Objectives (5 - 7 minutes)

1. Understand the concept of a second-degree function and its graphical representation, focusing on the identification of parabolas and their characteristics (concavity, vertex, axis of symmetry).

2. Master the ability to calculate the maximum or minimum value of a second-degree function, using the Bhaskara formula and the property of the axis of symmetry.

3. Develop the ability to solve practical problems involving the application of the second-degree function, such as determining maximum and minimum values in real contexts.

Secondary objectives:

• Stimulate active student participation in problem-solving, promoting critical thinking and decision-making skills.

• Foster collaboration among students, encouraging discussion and group work to solve the proposed exercises.

• Apply theory to practice, reinforcing the importance of mathematical content in daily life and in various areas of knowledge.

Introduction (10 - 12 minutes)

1. Review of previous content (3 - 4 minutes):

   • The teacher starts the lesson by reviewing the concepts of function, its characteristics, and graphical representation, which were covered in previous classes.
   • It is important for students to have a clear understanding of the definition of a function, how to identify the domain and range, and how to plot the graph of a linear function.

2. Problem situations (3 - 4 minutes):

   • The teacher proposes two situations that challenge students to think about the topic of the lesson:
     • A company needs to determine the production value that maximizes profit. How can the second-degree function help them solve this problem?
     • In an amusement park, an engineer needs to design a roller coaster that reaches the highest possible height, but with a safe incline. How can the second-degree function assist him in this task?

3. Contextualization (2 - 3 minutes):

   • The teacher contextualizes the importance of the second-degree function, explaining that it is a mathematical tool widely used in various areas such as engineering, economics, physics, and architecture, to model natural and social phenomena.
   • Additionally, the teacher may mention that the second-degree function also has practical applications in everyday problems, such as calculating the trajectory of a moving object or the shape of an arch in a bridge.

4. Introduction to the topic (2 - 3 minutes):

   • To spark students' interest, the teacher can share some curiosities about the second-degree function:
     • The Bhaskara formula for solving a quadratic equation is one of the oldest equations in the world, having been discovered in India in the 7th century.
     • The term 'quadratic' comes from the Latin 'quadratus', which means 'square', referring to the fact that the variable in the second-degree function is squared.
   • The teacher can also mention some modern applications of the second-degree function, such as the use of parabolas in creating special effects in movies and computer games.

5. Attention and engagement (1 minute):

   • To capture students' attention, the teacher can show a short video illustrating real-world applications of the second-degree function, such as modeling projectile trajectories in shooting games or determining the shape of a parabolic mirror.
   • The teacher can ask students if they can think of more examples of real-life situations that could be modeled by a second-degree function.

Development (20 - 25 minutes)

1. Parabola modeling activity (8 - 10 minutes):

   • Students will be divided into groups of 4 to 5 people. Each group will receive a gridded paper, a set of colored pens, and a ruler.
   • The task is to create a parabola on their paper, taking into account the following restrictions: the parabola must have its vertex at a specific point on the paper, and must pass through two determined points.
   • Students will calculate the coordinates of the vertex point, using the axis of symmetry formula and the property that the vertex of the parabola is exactly in the middle between the two points the parabola passes through.
   • After the calculation, students will mark the vertex point on the paper and then draw the parabola. They should use colored pens to differentiate the curve of the parabola, the axis of symmetry, and the determined points.
   • At the end of the activity, each group will present their parabola to the class, explaining how they made the calculations and the reasons for their choices. The teacher will assess the correctness of the parabolas and the students' understanding of the concept.

2. Problem-solving activity (8 - 10 minutes):

   • The teacher will distribute to each group a series of problems involving the application of the second-degree function. The problems can be from different contexts, such as economics, physics, engineering, among others.
   • Students will read and discuss each problem, identifying relevant information and the unknowns to be determined.
   • Next, students will model the problem as a second-degree function, writing the corresponding equation.
   • Finally, students will use the Bhaskara formula to solve the equation, determining the maximum or minimum value, and will present the solution to the class, justifying the reasoning used.
   • The teacher will circulate around the room, assisting groups that encounter difficulties and asking questions to promote reflection and understanding of the content.

