Objectives (5 - 7 minutes)

1. Understand the concept of permutation with repetition, applying it to practical and theoretical problems.

2. Develop the ability to calculate the number of possible permutations in situations where some elements are repeated.

3. Practice the application of the concept of permutation with repetition in everyday situations, promoting contextualization and the relevance of the content.

Secondary objectives:

• Encourage logical thinking and problem-solving through the study of combinatorial analysis.

• Promote interdisciplinarity, relating the content of combinatorial analysis to other disciplines, such as physics and chemistry, which also make use of this concept.

• Reinforce the practice of mathematical calculations, stimulating precision and attention to detail.

Introduction (8 - 10 minutes)

1. Review of Previous Content:

   • The teacher starts the lesson by reviewing the concepts of permutation and factorial, which were studied in previous classes.
   • They can propose problem situations involving the use of these concepts, so that students can review and apply them.

2. Problem Situations:

   • The teacher proposes two problem situations to instigate students' curiosity and introduce the new content.
   • The first situation may involve organizing a numerical password of 4 digits, where some digits are repeated.
   • The second situation may be the organization of a word with repeated letters, such as 'MATHEMATICS', in different orders.

3. Contextualization:

   • The teacher highlights the importance of combinatorial analysis in various areas, such as cryptography, probability, and genetics.
   • They can mention practical examples, such as the importance of permutations for creating secure passwords and encoding information in digital systems.

4. Introduction to the Topic:

   • The teacher introduces the topic of permutation with repetition, explaining that it is used when some elements of a set are repeated.
   • They can give a simple example, such as organizing 3 colors in a line, where 2 colors are repeated.
   • To spark students' interest, the teacher can mention that permutation with repetition is a fundamental concept in computing, where it is used to generate sequences, passwords, and cryptographic keys.

Development (20 - 25 minutes)

1. Activity 1: 'The Candy Store' (10 - 12 minutes)

   • Description: The teacher presents students with the situation of a candy store that wants to create packages with 6 candies each. In the store, there are 3 types of candies: A, B, and C. The challenge is to calculate how many different packages can be created, considering that it is possible to repeat the types of candies in the same package.
   • Step by step:
     1. The teacher distributes 3 cards of different colors to each group of students, representing the types of candies.
     2. The students, in their groups, must organize the cards to represent the possible combinations of candies in a package.
     3. After calculating the combinations manually, students must record their answers on an activity sheet.
     4. Then, the teacher asks a representative from each group to present their solution to the class, promoting discussion and understanding of the concept of permutation with repetition.

2. Activity 2: 'The Treasure Safe' (10 - 12 minutes)

   • Description: The teacher proposes a new problem situation, this time involving the creation of a password for a safe. The password must consist of 4 digits, with digits 1 and 2 not being used. The challenge is to calculate how many different passwords can be created.
   • Step by step:
     1. The teacher distributes 8 cards, numbered from 3 to 10, to each group of students.
     2. The students, in their groups, must organize the cards to represent the possible combinations of passwords.
     3. After calculating the combinations manually, students must record their answers on an activity sheet.
     4. Then, the teacher asks a representative from each group to present their solution to the class, promoting discussion and understanding of the concept of permutation with repetition.

3. Group Discussion and Conclusions (5 - 8 minutes)

   • After the presentations, the teacher facilitates a group discussion where students can compare their solutions and discuss the strategies used.
   • The teacher takes advantage of the discussion to reinforce the concepts of permutation with repetition and the importance of considering all possibilities in problem solving.
   • Finally, the teacher concludes the activity, highlighting the relevance of the content for daily life and other disciplines.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher starts the Return stage by promoting a group discussion about the solutions found by each team in the activities carried out.
   • They can ask one or two representatives from each group to share their solution, explaining the reasoning used.
   • During the discussion, the teacher should encourage students to ask questions and express their doubts or difficulties, promoting the exchange of ideas and the collective construction of knowledge.

2. Connection with Theory (2 - 3 minutes)

   • After the discussion, the teacher should make the connection between the solutions presented and the theory of permutation with repetition.
   • They can highlight the common points among the different solutions, reinforcing the theoretical concepts worked on during the lesson.
   • The teacher can also take the opportunity to clarify any possible misconceptions or misunderstandings that may have arisen during the discussion, correcting any errors of interpretation.

3. Individual Reflection (2 - 3 minutes)

   • The teacher suggests that students dedicate a minute to reflect individually on what was learned in the lesson.
   • They can ask guiding questions, such as: 'What was the most important concept learned today?' and 'What questions have not been answered yet?'.
   • After the minute of reflection, students can briefly share their answers, if they feel comfortable.
   • The goal of this stage is for students to internalize the lesson content and identify any gaps in their understanding, so they can seek clarification later if necessary.

4. Feedback and Closure (1 - 2 minutes)

   • Finally, the teacher requests general feedback from the class about the lesson, asking what they liked the most and what they found most challenging.
   • They can take note of these observations to guide the planning of future lessons.
   • The teacher concludes the lesson by reinforcing the main points learned and emphasizing the importance of the content for problem solving and understanding phenomena in various areas of knowledge.

Conclusion (5 - 7 minutes)

1. Summary and Recap (2 - 3 minutes)

   • The teacher starts the Conclusion by recalling the main points covered during the lesson.
   • They highlight the concept of permutation with repetition and its application in practical situations, such as organizing passwords and creating candy packages.
   • The teacher also gives a brief summary of the activities carried out, reinforcing the concepts learned and the strategies used to solve the problems.
   • For example, they can mention how students used cards of different colors to represent the types of candies in the 'The Candy Store' activity and the digits that could not be used in the 'The Treasure Safe' activity.

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

   • Next, the teacher emphasizes the importance of the interaction between theory, practice, and applications.
   • They highlight how the lesson allowed students to understand not only the theoretical concepts of permutation with repetition but also its usefulness and relevance in real situations.
   • For example, the teacher can mention how permutation with repetition is used in areas such as cryptography and genetics, contributing to students' understanding of the importance of mathematics in the world around them.

3. Extra Materials (1 - 2 minutes)

   • The teacher suggests some extra materials for students to deepen their studies on permutation with repetition.
   • These materials may include textbooks, educational websites, explanatory videos, and online exercises.
   • For example, the teacher can recommend a YouTube video that explains the concept of permutation with repetition clearly and simply, or a website with a variety of combinatorial analysis problems for students to practice.

4. Importance of the Subject (1 minute)

   • Finally, the teacher reinforces the importance of the content learned for daily life and other disciplines.
   • For example, they can mention how the ability to calculate the number of possible permutations is useful in everyday situations, such as organizing events, creating secure passwords, or solving probability problems.
   • The teacher can also highlight how combinatorial analysis relates to other areas of mathematics, such as probability and algebra, reinforcing the idea that mathematical knowledge is not composed of isolated topics, but of an interconnected set of concepts and skills.