Objectives (5 - 10 minutes)

1. Understand the concept of a second-degree function: The teacher must ensure that students understand what a second-degree function is and how it is represented. This includes identifying the coefficients a, b, and c in the equation and distinguishing between parabolas with concavity upwards and downwards.

2. Identify and calculate the vertex of a parabola: Students should be able to locate the vertex of a parabola and understand how this point represents the maximum or minimum of the second-degree function. This involves applying the vertex formula and solving practical problems.

3. Analyzing the variation of a second-degree function: Students should be able to determine if a second-degree function has a maximum or minimum, and how the presence of these points affects the shape of the parabola. This includes interpreting graphs and solving contextualized problems.

   Secondary objectives:

   • Develop critical thinking skills and problem-solving abilities.
   • Foster collaboration and discussion in the classroom through practical activities.

The teacher should start the lesson by presenting these Objectives to the students, so they know what to expect from the lesson and can mentally prepare for the content that will be covered.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by reviewing with the students the concepts of function, coefficients, and how to plot a graph on the Cartesian plane. This review is crucial for students to understand and apply the concepts of the second-degree function.

2. Problem situations: To spark students' interest, the teacher can present two problem situations that involve calculating maxima and minima. For example:

   • Situation 1: 'Imagine you are building a bridge and need to determine the highest point of the structure's curvature to ensure the safety of vehicles passing underneath. How can you use the second-degree function to solve this problem?'
   • Situation 2: 'Suppose you are a businessman and need to maximize your company's profit. You know that profit is modeled by a second-degree function. How can you use this function to determine the production level that maximizes profit?'

3. Contextualization: The teacher should explain to the students the importance of second-degree functions in different areas, such as engineering, economics, and physics. This can be done through practical examples, such as determining the trajectory of a projectile, modeling a natural phenomenon, or optimizing a production process.

4. Introduction to the topic: To introduce the topic, the teacher can share some interesting curiosities or applications of second-degree functions. For example:

   • Curiosity 1: 'Did you know that the shape of a soccer ball is modeled by a second-degree function? The parabola that describes the curvature of each panel of the ball is a quadratic function.'
   • Curiosity 2: 'Second-degree functions are also used in the process of animating characters in movies and games. The smooth and realistic movements we see on the screen are the result of complex mathematical functions, including second-degree functions.'

By the end of the Introduction, students should understand the relevance of the topic and be motivated to learn more about second-degree functions.

Development (20 - 25 minutes)

1. Theory: Second Degree Function and the Vertex of the Parabola (10 - 12 minutes):

   • Definition and Canonical Form: The teacher should start by explaining the definition of a second-degree function and the canonical form of the equation, f(x) = ax² + bx + c, where a ≠ 0. It is important to emphasize the roles of the coefficients a, b, and c in forming the parabola.
   • Vertex of the Parabola: Next, the teacher should teach how to identify the vertex of a parabola and the relationship of this point with the concavity of the curve. The vertex is given by the formula (-b/2a, f(-b/2a)) and represents the maximum or minimum point of the function.
   • Interpretation of the Vertex: The teacher should explain that if a > 0, the parabola has concavity upwards and the vertex represents the minimum point of the function. If a < 0, the parabola has concavity downwards and the vertex represents the maximum point of the function.

2. Practice: Calculating the Vertex (5 - 7 minutes):

   • The teacher should present students with a series of second-degree equations and ask them to calculate the vertex of each parabola. This can be done in groups to promote collaboration and discussion. The teacher should move around the classroom, offering help and clarifying doubts as needed.

3. Theory: Analysis of Second Degree Functions (5 - 6 minutes):

   • Maxima and Minima: The teacher should teach students how to determine if a second-degree function has a maximum or minimum, based on the value of the coefficient a. If a > 0, the function has a minimum. If a < 0, the function has a maximum.
   • Function Variation: The teacher should explain that if the function has a maximum, it decreases to the maximum and then increases. If the function has a minimum, it increases to the minimum and then decreases.

4. Practice: Analysis of Second Degree Functions (5 - 7 minutes):

   • The teacher should present students with a series of graphs of second-degree functions and ask them to identify the vertex, the concavity, and whether the function has a maximum or minimum. This can also be done in groups to promote collaboration and discussion. The teacher should move around the classroom, offering help and clarifying doubts as needed.

By the end of the Development, students should be able to calculate the vertex of a parabola, interpret the variation of a second-degree function, and analyze a graph to determine if the function has a maximum or minimum. They should also understand the role of the coefficients a, b, and c in forming the parabola.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher should facilitate a group discussion so that students can share their answers and solutions to the practical activities. Each group should briefly present their conclusions, explaining how they arrived at them and which strategies they used. The teacher should encourage students to ask questions and make comments to promote interaction and reflection.

2. Connection with Theory (3 - 5 minutes): After the group presentations, the teacher should summarize the discussions, highlighting the main points and connecting them with the theory presented. The teacher should reinforce key concepts, emphasizing the importance of the vertex in the analysis of second-degree functions.

3. Individual Reflection (2 - 3 minutes): The teacher should suggest that students make a brief individual reflection on what they learned in the lesson. They should think about answers to the following questions:

   1. What was the most important concept you learned today?
   2. What questions have not been answered yet?

   Students should write down their answers, which can be shared with the class or handed in to the teacher. This activity aims to help students consolidate their learning and identify possible gaps in their understanding.

4. Feedback and Closure (1 - 2 minutes): The teacher should thank the students for their participation and effort, and encourage them to continue studying the topic. The teacher can provide general feedback on the class's performance, highlighting strengths and areas that need more practice or study.

By the end of the Return, students should have a solid understanding of the concept of a second-degree function and how to calculate the vertex of a parabola. They should also be able to interpret the variation of a second-degree function and analyze a graph to determine if the function has a maximum or minimum. Additionally, students should have had the opportunity to reflect on what they learned and identify areas that need further study.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion by giving a brief summary of the main points covered in the lesson. This includes the definition of a second-degree function, the canonical form of the equation, the vertex formula, the interpretation of the vertex, the analysis of second-degree functions, and the variation of a function. The teacher should ensure that students understand and remember these fundamental concepts.

2. Connection between Theory and Practice (1 - 2 minutes): Next, the teacher should reinforce how the lesson connected theory, practice, and the application of the content. The teacher can highlight how solving practical problems and analyzing real situations helped students better understand the theory and the importance of the topic.

3. Extra Materials (1 - 2 minutes): The teacher should suggest extra materials for students who want to deepen their knowledge of second-degree functions. These may include books, websites, videos, games, and apps that offer additional explanations, examples, exercises, and practical applications of the topic. The teacher should emphasize the importance of autonomous study and regular practice for effective learning.

4. Applications in Everyday Life (1 - 2 minutes): Finally, the teacher should illustrate how the knowledge acquired in the lesson can be applied in everyday life. This can be done through practical examples, such as modeling natural phenomena, optimizing industrial processes, predicting economic trends, among others. The teacher should emphasize that second-degree functions are powerful tools for problem-solving and decision-making in different areas.

By the end of the Conclusion, students should have a clear and comprehensive understanding of the lesson topic, including the main concepts, the connection between theory and practice, the applications in everyday life, and the resources available to deepen learning. The teacher should conclude the lesson by reinforcing the importance of continuous study and regular practice, and encouraging students to explore the topic autonomously and creatively.