Objectives (5 - 7 minutes)

1. Understand the definition of an exponential function and its importance for calculating growth rates: The teacher should clearly explain what an exponential function is, how it is mathematically represented, and why it is an essential tool for calculating the exponential growth of natural phenomena, economic factors, among others.

2. Identify and interpret the graph of an exponential function: The teacher should teach students how to identify an exponential function from a graph and how to interpret this graph, relating it to the phenomenon it represents. This will help students visualize and better understand the concept of exponential growth.

3. Apply the acquired knowledge to solve practical problems: The teacher should present students with a series of problem situations involving exponential functions and ask them to apply what they have learned to solve them. This will help consolidate students' understanding of the subject and develop their problem-solving skills.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing basic mathematical concepts such as exponentiation and exponential equations. This review is essential for students to be able to follow the new content that will be presented. Additionally, the teacher should reinforce the importance of analyzing graphs in the study of Mathematics.

2. Problem Situation: The teacher can propose two problem situations to arouse students' interest in the subject. The first one could be: 'How can we predict the growth of a population over time?'. The second one: 'How can we calculate the value of an investment that grows with a compound interest rate?'. Both situations involve the use of exponential functions, and the teacher should encourage students to think about how these questions can be solved.

3. Contextualization of the Subject: The teacher should explain to students that exponential functions have practical applications in various fields, such as demography, economics, biology, among others. For example, they can be used to model population growth, radioactive decay, the appreciation of an investment, among other phenomena. This will help students understand the importance of the subject beyond the school environment.

4. Introduction to the Topic: The teacher should introduce the topic of the lesson by explaining that they will study how exponential functions are graphically represented. They can mention that visualizing the graph of an exponential function can help better understand the phenomenon it represents. To arouse students' curiosity, the teacher can show some examples of exponential function graphs and ask students what they think these graphs represent.

Development (20 - 25 minutes)

1. 'Population Growth' Activity (10 - 12 minutes): In this activity, students will be challenged to model the population growth of a fictional city using exponential functions. The teacher should divide the class into groups of 4 to 5 students and provide each group with a gridded paper and colored markers. Students should be instructed to draw the graph of population growth over 50 years, considering an exponential growth rate. The teacher should provide the growth rate (for example, 5% per year) and the initial number of inhabitants. Students should then calculate the number of inhabitants for each year and mark these points on the graph. At the end of the activity, each group should present their graph and explain how it represents population growth.

   1. Activity Step-by-Step:
      1. The teacher should divide the class into groups and provide each group with gridded paper and colored markers.
      2. The teacher should provide the growth rate and the initial number of inhabitants for each group.
      3. Students should calculate the number of inhabitants for each year and mark these points on the graph.
      4. Each group should present their graph and explain how it represents population growth.

2. 'Financial Investment' Activity (10 - 12 minutes): In this activity, students will model the growth of a financial investment over time. The teacher should provide each group with gridded paper, colored markers, and the compound interest rate. Students should calculate the value of the investment for each year and mark these points on the graph. At the end of the activity, each group should present their graph and explain how it represents the investment growth.

   1. Activity Step-by-Step:
      1. The teacher should provide each group with gridded paper, colored markers, and the compound interest rate.
      2. Students should calculate the value of the investment for each year and mark these points on the graph.
      3. Each group should present their graph and explain how it represents the investment growth.

3. Group Discussion (5 - 8 minutes): After the presentations, the teacher should promote a group discussion where students can compare the graphs and discuss the similarities and differences between them. The teacher should guide the discussion by asking questions to direct students' attention to relevant aspects of the graph. For example, the teacher can ask: 'What are the inflection points on the graph? What do they represent in the context of population growth/financial investment?' or 'How does the growth rate/investment affect the curve of the graph?'.

Return (8 - 10 minutes)

1. Group Discussion on Solutions (3 - 4 minutes): The teacher should gather all students and promote a joint discussion on the solutions found by each group in the 'Population Growth' and 'Financial Investment' activities. Each group should briefly present their graph and explain how they arrived at the solution. During the presentations, the teacher should encourage other students to ask questions and make comments, fostering a collaborative learning environment. The objective of this stage is for students to see different approaches to the same problem and reflect on the effectiveness of each one.

2. Correlation with Theory (2 - 3 minutes): After the presentations, the teacher should briefly recap the theoretical concepts discussed at the beginning of the lesson and show how they apply to the solutions found by the students. For example, the teacher can explain how the growth rate was represented in the graph of the exponential function, or how the initial number of inhabitants/investment affected the position of the graph on the Cartesian plane. This step is important to reinforce students' understanding of the theory and to show the relevance of the concepts studied.

3. Reflection on Learning (2 - 3 minutes): To conclude the lesson, the teacher should propose that students reflect on what they have learned. The teacher can ask questions like: 'What was the most important concept you learned today?' or 'What questions have not been answered yet?'. Students should write down their answers on a piece of paper, which will be handed to the teacher. This reflection activity is important for students to consolidate what they have learned and identify any doubts or difficulties they may have.

   1. Activity Step-by-Step:
      1. The teacher should gather all students and promote a joint discussion on the solutions found by each group.
      2. Each group should briefly present their graph and explain how they arrived at the solution.
      3. The teacher should briefly recap the theoretical concepts and show how they apply to the solutions found by the students.
      4. The teacher should propose that students reflect on what they have learned and write down their answers on a piece of paper.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should summarize the main points covered during the lesson. This includes the definition of an exponential function, how to identify and interpret the graph of an exponential function, and how to apply the knowledge of these functions to solve practical problems. The teacher can reinforce these concepts through practical examples, such as population growth and financial investment, which were used during the group activities.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should explain how the lesson connected theory, practice, and applications. They can mention how the initial theoretical discussion prepared students for practical activities, and how these activities allowed them to apply what they learned to solve real-world problems. The teacher can also highlight how visualizing the graph of an exponential function helped students better understand the phenomenon it represented.

3. Extra Materials (1 minute): The teacher should suggest additional materials for students who wish to deepen their knowledge of exponential functions. This may include math books, online teaching websites, explanatory videos, among others. For example, the teacher can recommend the book 'Matemática: Ciência e Aplicações' by Gelson Iezzi, for those who want a more in-depth study of the subject.

4. Relevance of the Subject (1 - 2 minutes): Finally, the teacher should emphasize the importance of the subject for daily life and other disciplines. They can explain, for example, how understanding exponential functions can help students better understand the world around them, from population growth to the appreciation of an investment. Additionally, the teacher can mention that exponential functions are widely used in other disciplines, such as Physics, Chemistry, and Biology, and therefore, mastering this subject can be useful in various contexts.