Objectives (5 minutes)

1. Understanding the Concept of Arithmetic Progression: The teacher must ensure that students clearly understand what an arithmetic progression is, how it is formed, and what its main characteristics are. This should include identifying the common difference between terms and the ability to predict the next terms.

2. Identification of Terms in an Arithmetic Progression: Students should be able to identify and name the terms of an arithmetic progression, as well as the position of each term in the sequence. This requires a clear understanding of how the common difference is used to calculate each term.

3. Calculation of Terms in an Arithmetic Progression: Students should be able to calculate the terms of an arithmetic progression, given the position of the term and the common difference. This requires the application of mathematical formulas and the ability to work with numbers.

Secondary Objectives:

• Development of Problem-Solving Skills: Through the study of arithmetic progressions, students will also develop their problem-solving skills, as solving mathematical problems requires the application of concepts and formulas in different contexts.

• Application of Knowledge in Practical Situations: The teacher should also encourage students to apply the knowledge gained in practical situations, such as calculating the time required to travel a distance at a constant speed.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by briefly reviewing the concepts of numerical sequences and common differences. These are fundamental concepts for understanding arithmetic progressions and should therefore be revisited before proceeding. The teacher can do this through a quick classroom discussion or a brief quiz to assess students' prior knowledge.

2. Presentation of Problem Situations: To arouse students' interest and demonstrate the relevance of the topic, the teacher can present two problem situations:

   • The first situation may involve calculating the value of an item in a series of progressive discounts.
   • The second situation may involve calculating the time required to travel a distance at a constant speed.

3. Contextualization of the Subject: The teacher should then explain how arithmetic progressions are used in everyday situations. For example, they are often used in finance to calculate the growth of investments over time. Additionally, they are also used in physics to calculate uniform motion.

4. Introduction to the Topic: To capture students' attention, the teacher can share curiosities or interesting facts about the topic. For example:

   • The most famous arithmetic progression is the Fibonacci sequence, which is a series of numbers where each number is the sum of the two previous ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
   • The arithmetic progression is one of the first numerical sequences that mathematicians study, as it has many interesting properties and is relatively easy to understand.

5. Lesson Objectives: At the end of the Introduction, the teacher should summarize the lesson objectives, reminding students that they will learn to identify, name, and calculate the terms of an arithmetic progression. Additionally, they will develop their problem-solving skills and learn to apply knowledge in practical situations.

Development (20 - 25 minutes)

1. Theory - Concept of Arithmetic Progression (5 - 7 minutes): The teacher should start the theoretical part by explaining the concept of arithmetic progression. They should emphasize that an arithmetic progression is a sequence of numbers where the difference between each pair of consecutive terms is always the same, called the common difference. The teacher can use visual examples, such as a slanted line, to illustrate the idea of a constant difference. They should then present the general formula for the nth term of an arithmetic progression: a_n = a_1 + (n - 1)d, where a_n is the nth term, a_1 is the first term, n is the term's position, and d is the common difference.

2. Theory - Identification of Terms in an Arithmetic Progression (5 - 7 minutes): The teacher should then explain how to identify and name the terms of an arithmetic progression. They should demonstrate that, given the first term and the common difference, it is possible to calculate any term of the sequence using the general formula. The teacher can do this through practical examples, such as calculating the 10th term of an arithmetic sequence with a first term of 2 and a common difference of 4.

3. Practice - Calculation of Terms in an Arithmetic Progression (5 - 7 minutes): After the theoretical explanation, the teacher should guide students through several practical exercises of calculating terms in an arithmetic progression. They should start with simple examples and gradually increase the complexity of the exercises. The teacher should provide constant feedback and correct any errors immediately to ensure that students are understanding the material.

4. Theory - Infinite Arithmetic Progressions (3 - 5 minutes): The teacher should then explain that an arithmetic progression can have an infinite number of terms. They should demonstrate this using the general formula and explain that even though the common difference is a finite number, the sequence of terms will continue forever.

5. Application - Practical Examples (2 - 3 minutes): To conclude the theoretical part, the teacher should present some practical examples of how arithmetic progressions are used in everyday life. For example, they can show how they are used to calculate the growth of investments over time or to calculate the time required to travel a distance at a constant speed.

6. Practice - Problem Solving (3 - 5 minutes): Finally, the teacher should propose some problems for students to solve in the classroom. These problems should involve applying the concepts learned to solve real-life situations. The teacher should encourage students to work together and discuss their problem-solving strategies. They should also be available to answer any questions and provide guidance as needed.

Feedback (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher should initiate a group discussion, allowing students to share their solutions and approaches to the proposed problems. This not only helps reinforce the concepts learned but also promotes collaboration and critical thinking. During the discussion, the teacher should ask open-ended questions to stimulate students' reflection and ensure that everyone is engaged in the conversation.

2. Connection to Theory (3 - 5 minutes): After the discussion, the teacher should make the connection between practice and theory, highlighting how the concepts and formulas that were learned were applied to solve the problems. This helps students understand the relevance of what they have learned and how they can apply this knowledge in different contexts.

3. Doubt Resolution (2 - 3 minutes): The teacher should then give students the opportunity to clarify any doubts that may have arisen during the discussion. They should encourage students to ask questions and share their difficulties, and should respond to all questions clearly and concisely. If there are questions that the teacher does not know how to answer, they should commit to researching the answer and bringing it to the next class.

4. Individual Reflection (5 minutes): Finally, the teacher should suggest that students reflect individually on what they learned in the lesson. They can do this by asking questions such as:

   • What was the most important concept you learned today?
   • What questions have not been answered yet?
   • How can you apply what you learned today in everyday situations? Students should write down their answers, and the teacher can collect these notes to assess students' understanding and to plan future lessons.

5. Teacher Feedback (2 - 3 minutes): To conclude the lesson, the teacher should provide feedback to students on their performance. They should praise strengths and offer constructive suggestions for improvements. Additionally, the teacher should reinforce the most important concepts of the lesson and remind students of what will be learned in the next class.

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should start the Conclusion of the lesson by summarizing the main topics covered. They should review the concepts of arithmetic progression, term identification, and the formula for calculating terms. The teacher can do this through a brief quiz or classroom discussion to check students' retention.

2. Connection between Theory, Practice, and Applications (2 - 3 minutes): Next, the teacher should explain how the lesson connected theory, practice, and applications of the topic. They should reinforce that through theory, students were able to understand and apply the formula of an arithmetic progression. Practice, in turn, allowed them to enhance their calculation and problem-solving skills. And finally, the discussion of applications showed students how this knowledge is relevant to everyday situations.

3. Additional Materials (1 - 2 minutes): The teacher should then suggest some additional study materials for students who wish to deepen their understanding of the topic. This may include math books, educational websites, explanatory videos, and online exercises.

4. Relevance of the Topic (1 - 2 minutes): The teacher should conclude the lesson by emphasizing the importance of the topic for students' daily lives. They should explain that although arithmetic progressions may seem abstract, they are used in many aspects of life, from predicting financial trends to calculating the time required to travel a distance at a constant speed.

5. Encouragement for Continued Study (1 - 2 minutes): Finally, the teacher should encourage students to continue studying the topic on their own. They should remind them that mathematics is a cumulative discipline, and mastery of a topic depends on understanding previous topics. Therefore, the teacher should suggest that students review the concepts learned and practice calculations at home to ensure they are prepared for future classes.