Objectives (5 - 7 minutes)

1. Understand the concept of Combinatorial Analysis and its applications: The teacher must ensure that students understand what Combinatorial Analysis is and how it can be used to solve counting problems. Students need to understand that Combinatorial Analysis is the mathematics behind counting and probability.

2. Develop problem-solving skills with Combinatorial Analysis: The teacher should guide students in applying the concepts learned to solve practical problems. Students should be able to identify the type of problem they are facing and apply the correct Combinatorial Analysis technique to solve it.

3. Apply Combinatorial Analysis to determine the number of positive integer solutions: The teacher should help students understand how to apply Combinatorial Analysis to determine the number of positive integer solutions in a problem. Students should be able to identify when and how to use this specific technique.

   Secondary objectives:

   • Promote critical thinking and problem-solving: Encourage students to think critically about the problems presented and develop their own problem-solving strategies.

   • Foster collaboration and group discussion: Promote classroom discussion, encouraging students to share their ideas and problem-solving strategies with their peers.

Introduction (10 - 15 minutes)

1. Review of previous content:

   • The teacher should briefly review the concepts of counting and probability, as they are fundamental to understanding Combinatorial Analysis.
   • Additionally, the teacher should remind students about what positive integers are, so they can understand the focus of the lesson.

2. Problem situations:

   • The teacher can start the lesson with two problem situations to spark students' interest:
     • The first one: 'How many three-digit numbers can be formed using the digits 1, 2, and 3, if no digit can be repeated?'
     • The second one: 'How many four-digit numbers can be formed using the digits 1, 2, 3, and 4, if repetitions are allowed?'
   • These questions should be proposed without an explanation of how to solve them, so that students think about possible resolution strategies.

3. Contextualization of the theme:

   • The teacher should show students the importance of Combinatorial Analysis, explaining that this mathematical tool is used in various areas, such as computer programming, statistics, physics, among others.
   • One way to contextualize the theme is by presenting real examples, such as counting possible combinations in a card game or predicting outcomes in a scientific experiment.

4. Introduction to the topic:

   • To introduce the topic and capture students' attention, the teacher can tell the story of Blaise Pascal, a 17th-century French mathematician and physicist known for his contributions to Combinatorial Analysis.
   • Another interesting curiosity is the origin of the term 'Combinatorial Analysis,' which comes from the Latin 'combinare,' meaning 'to join, to gather.' This can be used to explain that Combinatorial Analysis is the mathematics behind how to 'join' and 'gather' elements in different ways.

Development (20 - 25 minutes)

1. Activity 1 - Secret Number Game: (10 - 12 minutes)

   • Activity description:
     • The teacher will divide the class into groups of 4 or 5 students and give each group a sheet of paper with spaces to fill in with numbers.
     • Then, the teacher will explain that each group must fill in the spaces with positive integers so that the sum of the numbers equals a secret number, which will be set by the teacher.
     • The challenge is for the groups to find all possible combinations of numbers that add up to the secret number.
     • At the end of the activity, the group that finds the highest number of correct combinations will be the winner.
   • Step by step:
     1. The teacher sets the secret number (for example, 10).
     2. The groups start filling in the spaces with numbers (for example, 1 + 2 + 3 + 4).
     3. The groups must find all possible combinations of numbers that add up to the secret number.
     4. The teacher circulates around the room, helping groups that are having difficulties and encouraging discussion.
     5. At the end of the activity, the teacher checks the groups' answers and declares the winner.

2. Activity 2 - Color Challenge: (10 - 12 minutes)

   • Activity description:
     • The teacher gives each group a set of colored cards, each color representing a different number.
     • The teacher then proposes a challenge: the groups must form sequences of colors that add up to a specific number, for example, 10.
     • The difficulty is that each sequence must start with the red color (representing the number 1) and there can be no repetition of colors.
   • Step by step:
     1. The teacher establishes the number to be formed (for example, 10).
     2. The groups start forming color sequences, respecting the rules.
     3. The groups must find all possible sequences that add up to the established number.
     4. The teacher circulates around the room, helping groups that are having difficulties and encouraging discussion.
     5. At the end of the activity, the teacher checks the groups' answers and declares the winner.

