Objectives (5-7 minutes)

1. Understand the concept and application of the independent term of x in Newton's binomial theorem.

   • Students should be able to define the independent term of x in a Newton binomial.
   • They should understand how to identify and calculate the independent term of x in a binomial.

2. Develop skills in calculating the independent term of x in Newton's binomial theorem.

   • Students should be able to use the general formula to determine the independent term of x.
   • They should practice calculating the independent term of x through multiple examples.

3. Apply the concept of the independent term of x to real-world situations.

   • Students should be able to identify and use the concept of the independent term of x in real-world problems.
   • They should be able to connect the concept of the independent term of x to everyday situations.

Secondary objectives:

1. Foster critical thinking and problem-solving skills.

   • Students should be encouraged to think critically about how and when to apply the independent term of x.
   • They should be encouraged to solve problems involving the calculation of the independent term of x.

2. Promote collaboration and teamwork.

   • Students should be encouraged to work together to solve problems, share ideas, and strategies.
   • They should be encouraged to help each other understand the concept of and how to apply the independent term of x.

Introduction (10-15 minutes)

1. Review of prior knowledge:

   • The teacher should review the concepts of the Newton binomial theorem, general terms, and binomial expansion previously covered in class.
   • Ensuring that all students have a solid understanding of these concepts is important because they will be the foundation for understanding the current topic.

2. Problem situations:

   • The teacher can present two initial situations to arouse students' interest. For example:
     • Problem 1: "Imagine you are expanding (2x + 3)². How can you find the independent term of x?"
     • Problem 2: "Suppose you have a complex mathematical formula that involves the expansion of a binomial. How can you identify and calculate the independent term of x to simplify the expression?"

3. Contextualization:

   • The teacher should explain the importance of the topic by showing how calculating the independent term of x is applied in various fields such as physics, engineering, economics, and computer science.
   • Practical examples can be given of how this concept is used, such as in predicting natural phenomena, designing structures, modeling economic systems, and so on.

4. Introduction to the topic:

   • To pique students' interest, the teacher can share some interesting facts or applications of the topic. For example:
     • Fun fact 1: "Did you know that the independent term of x in Newton's binomial theorem can be calculated using the binomial coefficient? This was discovered by the German mathematician Gottfried Wilhelm Leibniz in the 17th century."
     • Fun fact 2: "Calculating the independent term of x is very important in mathematics because it allows us to simplify complex expressions and solve equations more efficiently. It's like a powerful tool we use to 'cut off' the part of the expression that we don't want or need."
   • The teacher should then state the objective of the lesson: to understand the concept and application of the independent term of x in Newton's binomial theorem and to develop skills in calculating it.

Development (20-25 minutes)

1. Activity "Exploring the Formula" (10-12 minutes)

   • The teacher should divide the class into groups of 4-5 students each and provide each group with a paper with the general formula for the Newton binomial and the task of calculating the independent term of x for different binomials.
   • Students should work together to identify the binomial coefficients and the terms with the variable x raised to a power, and then calculate the independent term of x.
   • The teacher should circulate around the room, assisting groups that are struggling and asking guiding questions to encourage critical thinking and conceptual understanding.
   • After the allotted time, each group should present one of their solutions to the class, explaining the process they used and any challenges they faced.

2. Activity "The Independent Term Game" (10-12 minutes)

   • Still in their groups, students should be given a series of problems involving the calculation of the independent term of x.
   • The problems can vary in difficulty, from simple binomial expansions to solving equations involving the independent term of x.
   • The goal is for students to apply what they have learned about the independent term of x in a more practical and contextualized way.
   • The teacher should encourage group members to discuss, share ideas, and collaboratively solve the problems.
   • At the end of the activity, the teacher should select a few problems to solve together as a class, promoting participation from everyone and reinforcing the learning.

3. Discussion and Reflection (5-7 minutes)

   • After the activities, the teacher should lead a class discussion to solidify what was learned.
   • Students should be encouraged to share their solving strategies, the difficulties they faced, and how they overcame them.
   • The teacher should emphasize the importance of the independent term of x in simplifying expressions and solving equations.
   • The teacher should also reinforce the applications of the independent term of x in different fields, showing students how mathematics is present in everyday situations.
   • Finally, the teacher should have students take a minute to reflect on the lesson, thinking about one question that was not answered and one concept they feel they learned well. Students can share their responses with the class if they wish.

Feedback (8-10 minutes)

1. Group Discussion (3-4 minutes)

   • The teacher should gather the attention of the whole class for a group discussion. Each group will have up to 2 minutes to share their solutions, strategies, and takeaways from the activities done during class.
   • It is important for the teacher to encourage students to listen attentively to other groups' presentations so that they can learn from different approaches and perspectives.
   • The teacher can ask questions to each group to make sure that all aspects of the activity have been understood and discussed. For example: "How did you identify the independent term of x in this problem?", "What steps in the calculation were the most challenging?"
   • The purpose of this discussion is to promote reflection on what was learned and to reinforce their understanding of the concept of the independent term of x.

2. Theory Connection (2-3 minutes)

   • After the group presentations, the teacher should summarize the content, connecting the activities done to the theory presented in the Introduction of the lesson.
   • The teacher should reassert the importance of the independent term of x in simplifying expressions and solving equations, and how it applies to real-world situations.
   • It is important for the teacher to check whether all students are able to make the connection between the theory and the activities, clarifying doubts and reinforcing concepts if necessary.

3. Final Reflection (2-3 minutes)

   • To wrap up the lesson, the teacher should have students take a minute to reflect on what they learned.
   • The teacher can ask guiding questions for their reflection, such as: "What was the most important concept learned today?", "What questions still remain unanswered?"
   • After the reflection, students who wish to can share their answers with the class. The teacher should listen attentively to the answers because they can indicate points that need to be reinforced in future lessons.

4. Feedback and Closure (1 minute)

   • The teacher should thank the students for their participation and remark on the importance of the independent term of x in mathematics and in various fields of knowledge.
   • The teacher can also give brief feedback on the lesson, commending students' efforts, pointing out areas for improvement, and encouraging them to continue studying and practicing what was learned.
   • Finally, the teacher should preview the topic of the next lesson and any expectations students should come prepared with.

Closure (5-7 minutes)

1. Recap (2-3 minutes)

   • The teacher should begin the Closure phase by recapping the main points covered during the lesson.
   • This includes defining the independent term of x in a Newton binomial, the general formula to calculate it, and the importance of this calculation in simplifying expressions and solving equations.
   • The teacher can ask one or two students to summarize what they learned, reinforcing their grasp of the concepts.

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

   • Next, the teacher should highlight how the lesson connected the theory of the independent term of x with practice, through the activities done in class.
   • The teacher should emphasize how these concepts and skills can be applied in real-world situations, such as solving complex mathematical problems, simplifying expressions in various fields of science and engineering, and so on.
   • The teacher can give concrete examples of how the independent term of x is applied in different contexts, reinforcing the relevance of what was learned.

3. Extra Materials (1 minute)

   • The teacher should suggest additional study materials for students who wish to delve deeper into the independent term of x.
   • This can include math textbooks, educational websites, explanatory videos, and so on.
   • The teacher can provide a list of these resources or make them available on an online platform so that students can easily access them.

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

   • Finally, the teacher should summarize the importance of the independent term of x as a fundamental concept in mathematics and many other fields of knowledge.
   • The teacher should explain that, despite being a specific topic, the concept of the independent term of x is highly relevant and has a wide range of applications.
   • The teacher should encourage students to continue studying and practicing, not only to do well on tests and exams but also to develop skills that will be useful in their academic and professional lives.