Introduction

Second Degree Inequalities are mathematical expressions that present an imbalance, represented by inequality symbols: greater (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). They are called second degree because their highest power is the exponent 2. Working with inequalities, it is possible to understand the fundamental principles of real numbers and their relationships, which is essential for solving more complex mathematical problems.

Inequalities are part of the high school mathematics curriculum and are a fundamental concept for the understanding of various disciplines, including Physics, Chemistry, Economics. Being able to understand how they work and how to solve them will be very useful not only for these disciplines, but also for exams like the National High School Exam (ENEM) and university entrance exams.

Contextualization

Second Degree Inequalities have numerous practical applications in everyday life. For example, imagine that you need to calculate the necessary amount of an ingredient for a recipe, the area needed to plant a garden, the time needed to travel at a certain speed, among others. In practical situations in our daily lives, we often need to deal with quantities that are not exact but must be within certain limits. For these situations, inequalities are essential.

Understanding the concept of inequality and knowing how to solve a second degree inequality is crucial for many professions, especially those involving complex mathematical calculations such as Engineering, Economics, Physics, among others. Even outside the academic world, the logic behind solving inequalities can help develop important skills, such as problems solving and creative thinking.

Practical Activity

Activity Title: Conquest of the Quadratic Planet

Project Objective

The objective of the project is not only to deepen students' knowledge of second degree inequalities but also to develop their problem-solving skills and teamwork. In addition, the effective use of digital technology, such as graphing tools and online research, will be required.

Detailed Project Description

Groups will be challenged to discover a fictional planet, the Quadratic Planet, using their knowledge of second degree inequalities. They will have to solve missions involving inequalities to ensure the survival of astronauts on this unknown planet.

The missions will involve scenarios such as:

1. Finding the best location to land the spacecraft, where the topography is indicated by a second degree inequality.
2. Calculating the safe amount of a resource that can be collected, represented by an inequality.
3. Determining the safe zone to explore, where the temperature is dictated by a second degree inequality.

Groups will have to present their findings and mathematical processes in a clear and concise manner, using digital tools to create graphs that aid in the understanding of the solutions for each scenario.

Necessary Materials

1. Notebooks for notes and calculations
2. Computers with internet access and software/graphing tools (such as GeoGebra, Desmos)
3. Pencils and erasers
4. Ruler or compass for precise drawings

Detailed Step-by-Step for Activity Execution

1. Form groups of 3 to 5 students.
2. Introduce the Quadratic Planet to the groups and each mission they will have to accomplish.
3. Each group should start by discussing and planning how they will approach each mission. Encourage collaboration and teamwork.
4. Groups should perform calculations and graphs for each mission, and document the entire resolution process.
5. At the end of each mission, groups should review the solutions found and discuss the implications and practical applicability.
6. Finally, groups will have to prepare a report containing the processes used to solve each mission, the graphs, and how these solutions would affect the survival of astronauts on the Quadratic Planet.

This project should take place over 2 weeks, allowing enough time for resolutions, discussions, and report preparation.

Project Deliverables

Each group must present:

1. A report documenting the resolution process of each mission, including:

   • Introduction: Explain the relevance and application of second degree inequalities in each scenario.
   • Development: Describe in detail the theory behind second degree inequalities and how each mission was approached and solved. Show the graphs and discussions made by the group.
   • Conclusion: Recap the main learnings, challenges, and conclusions drawn from the project.
   • Bibliography: List the sources consulted during the project.

2. An oral presentation of the report, including the graphs elaborated for each mission. Students can use programs like PowerPoint or Google Slides for this presentation.

By working on this project, students will not only deepen their understanding of second degree inequalities but also learn how to apply them in practical situations, while developing skills such as teamwork, communication, problem-solving, creative thinking, and time management.