3. Discussion and Synthesis: (5 - 7 minutes)

   • The teacher gathers the class and promotes a discussion about the strategies used by the groups to solve the challenges.
   • The teacher highlights the importance of Combinatorial Analysis for solving these problems, explaining that students used this mathematical tool even without realizing it.
   • The teacher also takes the opportunity to reinforce the concepts of counting and probability, relating them to the activities carried out.
   • Finally, the teacher summarizes what was learned, reinforcing the lesson's objectives and addressing any possible student doubts.

Return (8 - 10 minutes)

1. Group Discussion: (4 - 5 minutes)

   • The teacher should promote a group discussion, where each team shares their strategies for solving the proposed challenges.
   • The goal of this discussion is for students to see the different ways to approach a problem and the effectiveness of each strategy.
   • The teacher should guide the discussion by asking questions to stimulate critical thinking and students' reflection. For example: 'Why did you choose this strategy?' or 'How could you improve the effectiveness of this strategy?'.

2. Connection with theory: (2 - 3 minutes)

   • After the discussion, the teacher should connect the activities carried out with the theory of Combinatorial Analysis.
   • The teacher can start by asking students how they believe the activities are related to the concepts of counting and probability.
   • Then, the teacher should explain how Combinatorial Analysis is used to solve counting problems, like the ones proposed in the activities.
   • The teacher should also reinforce the concept of the number of positive integer solutions, explaining that this is one of the main topics of Combinatorial Analysis.

3. Individual Reflection: (2 - 3 minutes)

   • Finally, the teacher should suggest that students reflect individually on the lesson.
   • The teacher can ask guiding questions for this reflection, such as: 'What was the most important concept you learned today?' or 'What questions have not been answered yet?'.
   • The teacher should give a minute for students to think about these questions and then may ask some students to share their answers with the class.
   • This final reflection is important for students to consolidate what they have learned and identify any gaps in their understanding, which the teacher can address in future classes.

4. Additional Materials:

   • The teacher can suggest additional study materials for students, such as explanatory videos, interactive math websites, or Combinatorial Analysis exercises to solve at home.
   • These materials can help students review what was learned in the lesson, deepen their understanding of the topic, and practice applying the concepts in different contexts.

Conclusion (5 - 7 minutes)

1. Summary of Contents:

   • The teacher should start the Conclusion of the lesson by recapping the key points of Combinatorial Analysis and the application of counting and probability concepts to solve practical problems.
   • It is important for the teacher to highlight how students applied these concepts in the activities carried out, reinforcing the idea that mathematics is not just an abstract theory but a useful and powerful tool for solving real problems.

2. Connection between Theory, Practice, and Applications:

   • The teacher should explain how the lesson connected the theory of Combinatorial Analysis with practice, through the proposed fun and challenging activities.
   • Additionally, the teacher should emphasize the practical applications of Combinatorial Analysis, showing students how this mathematical tool is used in various areas of knowledge and everyday life.

3. Extra Materials:

   • The teacher should suggest extra study materials for students, such as books, videos, websites, and math apps that offer further explanations and exercises on Combinatorial Analysis.
   • These resources can help students review what was learned in the lesson, deepen their understanding of the topic, and practice applying Combinatorial Analysis concepts.

4. Importance of the Subject:

   • Finally, the teacher should reinforce the importance of Combinatorial Analysis, explaining that this area of mathematics is essential for solving counting and probability problems in various everyday situations and in different professional careers.
   • The teacher can give examples of how Combinatorial Analysis is used in different contexts, such as computer programming, statistics, physics, engineering, among others.
   • In this way, the teacher concludes the lesson by motivating students to continue studying and applying Combinatorial Analysis concepts, as they will be useful not only for the Mathematics discipline but also for life.