Objectives (5 - 7 minutes)

1. Understand the definition of even and odd functions, recognizing their distinct characteristics.

   • Identify the even function as the one where f(x) = f(-x) for any value of 'x' in the function's domain.
   • Identify the odd function as the one where f(-x) = -f(x) for any value of 'x' in the function's domain.

2. Apply the acquired knowledge in the analysis of function graphs, identifying if they are even, odd, or neither.

   • Develop the ability to recognize even or odd symmetry in a graph.

3. Solve practical problems involving even and odd functions, using the properties of these functions to facilitate the resolution.

   • Develop the ability to apply the concept of even and odd functions in everyday situations.

Secondary Objectives:

• Promote students' logical and analytical reasoning skills, encouraging structured problem-solving.
• Stimulate teamwork and active participation of students during discussions and proposed activities in the lesson.

Introduction (10 - 15 minutes)

1. Review of Previous Content:

   • The teacher should start the lesson by reviewing important concepts that are prerequisites for understanding the current topic, such as the concept of function, domain and range, and the analysis of function graphs. (2 - 3 minutes)

2. Problem Situations:

   • Present two problem situations that will be solved throughout the lesson, both involving even and odd functions:
     • Situation 1: 'Imagine you are drawing the profile of a roller coaster on a graph, where the x-axis represents time and the y-axis represents height. If the roller coaster is symmetrical with respect to the y-axis, what type of function does it represent?'
     • Situation 2: 'If we have the function f(x) = x^3 - 3x, how can we determine if it is even, odd, or neither?' (5 - 7 minutes)

3. Contextualization:

   • Explain the importance of studying even and odd functions, emphasizing their applicability in various areas, such as physics (when studying the parity of physical quantities), economics (in cost and revenue analysis), and even in cryptography (where they are used in public key encryption algorithms). (2 - 3 minutes)

4. Engaging Students' Attention:

   • Curiosity 1: 'Did you know that even and odd functions are related to the symmetry of our body? If we draw a vertical line in the middle of our body, we realize that our arms and legs are symmetrical, meaning that if we invert the sign of one coordinate, the other also inverts. This is a characteristic of even functions! On the other hand, our face, which is not symmetrical, is a characteristic of odd functions!'
   • Curiosity 2: 'Did you know that even and odd functions are also present in nature? The symmetry of the leaves of many plants, such as ferns, is an example of an even function. On the other hand, the shape of ocean waves, which are not symmetrical, is an example of an odd function!' (3 - 5 minutes)

Development (20 - 25 minutes)

1. Theory: Definition and Characteristics of Even and Odd Functions (8 - 10 minutes)

   • The teacher should explain clearly and objectively what an even and odd function is, using mathematical notation and practical examples.
   • Definition of Even Function: f(x) = f(-x) for any value of 'x' in the function's domain.
   • Definition of Odd Function: f(-x) = -f(x) for any value of 'x' in the function's domain.
   • Characteristics of Even and Odd Functions: Symmetry with respect to the y-axis (even function) and origin (odd function).
   • Examples: f(x) = x^2 (even function), f(x) = x^3 (odd function), f(x) = x (neither even nor odd).

2. Reflection Activity: Connection with the Real World (5 - 7 minutes)

   • The teacher should lead a classroom discussion, questioning students about possible applications of even and odd functions in everyday situations or in other disciplines.
   • For example, students can be asked how even and odd functions can be useful in areas such as physics, economics, or cryptography.
   • Students should be encouraged to share their ideas and think critically about the relevance of the subject.

3. Theory: Identification of Even and Odd Functions in Graphs (5 - 7 minutes)

   • The teacher should explain how to identify if a function is even or odd by looking at its graph.
   • Even Function: the function is symmetric with respect to the y-axis, meaning that if we reflect the graph with respect to this axis, we obtain exactly the same graph.
   • Odd Function: the function is symmetric with respect to the origin (0, 0), meaning that if we reflect the graph with respect to this point, we obtain exactly the same graph.
   • Examples: Graphs of the functions f(x) = x^2 (even), f(x) = x^3 (odd), f(x) = x (neither even nor odd).

