Objectives (5 - 10 minutes)

1. Understanding the Binomial Theorem: Students should be able to understand what the binomial theorem is, how it is formulated, and how it is represented mathematically. This includes identifying the terms and the binomial coefficients.

2. Identifying Patterns in the Expansion of the Binomial Theorem: Students should be able to identify the patterns that appear when expanding the binomial theorem, such as Pascal's Triangle. This includes an understanding of how binomial coefficients are calculated.

3. Applying the Binomial Theorem to Problem Situations: Students should be able to apply what they have learned to solve problems that involve the binomial theorem. This includes expanding binomials raised to a certain power and solving equations that involve binomial expressions.

Secondary objectives:

• Developing critical and analytical thinking: By solving problems that involve the binomial theorem, students will be encouraged to develop their critical and analytical thinking skills.

• Developing mathematical communication skills: Students will be encouraged to explain and discuss their solutions, helping them to develop their mathematical communication skills.

Introduction (10 - 15 minutes)

1. Review of Prerequisites: The teacher begins the lesson by reminding students about the concept of a binomial, explaining that it is an algebraic expression that has two terms separated by an addition or subtraction sign. The teacher then briefly reviews the exponent laws, focusing on the idea that a number or variable raised to an exponent can be written as a product of equal factors. This review is essential for students to understand the concept of the binomial theorem. (3 - 5 minutes)

2. Contextualized Problem Situations: The teacher introduces two problem situations. The first one could be a question that involves calculating the number of possible paths on a chessboard, which can be solved using Pascal's Triangle, a tool that arises in the expansion of the binomial theorem. The second one could be a question that involves expanding a binomial raised to a certain power, such as (a + b)³. These situations are presented to trigger students' curiosity about the topic and the need to learn the binomial theorem to solve them. (3 - 5 minutes)

3. Contextualizing the Importance of the Binomial Theorem: The teacher explains that the binomial theorem is an important tool in various areas of mathematics and physics. For example, in statistics, the binomial theorem is used to calculate the probability of a certain number of successes in a certain number of independent trials. In physics, the binomial theorem is used in several formulas, such as the formula that calculates the net force on an object that is moving in a circle. (2 - 3 minutes)

4. Curiosity and Introduction to the Topic: To spark students' interest, the teacher can share some curiosities about the binomial theorem. For example, the binomial theorem was discovered by the English mathematician Isaac Newton, who is best known for his laws of motion and his theory of gravity. Also, the binomial theorem is closely related to the famous Pascal's Triangle, which has many interesting properties and is used in many other topics in mathematics. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Inquiry Activity - Binomial Theorem and Pascal's Triangle (10 - 15 minutes)

   • The teacher divides the class into groups of up to 5 students and gives each group a piece of paper with a partially filled Pascal's Triangle. The teacher explains that Pascal's Triangle is a geometrical representation of the binomial theorem, where each number in the triangle is the sum of the two numbers above it.

   • The teacher then instructs the students to complete Pascal's Triangle, starting from the third row. They should observe the relationships between the numbers and how they relate to the Binomial Expansion.

   • After the completion of the activity, the teacher asks a representative from each group to share the group's observations and conclusions with the class. This promotes discussion and exchange of ideas among the students.

2. Practical Activity - Expanding the Binomial Theorem (10 - 15 minutes)

   • The teacher provides each group with a series of binomial expressions to expand, such as (a + b)², (x - y)³, (m + n)⁴, etc. The teacher explains that expanding a binomial is a process that involves multiplying each term of the first binomial by the second binomial and then simplifying like terms.

   • The students work in their groups to expand the binomial expressions, using Pascal's Triangle to find the binomial coefficients. They should discuss and explain the process to each other, with guidance from the teacher if needed.

   • After the completion of the activity, the teacher asks a representative from each group to share the expansion of the binomial theorem they worked on with the class. This allows students to see different approaches to the same problem and helps reinforce the concept and the process of binomial expansion.

