Contextualization

Theoretical Introduction

Solving systems of equations is a fundamental skill in mathematics that helps understand the intersection of different sets of equations. Solving systems of equations allows determining where two or more equations are simultaneously true.

A system of equations is a set of two or more equations with the same set of variables. In many cases, we deal with systems of two equations with two unknowns, as they are simpler and have more immediate practical applications. The solution of the system is the set of values that satisfy all the equations simultaneously.

Studying the number of solutions of the system allows us to understand if the system is consistent (has at least one solution) or inconsistent (has no solution). In addition, a system can have a unique solution, infinite solutions, or no solution. These three cases provide valuable information about the characteristics of the system and the relationships between the equations.

Rich Contextualization

Systems of equations are a constant part of the world we live in. They are used in various areas such as physics, chemistry, economics, engineering, among others. For example, in economics, systems of equations are used in optimization problems to find the best way to use limited resources, such as budget or time.

Understanding the number of solutions of the system of equations is important to understand if a particular problem has a single solution, multiple solutions, or no solution. This can help make informed decisions and develop appropriate strategies in various real-life contexts.

Activity

Activity Title: "Building Models and Solving Systems"

Project Objective:

The main objective of this project is to allow students to apply the theoretical knowledge acquired about systems of first-degree equations with two unknowns and the interpretation of the number of their solutions (unique, infinite, or none) in a practical and playful context. Additionally, it aims to promote socio-emotional skills such as collaboration, time management, communication, and creative thinking.

Detailed Project Description:

Students will be divided into groups of 3 to 5 participants. Each group will receive a specific scenario involving everyday situations that can be represented by a system of equations. Each group must represent the described situation through a system of first-degree equations with two unknowns and define the number of possible solutions for this system.

The activity will be divided into two parts. The first part involves the creation and resolution of the system of equations. The second part involves the preparation of a report documenting the process and the results.

The activity will last approximately 5 to 10 hours per student, and the project delivery time will be one month.

Required Materials:

• Paper and pencil.
• Computer with internet access (for research and report preparation).
• Relevant mathematics books and educational materials on the topic.

Detailed Step-by-Step Guide for Activity Execution:

1. Firstly, the groups will receive their specific scenario and discuss how to represent this situation through a system of equations.

2. Next, the students should create the system of equations that represents the proposed scenario.

3. After constructing the system, the students must solve the system of equations and determine the number of possible solutions.

4. The students should then discuss the results obtained and what they mean in terms of the proposed scenario.

5. Finally, each group will write a detailed report describing the entire process, from representing the scenario through the system of equations, obtaining the solutions, and interpreting the results. The report should be structured into the following topics: Introduction, Development, Conclusions, and Bibliography used.

Project Deliverables:

Upon completing the project, the groups must deliver:

1. The system of equations developed from the proposed scenario.

2. The solution of the system of equations and the determination of the number of solutions.

3. An oral or written presentation (depending on the school's possibilities) of their system of equations, its resolution, and its conclusions.

4. A written report following the proposed structure: Introduction (contextualization of the theme, its relevance and application in the real world, as well as the description of the project's objective), Development (the theory behind the central topic of the project, the activity in detail, the methodology used, and the results obtained), Conclusions (the learnings and conclusions drawn about the project), and Bibliography (the sources they relied on to work on the project).

This report is a fundamental part of the project, as through it, students can demonstrate their understanding of the subject, their ability to apply the acquired knowledge in a practical context, and their writing and presentation skills.