Objectives (5 - 7 minutes)

1. Understand the concept of the general term of an arithmetic progression (AP):

   • Students should be able to identify and understand the formula that allows them to find the value of any term in an AP.
   • They should be able to apply this formula to different examples of AP, thus demonstrating their understanding of the concept.

2. Calculate the value of a term in an AP based on its position:

   • In addition to understanding the general term formula, students should be able to calculate the value of a specific term in an AP based on its position in the sequence.
   • They should be able to solve problems involving the identification and calculation of AP terms, thus demonstrating their ability to apply the concept in practice.

3. Recognize the importance and application of the concept of AP terms in everyday situations:

   • To consolidate learning, students should be able to identify real-life situations where the concept of AP terms can be applied.
   • They should be able to explain how understanding and applying this concept can be useful in solving everyday problems.

Secondary Objectives:

• Develop critical thinking and problem-solving skills:
  • Through solving practical exercises and discussing real-life situations, students should be able to develop their critical thinking and problem-solving skills.
  • They should be able to analyze different scenarios, apply the acquired knowledge, and arrive at logical and coherent solutions.

Introduction (10 - 15 minutes)

1. Review of important concepts (3 - 5 minutes)

   • The teacher should start the lesson by reminding students about the concept of numerical sequences, particularly arithmetic sequences. This is a crucial starting point for understanding arithmetic progression and sequence terms.
   • The review should include the definition of an arithmetic sequence, where each term is the sum of the previous term and a constant (the ratio), and also how to calculate the ratio of an arithmetic sequence.

2. Problem-solving situations (3 - 5 minutes)

   • The teacher can then present two problem-solving situations that will serve as a 'hook' for the topic Introduction. For example, 'If you had to calculate the value of the 20th term of an arithmetic sequence with a ratio of 3, how would you do it?' or 'If you knew the 5th and the 10th position of an arithmetic sequence, how would you find the ratio and the 1st term?'
   • These questions should stimulate students' curiosity and prepare them for the new content.

3. Contextualization (2 - 3 minutes)

   • The teacher should then contextualize the importance of arithmetic progression and sequence terms, explaining how they are applied in various areas such as finance, physics, and computer science.
   • For example, the teacher may mention how the idea of arithmetic progression is used to calculate compound interest in finance, or how it is used to model uniform motion in physics.

4. Introduction to the topic (2 - 3 minutes)

   • Finally, the teacher should introduce the topic of the lesson - the general term of an arithmetic progression - explaining that this is the next step to fully understand arithmetic sequences.
   • The teacher can share some curiosities or interesting applications of this topic to capture students' attention. For example, 'Did you know that the Italian mathematician Fibonacci used arithmetic progressions to model the population growth of rabbits?' or 'In programming, arithmetic sequences are used to create loops that repeat an action a fixed number of times.'
   • This Introduction should prepare students for the lesson content, arousing their interest and showing the relevance of the topic.

Development (20 - 25 minutes)

1. Simulation Activity (10 - 12 minutes)

   • To start this activity, the teacher should divide the class into small groups of 3 to 5 students.
   • Each group will receive a set of numbered cards from 1 to 10. The difference between the numbers in each set of cards should be the same for all groups. For example, one group may receive cards from 1 to 10, with a difference of 1 between the numbers, while another group may receive cards from 5 to 50, with a difference of 5 between the numbers.
   • The challenge for students is to discover the 'rule' that determines the sequence of numbers in their sets of cards. They should work together to identify the difference between the numbers (the ratio) and the first number of the sequence (the initial term).
   • Once students have identified the 'rule' for the sequence of their cards, they should use it to predict the next number in the sequence (the next term of the arithmetic progression).
   • After a period of discussion and calculations, each group should present their conclusions to the class, explaining their 'rule' and justifying their prediction for the next term of the sequence.
   • This activity allows students to explore the concept of the general term of an AP in a playful and interactive way, promoting discussion and collaboration among group members.

