Contextualization

The number of non-negative integer solutions is a recurring theme in discrete mathematics and, therefore, one of the essential skills that you, as 2nd-year high school students, must develop.

Let's start by understanding what 'non-negative integer solutions' actually means. The word 'integer' refers to a number that is not a fraction, that is, it does not have decimal places, while 'non-negative' means that this number must be equal to or greater than zero. For example, if we have the equation x + y = 10, and we ask how many non-negative integer solutions exist for this equation, we are looking for combinations of 'x' and 'y' that satisfy the equation, and at the same time, both variables are integers greater than or equal to zero.

Now that we have a clear understanding of what 'non-negative integer solutions' are, the question that remains is how to find these solutions? Is there a formula for this or do we need to use complex equation-solving techniques? The answer is simpler than you might think! By using combinations and a technique called 'stars and bars combinatorics,' we can solve these questions systematically and effectively.

In real-world applications, non-negative integers are of great importance in various disciplines such as computer science, physics, and engineering. For example, in programming, we have to count how many solutions exist for a specific problem, and often these solutions must be integers and non-negative, as it is impossible to have, for example, -2 results or 3.5 results. Similarly, in physics and engineering, when we have to solve problems involving discrete quantities, such as the number of particles in a system or the number of lanes on a highway, the need for non-negative integer solutions becomes clear.

To delve deeper into this topic, here are some reliable sources:

• Book: 'Discrete Mathematics and Its Applications' by Kenneth H. Rosen.
• YouTube Video: Matemática Rio with Prof. Rafael Procopio.
• Website: Brilliant.org: An interactive learning platform that has excellent resources on Discrete Mathematics.

Practical Activity

Activity Title: Guessing Game - Non-Negative Integer Solutions

Project Objective

The objective of this project is to apply and understand the theory of non-negative integer solutions through a guessing game in which the group creates their own equations, solves them using combination and stars and bars tools, and then challenges other groups to find the correct solutions.

Detailed Project Description

The activity consists of two main steps:

1. Each group must create and solve their own equations, which contain 3 to 5 variables and whose sum of values must be a number chosen by the group (between 10 and 20). Each created equation must have at least 3 non-negative integer solutions.

2. After creating and solving the equations, each group presents their equations (without the solutions) to the other groups, who must find the correct solutions.

Required Materials

• Paper and pencil for each group.

Detailed Step-by-Step

1. Divide the class into groups of 3 to 5 students. Each group should have the same number of students.

2. Ask each group to think and create a linear equation (involving 3 to 5 variables) whose sum of the variables is a number between 10 to 20. For example: a+b+c+d=10.

3. The groups must now calculate the non-negative integer solutions for their equations using the combination and stars and bars method. Each equation must have at least three solutions.

4. Once the groups have solved their equations, each group must challenge the others to find the correct solutions for their equations, without revealing the solutions.

5. Each group must try to solve the equations presented by the others. For each correctly solved equation, the group earns a point. The group with the most points is the winner.

6. At the end of the activity, each group must write a report on the activity carried out, explaining the theory of non-negative integer solutions, the stars and bars method, and giving details on how they created their equations, solved them, and their experiences during the game.

Project Deliverables

At the end of this activity, each group must deliver the following:

1. The equations created by the group as well as their respective solutions.

2. A list of the equations presented by the other groups and the solutions that your group found for each of them.

3. A detailed written report containing four main sections: Introduction, Development, Conclusions, and Bibliography used.

• In the Introduction, students should present the theory of non-negative integer solutions, its importance, and the objective of this project.
• In the Development, the group should explain the activity carried out, the methodology used (specifically how they used combination and stars and bars to solve the equations), the equations they created, and how they found their solutions. They should also include the experiences during the challenge of the equations with the other groups.
• In the Conclusion, students should highlight the main points of the project, what they learned about the theory of non-negative integer solutions through this activity, and the conclusions they drew about the project.
• In the Bibliography, books, web pages, videos, and other sources used to work on the project should be cited.

This set of deliverables will be proof of the work carried out during the activity and the learning resulting from it, as well as the collaboration and teamwork of the group.