Objectives (5 - 7 minutes)

1. Understanding Geometric Progression:

   • Students should be able to understand the concept of geometric progression and how it differs from an arithmetic progression.
   • They should be able to identify the general formula of a geometric progression and apply it correctly.

2. Identification of Terms in a Geometric Progression:

   • Students should be able to identify the first term, the ratio, and the nth term of a geometric progression.
   • They should be able to calculate the value of a specific term in a geometric progression.

3. Practical Application of Geometric Progressions:

   • Students should be able to solve real-world problems involving the use of geometric progressions.
   • They should be able to recognize situations where a geometric progression can be applied and correctly apply the concepts learned.

Secondary Objectives:

• Foster active participation of students in the class, encouraging questions and discussions on the subject.
• Develop students' critical thinking and problem-solving skills through the application of geometric progressions in practical contexts.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts:

   • The teacher should start the lesson by reviewing the concepts of numerical sequences, particularly arithmetic sequences, that were previously studied. This review is essential for students to understand the difference between arithmetic and geometric progressions.
   • The teacher can propose some simple examples of numerical sequences for students to identify whether they are arithmetic, geometric, or neither.

2. Problem Situation:

   • The teacher can present two problem situations to stimulate students' thinking:
     1. "A virus multiplies every hour, and at the end of each hour, the amount of virus is double the previous hour. If we have 1 virus initially, how many viruses will we have after 5 hours?"
     2. "A bacterium reproduces every 30 minutes, and with each reproduction, the number of bacteria triples. If we have 2 bacteria initially, how many bacteria will we have after 2 hours?"

3. Contextualization:

   • The teacher can explain how geometric progressions are used in various areas of knowledge, such as biology (in population growth), economics (in compound interest calculations), and physics (in the study of radioactive decay).
   • The teacher can also mention how understanding geometric progressions can be useful in everyday life, for example, to calculate the amount of money we will have in the future if we invest a certain amount with a fixed interest rate.

4. Introduction to the Topic:

   • The teacher should introduce the concept of geometric progression, explaining that it is a sequence of numbers in which each term, starting from the second, is obtained by multiplying the previous term by a constant called the ratio.
   • The teacher can present the general formula of a geometric progression: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the ratio, and n is the number of the term we want to calculate.
   • The teacher can also mention that, unlike arithmetic sequences, where the difference between terms is always the same, in geometric sequences, the ratio between terms is always the same.

Development (20 - 25 minutes)

1. Activity "The Ratio Game" (10 - 12 minutes):

   • The teacher divides the class into groups of 4 to 5 students.
   • For this activity, the teacher prepares cards with numerical sequences, some arithmetic and others geometric. The sequences should be random, without an obvious pattern.
   • Each group receives a set of cards, and the goal is to correctly classify each sequence as arithmetic or geometric, identifying the ratio or the difference between the terms.
   • The teacher circulates around the room, assisting groups that encounter difficulties and correcting errors.
   • At the end of the activity, the teacher promotes a classroom discussion where each group presents their answers and justifies their classifications. The teacher complements the discussions, clarifying doubts, and reinforcing the concepts.

2. Activity "Population Challenge" (10 - 12 minutes):

   • The teacher presents a scenario where a population of animals, for example, rabbits, multiplies geometrically, meaning that each generation, each pair of rabbits has a new pair.
   • The teacher presents to the students the formula to calculate the number of rabbits in a certain generation: an = a1 * 2^(n-1), where a1 is the initial number of rabbits, n is the generation we want to calculate, and an is the number of rabbits in generation n.
   • The teacher proposes a challenge: "If we have 2 rabbits initially, how many rabbits will we have after 5 generations? And after 10 generations?"
   • Students must apply the formula and calculate the number of rabbits for each generation. The teacher circulates around the room, assisting students who encounter difficulties.
   • Students should realize that the rabbit population is growing rapidly and that the progression is geometric. This helps illustrate the idea of exponential growth.
   • The teacher promotes a classroom discussion where students share their answers and discuss the differences between the rabbit populations in each generation. The teacher reinforces the concepts of geometric progression and exponential growth.

3. Activity "Profitable Investment" (5 - 7 minutes):

   • The teacher presents a new scenario: a young person decides to invest a certain amount of money in an investment fund that promises a fixed interest rate of 10% per month.
   • The teacher presents the formula to calculate the investment value after a certain number of months: an = a1 * 1.1^(n-1), where a1 is the initial investment amount, n is the number of months, and an is the investment value after n months.
   • The teacher proposes the challenge: "If the young person invests R$ 1,000.00, how much will he have after 5 months? And after 10 months?"
   • Students must apply the formula and calculate the investment value for each month. The teacher circulates around the room, assisting students who encounter difficulties.
   • Students should realize that the investment value is growing rapidly and that the progression is geometric. This helps illustrate the idea of compound interest.
   • The teacher promotes a classroom discussion where students share their answers and discuss the importance of compound interest in investments.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher invites each group to share the solutions or conclusions they reached during the activities.
   • Each group has a maximum of 3 minutes to present. During the presentations, the teacher should encourage other students to ask questions and express their opinions.
   • The teacher should reinforce the idea that there is no single correct way to solve problems, and that different approaches can lead to equally valid solutions.

2. Connection to Theory (2 - 3 minutes):

   • After the presentations, the teacher should revisit the theoretical concepts discussed at the beginning of the lesson and make the connection with the practical activities carried out.
   • For example, the teacher can explain how the activity "The Ratio Game" helped reinforce students' understanding of the difference between arithmetic and geometric progressions, and how the activities "Population Challenge" and "Profitable Investment" illustrated the practical application of geometric progressions.

3. Individual Reflection (1 - 2 minutes):

   • The teacher suggests that students reflect individually on what they learned in the lesson.
   • The teacher can ask questions like: "What was the most important concept you learned today?" and "What questions have not been answered yet?".
   • The teacher should give a minute for students to think about these questions.

4. Feedback and Closure (2 - 3 minutes):

   • The teacher invites students to share their reflections, if they feel comfortable.
   • The teacher takes this opportunity to clarify any remaining doubts and to reinforce the most important points of the lesson.
   • The teacher thanks the students for their participation and concludes the lesson, indicating what will be covered in the next lesson and if there is any homework to be done.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes):

   • The teacher should start the Conclusion by recalling the main points discussed during the lesson. This includes the concept of geometric progression, the difference between arithmetic and geometric progressions, the general formula of a geometric progression, and how to identify and calculate the terms of a geometric progression.
   • The teacher can briefly review the practical activities carried out, highlighting the main learnings from each of them and how they helped reinforce the theoretical concepts.

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

   • The teacher should emphasize how the lesson connected the theory of geometric progressions with practice through the activities carried out.
   • The teacher can explain again how geometric progressions are applied in different contexts, such as biology and economics, and how understanding these concepts can be useful in everyday life.

3. Extra Materials (1 minute):

   • The teacher can suggest additional reading materials or online resources for students who wish to deepen their knowledge of geometric progressions. This may include math books, explanatory videos, educational websites, among others.

4. Importance of the Topic (1 - 2 minutes):

   • To conclude, the teacher should reinforce the importance of the topic covered for learning mathematics and for students' everyday lives.
   • The teacher can highlight how the ability to recognize and work with geometric progressions can be useful in various situations, from solving complex mathematical problems to making smart financial decisions.
   • The teacher should end the lesson by reinforcing that, despite seeming like an abstract mathematical concept, geometric progressions have real practical applications and are fundamental for understanding many natural and social phenomena.