4. Practical Activity: Analysis of Graphs (2 - 3 minutes)

   • The teacher should provide students with a series of function graphs and ask them to identify if each function is even, odd, or neither.
   • Students should work in groups to solve the activity, discussing their answers and justifying their conclusions.
   • The teacher should move around the classroom, guiding students and clarifying doubts.

5. Theory: Solving Practical Problems with Even and Odd Functions (5 - 7 minutes)

   • The teacher should explain how to use the properties of even and odd functions to solve practical problems.
   • Examples of Problems: Determine if a function is even, odd, or neither based on its algebraic expression. Determine if a function is even, odd, or neither based on its graph.
   • The teacher should solve some of these problems on the board, step by step, so that students can follow and understand the process.

6. Practical Activity: Problem Solving (2 - 3 minutes)

   • The teacher should provide students with a series of problems involving even and odd functions and ask them to solve them.
   • Students should work in groups to solve the problems, discussing their solutions and justifying their reasoning.
   • The teacher should move around the classroom, guiding students and clarifying doubts.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes)

   • The teacher should promote a classroom discussion where each group shares their solutions or conclusions from the practical activities carried out.
   • Each group will have a maximum of 3 minutes to present their conclusions. During the presentations, other groups and the teacher can ask questions or make comments.
   • The teacher should guide the discussion, highlighting the main points and correcting misconceptions, if necessary. It is important for the teacher to encourage the participation of all students, valuing their contributions.

2. Connection with Theory (3 - 5 minutes)

   • After the presentations, the teacher should summarize the main ideas discussed, connecting them with the theory presented at the beginning of the lesson.
   • The teacher should emphasize how the theory of even and odd functions was applied in solving the proposed problems, reinforcing the importance of theoretical knowledge for solving practical problems.
   • The teacher can ask targeted questions to students to stimulate reflection and verify their understanding of the content. For example: 'How did you apply the definition of even and odd functions to solve problem X?' or 'Can what we learned today be useful in everyday situations or in other disciplines?'.

3. Individual Reflection (2 - 3 minutes)

   • The teacher should suggest that students make a brief individual reflection on what was learned in the lesson.
   • Students should think for a minute and then share aloud an idea or concept they found most interesting or challenging.
   • The teacher should value students' contributions, reinforcing the importance of reflection and self-awareness in the learning process.

4. Feedback and Closure (1 - 2 minutes)

   • The teacher should provide overall feedback on the lesson, highlighting the positive points and areas that can be improved.
   • The teacher should thank the students for their participation and effort, encouraging them to continue studying and dedicating themselves.
   • The teacher should remind students about the content of the next lesson and any homework that may be necessary to prepare them for the next topic.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes)

   • The teacher should give a brief summary of the main points covered during the lesson, reiterating the definition of even and odd functions, their characteristics, and how to identify them in graphs.
   • For example, the teacher can review the difference between even and odd functions, highlighting even symmetry with respect to the y-axis and odd symmetry with respect to the origin (0, 0).
   • The teacher should reinforce the importance of understanding these concepts for solving practical problems and their applicability in various contexts, from cryptography to physics.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes)

   • The teacher should explain how the lesson connected the theory, practice, and applications of the concept of even and odd functions.
   • The teacher can mention how the theory was used to solve practical problems and how these problems have applications in everyday life and other disciplines.
   • The teacher should emphasize that understanding the theory is essential to be able to apply it correctly and solve problems effectively.

3. Extra Materials (1 - 2 minutes)

   • The teacher should suggest some extra materials for students who wish to deepen their knowledge of even and odd functions.
   • Extra materials may include explanatory videos, interactive math websites, additional exercises, and textbooks.
   • The teacher should encourage students to explore these materials on their own, emphasizing that learning is not limited to what is taught in the classroom.

4. Importance of the Subject (1 minute)

   • Finally, the teacher should summarize the importance of the subject presented for daily life and other disciplines.
   • For example, the teacher can mention how the ability to identify even and odd functions in real-world problems can be useful in various professions, from engineering to economics.
   • The teacher should end the lesson by reiterating the relevance of the subject and motivating students to continue studying and making an effort.