Objectives (5 - 7 minutes)

1. Understand the concept of circular permutations: By the end of the lesson, the teacher should ensure that the students have a clear understanding of what a circular permutation is. This includes being able to identify a situation in which a permutation is circular and to correctly apply the formula for calculating the number of possible circular permutations.

2. Solve circular permutation problems: Students should be able to apply the knowledge they have gained to solve problems involving circular permutations. This includes being able to correctly identify the number of elements in a circular permutation and using the appropriate formula to calculate the number of different ways in which the elements can be arranged.

3. Relate the concept of circular permutations to real-world situations: The teacher should encourage students to think critically about how the concept of circular permutations applies to the real world. This can be done through examples and problems that reflect everyday situations, such as arranging people in a circle or permuting positions in a game.

Secondary Objectives:

• Develop logical and critical thinking skills: In addition to focusing on the specific mathematical concept, the teacher should encourage students to think critically about how and why things work the way they do. This can be done through questioning and classroom discussions.

• Promote mathematical reasoning: By solving circular permutation problems, students will have the opportunity to apply and enhance their mathematical reasoning skills.

• Encourage classroom collaboration: The teacher should promote collaboration among students, encouraging them to work together to solve problems. This will not only help to reinforce the concept of circular permutations, but it will also foster important social skills such as teamwork and effective communication.

Introduction (10 - 12 minutes)

1. Review of previous concepts: The teacher should begin the lesson by reviewing the concepts of permutation and factorial, which were studied in previous lessons and are fundamental to understanding circular permutations. The idea that a permutation is the arrangement of elements of a set in different orders should be reviewed, and the factorial should be reviewed as the product of all positive integers less than or equal to a given number. This review can be done through quick exercises or questions to students to activate prior knowledge. (3 - 5 minutes)

2. Problem situations: The teacher could propose two problem situations to initiate the discussion on circular permutations. The first could be the arrangement of people in a circle for a group activity. The teacher can ask students how many different ways they could be arranged. The second situation could be the permutation of positions in a circular board game, and again, the teacher can ask students how many different ways this could be done. These situations will help to contextualize the importance of circular permutations. (3 - 4 minutes)

3. Contextualization of the topic: The teacher can explain that circular permutations are a useful mathematical tool in various fields, including computer science (for example, in cryptography and password generation), physics (for example, in modeling particles in circular motion) and even in music (for example, in string theory). These examples will help students understand the relevance of the topic. (2 - 3 minutes)

4. Introduction to the topic: Finally, the teacher can introduce the topic of circular permutations, explaining that, unlike linear permutations in which elements are arranged in a straight line, in circular permutations the elements are arranged in a circle. The teacher could use a diagram to illustrate this. (1 - 2 minutes)

Development (20 - 25 minutes)

1. Theory (10 - 12 minutes): The teacher should begin explaining the theory behind circular permutations, including the formula for calculating the number of possible circular permutations. This can be done by following these steps:

   • Definition of circular permutation (2 - 3 minutes): The teacher should explain that, in a circular permutation, the elements are arranged in a circle. A circular permutation is considered unique if, upon rotating the circle, the relative positions of the elements do not change.

   • Counting the elements in a circular permutation (2 - 3 minutes): The teacher should explain that, in a circular permutation, the number of elements is always the same as the number of positions. This is because, as the circle is rotated, the elements retain their relative positions.

   • Formula for circular permutations (2 - 3 minutes): The teacher should present the formula for calculating the number of possible circular permutations, which is (n-1)!. The teacher should explain that the formula is derived from the one for linear permutations, but with an adjustment to take into consideration that circular permutations are considered unique if the relative positions of the elements do not change.

   • Examples of applying the formula (2 - 3 minutes): The teacher should provide a few examples of how to apply the formula to calculate the number of possible circular permutations. This can be done by solving problems such as the number of ways to arrange people in a circle or the number of ways to permute positions in a circular board game, step-by-step.

