Contextualization

Mathematics is a crucial tool in a wide range of areas, from physics to computer science, through economics, engineering, and many others. One of these crucial mathematical tools is solving systems of linear equations, which allows us to find solutions to multiple simultaneous problems. This ability is vital for many scientific and technological fields, as well as for decision-making in everyday life. Therefore, let's delve into the method of scaling to solve systems of linear equations.

A linear system is a set of two or more linear equations with the same variables. Solving these systems is a technique that allows us to find the values that simultaneously satisfy all the equations of the system. When we use the scaling method, we transform the original system into a simpler equivalent, which can be solved more directly.

The essence of this project is to explore the scaling method to solve systems of linear equations. We want you to become proficient in this technique, as well as understand its importance and practical application.

Importance and Applications of Scaling

Have you ever wondered how the most efficient route is determined by your GPS application? Or how a business analyst decides on the best combination of investments to maximize profit and minimize risk? Answering these questions - and many others - requires solving systems of linear equations.

Scaling plays a crucial role in these calculations. For example, routing algorithms, like the one used by Google Maps, often rely on solving systems of linear equations. Similarly, optimizing investment portfolios may also involve solving such systems.

Therefore, mastering the scaling method is an essential skill for those intending to pursue careers in the so-called STEM (Science, Technology, Engineering, and Mathematics) fields, as well as in the business world.

Practical Activity

Activity Title: "Scaling Paths: Optimizing Travel Routes"

Project Objective

In this project, students will apply the scaling method to optimize a travel route. The groups will design a travel plan for a hypothetical music band tour. The band needs to visit several cities, and it is up to the students to identify the most efficient route, considering time and cost.

Detailed Project Description

Students will be divided into groups of 3 to 5 people. Each group will receive a list of cities that the band needs to visit, as well as the distance and cost between each pair of cities. The goal is to find the route that minimizes the total travel time and cost, using the scaling method.

Additionally, the project will require students to apply concepts of Mathematics and Geography, as they will need to understand and interpret maps, as well as perform calculations based on the provided distances and costs.

The project should be completed in approximately 2 weeks, with each student dedicating an average of 12 hours of work.

Necessary Materials

1. List of cities and distances/costs between them (provided by the teacher).
2. Physical or digital maps (Google Maps, for example)
3. Calculator and calculation software (such as Excel)
4. Writing material for sketches and planning
5. Computer with internet access

Detailed Step-by-Step for Activity Completion

1. Problem Understanding: Students must understand the proposed problem by analyzing the list of cities and the associated distances/costs between them.

2. Preliminary Analysis: Students should conduct an initial analysis of the problem to identify which paths seem more practical and economical.

3. Mathematical Modeling: At this point, the problem should be translated into a system of linear equations. The number of equations will depend on the number of cities involved.

4. System Resolution: Once the system has been established, students must use the scaling method to find the solution that minimizes time and cost.

5. Verification: The solution found must be verified, analyzing if it indeed corresponds to the lowest possible time and cost.

6. Report Writing: After completing the work, students will write a report detailing the resolution process and its results.

Project Deliverables

Written Document

Students must deliver a detailed report in the form of a scientific article containing the following topics:

1. Introduction: Here, students should contextualize the theme, show its relevance, and explain the project's objective.
2. Development: Here, students should explain: a) The theory behind scaling. b) The methodology used to solve the problem (including a detailed description of the practical activity) c) The results obtained.
3. Conclusion: In this part, students should discuss the acquired learnings, highlight the challenges encountered, and the skills developed during the project.
4. Bibliography: Students must indicate the sources used for theoretical basis and for the project's completion.

Presentation

Students must give a presentation to the class arguing the viability of the chosen route, showing their decision-making process, and how they solved encountered challenges. The presentation will be a way to assess the students' communication competence, as well as collaboration and teamwork.