1. Introduction and Contextualization

Introduction

Linear equations are the essence of Algebra. They are the mathematical representation of an equality relationship that includes one or more variables. By solving a linear equation, we are looking for the value that makes the equality true - the value that, when inserted in place of the variable in the equation, makes both sides of the equality equal.

The comparison of linear equations, in turn, is a key concept in many areas of mathematics and is a fundamental skill that you need to acquire to advance in your math studies. Through the comparison of equations, you can determine if two equations are identical, different, or if one is a multiple of the other. Furthermore, comparing equations also allows you to determine if two lines, represented by two equations on a Cartesian plane, are parallel, coincident, or intersect.

Given the importance of this fundamental topic in mathematics, our project will focus on the theme of 'Comparison between linear equations'.

Contextualization

Linear equations permeate our daily lives in ways that we are not always aware of. Whether going to school by bus, calculating the time it takes for a bike trip, or planning an event, we are constantly dealing with situations that can be described through linear equations.

The ability to compare linear equations is particularly useful for solving problems where there are two or more related variables. For example, in situations involving financial management or the production of goods, there are often multiple interrelated factors that must be taken into account. In these contexts, comparing linear equations allows us to predict future behaviors and make informed decisions.

2. Practical Activity

Activity Title

'Unveiling the Treasure: A Journey of Linear Equations'

Project Objective

Through solving problems related to students' closest context, this project aims to stimulate the understanding and application of concepts of linear equations and systems of equations, especially the comparison of linear equations. Additionally, it also aims to foster collaborative, communication, and time management skills.

Detailed Project Description

In this project, students will be divided into groups of 3 to 5 participants and will have the challenge of solving a series of problems that require the comparison of linear equations to be solved. Each problem will lead to a 'clue' for the location of a 'treasure'.

Throughout the project, they will encounter situations where they need to:

1. Solve and develop problems related to their closest context, which can be represented by systems of first-degree equations with two unknowns and interpret them, using the Cartesian plane as a resource.

2. Compare two or more linear equations in order to verify when both will have the same value for a certain variable or the value of one variable when the other has a fixed value.

The groups should also research the real-life applications of linear equations and creatively include these applications in their final presentations.

Required Materials

1. Notebook for notes and calculations
2. Computer with internet access
3. Mathematics textbooks
4. Presentation software (e.g. PowerPoint, Google Slides, etc.)

Detailed Step-by-Step for Activity Execution

1. Group Formation and Initial Study (3 hours): The teacher will form groups of 3 to 5 students and provide general project instructions. Students should study the theme 'Linear equations and systems of equations' using the suggested resources and educational materials.

2. Problem Solving (6 hours): The teacher will provide each group with a series of problems related to linear equations. Each correct solution will provide a clue to the location of the 'treasure'.

3. Research and Presentation (3 hours): Students should research the real-life applications of linear equations and systems of equations and prepare a final presentation, creatively incorporating the real applications of linear equations in their presentations.

The groups should document the entire work process, including the reasoning for solving the problems, the difficulties encountered, the strategies used to overcome them, as well as reflections on the importance and applicability of the concepts learned.

Project Deliverables

1. Problem Solutions: The solutions to the problems must be recorded in a document detailing the reasoning used to solve the equations and systems. This document will serve as the basis for the final report.

2. Final Report: Each group must produce a report describing the work done. The report should include:

• Introduction: The introduction should contextualize the theme of 'Comparison between linear equations', its relevance and real-world application, the project's objective, and an overview of what was accomplished.

• Development: The theory behind the project's main theme should be explained, the activities should be detailed, the methodology used in solving the problems should be indicated, and finally, the results obtained should be presented and discussed.

• Conclusion: The conclusion should summarize the main points of the work, explaining the lessons learned and the conclusions drawn about the project, as well as reflections on the importance and applicability of linear equations concepts.

• Bibliography: Students should indicate all sources used to support the project: books, websites, videos, etc.

3. Presentation: The groups will make a final project presentation to the entire class, explaining the steps they followed, the problem-solving process, the real-life applications of linear equations, and the main conclusions.