3. Problem Solving Activity - Applying the Binomial Theorem (5 - 10 minutes)

   • The teacher presents the students with a series of problems that involve the binomial theorem, such as "In how many different ways can a pawn move on a chessboard if it can only move right and up, and at each move it has the same probability of moving right or up?" and "What is the binomial coefficient in the expansion of (a + b)⁵?".

   • The students work in their groups to solve the problems, applying what they have learned about the binomial theorem. They should discuss their solutions and explain their reasoning to each other, with guidance from the teacher if needed.

   • After the completion of the activity, the teacher asks a representative from each group to share their solutions and explanations with the class. This promotes discussion and exchange of ideas among the students, and helps reinforce the application of the binomial theorem to solve problems.

Debrief (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes): The teacher facilitates a group discussion with all the students, where each group shares their solutions and their conclusions from the activities that were carried out. During this discussion, the teacher can:

   • Ask each group what were the most important observations they made during the Pascal's Triangle inquiry activity and how they relate to the binomial theorem.
   • Ask each group what strategies they used to expand the binomial expressions and how they used Pascal's Triangle to find the binomial coefficients.
   • Ask each group how they applied the binomial theorem to solve the problems proposed in the problem solving activity.
   • Encourage students to ask each other questions and to answer their peers' questions.

2. Connection to Theory (3 - 5 minutes): After the group discussion, the teacher reviews the theoretical concepts that were applied in the hands-on activities. The teacher emphasizes the connection between theory and practice, highlighting how the expansion of the binomial theorem and the use of Pascal's Triangle are direct applications of the binomial theorem concept. The teacher also clarifies any doubts that may have arisen during the discussion.

3. Individual Reflection (2 - 3 minutes): To conclude the lesson, the teacher asks the students to reflect individually on what they have learned. The teacher asks the following questions:

   1. What was the most important concept you learned today?
   2. What questions do you still have?
   3. How can you apply what you learned today to real-life situations or to other subjects?

4. Sharing of Reflections (2 - 3 minutes): The teacher invites a few students to share their answers with the class. This allows the students to see different perspectives and ideas, and helps to reinforce what was learned. The teacher can also use this opportunity to provide feedback and additional guidance if needed.

5. Feedback (1 - 2 minutes): The teacher asks the students to provide feedback on the lesson, asking what they liked the most, what they found most challenging, and what they would like to learn more about. The students' feedback is valuable for the teacher to adjust and improve their future lessons.

Conclusion (5 - 7 minutes)

1. Summary of Content (2 - 3 minutes): The teacher recaps the main points that were covered in the lesson, including the definition of the binomial theorem, the use of Pascal's Triangle in expanding binomial expressions, and how to apply the binomial theorem to solve problems. The teacher reinforces that the binomial theorem is a valuable tool in mathematics and in several other areas, and that its understanding and application are fundamental for success in future topics.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher reinforces how the lesson was structured to connect theory, practice, and applications. The teacher highlights how the inquiry activity with Pascal's Triangle allowed the students to discover the patterns in the expansion of binomial expressions, and how the problem solving activity allowed the students to apply what they learned to solve real-world problems. The teacher also emphasizes that, although the lesson focused on mathematics, the binomial theorem has applications in many other areas, such as physics and statistics.

3. Extra Materials (1 minute): The teacher suggests some extra materials for the students who wish to deepen their understanding of the binomial theorem. These materials could include explanatory videos, interactive websites that allow students to explore Pascal's Triangle and the binomial theorem, and additional problems for students to practice. The teacher can provide a list of these resources in a review sheet that the students can take home.

4. Importance of the Topic (1 minute): Finally, the teacher emphasizes the importance of the binomial theorem, not only in mathematics but also in many other fields of knowledge. The teacher highlights that the binomial theorem is a powerful tool for describing and predicting patterns, and that the ability to work with the binomial theorem can open many academic and professional doors for the students. The teacher encourages the students to continue exploring and applying what they have learned, and to not hesitate to ask for help if they encounter any difficulties.