Contextualization

The first-degree inequality may seem like an isolated topic in a purely mathematical context, but it is surprising how deeply it is incorporated into our daily lives. Have you ever thought about how real-world situations, such as controlling expenses, applying discount rates, optimizing time and resources, can be modeled through first-degree inequalities?

Mathematics, as a whole, is not a set of abstract formulas and calculations, but rather a powerful tool that allows us to understand and interact with the world around us. Specifically, the first-degree inequality is an important part of this understanding, providing a mathematical structure for solving complex problems involving limits, constraints, and conditions.

Theoretical Introduction

Inequalities are order relations that can be established between any members of the set of real numbers. They are quite useful for representing constraints that frequently appear in problems from various areas. First-degree inequalities, in particular, have a general form ax + b > 0 or ax + b < 0, where a and b are constants and x is the variable to be determined.

Studying first-degree inequalities allows us to determine the possible values for a variable, given the limits imposed by a problem. This topic is closely related to the study of linear equations and geometry, as the graphs of these inequalities produce lines and half-lines in the Cartesian plane, hectares.

In this project, we will explore the theory behind first-degree inequalities and apply them in a practical context, enriching our mathematical knowledge and developing valuable skills for creative problem-solving.

Practical Activity Suggestion

Activity Title: The Company of Dreams

Project Objective:

This project aims to engage students in the practice of solving first-degree inequalities and interpreting their results, applying them in a realistic and challenging scenario. Additionally, it is expected that students develop collaborative and socio-emotional skills, such as teamwork, time management, and problem-solving.

Project Description:

Students will be divided into groups of 3 to 5 members. Each group will have the task of creating a plan to open a company, taking into account various factors such as initial budget, fixed and variable costs, and expected monthly revenue.

They will use first-degree inequalities to model situations such as profit goals, spending limits, and production needs. The goal is to optimize the company's resources within the existing constraints.

Necessary Materials:

1. Paper and pen
2. Calculator
3. Computer with internet access
4. Spreadsheet program (e.g., Excel, Google Sheets)

Step by Step:

1. Company Definition: Each group must decide what type of company they would like to open and what products or services it will offer.

2. Case Study: Based on their choice, students should research the associated costs (fixed costs, variables, taxes), the current market for the product/service, average prices, etc.

3. Mathematical Modeling: With the collected data, students will create mathematical models using first-degree inequalities to represent and solve real problems, such as optimizing production to achieve profit goals, finding the break-even point, among others.

4. Simulation: Using a spreadsheet program, students will simulate the operation of their company during the first year of operation, based on the mathematical models they built.

5. Final Report: After completing the simulation, the group must write a report detailing the entire process, from data collection to the final interpretation of the results.

Project Deliverables:

Two materials should be delivered at the end of the project:

1. Spreadsheet: containing all simulations performed and the mathematical models used.

2. Written report containing:

   • Introduction: contextualization of the chosen company, explanation of the relevance of first-degree inequalities for its planning and management.

   • Development: detailed description of the entire process, justifying the choices and presenting the mathematical models used. In this topic, students must clearly explain how they applied first-degree inequalities to solve the problems they encountered.

   • Conclusions: general comment on the results obtained, the learnings during the project, and any relevant conclusion that can be drawn from it.

   • Bibliography: list of references used in the studies and data collection.

The quality of the report will be evaluated based on how well it aligns with what was developed in the practical activity, and how well it demonstrates the correct understanding and application of the concepts related to first-degree inequalities.