Objectives (5 - 7 minutes)

1. Understand the concept of a function, identifying the relationship between the elements of the domain and the elements of the codomain.

2. Analyze and interpret function graphs, identifying fundamental characteristics (such as growth, decay, maximums, and minimums) and the relationships between the elements of the domain and the codomain.

3. Solve practical problems involving functions, applying the concepts learned in a contextualized and coherent manner.

Secondary objectives:

• Develop critical and analytical thinking skills in solving mathematical problems.
• Foster the ability to interpret and analyze data through graphs.
• Stimulate autonomy and responsibility in the learning process.

Introduction (10 - 15 minutes)

1. Review of previous content: (2 - 3 minutes)

   • The teacher starts the lesson by reviewing concepts of variables, equations, and inequalities that were covered in previous classes, as these concepts are fundamental for understanding the current topic - functions.

2. Presentation of the problem situation: (3 - 4 minutes)

   • The teacher presents two problem situations to initiate the discussion about functions. The first situation may involve, for example, the price of a product as a function of the quantity purchased. The second situation could be the speed of a car as a function of time. Both situations are examples of how functions can be used to model and predict behaviors.

3. Contextualization of the importance of the subject: (2 - 3 minutes)

   • The teacher explains that functions are widely used in various areas of knowledge, such as Physics, Economics, Engineering, among others. He emphasizes that the ability to understand and work with functions is essential for solving problems in various areas.

4. Curiosities and practical applications: (3 - 5 minutes)

   • The teacher shares some curiosities and practical applications of using functions. For example, he may mention that the trajectory of a projectile, such as a soccer ball, can be described by a quadratic function. Or that the population growth of a city can be modeled by an exponential function. These examples serve to illustrate the importance and relevance of the subject.

5. Introduction to the topic: (1 - 2 minutes)

   • The teacher formally introduces the concept of a function, explaining that a function is a mathematical relationship between two sets, the domain and the codomain, where each element of the domain is associated with a unique element of the codomain. He emphasizes that the function is a powerful tool for the representation and modeling of real phenomena.

Development (20 - 25 minutes)

1. Presentation of the theory: (8 - 10 minutes)

   1. Function definition: (2 - 3 minutes)
      • The teacher begins the theory presentation by formally defining what a function is. He explains that a function is a relationship between two sets, called domain and codomain, so that each element of the domain is associated with a unique element of the codomain.
   2. Elements of a function: (2 - 3 minutes)
      • The teacher continues the explanation, detailing the elements that make up a function: domain, codomain, numerical value, and image.
   3. Representation of a function: (2 - 3 minutes)
      • The teacher then explains the different ways to represent a function: through a table, a mathematical formula, and a graph. He emphasizes that all these representations are equivalent and that the choice of one depends on the context and the convenience of the problem at hand.
   4. Characteristics of a function: (2 - 3 minutes)
      • The teacher discusses the characteristics of a function, such as monotonicity, concavity, local and global extremes, among others. He explains that these characteristics can be identified through the study of the function's graph.

2. Practical examples: (8 - 10 minutes)

   • The teacher presents some practical examples to illustrate the theoretical concepts discussed. For example, he can show how to represent the function 'price of a product as a function of the quantity purchased' through a table, a formula, and a graph. He can also demonstrate how to interpret the graph of a function, identifying its characteristics and making predictions from it.
   • The teacher invites students to actively participate in the lesson, discussing and proposing solutions to the examples presented. He clarifies doubts and provides constant feedback, ensuring that students understand the concepts presented.

3. Fixation activities: (4 - 5 minutes)

   • The teacher proposes some fixation activities so that students can apply the concepts learned. For example, he may ask students to graphically represent the function 'height of a ball as a function of time' or solve a problem involving the interpretation of a function graph.
   • The teacher circulates around the classroom, assisting students, clarifying doubts, and providing feedback. He encourages collaboration among students, encouraging them to discuss solutions among themselves.

4. Theory closure: (2 - 3 minutes)

   • The teacher summarizes the concepts presented in the lesson, reinforcing the importance of understanding and being able to work with functions. He emphasizes that functions are a powerful tool for the representation and modeling of real phenomena and are widely used in various areas of knowledge.

Return (8 - 10 minutes)

1. Group discussion: (3 - 4 minutes)

   • The teacher organizes a group discussion where students are invited to share their conclusions and solutions to the activities carried out. Each group will have a maximum of 3 minutes to present their answers.
   • During the presentations, the teacher promotes interaction between the groups, encouraging questions and comments. He also takes the opportunity to make connections between the solutions presented and the theoretical concepts discussed, reinforcing the students' learning.

2. Learning verification: (2 - 3 minutes)

   • After the presentations, the teacher conducts a brief verification of learning by asking questions about the main concepts covered in the lesson. These questions can be answered orally by the students or in writing, depending on the available time.
   • The objective of this activity is to verify if students have understood the concepts and can apply them coherently and contextually.

3. Individual reflection: (2 - 3 minutes)

   • The teacher suggests that students make an individual reflection on what they learned in the lesson. He presents some guiding questions, such as: 'What was the most important concept you learned today?', 'What questions have not been answered yet?', 'How can you apply what you learned in the lesson in everyday situations?'.
   • Students have a minute to think about these questions. After that time, they are invited to share their answers with the class, if they feel comfortable.

4. Lesson closure: (1 minute)

   • To conclude the lesson, the teacher thanks the students for their participation and effort. He reinforces the importance of studying and practicing the concepts learned, and of asking questions whenever necessary. He also informs about the topic of the next lesson, encouraging students to prepare in advance.

This Return is essential for the teacher to assess the effectiveness of the lesson, identify possible gaps in students' understanding, and plan future interventions. Additionally, it promotes reflection and metacognition, skills that are essential for the development of critical and autonomous thinking.

Conclusion (5 - 7 minutes)

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

   • The teacher recaps the main points covered during the lesson. He reinforces the concept of a function as a relationship between two sets, the domain and the codomain, where each element of the domain is associated with a unique element of the codomain. He also recalls the different ways to represent a function and the characteristics of functions, such as monotonicity, concavity, local and global extremes.
   • To make the recapitulation more interactive, the teacher may ask students to recall the concepts by answering questions or completing sentences.

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

   • The teacher highlights how the lesson connected theory to practice, showing how the theoretical concepts of functions were applied in solving practical problems. He reinforces the importance of understanding the theory to be able to solve problems efficiently and coherently.

3. Supplementary materials: (1 - 2 minutes)

   • The teacher suggests some materials for students to deepen their knowledge about functions. These materials may include textbooks, explanatory videos, math websites, among others. He emphasizes that autonomous study is an essential part of the learning process and that these materials can be useful for reviewing the concepts learned or for exploring related topics in more depth.

4. Relevance of the subject: (1 minute)

   • Finally, the teacher reinforces the importance and relevance of the subject. He refers to the curiosities and practical applications presented during the lesson, highlighting how functions are useful and necessary in various everyday situations and in various areas of knowledge.
   • The teacher may also encourage students to observe and recognize functions in their environment, such as the function that describes the movement of a clock's hand, the function that determines the amount of light emitted by a lamp as a function of time, among others. This activity can help consolidate students' understanding of the topic and the applicability of the concepts learned.

The Conclusion serves to consolidate the knowledge acquired during the lesson, to connect theory to practice, and to motivate students to continue studying and delving deeper into the subject. Additionally, it reinforces the relevance of the subject, helping students to realize the importance and usefulness of what they have learned.