Contextualization

Second-degree equations are one of the oldest and most powerful tools in mathematics. Originating in ancient Mesopotamia and Babylon to solve practical problems, such as land measurement and geometry calculation, they are one of the fundamental concepts in many disciplines, from natural sciences to economics and engineering. They appear in many different contexts, making them indispensable tools in many areas of the real world.

In a second-degree equation, the highest degree term is the square of the unknown (i.e., the variable we are trying to solve). It is given in the general form ax² + bx + c = 0, where a ≠ 0.

Importance and Applications in the Real World

Second-degree equations are fundamental to mathematics and its various applications in the real world. For example, through them, it is possible to calculate the area of a terrain, the time of an object in free fall, predict gains and losses in a company, among many other situations. Understanding the second degree is an important step towards more advanced mathematics, which will be used in future studies and in many careers.

Furthermore, the analytical thinking involved in solving second-degree equations helps develop critical thinking and problem-solving skills, which are highly valued in many fields and professions.

Practical Activity

Activity Title: Quadratic Adventure

Project Objective

Students will be challenged to design and solve a series of problems that can be expressed in quadratic equations. This activity aims to allow students to apply the knowledge acquired in theory, as well as develop their critical thinking and teamwork skills.

Detailed Project Description

Divide into groups of 3 to 5 students. Each group will be responsible for creating and solving 5 different scenarios involving the application of second-degree equations. These scenarios can be inspired by daily situations, business challenges, environmental issues, physics, etc.

Required Materials

1. Paper and pens/pencils for notes and drafts.
2. Internet for research and digitization of the final work.
3. Computer with word processing software and spreadsheet for the elaboration of the report and respective calculations.

Detailed Step-by-Step for Activity Execution

1. Review the theory of second-degree equations using the suggested resources above.
2. Decide among yourselves the division of work within the group.
3. Create 5 different scenarios where the second-degree equation could be applied to solve a specific problem or situation.
4. For each created scenario, formulate a second-degree equation that represents the problem.
5. Find the solutions for each of the created equations.
6. Prepare a written report containing all the data, calculations performed, and solutions found.

Project Deliverables

1. Each group must submit a written report. With the following topics:
   • Introduction: Present the created scenarios and justify the importance of the second-degree equation for solving these problems.
   • Development: Present the created equations and detail how they were solved. Include the calculations and results obtained.
   • Conclusions: Explain what the learnings were during the project. Consider both the mathematical concepts and the socio-emotional skills developed.
   • Bibliography: References to the materials and resources used for the project execution.
2. The reports must be submitted in digital format and should average between five and seven pages.
3. In addition to the report, each group must prepare to present one of the solutions to the class, explaining the scenario, the equation used, and how they arrived at the solution.

Remember, the goal of the project is for you to learn to apply mathematics to solve practical problems in a fun and meaningful way. Have fun while you learn!