Contextualization

Mathematics is a universal language, capable of describing and solving everyday situations. Within it, number theory is one of the most fundamental and fascinating branches, including the study of the properties of integers. This project will address a specific topic in this theory: the number of non-negative integer solutions. This concept refers to the quantity of non-negative integer solutions (that is, integers greater than or equal to zero) that a certain equation can have.

Understanding this topic is important as it is applied in various areas such as Statistics, Physics, Information Technology, among others. Non-negative integer solutions are important in solving integer programming problems, a type of optimization problem where some or all variables are restricted to be integers. In Statistics, they are used in combinatorial analysis and probability.

Integers and their properties are the foundation for many other fields of Mathematics. Understanding these properties allows you to advance to more complex and applicable studies, such as cryptography, prime number theory, among others. Do not be mistaken, integers are more than just counting; they are present in various aspects of our daily lives and are an important tool for problem-solving.

Practical Activity

Activity Title: "Discovering Non-Negative Integer Solutions"

Project Objectives:

This project aims to teach the theory and practice of calculating non-negative integer solutions through fun and collaborative activities, with the development of a mathematical mini-game involving the concept.

Detailed Project Description:

As a team of three to five students, you will create a game board and cards. The board will have numbered spaces from 0 to 10. The cards are numbered from 1 to 10, with each number repeated 3 times.

At the beginning of the game, players must choose 3 cards randomly, without showing them to others. The goal is, through additions, deductions, and multiplications, to create an equation whose solution is present on the board. The solution of the equation must be a non-negative integer.

Players will take turns, where each will have the chance to form an equation. If correct, the player will move a piece to the space corresponding to the solution of the equation on the board. The game ends when one of the players reaches space number 10.

Students should create and define additional rules as they find interesting for the game, always focusing on non-negative integer solutions.

Materials Needed:

1. Cardboard or cardboard to create the game board.
2. Numbered cards or card paper to make them.
3. Game pieces (can be small objects like buttons, clips, etc).
4. Pen or marker to draw on the board and create the cards.

Step-by-Step Guide for the Activity:

1. Divide into groups of 3 to 5 people.
2. Each group should create their game board and cards, according to the specifications above.
3. Define the game rules. Remember that the goal is to practice calculating non-negative integer solutions.
4. Play a few rounds to test and adjust the rules, if necessary.
5. Present the game to the class, explaining the rules and demonstrating a match.
6. Write a detailed report on the creation process and the learnings acquired.

Project Delivery and Connection with Activities:

After playing the game, students should write a document in the form of a report containing the following topics:

1. Introduction: The student must contextualize the theme, its relevance, and real-world applications, as well as the objective of this project.
2. Development: The student must explain the theory behind the number of non-negative integer solutions, describe the game in detail (including the rules), the methodology used to create the game, and finally present and discuss the results obtained by the group during the matches.
3. Conclusion: The student must conclude the work by summarizing its main points, explaining the learnings obtained, and drawing conclusions about the project.
4. Bibliography: The student must indicate the sources they relied on to work on the project, such as books, web pages, videos, etc.

The report must be submitted within a month after the start of the project. Each group member must contribute to the report writing, and all parts must be well connected and cohesive, reflecting teamwork and collaboration throughout the project.