Objectives (5 - 10 minutes)

1. Understanding the concept of exponential function and its behaviour: Students should be able to understand the definition of exponential function, including the role of the base and the exponent. They should also understand how the exponential function behaves and how it can be represented graphically.

2. Identifying the inputs and outputs of an exponential function: Students should be able to identify and differentiate the inputs and outputs of an exponential function. This includes the ability to identify the independent variable (input) and the dependent variable (output) in a function.

3. Applying the knowledge of exponential function to solve practical problems: Students should be able to apply the concept of exponential function to solve real-world problems. This includes the ability to model situations with an exponential function and to use the function to predict or analyze behaviors.

Secondary objectives:

• Developing critical thinking and problem-solving skills: Through the study of exponential function, students should be encouraged to develop critical thinking and problem-solving skills. They should be able to analyze situations, identify patterns and trends, and use mathematics to solve complex problems.

• Promoting the understanding of the importance of mathematics in the real world: By applying the exponential function to solve real-world problems, students should develop an appreciation of the relevance and usefulness of mathematics in their everyday lives.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the class by recalling the concepts of powers and exponentiation, already studied previously. This is crucial for students to fully understand the concept of exponential function. The teacher can do this through direct questions to students, recalling the definition of base and exponent, and asking for examples of power calculations.

2. Problem situations: The teacher should then present two problem situations that will be solved throughout the lesson. The first one can be a question involving the growth of a bacteria population, and the second one can be a question about the depreciation of an asset over time. These situations will serve to contextualize the study of exponential functions and to show students how this concept can be applied to solve real-world problems.

3. Contextualizing the importance of the subject: The teacher should then contextualize the importance of exponential functions, explaining how they are used in various areas of science, economics, and engineering. For example, exponential functions can be used to model population growth, radioactive decay, the growth of financial investments, among others. The teacher can also mention how understanding exponential functions can be useful for students in their daily lives, for example, to understand how compound interest works in loans and investments.

4. Introduction of the topic: Finally, the teacher should introduce the topic of the lesson - exponential function: inputs and outputs. The teacher can start by explaining that an exponential function is a function in which the independent variable (input) appears in the exponent. The teacher can then give an example of an exponential function and explain how to identify the inputs and outputs. The teacher can also mention that, in an exponential function, the outputs grow or decrease rapidly as the inputs increase, due to the exponential nature of the growth or decrease.

Development (20 - 25 minutes)

1. Presentation of the theory (10 - 15 minutes): The teacher should present the theory about exponential function, explaining the concept, the general formula and the main characteristics. The presentation can follow the following steps:

   • Definition of exponential function: The teacher explains that an exponential function is a function in which the independent variable (input) appears in the exponent. The teacher can give examples of exponential functions, such as the function (f(x) = 2^x) and the function (f(x) = 3^{2x}).

   • The general formula of an exponential function: The teacher should present the general formula of an exponential function, which is (f(x) = a \cdot b^x), where (a) is the proportionality constant and (b) is the base of the exponential function. The teacher should explain that the base determines the rate of growth or decrease of the function.

   • Characteristics of an exponential function: The teacher should explain that, in an exponential function, the inputs (independent variable) can be any real numbers, while the outputs (dependent variable) are always positive numbers (except in the case of the constant function). The teacher should also emphasize that, in an exponential function, the outputs grow or decrease rapidly as the inputs increase, due to the exponential nature of the growth or decrease.

2. Solving the problem situations (10 - 15 minutes): The teacher should now solve the problem situations presented in the Introduction, using the concept of exponential function. The teacher should do this step by step, explaining each step of the solving process. The teacher should emphasize that the key to solving these problems is to model the situation with an exponential function, identifying the proportionality constant and the base correctly.

   • Solving the first problem situation: The teacher should start by solving the first problem situation, which involves the growth of a bacteria population. The teacher should explain how to model this situation with an exponential function, identifying the proportionality constant and the base correctly. The teacher should then use the exponential function to predict the population of bacteria at a certain moment in the future.

   • Solving the second problem situation: The teacher should then solve the second problem situation, which involves the depreciation of an asset. The teacher should explain how to model this situation with an exponential function, identifying the proportionality constant and the base correctly. The teacher should then use the exponential function to predict the value of the asset at a certain moment in the future.

3. Discussion and clarification of doubts (5 - 10 minutes): After solving the problem situations, the teacher should promote a discussion about the solutions, emphasizing how the exponential function was used to solve the problems. The teacher should also clarify any questions that students may have about the concept of exponential function, the general formula, the characteristics, and the problem solving.

Return (10 - 15 minutes)

1. Connection with the real world (5 - 7 minutes): The teacher should now revisit the problem situations discussed during the lesson and make the connection with the real world. The teacher can do this by explaining how the exponential function is used in various areas of science, economics, and engineering to model phenomena and predict behaviors. The teacher can give examples of how the exponential function is used to model population growth, radioactive decay, the growth of financial investments, among others. The teacher can also mention how understanding exponential functions can be useful for students in their daily lives, for example, to understand how compound interest works in loans and investments.

2. Review of the concepts (2 - 3 minutes): The teacher should then review the main concepts covered during the lesson, such as the definition of exponential function, the general formula, the characteristics, and the problem solving. The teacher can do this interactively, asking students to explain the concepts in their own words.

3. Reflection on the learning (3 - 5 minutes): Finally, the teacher should propose that students reflect on what they have learned during the lesson. The teacher can do this by asking students to answer questions like:

   • What was the most important concept you learned today?
   • What questions have not yet been answered?
   • How can you apply what you learned today in your daily life?

   The teacher should then give the students the opportunity to share their reflections with the class, promoting an open and respectful discussion.

4. Teacher feedback (1 minute): To conclude, the teacher should provide general feedback to the class, praising the students' effort and participation, and reinforcing the importance of the concepts learned. The teacher should also make it clear that full understanding and application of these concepts requires continuous practice and individual effort, and encourage students to continue studying and striving.

Conclusion (5 - 10 minutes)

1. Summary of the contents (2 - 3 minutes): The teacher should start the Conclusion of the lesson by summarizing the main contents covered. This includes the concept of exponential function, the general formula, the characteristics, and the problem solving. The teacher can do this by recalling the key points of each topic and highlighting the interconnection between them.

2. Connection between theory, practice and applications (1 - 2 minutes): The teacher should then reinforce how the lesson connected theory, practice, and applications. The teacher can mention how the theoretical explanation was followed by the practical solving of problem situations, and how these theoretical and practical concepts can be applied to solve real-world problems. The teacher should emphasize that mathematics is not just a set of abstract rules and formulas, but a powerful tool to understand and analyze the world around us.

3. Extra materials (1 - 2 minutes): The teacher should then suggest extra materials for students who want to deepen their understanding of exponential function. These materials can include textbooks, educational videos, interactive mathematics websites, among others. The teacher can also suggest additional exercises for students to practice at home.

4. Importance of the subject (1 - 2 minutes): Finally, the teacher should summarize the importance of the subject covered for everyday life. The teacher can recall how exponential functions are used in various areas of science, economics, and engineering to model phenomena and predict behaviors. The teacher can also mention examples of how understanding exponential functions can be useful for students in their daily lives, for example, to understand how compound interest works in loans and investments.

5. Closing (1 minute): To conclude, the teacher should thank the students for their participation and effort, reinforce the importance of continuous study and encourage students to ask questions and seek clarification whenever necessary. The teacher should then close the lesson, wishing everyone a good day and recalling the date of the next lesson.