Objectives (5-7 minutes)

1. To provide students with a clear and concise understanding of the concept of Newton's Binomial, its applications, and its importance in mathematics.

2. To develop students' ability to solve problems involving Newton's Binomial, through practical examples and exercises.

3. To stimulate students' critical thinking, encouraging them to apply the acquired knowledge to solve real-world problems that can be modeled using Newton's Binomial.

Secondary objectives:

• Promote interaction among students, through group activities and discussions, to enhance their communication and collaboration skills.

• Encourage research and autonomous study, by providing additional resources for students who wish to deepen their understanding of the topic.

Introduction (10-12 minutes)

1. Review of previous content (3-5 minutes): The teacher should begin the lesson by reminding students about the concepts of exponentiation, factorial, and simple combination. These are essential for understanding Newton's Binomial. The teacher can use simple examples to reinforce these concepts.

2. Problem situations (3-4 minutes): After the review, the teacher can propose two situations that involve Newton's Binomial. For example, how to calculate the value of (a+b)^2 or (a-b)^3. The goal here is to prepare students for the content that will be covered, by showing them the practical relevance of Newton's Binomial.

3. Contextualization (2-3 minutes): The teacher should explain to students that Newton's Binomial has applications in various areas of science, such as physics and engineering, especially in situations involving the expansion of a polynomial. This will help students understand the importance of the topic.

4. Introduction to the topic (2-3 minutes): The teacher should introduce the topic by explaining that Newton's Binomial is a formula used to calculate the expansion of an expression of the type (a+b)^n, where 'n' is a natural number. The teacher can share curiosities about the mathematician Isaac Newton, who gave his name to this topic. For example, that Newton is known for his contributions not only to mathematics but also to physics and astronomy.

5. Grabbing students' attention (1-2 minutes): To conclude the Introduction, the teacher can propose a challenge involving Newton's Binomial. For example, ask students to try to find the expansion of (a+b)^4. This challenge will pique students' curiosity and prepare them for the Development of the topic.

Development (20-25 minutes)

1. Presentation of the theory (8-10 minutes): The teacher should start by explaining what Newton's Binomial is and how it is represented mathematically. It should be emphasized that the general formula of Newton's Binomial is given by:

(a + b)^n = Cn0 * a^n * b^0 + Cn1 * a^(n-1) * b^1 + ... + Cnn * a^0 * b^n

Where Cnr is the binomial coefficient, given by Cnr = n! / (r! * (n-r)!), a is the first term of the binomial, b is the second term, and n is the exponent.

The teacher should explain that the formula allows calculating each term of the expansion, without the need to expand it completely.

2. Detailed explanation of each element of the formula (5-7 minutes): The teacher should detail each element of the formula, explaining what the binomial coefficient is and how to calculate it, what the term a^n represents, the term b^0, and so on.

The teacher should emphasize that, when the exponent of a decreases, the exponent of b increases, and vice versa.

The teacher can use numerical examples to facilitate understanding, performing the calculations step by step and explaining each stage.

3. Solving practical examples (5-7 minutes): The teacher should then solve practical examples, using the Newton's Binomial formula.

The examples should vary in difficulty, starting with simple examples and gradually progressing to more complex ones.

The teacher should explain each step of the solution, always reinforcing the application of the formula and the importance of each element.

4. Guided practice (2-3 minutes): After solving the examples, the teacher should propose that the students solve a similar problem, but with the teacher's guidance.

The teacher should walk around the room, assisting students as needed and clarifying doubts.

This activity serves to consolidate learning and check students' understanding.

5. Discussion and clarification of doubts (2-3 minutes): Finally, the teacher should open up for discussion and clarification of doubts.

Students should be encouraged to share their insights, difficulties, and resolution strategies.

The teacher should clarify any remaining doubts and provide feedback to the students.

This Development of the class will allow students to understand the concept of Newton's Binomial, know how to apply the formula, and solve problems that involve its use. In addition, guided practice and discussion will help consolidate learning and develop critical thinking and problem-solving skills.

Review (8-10 minutes)

1. Review of the Lesson (3-4 minutes): The teacher should begin the Review phase by reviewing the main points covered in the lesson. This will help students consolidate the knowledge acquired and see the connection between the different aspects of Newton's Binomial. The teacher can ask review questions, such as "What is Newton's Binomial?" "What is the formula for Newton's Binomial?". Students should be encouraged to participate, answering the questions and sharing their own reflections.

2. Connection between Practice and Theory (2-3 minutes): The teacher should then explain how the lesson connects the theory of Newton's Binomial with its practical application. The teacher can use examples of problems solved in class to illustrate how the theory is applied in practice. For example, the teacher can show how the Newton's Binomial formula was used to solve a specific problem. This will help students see the relevance of what they have learned and better understand how to apply Newton's Binomial in real-world situations.

3. Individual Reflection (2-3 minutes): The teacher should then propose that students make an individual reflection on what they have learned. Students should be encouraged to think about the following questions:

• What was the most important concept you learned today?
• What questions do you still have about Newton's Binomial?
• How can you apply what you learned in your daily life or in other disciplines?

Students should have a minute to think about their answers. After this time, the teacher can ask some students to share their reflections. The goal of this activity is to encourage students to think critically about what they have learned and to identify possible areas for improvement. Also, by reflecting on how they can apply what they have learned, students will be developing valuable skills for life beyond the classroom.

4. Feedback and Closure (1-2 minutes): Finally, the teacher should thank the students for their participation and effort during the lesson. The teacher can then give general feedback on the lesson, highlighting the strengths and areas that can be improved. The teacher should encourage students to continue studying Newton's Binomial, providing additional resources, such as extra homework problems or links to explanatory videos.

This Review phase is crucial for consolidating learning, assessing the effectiveness of the lesson, and preparing students for independent study. By reflecting on what they have learned and identifying possible areas for improvement, students will become more aware of their own learning process and will be better prepared for future lessons.

Conclusion (5-7 minutes)

1. Summary of the Content (2-3 minutes): The teacher should recap the main points covered during the lesson. This includes the definition of Newton's Binomial, the general formula for the expansion of a binomial, and the importance of binomial coefficients. The teacher can use diagrams or diagrams to visually reinforce these concepts.

2. Theory-Practice Connection (1-2 minutes): Then, the teacher should highlight how the lesson connected theory with practice. The teacher can recall examples of problems that were solved during the lesson and show how the Newton's Binomial formula was applied in practice. This will allow students to understand the relevance of what they have learned and how they can use this knowledge to solve real-world problems.

3. Suggestion of Extra Materials (1-2 minutes): The teacher should then suggest extra materials for students who wish to deepen their understanding of Newton's Binomial. This may include math books, educational websites, online videos, and math apps. The teacher may, for example, recommend a video that explains the Newton's Binomial formula in a different way or a website that offers interactive exercises for practice.

4. Application in Everyday Life (1 minute): Finally, the teacher should briefly explain how Newton's Binomial can be applied in everyday life. The teacher can mention that this formula is used in various areas of science and engineering to model and solve problems. For example, in physics, the Newton's Binomial formula can be used to calculate the trajectory of an object in a gravitational field. By highlighting these applications, the teacher can help students see the relevance of what they have learned and motivate them to continue studying.

The Conclusion of the lesson is an important opportunity to consolidate learning, reinforce the connection between theory and practice, and prepare students for independent study. By summarizing the content, suggesting extra materials, and discussing the applications of Newton's Binomial, the teacher can help students solidify their understanding and develop a lasting interest in the subject.