2. Research and Application Activity (10 - 12 minutes)

   • In this activity, groups should research and find examples of how arithmetic progressions are applied in different contexts (e.g., finance, physics, computer science, sports, music, etc.).
   • Each group should select an application they find particularly interesting and create a short presentation to share with the class. The presentation should include a brief explanation of how the arithmetic progression is used in the chosen application and a numerical example.
   • For example, a group may choose the application of arithmetic progressions in music (e.g., the sequence of notes in a scale or the sequence of chords in a song) and create a brief musical piece illustrating the arithmetic progression.
   • This activity allows students to see the relevance and applicability of the concept of AP terms in real-world situations, as well as promoting research, creativity, and presentation skills.

3. Group Discussion (5 - 6 minutes)

   • After the presentations, the teacher should facilitate a group discussion where students can share their findings, make connections with the theory, and clarify any doubts.
   • The teacher should encourage students to reflect on how the activities relate to the concept of AP terms and how they can apply this knowledge in their daily lives.
   • This discussion will allow students to consolidate their understanding of the concept, clarify any misunderstandings, and reflect on what they have learned.

Return (10 - 12 minutes)

1. Sharing and Group Discussion (5 - 6 minutes)

   • The teacher should gather all students and promote a group discussion on the solutions or conclusions found by each team during the simulation and research activities.
   • Each group should have the opportunity to share their findings, explain the 'rule' they identified for the sequence of their cards, and present the application they chose.
   • During this discussion, the teacher should encourage students to make connections with the theory, explain how they applied the general term concept of an AP to solve the proposed problems, and give examples of how they can apply this knowledge in other situations.
   • The teacher should also facilitate the discussion by asking questions that lead students to reflect on what they have learned and identify any difficulties or misunderstandings they may have.
   • This group discussion will allow students to see different approaches to problem-solving, promote the exchange of ideas, and clarify any doubts they may have.

2. Learning Verification (3 - 4 minutes)

   • After the discussion, the teacher should briefly review the lesson content, highlighting the key points and concepts that were covered.
   • The teacher should then conduct a quick learning verification by proposing one or two simple problems that students must solve individually.
   • The proposed problems should involve identifying the general term of an AP and calculating the value of a term based on its position.
   • The teacher should circulate around the room, observing students as they solve the problems and offering help or guidance if needed.
   • This learning verification will allow the teacher to assess how well students understood the concept and application of the general term of an AP and identify any areas that may need reinforcement or review.

3. Final Reflection (2 - 3 minutes)

   • To conclude the lesson, the teacher should propose that students make a brief reflection on what they have learned.
   • The teacher can ask questions such as: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • Students should have a minute to think about these questions and then will be invited to share their answers with the class, if they wish.
   • This final reflection will allow students to consolidate their learning, identify any areas they still do not fully understand, and prepare for the next lesson.

Conclusion (3 - 5 minutes)

1. Summary of Contents (1 - 2 minutes)

   • The teacher should start the Conclusion by summarizing the key points discussed during the lesson. This may include the definition of arithmetic progression, the concept of the general term, and how to calculate the value of a term based on its position.
   • The teacher should emphasize the importance of understanding and correctly applying the general term formula, as this will allow students to solve a variety of problems involving arithmetic sequences.

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

   • Next, the teacher should highlight how the practical and research activities carried out during the lesson helped reinforce students' theoretical understanding of the concept of AP terms.
   • The teacher should emphasize how the applications of the concept of AP terms in different contexts (such as finance, physics, computer science, music, etc.) helped make the subject more relevant and engaging for students.

3. Supplementary Materials (1 minute)

   • The teacher should then suggest some readings, videos, or websites that students can explore to deepen their understanding of arithmetic progressions and AP terms.
   • This may include online resources such as Khan Academy or YouTube videos, interactive math websites like Wolfram Alpha, or recommended math books.

4. Importance of the Subject (1 minute)

   • To conclude, the teacher should emphasize the importance of the concept of AP terms in various areas of knowledge and everyday life.
   • For example, the teacher may mention how understanding and applying arithmetic progressions can help solve practical problems, make more informed financial decisions, or better understand complex concepts in other disciplines.
   • This final discussion will help motivate students to continue studying and applying what they have learned, as they realize the relevance and usefulness of the subject.