Objectives (5 - 10 minutes)

1. Understanding the Quadratic Equation: The main objective is for students to understand what a quadratic equation is and how it differs from equations of lower or higher degrees. They should understand that the quadratic equation is a mathematical expression that includes squared, first degree terms, and constants.

2. Identifying Coefficients and Terms: Students should be able to identify the coefficients (a, b, c) and the terms (ax² + bx + c) in a quadratic equation. They should understand that the variable x represents an unknown value, and the coefficients and terms represent known values.

3. Solving Quadratic Equations with Real Roots: Students should be able to solve quadratic equations using the quadratic formula, identifying the real roots. They should be able to apply the formula correctly and simplify the equation to find the solutions.

   Secondary Objectives:

   • Applying the Knowledge to Problem Situations: Students should be able to apply the knowledge acquired to solve problems involving quadratic equations in real life. They should be able to interpret the problem, identify the appropriate equation, and solve it.

   • Developing Critical Thinking and Problem Solving Skills: Solving quadratic equations requires critical thinking and problem solving skills. Students must be able to analyze the situation, identify the problem, develop a solving strategy, and review their solution.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should begin the class by reviewing some basic algebra concepts necessary for understanding the quadratic equation. This may include defining an equation, the difference between variables and constants, and the importance of coefficients. In addition, it will be helpful to review the properties of exponents and the simplification of algebraic expressions. (3 - 5 minutes)

2. Problem Situation 1 - Finding the Area of a Rectangular Plot with Partial Information: The teacher should present a problem situation that involves solving a quadratic equation. For example, "Suppose you have a rectangular plot whose length is twice its width. If the area of the plot is 36 square meters, how can you determine the dimensions of the plot?" The teacher should then explain that this problem can be solved using a quadratic equation. (3 - 5 minutes)

3. Contextualization of the Importance of the Quadratic Equation: The teacher should explain that the quadratic equation is an important mathematical tool that has numerous applications in real life. It is used in areas such as physics, engineering, economics, social sciences, and even in games of chance, such as the lottery. The ability to solve quadratic equations is therefore an essential skill that can be applied in many everyday situations. (2 - 3 minutes)

4. Introduction to the Topic with Interesting Facts: To gain students' attention, the teacher can share some interesting facts or applications of the quadratic equation. For example, he could mention that the quadratic equation is the simplest equation that can have two different solutions. In addition, he could cite the example of the quadratic formula, which is widely used to solve quadratic equations and is named after the Indian mathematician who discovered it, Bhaskara. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Activity 1 - "The Mystery Equations Game": (10 - 15 minutes)

   • Preparation: The teacher should divide the class into groups of 4 to 5 students. Then, distribute to each group a series of cards, each card containing a different quadratic equation. These equations should contain only integers and the roots should be real. In addition, the teacher should also provide each group with a blank sheet of paper and a marker.

   • Description: The goal of the game is for each group to solve the quadratic equations they have received as quickly as possible. To do this, they must work together, discussing and applying the quadratic formula to find the solutions. The first group to solve all of their equations correctly and on time is the winner.

   • Implementation: The students should then begin solving the equations, writing the solutions on the sheet of paper. The teacher should circulate around the room, assisting groups that encounter difficulties and checking if the answers are correct. At the end of the allotted time, the teacher should stop the activity and check the answers of each group. The group that has solved all the equations correctly is declared the winner.

2. Activity 2 - "Solving Real-Life Problems": (10 - 15 minutes)

   • Preparation: The teacher should prepare a list of real-life problems that can be solved using quadratic equations. These problems should be varied and interesting, such as determining the time it takes an object to fall from a certain height, calculating the trajectory of a ball kicked on a soccer field, or determining the price of an item on sale.

   • Description: The teacher should present one problem at a time to the class and ask for help from everyone to solve them. The students should then work together to identify the appropriate equation, substitute the values, and solve. The teacher should guide the discussion, asking questions to direct the students and correcting possible errors.

