Objectives (5 - 7 minutes)

1. Understand the concept of Newton's Binomial: The teacher must ensure that students understand what a binomial is and the central idea behind Newton's formula. This can be done through practical examples and illustrations.

2. Apply Newton's Formula in combination problems: Students should be able to solve problems involving combinations using Newton's formula. This includes identifying binomial coefficients and applying them correctly.

3. Recognize the importance of Newton's Binomial in Number Theory: The teacher should highlight how this formula is widely used in various branches of Mathematics, including Number Theory. Students should be able to recognize the formula in different contexts and understand its relevance.

Secondary objectives:

• Develop problem-solving skills: By working with Newton's formula, students will have the opportunity to improve their problem-solving skills, including logical and analytical thinking.

• Promote active participation and discussion in the classroom: The teacher should encourage students to actively participate in the class by asking questions, discussing solutions, and sharing their own problem-solving strategies.

Introduction (10 - 15 minutes)

1. Recalling previous concepts: The teacher should start the lesson with a brief review of mathematical concepts that are fundamental to understanding the lesson topic. This may include the concept of factorial, combinations, and the Binomial Theorem. The teacher can ask quick questions to verify if students still remember these concepts.

2. Problem situations: Next, the teacher should present two problem situations that will be the starting point for the Introduction to Newton's Binomial. For example:

   • "If I have a box with 5 red balls and 3 blue balls, in how many different ways can I choose 2 balls?"
   • "If I have an algebraic expression like (x + y)^3, how could I expand it without the need to perform all the multiplications?"

3. Contextualization: After presenting the problem situations, the teacher should explain how Newton's Binomial is a very useful mathematical tool in various areas, such as probability, physics, and economics. The teacher can give practical examples of these applications to make the topic more relevant and interesting for students.

4. Introducing the topic: Finally, the teacher should introduce the lesson topic, explaining that Newton's Binomial is a formula that allows us to expand an expression of the type (a + b)^n quickly and efficiently, without the need to perform all the multiplications. The teacher can show the formula on the board and briefly explain the meaning of each term.

   • "Today, we will learn about Newton's Binomial, a powerful tool that allows us to expand binomial expressions quickly and efficiently. With it, we will no longer need to perform all the multiplications, saving a lot of time and effort. In addition, Newton's Binomial has several practical applications, such as solving combination problems and expanding expressions in Number Theory."

5. Spark interest: To spark students' interest, the teacher can share curiosities about Newton's Binomial. For example, the fact that Isaac Newton was not the first to discover this formula, but rather Blaise Pascal, a French mathematician and philosopher, who described it in a book published in 1654. Another curiosity is that Newton's Binomial has a direct relationship with Pascal's Triangle, a very interesting mathematical figure.

Development (20 - 25 minutes)

1. Newton's Binomial Theory (10 - 12 minutes): The teacher should start by explaining the theory behind Newton's formula. It should be emphasized that the formula is used to expand a binomial expression raised to any power. The teacher can use the board to write the formula and explain each element.

   • "Newton's formula is represented by (a + b)^n, where a and b are the terms of the binomial and n is the exponent. To expand this expression, we use binomial coefficients, which are the numbers that appear in the expansion terms."

   • "Binomial coefficients are calculated using Pascal's Triangle, where each number is the sum of the two numbers above it. The first and last number of each line are always 1."

   • "Let's see an example: (a + b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3. Here, the coefficients are 1, 3, 3, and 1, which are the numbers from the third line of Pascal's Triangle."

2. Practice with Examples (5 - 7 minutes): After explaining the theory, the teacher should move on to practice. The teacher can use the whiteboard to solve some examples step by step, showing students how to apply Newton's formula. It is important for the teacher to explain each step in detail to ensure that students understand the process.

   • "Let's solve an example together: (2x - 3y)^2. First, we use the formula (a + b)^n. In this case, a is 2x, b is -3y, and n is 2. Now, let's apply the formula."

   • "1(2x)^2 + 2(2x)(-3y) + 1(-3y)^2. Now, let's simplify. 4x^2 - 12xy + 9y^2."

   • "See how the formula allows us to expand the expression quickly and efficiently, without the need to perform all the multiplications."

3. Discussion and Clarification of Doubts (5 - 6 minutes): After solving some examples, the teacher should open up for questions and discussion. Students can share their problem-solving strategies, and the teacher can clarify any doubts that may arise. It is important for the teacher to encourage everyone's participation and create a collaborative learning environment.

   • "Did you understand how to apply Newton's formula? Does anyone have any questions or would like to share a strategy they used to solve the problem?"

   • "Remember, Newton's Binomial is a very useful tool that allows us to save time and effort in solving combination problems and expanding expressions in Number Theory. So, it's important that you practice a lot to become familiar with the formula."

4. Practical Activity (5 minutes): To reinforce the concept, the teacher can propose a practical activity for students. For example, they can be divided into groups and each group receives a binomial expression to expand. The groups can share their solutions at the end, and the teacher can provide feedback on their work.

   • "Let's do a practical activity to consolidate what we've learned. I will divide you into groups, and each group will receive a binomial expression to expand. You will have 5 minutes to complete the activity, and then we will share the solutions."

5. Review (2 - 3 minutes): At the end of the Development, the teacher should review the main points of the lesson, summarizing the theory and highlighting the practical examples. The teacher can ask students to repeat Newton's formula and explain how it is used. This will help ensure that students have understood the lesson topic.

   • "Let's do a quick review. Who can repeat Newton's formula? And how is the formula used to expand a binomial expression?"

   • "Great! It seems like everyone understood. Now, let's continue with the lesson and explore more about Newton's Binomial."

Note: The suggested time for each stage of the Development may vary depending on the dynamics of the classroom and the students' level of understanding. The teacher should monitor the students' progress and adapt the pace of the lesson as needed.