Objectives (5 - 7 minutes)

1. Understand what a geometric progression is:

   • Students should be able to define the concept of geometric progression, understanding that it is a numerical sequence in which each term is found by multiplying the previous term by a constant, called the ratio.
   • It should be emphasized that in a geometric progression, the difference between the terms is always the same.

2. Identify and calculate terms in a geometric progression:

   • Students should be able to identify a geometric progression in a set of numbers, identifying the first term and the ratio.
   • They should be able to calculate any term in a geometric progression, using the general formula for the term of a geometric progression.
   • Students should be able to verify if a number belongs to a geometric progression, finding the ratio between the terms.

3. Solve problems involving terms of a geometric progression:

   • Students should be able to apply the acquired knowledge to solve practical problems involving the calculation of terms in a geometric progression.
   • They should be able to apply the general formula for the term of a geometric progression in various contexts, such as the population of a colony of bacteria, the depreciation of a car, etc.

Secondary Objectives:

• Promote understanding of the content in a playful and contextualized way:
  • The teacher should seek practical examples and everyday situations to illustrate the application of geometric progressions, encouraging students to do the same.
  • Active participation of students should be encouraged, promoting discussions and questioning about the subject.

Introduction (10 - 15 minutes)

1. Review of fundamental concepts:

   • The teacher should start the lesson by reviewing the concepts of numerical sequences and arithmetic progressions, considering that geometric progression is a special sequence that differs from arithmetic progressions by not having a constant difference between terms.
   • A brief practical exercise can be done, asking students to identify if a sequence of numbers is arithmetic or not, and if it is not, what type of sequence it is.

2. Problem situations:

   • The teacher should present two problem situations to arouse students' interest and contextualize the importance of studying geometric progressions:
     1. "Imagine you have a colony of bacteria that multiplies every hour, so that each hour the number of bacteria is three times the number of bacteria from the previous hour. How can we calculate the number of bacteria at a certain moment?"
     2. "Consider that you are buying a new car whose value depreciates by 20% per year. How can we calculate the value of the car in 5 years?"

3. Contextualization of the subject's importance:

   • The teacher should explain that geometric progressions are widely used in various areas, such as biology, economics, physics, among others.
   • Examples can be given on how geometric progression is used to model the growth of a species' population, the depreciation of a property over time, the variation of a stock's value in the stock market, etc.

4. Curiosities and practical applications:

   • To arouse students' curiosity, the teacher can share some curiosities and practical applications of geometric progressions:
     1. "Did you know that the Fibonacci sequences, which are one of the most famous numerical sequences, are actually a geometric progression?"
     2. "Geometric progressions are used in cryptography, a security technique used to protect confidential information on the internet. Without geometric progressions, it would be much more difficult (and time-consuming) for computers to perform complex mathematical operations, such as those used in cryptography."

At the end of this stage, students should be prepared and motivated to learn about geometric progressions, understanding the relevance and application of this content.

Development (20 - 25 minutes)

1. Presentation of theory and concepts (10 - 12 minutes):

   • The teacher should introduce the concept of geometric progression, explaining that it is a sequence of numbers in which each term, from the second one, is the multiplication of the previous term by a constant, called the ratio.
   • It should be emphasized that, in a geometric progression, the ratio is always the same, which means that the difference between the terms is also always the same.
   • The teacher should present the general formula for the term of a geometric progression: a_n = a_1 * r^(n-1), where a_n is the n-th term of the progression, a_1 is the first term, r is the ratio, and n is the number of the term to be found.
   • Each element of the formula should be discussed, explaining what it represents and how it is calculated.
   • The teacher should illustrate the explanation with practical examples, such as calculating the 10th term of a geometric progression with the first term being 2 and the ratio being 3, or finding the ratio of a geometric progression knowing the first term and the fifth term.

2. Resolution of examples (5 - 7 minutes):

   • After presenting the theory, the teacher should solve some examples step by step, so that students can see how to apply the concepts in practice.
   • The examples should be varied, including problems of calculating terms, ratios, and verifying belonging to a geometric progression.
   • The teacher should explain each step of the resolution, showing how to apply the general formula and how to simplify the algebraic expressions.

