practical and fun way about the concept of G.P and its applications.
