Objectives (5 minutes)

1. Understand the concept of second-degree function and its graphical representation, including the identification of the vertex, axis of symmetry, and concavity.
2. Learn to determine the maximum and minimum points of a second-degree function through the application of the Bhaskara's formula.
3. Develop skills to solve practical problems involving the determination of maxima and minima of second-degree functions, applying the concepts learned.

Secondary Objectives:

• Develop critical and analytical thinking skills when solving complex mathematical problems.
• Promote understanding of how mathematics is applied in various contexts, strengthening the importance of learning the discipline.
• Encourage active student participation through practical activities and group discussions.

During the establishment of the Objectives, the teacher should briefly review the necessary previous concepts for understanding the lesson topic, such as the Bhaskara's formula, and the graphical representation of second-degree functions.

Introduction (10 - 15 minutes)

1. Content Review: The teacher should start the lesson by reviewing the basic concepts of second-degree functions, such as the general form of the equation, the concept of concavity, and the location of the vertex. This is a crucial moment to ensure that all students are on the same page before moving on to more complex content. (5 minutes)

2. Problem Situation: Next, the teacher can present two problem situations involving the determination of maxima and minima of second-degree functions. For example:

   • 'A manufacturing company has a monthly fixed cost of R$ 1,000.00 and a variable cost of R$ 50.00 per unit produced. Each unit is sold for R$ 100.00. What should be the number of units produced for the company to maximize its profits?'
   • 'An object is thrown upwards with an initial velocity of 20 m/s from a height of 5 m. What will be the maximum height reached by the object and how long will it take for the object to reach the ground?' (5 minutes)

3. Contextualization: The teacher should then explain how solving these problem situations is directly related to the second-degree function and the determination of maxima and minima. For example, in the first problem, the second-degree function represents the company's profit as a function of the number of units produced. In the second problem, the second-degree function represents the object's height as a function of time. (2 minutes)

4. Engaging Students' Attention: To spark students' interest, the teacher can share some curiosities or interesting applications of second-degree functions. For example:

   • 'Did you know that parabolic trajectories, described by second-degree functions, are used in sports such as dart throwing and gymnastics?'
   • 'And that the Bhaskara's formula, which we will use to determine the maxima and minima, was developed in India long before it was taught in our schools?' (3 minutes)

5. Introduction to the Topic: Finally, the teacher should introduce the lesson topic, explaining that students will learn to determine the maximum and minimum points of a second-degree function through the application of Bhaskara's formula. (2 minutes)

Development (20 - 25 minutes)

1. Practical Activity: 'Calculating Profits' (10 - 12 minutes)

   • Divide students into groups of 3 to 4.
   • Provide each group with a set of cards with different values for fixed cost, variable cost, and selling price of a product.
   • Each group should choose a set of cards and then use these values to create a second-degree function representing the company's profit.
   • After creating the function, students should identify the vertex of the parabola (which will represent the maximum or minimum profit, depending on the context) and calculate the corresponding value of x.
   • Finally, students should interpret the result, explaining what it would mean in practical terms for the company. For example, if the value of x is negative or greater than the number of units the company can produce, this would mean that the company could not maximize its profits.

2. Modeling Activity: 'Projectile Launch' (10 - 12 minutes)

   • Still in groups, students should analyze the situation of an object being thrown upwards with an initial velocity and initial height.
   • The task is to model the situation with a second-degree function representing the object's height as a function of time.
   • They should then determine the vertex of the parabola, which represents the maximum height reached by the object, and the corresponding value of x, which represents the time it will take for the object to reach the ground.
   • After determining, students should discuss the result, explaining what it would mean for the object's launch. For example, if the value of x is negative, this would mean that the object has already reached the ground.

3. Group Discussion: 'Application in Real Situations' (5 - 6 minutes)

   • After the conclusion of the activities, the teacher should promote a group discussion for students to share their solutions and interpretations.
   • The objective of this discussion is to reinforce students' understanding of the practical application of the concepts of second-degree function and determination of maxima and minima.
   • The teacher should facilitate the discussion by asking questions to stimulate critical thinking and deepen students' understanding. For example: 'How do changes in fixed cost, variable cost, and selling price values affect the vertex of the parabola?' or 'How could you use the function you created to predict the company's profits in different scenarios?'.

Return (10 - 15 minutes)

1. Group Discussion: 'Connections to the Real World' (5 - 7 minutes)

   • The teacher should start a group discussion where students will have the opportunity to share their insights on the activities carried out and the real-world applications of second-degree functions.
   • Each group should briefly present their solutions and conclusions, emphasizing how the concepts of second-degree function and determination of maxima and minima were applied and how they were able to solve the proposed problems.
   • The teacher should encourage students to make connections with real-life situations, such as the example of the manufacturing company and the projectile launch, discussed during the activities. Students can share examples of everyday situations where understanding these concepts can be useful, such as in personal finance, entrepreneurship, physics, among others.
   • The teacher should facilitate the discussion by asking questions that encourage students to reflect on the practical applications of the concepts learned. For example: 'How could you apply the concept of the vertex of a second-degree function in your lives?' or 'In what other contexts could you use the Bhaskara's formula?'.

2. Individual Reflection: 'Learnings and Doubts' (3 - 5 minutes)

   • The teacher should suggest that students spend a minute in silence thinking about what they learned in the lesson and what questions have not yet been answered.
   • Then, students should share with the class a concept or idea they consider to have learned well and a doubt they still have.
   • The teacher should take note of the students' doubts to address them in the next lesson or during the remainder of the lesson, if time allows.
   • This step is important for the teacher to assess the effectiveness of the lesson and identify areas that may need reinforcement or additional clarification.

3. Feedback and Closure (2 - 3 minutes)

   • To conclude the lesson, the teacher should reinforce the main concepts and skills covered, highlighting their importance for understanding second-degree functions and determination of maxima and minima.
   • The teacher should also thank the students for their participation and encourage them to continue practicing and exploring the concepts learned.
   • Finally, the teacher should ask for feedback from the students about the lesson, asking what they liked and what they would like to see more of in future lessons. This will help the teacher adjust and improve the planning and conduct of future lessons.

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 during the lesson. This includes the concept of second-degree function, the determination of maxima and minima, Bhaskara's formula, and the graphical representation of second-degree functions.
   • The teacher should emphasize the importance of each of these points and how they relate to each other to enable the resolution of complex problems involving second-degree functions.

2. Connection to Practice (1 - 2 minutes)

   • Next, the teacher should explain how the lesson connected theory with practice. It should be highlighted that practical activities, such as modeling the situation of the manufacturing company and the projectile launch, allowed students to apply theoretical concepts concretely and understand their relevance in real situations.
   • The teacher should reinforce that mathematics is not just a set of rules and formulas, but a powerful tool for understanding and solving real-world problems.

3. Complementary Materials (1 - 2 minutes)

   • The teacher should then suggest some complementary materials for students who wish to deepen their knowledge of second-degree functions and determination of maxima and minima. This may include math books, educational websites, explanatory videos, among others.
   • The teacher should also encourage students to practice more by solving additional problems and exploring different types of second-degree functions.

4. Relevance of the Subject (1 - 2 minutes)

   • Finally, the teacher should emphasize the importance of the lesson subject for daily life and other areas of knowledge. It should be highlighted that the ability to analyze and interpret second-degree functions and determine their maxima and minima is useful in many professions and fields of study, including engineering, natural sciences, economics, finance, among others.
   • The teacher should end the lesson by reinforcing the relevance of effort and dedication in studying mathematics, reminding students that, although it can be challenging at times, mathematics is a discipline that rewards persistence and critical thinking.