Contextualization

Linear equations are a fundamental area of mathematics that allow us to understand and solve various everyday problems involving proportions and relationships between quantities. This field of study is essential to build a solid foundation in mathematics, which will be used in many other subjects as well as in various other disciplines throughout your educational journey and, later on, in your professional career.

The term 'linear' comes from the word 'line', as the graphical representation of a linear equation is a straight line. A linear equation in one variable, in the form $ax + b = 0$, is such that the graph of $y = ax + b$ is a straight line. It can be used to represent a directly proportional relationship between two quantities, expressed in $x$ and $y, where the coefficient $a$ represents the rate of change between them, and $b$ is the initial value or the linear coefficient.

Linear equations are extremely useful in various real-life situations, from simple household management tasks to making complex decisions in business contexts. For example, creating a family budget, where we have a fixed amount of money to spend and many different desires and needs to balance. Every decision we make affects all the others, and a linear equation can help represent the relationship between our income, our expenses, and our savings needs.

Linear equations are also essential for many professional careers, including engineering, computer science, economics, physics, among others. Therefore, it is crucial to have a good understanding of this concept that will boost your mathematical knowledge and open many doors in the future.

To deepen your knowledge of linear equations, we recommend the following resources:

• Khan Academy - Linear Equations
• Book: 'Fundamentos de Matemática Elementar' by Gelson Iezzi (Volume 1 - Sets and Functions)
• Poliedro Educational System - Linear Equations: Theory and Exercises

Practical Activity: 'Battle of the Lines'

Project Objective:

This project aims to help students understand and apply linear equations in a fun and practical way.

Group Size:

Each group should consist of 3 to 5 students.

Project Duration:

2 to 4 hours per student.

Detailed Project Description:

In this activity, students will create a board game where each move is determined by a linear equation. The board with various squares will represent the Cartesian plane, and the players' movements will be dictated by the linear equations that the students will have to solve.

Materials Needed:

• A chessboard or similar board
• Pieces or markers to represent the players
• Paper and pencil for calculations
• Cards to record the linear equations

Step-by-Step Guide for the Activity:

1. Students should first study the concept of linear equations, understand the meaning of the coefficients, and how to represent them graphically.
2. Next, students should create a set of linear equations, each representing a possible move on the game board.
3. Students should write each equation on a card. The solution of the equation will determine how much a player can move their marker.
4. On each turn, a player chooses a card, solves the equation, and moves their marker according to the solution.
5. The goal of the game is to reach a specific point on the board before the other players.

Project Deliverables and Written Document:

Upon completing the game, students should prepare a report containing the following topics:

1. Introduction: Students should contextualize what linear equations are, their importance and real-world applications, and the project's objective.
2. Development: Students should explain the theory of linear equations, detail the activity carried out, the methodology used, present and discuss the results obtained.
3. Conclusion: Students should summarize the main points of the project, reflect on the skills learned, and draw conclusions about the activity.
4. Bibliography: Students should list the sources used for understanding the concept and developing the project.

This report is a crucial part of the project, as it deepens students' understanding and consolidates learning about linear equations and their applications. Additionally, it contributes to the development of other important skills, such as written expression and critical reflection.