Objectives (5 - 7 minutes)

1. Understand the concept of Combination: The main objective is for students to understand what combination is and how this concept is applied. They should be able to identify problem situations where combination is the best tool to solve them.

2. Apply the Combination Formula: Students should learn the mathematical formula to calculate the number of possible combinations in a set of elements. They need to practice applying this formula to solve combination problems.

3. Solve Combination Problems: The final objective is for students to be able to solve complex combination problems. They should be able to interpret the problem, apply the formula correctly, and arrive at a solution.

   • Secondary Objectives: Develop logical and critical thinking skills, enhance problem-solving skills, and strengthen the understanding of the concept of probability.

At the end of the lesson, students should be able to recognize and solve problem situations involving combination, correctly applying the combination formula.

Introduction (10 - 12 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing the concepts of Probability, Permutation, and Factorial, which were studied earlier and are fundamental for understanding the topic of Combination. This can be done through a brief theoretical review, followed by some practice exercises. (3 - 4 minutes)

2. Problem Situation 1: The teacher should present the following situation: 'Suppose you go to a restaurant and have the option to create your own dish, choosing 3 ingredients out of a total of 10. How many possible dish combinations can you make?' The teacher should not give the answer immediately, but rather encourage students to think about how they could solve this problem. This situation will serve as a playful and practical introduction to the topic of Combination. (2 - 3 minutes)

3. Contextualization: The teacher should explain the importance of Combinatorial Analysis in everyday life, showing that it is present in various situations, such as choosing numbers in lotteries, assembling dishes in self-service restaurants, forming teams in sports competitions, among others. (1 - 2 minutes)

4. Problem Situation 2: The teacher should present a second situation: 'Suppose you have 5 T-shirts of different colors and want to choose 2 to wear. How many possible clothing combinations can you make?' This situation will help reinforce the concept of Combination and prepare students for the theoretical explanation that will follow. (2 - 3 minutes)

5. Engaging Students' Attention: To spark students' interest, the teacher can share some curiosities, such as the fact that Combinatorial Analysis was developed by a French mathematician named Blaise Pascal in the 17th century to solve problems related to gambling games. Another curiosity is that Combinatorial Analysis is one of the most applied areas of Mathematics in Computing, as it is essential for creating efficient algorithms. (1 - 2 minutes)

Development (20 - 25 minutes)

1. Theory - What is Combination? (5 - 7 minutes)

   • The teacher should start the theoretical explanation by defining what Combination is in Mathematics. They should emphasize that, unlike Permutation, in Combination, the order does not matter, meaning the grouping of elements is not relevant.
   • The concept should be illustrated with practical examples, such as the situation of choosing 3 ingredients for a dish out of 10 options, or the situation of choosing 2 T-shirts out of 5 to wear.

2. Theory - Combination Formula (5 - 7 minutes)

   • The teacher should introduce the Combination formula: C(n, p) = n! / (p! * (n - p)!), where n is the total number of elements, p is the number of chosen elements, and ! (factorial) is the product of all integers from 1 to n.
   • The formula should be explained step by step, showing the importance of the factorial and how the factorials cancel out in the formula to eliminate the order of elements.

3. Practice - Calculation of Combination (5 - 7 minutes)

   • The teacher should then demonstrate how to apply the Combination formula to calculate the number of possible combinations in a problem situation. They should start with simple examples and then move on to more complex ones.
   • The examples should be solved step by step, with the teacher explaining each stage of the process.

4. Practice - Problem Solving (5 - 7 minutes)

   • After students have understood the theory and the Combination formula, the teacher should propose some problems for them to solve individually. The problems should vary in difficulty to challenge the students.
   • The teacher should move around the classroom, assisting students who have difficulties and monitoring everyone's progress.

5. Review and Conclusion (2 - 3 minutes)

   • Finally, the teacher should review the main points of the lesson, recalling the concept of Combination, the Combination formula, and how to solve Combination problems.
   • The teacher should conclude the lesson by highlighting the importance of Combinatorial Analysis and how it can be applied in everyday situations. They should also encourage students to continue practicing and ask questions if they have any, reinforcing that Mathematics is a discipline that requires practice and persistence.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should start the group discussion by asking students to share the solutions or approaches they found to solve the proposed problems.
   • It is important for students to feel comfortable expressing their ideas and doubts, promoting a collaborative learning environment.
   • The teacher should encourage everyone to participate, reinforcing that there are no right or wrong answers, but rather different ways to approach and solve problems.

2. Connection with Theory (2 - 3 minutes)

   • After hearing the different resolutions from students, the teacher should make the connection between the solutions presented and the theory discussed in the lesson.
   • They should highlight how the Combination formula, for example, was applied in solving the problems and how understanding the concept of Combination was crucial to finding the solutions.

3. Individual Reflection (2 - 3 minutes)

   • The teacher should suggest that students reflect individually on what they learned in the lesson.
   • They should ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • Students should write down their reflections, which can be used as a basis for preparing the next lesson or for reviewing the content.

4. Feedback and Closure (1 - 2 minutes)

   • To end the lesson, the teacher should ask for feedback from students about the lesson. They can ask: 'What did you think of today's lesson?' and 'What could be improved?'.
   • The teacher should thank the students for their participation, reinforce the importance of continuous study and practice, and give a brief overview of what will be covered in the next lesson.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes)

   • The teacher should start the Conclusion by recalling the main points discussed in the lesson. This includes the definition of combination and the mathematical formula for calculating combinations.
   • They should reinforce how combination differs from permutation, and how the order of elements is not relevant in combination.
   • The teacher should also recap the types of problems that can be solved using the concept of combination, and how the combination formula is applied in solving these problems.

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

   • Next, the teacher should explain how the lesson connected theory and practice. They should emphasize how theory was used to understand the combination problems presented, and how practice helped solidify students' understanding of the concept of combination.
   • The teacher should also highlight how group discussion and problem-solving helped students apply theory in a practical and meaningful way.

3. Additional Materials (1 - 2 minutes)

   • The teacher should suggest some additional study materials for students who want to deepen their knowledge on the topic. This may include math books, educational websites, explanatory videos, and extra exercises.
   • They should encourage students to review the lesson content at home, using the additional materials to reinforce their understanding of the concept of combination and the combination formula.

4. Relevance of the Topic (1 minute)

   • Finally, the teacher should explain the importance of the topic for students' daily lives. They should highlight how combinatorial analysis is used in various everyday situations, from choosing clothes to wear to participating in gambling games.
   • The teacher should reinforce that, although mathematics may seem abstract and distant from reality, it is present in many aspects of our daily lives, and understanding mathematical concepts can help us make more informed decisions and solve problems more efficiently.