Contextualization

The quadratic equations are a fundamental part of the mathematics, which appear in varied contexts of the real world, from the physics until the economy. The coefficients of these equations are connected to important aspects of his solution, as his roots.

Quadratic equations have the form ax² + bx + c = 0, where a, b and c are the coefficients. They have a significant impact on the structure of the equation and in his solutions - the roots. In particular, the coefficient a gives the general form of the parabola, represented graphically by the equation, b gives the location of the axis of symmetry and c the point of intersection with the axis y.

The fundamental theorem of the algebra tells us that each polynomial equation of second degree, with real coefficients, has exactly two roots. These roots can be real or complex, depending on the value of the discriminant (b² - 4ac). When this quantity is positive, the roots are real; when it is zero, there is only one real root (the equation is called perfect); and when it is negative, the roots are complex.

Now, you can ask yourself, why study the quadratic equations and his coefficients? Which is the reason to learn about something so abstract and with complicated appearance? The answer is simple. The quadratic equations are omnipresent in our daily world and in almost all the scientific and technical disciplines.

These equations are used to solve practical problems in varied areas of the science, engineering, and economy. For example, in the physics, they are used to calculate trajectories of projectiles, in electronics for the design of filters and circuits, in the economy to optimize the production and the profit, and so on. In fact, any phenomenon that can be represented by a parabola can be modeled by a quadratic equation.

Here are some resources that you can use to explore more about the topic:

• Khan Academy (https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations): Site with classes and exercises on quadratic equations.
• Brasil Escola (https://brasilescola.uol.com.br/matematica/equacao-2-grau.htm): Texts on quadratic equations and his coefficients.
• Só Matemática (https://www.somatematica.com.br/emedio/funquad.php): More texts and exercises on the topic.

Practical Activity: "Conquering Lands with Quadratic Equations"

Project Objective

The objective of this project is to apply the theoretical knowledge on coefficients in quadratic equations to solve a playful and fun problem. The students will have the opportunity to apply his mathematical knowledge, to work in team and to develop socioemotional skills.

Detailed Project Description

The project will simulate the conquest of a territory by means of quadratic equations. Each group of students will be a "nation" trying to expand his territory. The territory is a Cartesian plane, and each nation will expand his territory by solving problems that involve quadratic equations.

The students will make use of his mathematical skills to solve the proposed challenges and to be able to "conquer" more territories. Besides, it is necessary that they are attentive to the territory of the other groups, once it will be possible to conquer the territories of the competitors by means of the strategic use of the quadratic equations.

This project relates the mathematics to the geography, once the students will have to interpret and to manipulate the space of the "battle map" (Cartesian plane).

Necessary Materials

1. Grid paper to serve as the battle map;
2. Colored pencils to identify the territories of each nation;
3. Cards with challenges that involve quadratic equations;
4. Calculators, if necessary.

Detailed Step-by-Step

1. Divide the students into groups of 3 to 5. Each group represents a "nation".
2. Provide each group with a grid paper, where they will draw the representation of the initial territory of his nation using a colored pencil.
3. Establish the expansion rules: to expand the territory, the students must solve correctly the quadratic equations that are in the challenge cards.
4. When starting the game, each group chooses a challenge card and tries to solve the proposed quadratic equation. If they solve it correctly, they can expand the territory of his nation.
5. During the game, the students can try to conquer territories of other nations. For this, they have to solve correctly more difficult challenges that involve the sum and the product of the roots, for example.
6. The game continues until all the challenge cards have been solved.
7. The nation with the largest territory at the end of the game is the winner.

Project Deliverables

After the practical activity, the students will have to deliver a detailed report on the project. This report must contain:

1. Introduction: Contextualization of the relevance of the study of the quadratic equations and his coefficients. The report must explain how the project contributes to the understanding of this subject.
2. Development: Detail of the theory of the quadratic equations and his coefficients. The report must also explain the methodology used to solve the challenges and the strategy used in the game to conquer territories. The students must present and to discuss the results obtained.
3. Conclusion: The students will have to present what they learned with the project, his difficulties, facilities and how the practical activity helped in the understanding of the content.
4. Bibliography: The report must include all the sources that the students used to study and to solve the challenges.