Introduction

One of the most fundamental concepts in mathematics is the First Degree Equation. This equation is a mathematical expression that relates two variables through basic operations, the main one being multiplication. In its simplest form, an equation of this type can be written as ax + b = 0, where 'a' and 'b' are numbers and 'x' is the variable we need to find.

In first degree equations, 'a' is the coefficient, and it cannot be zero. 'b' is the independent term and can be any real number. The solution to the equation is found by isolating the variable 'x', that is, by performing a series of manipulations on the equation until 'x' is alone on one side of the equality.

It is worth noting that the First Degree Equation is linear, which means that its graphical representation is a straight line. This characteristic greatly facilitates the visual interpretation of the concepts involved.

Contextualization

In the real world, first degree equations are widely used to solve everyday problems. Have you ever stopped to think about how many real-life problems can be solved with a first degree equation? From deciding how much time you have to prepare for a party to calculating the time needed to get somewhere, the first degree equation is present.

Furthermore, this type of equation is also very common in various areas of science and engineering. For example, in physics, to calculate the average speed of a moving object, we use the first degree equation. In economics, to determine a company's profit or loss, we also resort to it. Therefore, mastering the first degree equation is essential to understand and solve many practical problems.

To delve deeper into the subject, some recommended resources are the digital book 'Fundamentos de Matemática Elementar: Volume 7 - Equações de 1º e 2º graus, sistemas e problemas' by Osvaldo Doce and Gelson Iezzi, available in libraries and online bookstores, and the website Só Matemática (https://www.somatematica.com.br/emedio/sistemas/sistema1.php), which offers a complete and detailed explanation of first degree equations and systems of linear equations.

Practical Activity: 'Resolution of Real World Challenges with First Degree Equation'

Project Objective

The main objective of this project is to support students in understanding the theory of the first degree equation in a practical and exciting way. They should be able to connect theory to practice by solving real-world problems with the help of this mathematical formula.

Detailed Project Description

In this project, students, in groups of 3 to 5, will be challenged to find and solve at least four real-world problems that fit into a first degree equation. These problems can be related to everyday situations, such as trip planning or budget calculation, or to specific areas, such as physics or economics.

For each problem, the group must:

1. Clearly describe the problem.
2. Formulate the first degree equation that represents the problem.
3. Solve the equation.
4. Verify the answer in the context of the problem.

These challenges should be documented in detail, including equations, calculations, and clear explanations of the reasoning behind each solution. In addition, each group must prepare a 10-15 minute video presentation explaining each of the problems and their solutions.

Required Materials

1. Paper and pen for drafts and calculations.
2. Computer or smartphone to access online resources.
3. Presentation software (such as PowerPoint, Google Slides, etc.) to prepare the video presentation.
4. Textbooks and/or online resources for reference.

Detailed Step-by-Step Guide for the Activity

1. Each group should meet to discuss and choose four real-world problems that can be represented by a first degree equation.
2. For each problem, the group should create a detailed description of the problem, including the context and the formulation of the problem as a first degree equation.
3. The group then solves the equation for each problem, documenting each step of the resolution process.
4. For each problem solved, students should verify if the solution makes sense within the context of the problem.
5. Finally, the group prepares a video presentation explaining each problem, the formulation of the equation, and its resolution.
6. The project should take an average of about 12 hours per student and should be completed within a maximum of two weeks.

Project Deliverables

The group must produce a Written Report and a Video Presentation.

1. Written Report: Should include an introduction to the topic of first degree equations, a detailed description of the chosen problems, the formulation of the equations, the step-by-step resolution, and a conclusion with reflections on the project. The report should also contain a bibliography section listing the resources used in the project.
2. Video Presentation: The video should be a presentation of all project steps, explaining the choice of problems, the formulation of the equations, the resolution, and the lessons learned. The video should be 10 to 15 minutes long.

The Written Report and the Video Presentation are complementary and should reflect teamwork and the group's understanding of the first degree equation and its real-world applications.