Contextualization

Determinants are fundamental concepts in mathematics that have various applications, from solving systems of linear equations to calculating areas and volumes. In this project, we will focus on the determinant of 2x2 matrices, which is undoubtedly one of the pillars for understanding matrices of higher orders.

The determinant of a 2x2 matrix is the difference between the product of the elements on the main diagonal and the product of the elements on the secondary diagonal. It seems simple, but this formula hides many secrets and interesting applications that we will uncover throughout this project.

In certain situations, we need to decide whether a system of equations has a unique solution, infinite solutions, or no solution. This is where determinants come into play, as they provide us with a quick and effective way to make that decision without having to solve the entire system.

Importance of the Theme

Determinants are essential for many areas of mathematics and find applications in various disciplines such as physics, engineering, economics, among others. They are present, for example, in defining the inverse of a matrix, solving linear systems, calculating areas and volumes, analytical geometry, among others.

In this context, 2x2 determinants have a very special importance, as they are the building blocks for determinants of matrices of higher orders. Beyond practical applications, by studying 2x2 determinants, we will be taking an important step towards a deeper understanding of mathematics.

Practical Activity

Activity Title: "Conquering the World of 2x2 Determinants"

Project Objective:

The objective of this project is to strengthen students' understanding of 2x2 matrix determinants and their practical applications, while learning to work collaboratively and develop problemsolving skills.

Detailed Project Description:

Students will be divided into groups of 3 to 5 people. Each group will receive a series of real-world situations involving problems that can be solved using 2x2 matrix determinants. For example, a problem could be solving a simple system of linear equations or calculating the area of a polygon on a plane.

Required Materials:

• Computer with internet access for research and report preparation.
• Grid paper for manual calculations.
• Mathematics software (e.g., MATLAB, Mathematica, or even Excel) for verifying calculations.

Step-by-Step Guide for Activity Execution:

1. Group formation and distribution of problems to be solved.
2. Conduct research to understand and contextualize the given problem.
3. Solve the problem using the concept of 2x2 determinants.
4. Verify the solution using mathematics software.
5. Demonstrate problem-solving to the rest of the class.
6. Prepare the report, including theoretical explanation, problem solution, software usage for verification, and conclusions.

Project Deliverables:

Each group will be responsible for the following deliveries:

1. Problem Solution: Demonstration of how the problem was solved using 2x2 determinants, including all calculations involved and decisions made in the process.

2. Demonstration: A quick presentation to the rest of the class explaining the problem, how it was solved, and the results obtained.

3. Written Report: The report should follow the structure:

   • Introduction: Contextualization of the problem, explanation of the concept of 2x2 determinants and their relevance to problem solving.
   • Development: Detailed explanation of how the problem was solved, including the calculations performed and the use of mathematics software to verify the solution. It should also include a discussion on the methodology used and the results obtained.
   • Conclusion: Reflection on the learnings from the project, the difficulties encountered and how they were overcome, and the application of the concept of 2x2 determinants in the real world.
   • Bibliography: References used during the project, such as books, videos, websites, etc.

The report should be presented in digital format, with all parts clearly identified and with a clear focus on the discussion and analysis of the results.

Each group member should contribute to all parts of the project, and the division of work should be explicitly stated in the report. The goal is to facilitate collaboration among students and strengthen their ability to work in teams.