   • Implementation: The students should then start solving the problems, writing down the equations and solutions on the sheet of paper. The teacher should circulate around the room, assisting groups that encounter difficulties and checking if the answers are correct. At the end of the activity, the teacher should review the problems and solutions with the class, clarifying any doubts that may have arisen.

3. Discussion and Reflection: After completing the activities, the teacher should gather the class and lead a discussion about the solutions found. The teacher should ask students about the strategies they used to solve the equations and problems, and how they felt during the process. The goal is to get students to reflect on what they have learned and how they can apply that knowledge to other situations. (5 - 10 minutes)

Feedback (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes):

   • The teacher should ask each group to share their solutions or conclusions about the activities. Each group will have up to 3 minutes to present. They should explain the strategies they used to solve the equations and problems, and how they applied the quadratic formula.
   • During the presentations, the teacher should encourage the other groups to ask questions and make constructive comments. This can help promote a richer and more in-depth discussion on the topic.
   • The teacher, in turn, should ask questions to ensure that students have correctly understood the concepts and applications of the quadratic equation.

2. Connection with Theory (3 - 5 minutes):

   • After the presentations, the teacher should review the main theoretical concepts covered in the lesson and how they connect with the practical activities. This may include a recap of the quadratic formula, the identification of coefficients and terms, and the importance of simplifying the equation.
   • The teacher should highlight how these concepts were applied in solving the problems and answer any questions that may still exist.

3. Individual Reflection (2 - 3 minutes):

   • The teacher should then propose that students individually reflect on what they have learned in class. He could ask questions such as: "What was the most important concept you learned today?" and "What questions have not yet been answered?"
   • The teacher should give the students a minute to think and then ask some volunteers to share their answers. This can help the teacher identify any areas that may need review or clarification in future classes.

4. Feedback and Closure (1 - 2 minutes):

   • Finally, the teacher should ask students for feedback on the lesson. This may include questions such as: "What did you find most useful in today's class?" and "What could be improved?"
   • The teacher should thank the students for their participation and effort, and end the class by reminding them of the importance of ongoing study and practice for understanding and mastering the topic.

This Feedback is crucial for consolidating students' learning, allowing them to express any doubts or difficulties and providing valuable feedback to the teacher on the effectiveness of the lesson and possible improvements for future classes.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should begin the Conclusion by recapping the main points covered during the class. This includes the definition of a quadratic equation, the identification of coefficients and terms, the quadratic formula, and the solution of quadratic equations with real roots. He should reinforce these concepts, emphasizing their importance and how they apply to real-world problems.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should then explain how the lesson connected theory, practice, and applications. He should recall the activities carried out, such as the "Mystery Equations Game" and the solution of real-life problems, and how they allowed students to apply theory in a practical and meaningful way. He should emphasize that the ability to solve quadratic equations is a powerful tool that can be used to solve a variety of problems in various areas.

3. Supplementary Materials (1 - 2 minutes): The teacher should then suggest some supplementary materials for students who want to deepen their knowledge on the subject. This may include math textbooks, educational websites, YouTube videos, math learning apps, and additional problems to solve. He should encourage students to explore these resources at their own pace to further strengthen their skills in quadratic equations.

4. Relevance of the Topic (1 - 2 minutes): Finally, the teacher should highlight the importance of the topic to the students' everyday lives. He should explain that although solving quadratic equations may seem abstract, it has a wide range of practical applications. For example, it can be used to solve problems in physics, economics, engineering, social sciences, and even to better understand probability in games of chance. In addition, the teacher should emphasize that the ability to solve quadratic equations is a valuable skill that can improve students' ability to solve complex problems, develop critical thinking, and problem-solving skills.

This Conclusion is essential for consolidating students' learning, highlighting the importance of the topic, and motivating students to continue studying and practicing. In addition, by providing supplementary materials, the teacher is giving students the tools to become autonomous learners, capable of exploring and mastering mathematical concepts on their own.