Contextualization

Mathematics, often considered a rigid and theoretical discipline, actually permeates our daily lives in subtle and complex ways. A particularly powerful concept is the resolution of linear systems by scaling, a crucial tool in various disciplines ranging from engineering to economics. This project is dedicated to demystifying this technique and exploring it in a practical and meaningful way.

Scaling is an algebraic method for solving systems of linear equations. It involves the direct manipulation of equations in a system to create a scaled (or reduced) form matrix, from which the solution can be easily read. It is a technique that allows complex problems to be reduced to a series of simpler problems, facilitating the solution.

The importance of scaling and the resolution of linear systems is vast. In engineering, linear systems are used to model physical structures, design electrical circuits, and simulate dynamic systems. In economics, they are used to model relationships between different economic variables. In computer graphics, they are used to render three-dimensional images and animations.

These applications are just the tip of the iceberg. Understanding scaling and the resolution of linear systems opens the door to a wide range of technical fields and offers a window into how mathematics can be applied in a practical and impactful way.

To delve deeper into the subject, the following resources are recommended:

1. KAHN, Salman. Khan Academy: Systems of Linear Equations. A free online platform that offers a series of videos explaining in detail the scaling process.

2. IEZZI, Gelson; MURAKAMI, Carlos. Fundamentals of Elementary Mathematics: Matrices, determinants, linear systems. Ed. Atual. This book provides comprehensive coverage of basic and advanced techniques for solving linear systems.

Remember: the goal of this project is to learn, have fun, and apply mathematics in a way that makes sense to you. Let's do this together!

Practical Activity

Activity Title: "Scaling Towards the Solution"

Project Objective

This activity aims to engage students in solving everyday problems that require the application of the scaling method for linear systems. It promotes collaboration and teamwork, while allowing students the opportunity to apply and explore mathematical knowledge in a practical and fun context.

Detailed Project Description

Student groups will be challenged to create a fictional problem situation in which scaling is necessary to find the solution. This situation can involve various areas, such as economics, engineering, natural sciences, etc. It is important for students to be creative: they can, for example, create a story about resource distribution in a space trip, the balance of an ecosystem, the logistics of a company, among others.

After defining the problem, students should establish a system of linear equations that represents their problem situation. For this project, we recommend creating a system with three equations and three unknowns. Then, they should solve the system using the scaling method.

Next, they should produce an explanatory video, showing step by step the entire scaling process used to solve the system of equations.

Required Materials

• Pencil and notebook for developing the problem and the system of equations.
• Computer or smartphone with a camera for recording the video.
• Video editing software for finalizing the explanatory video (optional).

Detailed Step-by-Step for Carrying Out the Activity

1. Students, divided into groups of three to five, should create a problem situation that is solved through a system of linear equations.

2. Next, they should develop the system of equations that represents the problem situation. The system should consist of three linear equations with three unknowns.

3. After having the system, they should solve it using the scaling technique, showing all the resolution steps.

4. Once the scaling is completed, they should record an explanatory video describing their problem situation, the formation of the system of equations, and all the scaling steps taken, ending with the presentation of the solution to the system.

5. Finally, a written report must be submitted, containing: the description of the problem situation and its relevance; the created and justified system of linear equations; and a detailed description of the scaling process and the final solution obtained.

Project Deliverables

Groups should deliver:

1. The explanatory video, with a maximum duration of 10 minutes, showing the entire process carried out.

2. The written report, which must include four main topics:

• Introduction: The group must contextualize the chosen problem situation, its relevance and real-world application, and the objective of the activity.
• Development: The group must explain the theory of scaling, detail the system of equations developed, the methodology used for the resolution, and present and discuss the results obtained.
• Conclusion: The group must summarize the main points of the work, explain what they learned from the project, and draw conclusions regarding the problem situation and the system resolution.
• Bibliography: The group must indicate the sources used for the work.

In the report and video, students should make an effort to clearly explain all concepts and steps, so that a person with no prior knowledge of the subject can understand.