Contextualization

Mathematics is a fascinating field of study, full of mysteries and discoveries waiting to be unveiled. One of these intriguing concepts is the 'Equality between Two Members'. According to this principle, when two quantities are equal, that is, when they have the same measure, we can perform mathematical operations, such as addition, subtraction, multiplication, or division, on both members and the equality will remain the same.

For example, let's consider the equation 2x = 6. This equation states that 2 times some number (x) is equal to 6. If we divide both sides of the equation by 2 (a mathematical operation), we will get x = 3, which maintains the principle of equality, since 2 times 3 is equal to 6. Amazing, isn't it?

But why is this important and why should you, as a student, care to understand this concept? Well, the answer is: the equality between two members is one of the fundamental principles of mathematics and is present in almost all fields of this study, from basic operations to more advanced mathematical theories. Understanding this concept helps in problem-solving, understanding equations, and overall, in the development of logical skills.

Introduction

We now understand that this concept is a fundamental pillar in mathematics, but how can we explore it in a playful and in-depth way? For that, we propose a project called 'Discovering Equality'. In this project, you will have the opportunity to investigate this idea in a practical way and experience, in practice, the feeling of maintaining constant equality while manipulating the members of an equation.

The project will involve teamwork, allowing you to exercise skills such as communication, time management, and creative thinking. And in the end, you will be able to demonstrate that, indeed, by adding, subtracting, multiplying, or dividing each of the members of an equality by the same number, that equality is maintained.

To start, we recommend some readings that will enrich your knowledge on the subject. You can consult the following resources:

• 'Matemática: Volume Único' - Gelson Iezzi: This textbook addresses the concept of equality between two members and the principle of equality preservation.
• 'O jogo e a matemática no ensino médio' - Roberto Ribeiro da Silva: In this book, there are several playful activities that can help in the practical understanding of equality between members.

Explore the concept, apply it in practical activities, and discover the beauty of mathematical equality!

Practical Activity: 'The Equivalence Game'

Project Objective:

The project's objective is to understand the idea that any two members that are equal remain equal even if similar operations (addition, subtraction, multiplication, or division) are applied to both. This project is intended for groups of 3 to 5 students and should take about 5 to 10 hours for each student to complete.

Detailed Project Description:

The project consists of a practical board game called 'The Equivalence Game'. In the game, students must move pieces on a board, maintaining the equality of some values. Team members must collaborate and plan their actions to confirm that the operations performed on one side of the equality are replicated on the other side, thus maintaining equality.

Necessary Materials:

• A sheet of cardboard to create the board
• Colorful pens or markers
• Small pieces to move (can be pieces from old games, small stones, bottle caps, or whatever is convenient)
• Challenge cards (made of paper or cardboard)

Detailed Step-by-Step for Activity Execution:

1. Board Creation: The board should be square or rectangular in shape, divided into cells. The number of cells can vary according to the desired complexity of the game. Each cell should have a numerical value, representing a member of the equation.

2. Piece Creation: Each team should have a set of pieces, which should be placed in corresponding cells of equal numerical value, thus forming an equality.

3. Challenge Card Creation: The challenge cards should indicate a mathematical operation (addition, subtraction, multiplication, or division) and the value to be operated. For example: 'Add 2', 'Subtract 3', 'Multiply by 2', 'Divide by 3'. The number of cards should be sufficient to ensure a dynamic and challenging game.

4. Playing the Game: Students must play the game according to the established rules: draw a Challenge Card and apply the indicated operation on the pieces on both sides of the equality, moving the pieces to cells corresponding to the new value.

5. Reflection on the Game: After playing for a set time, students should reflect on the game experience, identifying how equality was maintained during the plays and which strategies were effective in ensuring the maintenance of equality.

At the end of the game, students should write a document where they discuss the experience. They should make an Introduction, presenting the objective of the game and the relevance of the concept of equality. In the Development, they should explain in detail how the game was set up and played, the rules, the strategies adopted, and how the results were obtained, always relating to the central mathematical concept: equality. In the Conclusions, they should revisit the experience, mention the learnings acquired, the difficulties encountered, and how they understood the concept of equality in practice. In Bibliography, they should list the references used to understand the concept and to develop the game.