3. Practical activity (5 - 6 minutes):

   • To consolidate learning, the teacher should propose a practical activity, in which students will have to solve some problems involving geometric progressions.
   • The problems should be contextualized so that students can see the practical application of what they are learning.
   • The teacher should walk around the classroom, assisting students who have difficulties and correcting errors.

4. Discussion and clarification of doubts (2 - 3 minutes):

   • At the end of the activity, the teacher should promote a discussion about the solutions to the problems, asking students to explain how they arrived at the results.
   • The teacher should take advantage of this discussion to clarify any doubts that may have arisen during the activity.
   • Students should be encouraged to ask questions and express their difficulties, so that the teacher can adjust the explanation and practice to each student's individual learning needs.

At the end of this stage, students should be able to identify, calculate, and solve problems involving terms of a geometric progression, understanding the importance and application of this content.

Return (8 - 10 minutes)

1. Group discussion (3 - 4 minutes):

   • The teacher should promote a group discussion about the solutions to the problems from the practical activity, asking students to share their answers and explain how they reached them.
   • The teacher should guide the discussion, asking questions to verify if students understood the concepts and can apply them in different contexts.
   • Students should be encouraged to comment on the strategies they used to solve the problems, so they can learn from each other and realize that there are different ways to reach the same result.

2. Connection with theory (2 - 3 minutes):

   • The teacher should then connect the practical activities with the theory presented, explaining how the theoretical concepts were applied to solve the problems.
   • The importance of understanding the theory to be able to apply it correctly in practice, and vice versa, should be highlighted.
   • Specific examples from the activity can be referenced to illustrate the connection between theory and practice.

3. Individual reflection (2 - 3 minutes):

   • The teacher should ask students to reflect individually on what they learned in the lesson.
   • Questions like: "What was the most important concept you learned today?" and "What questions have not been answered yet?" can be asked.
   • Students should be given a minute to think and write down their answers.
   • After a minute, the teacher should ask some students to share their reflections with the class, promoting a brief discussion about the answers.

4. Feedback and closure (1 minute):

   • The teacher should then end the lesson, thanking everyone for their participation and providing general feedback on the lesson.
   • This is an opportunity to briefly promote the content of the next lesson to maintain students' interest.

At the end of this stage, students should have consolidated their knowledge about geometric progressions, understanding the importance and application of this content, and will have the opportunity to reflect on their learning process.

Conclusion (5 - 7 minutes)

1. Summary of topics covered (2 - 3 minutes):

   • The teacher should summarize the main points discussed during the lesson, reinforcing the concepts of geometric progression, the general formula for the term of a geometric progression, and how to calculate and identify terms in a geometric progression.
   • The practical applications of geometric progressions should also be reviewed, highlighting the examples discussed during the lesson, such as the growth of a colony of bacteria and the depreciation of a car.
   • The teacher should remind students that geometric progression is a powerful tool for modeling phenomena that grow or decrease at a constant rate.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   • The teacher should emphasize how the lesson connected the theory, practice, and applications of geometric progressions, showing how the general formula and the learned calculation methods are used to solve real problems.
   • The practical examples and classroom discussions should also be reinforced as helping to illustrate and deepen the understanding of theoretical concepts.

3. Extra materials (1 minute):

   • The teacher should suggest some extra materials for students who wish to deepen their knowledge of geometric progressions.
   • These materials may include textbooks, math websites, educational videos, among others.
   • Students should be encouraged to explore these materials on their own, remembering that self-learning is an important skill for lifelong learning.

4. Relevance of the subject (1 - 2 minutes):

   • Finally, the teacher should emphasize the importance of geometric progressions in the real world, reminding students that this knowledge can be useful in various areas, such as science, economics, engineering, among others.
   • Examples can be given on how geometric progressions are used in practice, such as in predicting population growth, analyzing time series data, programming computers, etc.
   • It should be emphasized that, besides being useful, geometric progressions are also interesting in themselves, with many fascinating properties and surprising applications.

At the end of this stage, students should have consolidated their understanding of geometric progressions and be motivated to continue exploring this subject.