3. Group discussion (4 - 5 minutes):

   • After all groups have presented their solutions, the teacher will facilitate a group discussion where students can ask questions, give feedback, and comment on the solutions presented.
   • The teacher will reinforce important concepts, clarify possible doubts, and correct any errors. They may also present other possible solutions to show the diversity of approaches in problem-solving.
   • This step is crucial to consolidate learning, allowing students to see different perspectives and understand that there is not only one correct way to solve a problem.

Return (8 - 10 minutes)

1. Group discussion (3 - 4 minutes):

   • The teacher leads an open conversation with all students, where each group can share their solutions and conclusions.
   • During this discussion, the teacher must ensure that all groups have the opportunity to speak and that all students participate actively.
   • The teacher can ask guiding questions to stimulate students' reflection on the problem-solving process and the application of the second-degree function. For example: 'How did you decide which points to use to determine the parabola?' or 'How did you know that the solution you found was the maximum or minimum?'.

2. Connection with theory (2 - 3 minutes):

   • After hearing the groups' solutions, the teacher summarizes the most important points, relating them to the theory presented at the beginning of the lesson.
   • For example, the teacher can highlight how the Bhaskara formula was used to calculate the maximum or minimum value, or how the property of the axis of symmetry was used to determine the vertex of the parabola.
   • This step is crucial for students to understand the relevance of theory in solving practical problems and to consolidate their learning.

3. Individual reflection (2 - 3 minutes):

   • The teacher suggests that students reflect individually on what they learned in the lesson.
   • They can ask questions like: 'What was the most important concept you learned today?' or 'What questions have not been answered yet?'.
   • Students will have a minute to think about these questions and then will be invited to share their answers with the class.
   • The teacher should encourage students to express their opinions and doubts, and thank everyone for their participation and effort.

4. Teacher feedback (1 minute):

   • At the end of the lesson, the teacher will provide brief feedback on the students' performance, highlighting the positives and areas that need more attention.
   • The teacher should also reinforce the importance of the content learned, explaining how it relates to other topics in mathematics and practical applications.
   • This feedback will help students better understand their skills and difficulties, and prepare them for the next lessons.

5. Homework assignment (1 minute):

   • The teacher can assign homework that consists of solving other problems involving the second-degree function, or researching more about the applications of this topic in different areas of knowledge.
   • This homework assignment will allow students to practice what they have learned and deepen their understanding of the topic.

Conclusion (5 - 7 minutes)

1. Summary of contents (2 - 3 minutes):

   • The teacher should start the Conclusion by recapping the main points covered during the lesson. This includes the concept of the second-degree function, its graphical representation in the form of a parabola, the identification of its characteristics (concavity, vertex, axis of symmetry), and the ability to calculate the maximum or minimum value.
   • The teacher can give a brief summary of the Bhaskara formula and the property of the axis of symmetry, reinforcing their importance in solving problems involving second-degree functions.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   • Next, the teacher should emphasize how the lesson connected theory, practice, and applications. This may include discussing how theoretical concepts were applied in the practical activity of modeling with parabolas and in problem-solving.
   • The teacher can also reinforce the practical applications of the second-degree function, recalling the examples of real-life situations discussed in the Introduction of the lesson.

3. Extra materials (1 minute):

   • The teacher may suggest some extra materials for students who wish to deepen their knowledge on the topic. This may include math books, educational websites, explanatory videos, and online math games.
   • For example, the teacher may recommend the use of graphic design apps that allow students to plot parabolas, or reading popular science articles that explore the applications of the second-degree function in different areas.

4. Importance of the content (1 minute):

   • Finally, the teacher should emphasize the importance of the content learned for daily life and other disciplines.
   • The teacher can mention how the ability to model and solve problems with second-degree functions is useful in various areas such as engineering, physics, economics, and architecture.
   • Additionally, the teacher can reinforce that the ability to analyze and interpret graphs of second-degree functions is a valuable skill not only in mathematics but also in sciences and many professions.

5. Closure (1 minute):

   • The teacher thanks the students for their participation, reinforces the importance of continuous study and practice, and encourages students to bring their doubts to the next lesson.
   • The lesson ends with the expectation that students have acquired a good understanding of the second-degree function and solving problems involving maxima and minima.