Objectives (5 minutes)

1. Understand the concept of geometric progression (GP): Students should be able to identify and define a geometric progression and understand how it differs from other types of mathematical sequences.

2. Learn to calculate the sum of the terms of a finite GP: Students should be able to apply the formula to calculate the sum of the terms of a finite GP, using it to solve practical problems.

3. Practice applying the theory in exercises on sum of GP terms: Students should have the opportunity to apply what they have learned by solving a variety of problems involving the sum of GP terms.

Secondary Objectives

1. Develop problemsolving skills: Through practicing exercises on sum of GP terms, students should develop their problemsolving skills, including the ability to analyze a problem, identify the correct strategy, and apply it effectively.

2. Promote interaction and collaboration in the classroom: During the exercises, students should be encouraged to work in groups, promoting collaboration and discussion among them. This not only helps reinforce understanding of the material, but also develops teamwork and communication skills.

3. Stimulate critical thinking and curiosity: Throughout the lesson, students should be encouraged to ask questions, explore different approaches to problemsolving, and reflect on the learning process. This helps develop critical thinking and curiosity, essential skills for effective learning and solving complex problems.

Introduction (10 - 15 minutes)

1. Review of related content (5 minutes): The teacher should start the lesson by briefly reviewing the concepts of numerical sequences and arithmetic progressions, which were covered in previous classes. This review is necessary for students to understand the difference between a geometric progression and other types of sequences.

2. Problem situations (5 minutes): Next, the teacher should present two situations involving the sum of terms of a geometric progression (GP). For example:

   • "Imagine you have a plant that grows at a rate of 50% per day. After 5 days, how much will the plant have grown in total?"
   • "Suppose you have a debt that doubles in value every month. If you owe $100.00 in the first month, how much do you owe in total after 10 months?"

   These situations should be challenging enough to stimulate students' thinking, but not so complex that they cannot solve them without the theory.

3. Contextualization (3 minutes): The teacher should then explain the importance of geometric progression in the real world. They may mention that GP is used to model exponential growth in various areas, such as populations, financial investments, and chemical reactions.

4. Introduction to the topic (2 minutes): To introduce the topic in an interesting way, the teacher can share some curiosities or applications of geometric progression:

   • "Did you know that the famous Fibonacci sequence, which appears in many places in nature, is actually a geometric progression?"
   • "Geometric progression is used in cryptography to generate security keys, in gambling games to predict patterns, and even in music to create pleasant melodies!"

   These curiosities can help spark students' interest in the topic and show how mathematics is relevant and applied in the real world.

Development (20 - 25 minutes)

1. Activity 'Exponential Growth' (10 - 12 minutes): The teacher should divide the class into groups of 3 to 4 students and provide each group with a large sheet of paper, colored pens, and a calculator. Then, the teacher should explain the activity:

   • Each group should draw a graph representing the growth of a population of bacteria over time, considering that the population doubles every hour.
   • Students should calculate and mark on the graph the population at each hour, up to a total of 5 hours.
   • After drawing the graph, students should identify what type of sequence the graph represents (a geometric progression) and calculate the total population after 5 hours.
   • Finally, the groups should present their graphs and calculations to the class, explaining the reasoning behind their solutions.

   This practical activity allows students to visualize and understand the concept of a geometric progression and how to calculate the sum of the terms. Additionally, it promotes collaboration among students and critical thinking.

2. Activity 'Debt Challenge' (10 - 12 minutes): Still in groups, students will receive a new challenge:

   • "Suppose you have a debt that doubles in value every week. If you owe $100.00 in the first week, how much do you owe in total after 10 weeks?"
   • Students should use the formula for the sum of terms of a finite GP to calculate the total debt after 10 weeks.
   • After solving the problem, groups should develop a clear and concise explanation of how they arrived at the solution, which will be shared with the class.