2. Practice (10 - 13 minutes): After presenting the theory, students should be given the opportunity to practice what they have learned. The teacher can propose a series of problems that involve circular permutations and ask the students to try to solve them. The problems can range in difficulty, allowing the students to progress gradually from simpler to more complex problems. The teacher should circulate around the room, offering assistance where necessary and correcting any errors.

   • Direct application problems (4 - 5 minutes): The teacher should begin with problems that involve the direct application of the circular permutation formula. This will allow the students to become familiar with the formula and build confidence in their ability to use it.

   • Problems requiring more complex reasoning (4 - 5 minutes): As students gain confidence, the teacher can propose problems that require a little bit more reasoning. For example, the teacher can propose a problem in which students have to calculate the number of possible circular permutations, but with some constraints. This will help to develop the students' reasoning skills and encourage them to think more critically about the concept of circular permutations.

Closure (8 - 10 minutes)

1. Review and Reflection (3 - 4 minutes): The teacher should start the Closure by reviewing the main points covered in the lesson, reinforcing the concept of circular permutations, the formula for calculating the number of possible circular permutations and the practical application of this concept. The teacher can make a quick summary, highlighting the main points and concepts that the students should have learned. Then, a moment should be given for students to reflect on what was learned, by asking questions such as:

   • "What was the most important concept you learned today?"
   • "What questions do you still have about circular permutations?"
   • "How do you think you could apply what you learned today to real-world situations or to other disciplines?"

2. Connection with the real world (2 - 3 minutes): The teacher should then make the connection between the concept of circular permutations and real-world situations. This can be done through practical examples, such as arranging people in a circle for a group activity or permuting positions in a circular board game. The teacher could also mention some more complex applications, such as the use of circular permutations in computer science (for example, in cryptography and password generation) or in physics (for example, in modeling particles in circular motion).

3. Feedback from students (2 - 3 minutes): Finally, the teacher should ask for feedback from the students about the lesson. This can be done through a brief classroom discussion, where students have the opportunity to express what they liked about the lesson, what they found more challenging and any suggestions they may have to improve future lessons. The teacher should encourage the students to be honest in their feedback and to express any difficulty they may have had with the topic. This will help the teacher to identify areas that need more reinforcement and to adjust the lesson plan as necessary.

4. Homework (1 minute): The teacher should then assign a homework assignment that reinforces the concept of circular permutations. This could include solving additional problems, researching applications of circular permutations in other fields or preparing a presentation about the topic. The teacher should make sure that the homework is clear and that students know exactly what is expected of them.

This Closure stage is crucial for consolidating the students' learning and ensuring that they have understood the concepts presented. It also provides the teacher with an opportunity to evaluate the effectiveness of the lesson and to make adjustments as necessary.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes): The teacher should start the Conclusion by summarizing the main points covered during the lesson. This includes the definition of circular permutations, the formula for calculating the number of possible circular permutations and the application of this concept in everyday situations and different fields of knowledge. The teacher can use charts or diagrams to visually reinforce the concepts and should make sure that students have a clear understanding of each point.

2. Connection between Theory, Practice and Applications (1 - 2 minutes): Next, the teacher should explain how the lesson connected the theory, practice and applications of the concept of circular permutations. For example, the teacher can mention how the presentation of the theory was followed by the practice of solving circular permutation problems, and how real-world examples and situations were used to illustrate the application of the concept. The teacher should emphasize that the theory and the practice are not separate things, but rather integral parts of the learning process.

3. Extra Materials (1 - 2 minutes): The teacher can then suggest some extra materials for students who wish to deepen their understanding of the concept of circular permutations. This could include math textbooks, educational websites, online videos, math games or learning apps. The teacher should ensure that the suggested materials are accessible and appropriate for the students' skill level.

4. Importance of the Subject (1 minute): Finally, the teacher should summarize the importance of the concept of circular permutations. The teacher can explain that circular permutations are a valuable tool not only in mathematics, but also in many other areas of life. For example, the ability to think systematically and to calculate the number of possibilities can be useful in everyday situations, such as planning a party or solving a puzzle. In addition, circular permutations have practical applications in fields such as computer science and physics, which means that the knowledge acquired in the lesson could be useful in future studies or careers.