Contextualization

Polynomials are a highly relevant mathematical tool that spans various branches, from basic mathematics to more complex theories of algebra and calculus. They can be defined as the finite sum of monomials. Each monomial is composed of a number, called a coefficient, and one or more variables (usually represented by the letters x, y, z) raised to a natural power.

The division operation is, as we all know, one of the fundamental pillars of mathematics. Dividing polynomials is not as straightforward as dividing integers or rationals, but follows a logical process that reminds us of long division, which we learned in elementary school. We have all divided numbers before, but in this project, we will delve into polynomial division, a crucial topic in high school mathematics and later in university.

A key concept in polynomial division is the "remainder." The remainder is basically what is left after the complete division. In integer division, the remainder is a value smaller than the divisor. However, in polynomial division, the remainder is a polynomial of a lower degree than the divisor. Properly calculating the remainder in polynomial division is vital for solving many practical problems in exact sciences and engineering.

In the real world, these skills have a wide range of applications. Understanding polynomials and their remainders is essential for solving equations in physics, engineering, and economics. The remainder of a polynomial can be applied in polynomial interpolation, cryptography, among others.

I suggest the following sources for a better understanding of the topic:

• Book: Fundamentals of Elementary Mathematics vol. 1 - Sets and Functions (Gelson Iezzi and Carlos Murakami)
• Explanatory videos on the YouTube channel "Matemática Rio with Prof. Rafael Procópio"
• Page "Polynomials" on the website Just Mathematics (https://www.somatematica.com.br/emedio/poli/poli.php)

Remember that throughout this project, you will be developing valuable skills beyond mathematics itself, such as time management, teamwork, effective communication, and critical thinking.

Practical Activity

Activity Title: Developing the "Remainder Calculator" - A Polynomial Game

Project Objective

The objective of this project is to create an educational game called "Remainder Calculator," where players will have to correctly calculate the remainder of various polynomial divisions. The project aims not only at acquiring technical knowledge on the subject but also at developing socio-emotional skills such as teamwork, communication, time management, problem-solving, and creativity.

Detailed Project Description

Each group of 3 to 5 students will create a board game with challenge cards. These cards will contain polynomials to be divided, and players will have to calculate the remainder. The board can be constructed in various ways, and the game mechanics are free, as long as it involves the correct resolution of polynomial remainders.

Required Materials

• Cardboard or cardstock for the board and cards
• Colored pens
• Scissors
• Ruler
• Glue
• Dice or a spinner (optional)

Detailed Step-by-Step

1. Gather the group and divide the tasks. Some may be responsible for creating the board, others for creating the cards, and others for developing the game rules.
2. The board should be created with paths or squares. You can add special elements, such as bonus or penalty squares, the choice is up to the group.
3. The cards should contain divisions of polynomials of different degrees and difficulties. The important thing is that the correct answer is to calculate the remainder of the division.
4. The game rules should be clear and objective, and all group members should agree on them.
5. Once the game is created, group members should play it and make adjustments as necessary.
6. Finally, each group should present the game to the class and explain the rules. All groups will play all the games.
7. The final result of the project will be a written report containing the four main topics: Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables

The final project deliverable will be a written report and the board game "Remainder Calculator." The report should be prepared with the following topics:

• Introduction: The student must contextualize the theme, the relevance of polynomials and the calculation of their remainder, in addition to explaining the project's objectives.
• Development: Here, students must describe the theory behind polynomials and how to calculate their remainder. Additionally, it should be explained in detail how the game was built, the rules, and the methodology adopted by the group. Finally, students must present the results obtained during the games.
• Conclusions: In this part, students must highlight what they learned during the project, what were the main challenges and how they were overcome. It is also important to identify what worked well in the game and what can be improved.
• Bibliography: Students must indicate the sources used for the project. Include all books, web pages, videos, and other materials consulted during the project development.

The project should be presented to the class in a playful and engaging way, so that learning is attractive and effective!