   This activity provides students with the opportunity to apply the theory to a practical problem and to contextualize the concept of geometric progression in a real situation. Additionally, it helps develop problemsolving and communication skills.

3. Discussion and reflection (3 - 5 minutes): After the presentations, the teacher should lead a brief discussion with the class, encouraging students to reflect on what they have learned. Some questions that can be asked include:

   • "How did the activities help you understand the concept of geometric progression and the formula for calculating the sum of the terms of a finite GP?"
   • "Can you think of other everyday situations that can be modeled by a geometric progression?"
   • "What challenges did you face when solving the problems? How did you overcome them?"

   This final discussion allows the teacher to assess students' understanding of the topic and identify any gaps that need to be addressed in future classes.

Return (10 - 15 minutes)

1. Learning Verification (5 - 7 minutes): The teacher should start the Return by verifying students' learning. They can propose that each group briefly present their solutions or conclusions from the activities carried out. This will not only allow students to share their ideas and approaches, but also allow the teacher to assess the class's level of understanding of the topic.

2. Connection with Theory (3 - 5 minutes): Next, the teacher should make the connection between the practical activities carried out and the theory presented. They can point out how the activities illustrated the application of the formula for the sum of the terms of a geometric progression, and how understanding this formula allowed students to solve the proposed problems. The teacher can emphasize the importance of understanding the theory to solve practical problems and how practice helps consolidate understanding of the theory.

3. Individual Reflection (2 - 3 minutes): The teacher should then propose that students reflect individually on what they have learned. They can do this by asking students to write on a piece of paper their answers to questions such as:

   • "What was the most important concept you learned today?"
   • "What questions have not been answered yet?"

   This moment of individual reflection allows students to consolidate what they have learned and identify any doubts or areas of confusion they may have. Students' answers can also provide valuable feedback to the teacher on the effectiveness of teaching and the class's understanding of the subject.

4. Sharing Reflections (2 - 3 minutes): Finally, the teacher can suggest that some students share their reflections with the class. This can be done voluntarily, so that students feel comfortable sharing their ideas. The teacher should encourage students to be honest and to express any difficulties or doubts they may have. They should respond to these doubts as best as possible, or, if necessary, record the doubts to be addressed in future classes.

This Return moment is crucial to consolidate learning, clarify doubts, and assess the effectiveness of teaching. By allowing students to reflect on what they have learned and express their doubts, the teacher can identify and address any gaps in the class's understanding and adjust their teaching plan as needed.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion by recalling the main points covered in the lesson. They should emphasize the concept of geometric progression (GP), the formula for calculating the sum of the terms of a finite GP, and the importance of understanding the difference between a GP and other types of sequences. The teacher can use the blackboard or slides to visualize these points, ensuring that all students have a clear understanding of them.

2. Theory-Practice Connection (1 - 2 minutes): Next, the teacher should highlight how the lesson connected theory and practice. They can mention how the group activities allowed students to apply the theory in practical situations, helping to solidify their understanding of the topic. The teacher should emphasize that theory and practice are complementary and that understanding the theory is fundamental to solving practical problems.

3. Additional Materials (1 - 2 minutes): The teacher should then suggest additional materials for students who wish to deepen their understanding of the topic. These materials may include math books, educational websites, explanatory videos, and additional exercises. The teacher can provide a list of these materials at the end of the lesson plan, or send them by email to students after the lesson.

4. Everyday Applications (1 minute): Finally, the teacher should contextualize the importance of the topic for real life. They can mention that geometric progression is used in various areas, such as science, finance, and technology. For example, GP can be used to model population growth, predict market trends, optimize algorithms, and much more. The teacher can encourage students to look for examples of geometric progression in their daily lives, to reinforce the relevance of the topic.

This Conclusion will summarize the main points of the lesson, highlight the connection between theory and practice, and encourage students to continue learning about the topic. Additionally, by contextualizing the importance of the topic for real life, the teacher will help students see the relevance of mathematics and feel motivated to learn more.