Lesson plan of Combinatorial Analysis: Circular Permutation

Objectives (5 min)

1. Understand Circular Permutations: The main objective is for students to understand circular permutations, which is the arrangement of a set of elements in a closed sequence, where the first and last positions are considered adjacent. They should be able to distinguish circular permutations from linear permutations and identify when to apply circular permutations to a problem.

2. Solve Circular Permutation Problems: The second objective is for students to be able to apply circular permutations to solve mathematical problems. They should be able to identify when a problem involves circular permutations and use the appropriate formulas and concepts to solve it.

3. Apply Circular Permutation Knowledge to Real-World Situations: Finally, students should be able to apply their knowledge of circular permutations to real-world situations. This will help solidify their understanding of the concept and demonstrate the relevance of mathematics in everyday life.

Secondary Objectives:

• Develop Logical Thinking Skills: Solving circular permutation problems requires the application of logical thinking skills, which are essential in mathematics. Therefore, a secondary objective is for students to develop and enhance these skills.

• Promote Collaborative Learning: Flipped classrooms provide an excellent opportunity to promote collaborative learning. Students should be encouraged to work in teams, discuss concepts, and solve problems together, which also helps strengthen communication and collaboration skills.

Introduction (10 - 15 min)

1. Review of Prior Knowledge: The teacher begins the class by reviewing the concepts of linear permutations, which should have been taught previously. They can present a simple linear permutation problem and ask students to solve it at their desks as a review. This will help refresh prior knowledge and set the stage for the introduction of circular permutations.

2. Problem Situations: Next, the teacher presents two problem situations that involve circular permutations. The first one could be: "Imagine you are on a carousel with four numbered seats. How many different ways can you and your friends sit, considering that the order of the seats matters?" The second one could be: "Suppose you have a necklace with four beads of different colors. How many different ways can you wear the necklace, considering that you can rotate it?" The teacher can ask students to think about these situations and share their ideas.

3. Contextualization: The teacher then explains that circular permutations are an important concept in mathematics and have many real-world applications. For example, in computer programming, circular permutations are used to generate random sequences. In board games like Rubik's Cube, circular permutations are used to determine the different possible configurations. The teacher can mention other applications and ask students to think of more examples.

4. Capture Students' Attention: To capture students' attention, the teacher can share some interesting facts about circular permutations. For instance, they can mention that the concept of circular permutations was first used in Ancient Greece to solve combinatorial problems. They can also mention that circular permutations are an example of a cyclic group, which is an important concept in abstract algebra. The teacher can then ask students if they have any curiosities or interesting facts to share about circular permutations.

Development (20 - 25 min)

1. Hands-on Activity 1: "The Carousel" (10 - 12 min): The teacher divides the class into groups of four. Each group receives a set of cards, each with a number from 1 to 4. Students have to arrange the cards in a way that represents all possible permutations of the seats on a carousel (i.e., a circular permutation of four elements). They should record their answers on a piece of paper. The teacher circulates around the room, providing assistance and clarifying doubts as needed. After the completion of the activity, the teacher asks one representative from each group to share their answers with the class. The teacher then explains the differences between a circular permutation and a linear permutation, and how circular permutations can be represented using an arrow diagram.

   Materials Required: Set of cards numbered 1 to 4 for each group, sheets of paper, and pencils.

2. Hands-on Activity 2: "The Necklace" (10 - 12 min): The teacher continues with the same groups from the previous activity. This time, each group receives a set of beads of different colors and a string. Students have to create a necklace, arranging the beads in a way that represents all possible permutations of the beads (i.e., a circular permutation of four elements). They should record their answers on a piece of paper. The teacher circulates around the room, providing assistance and clarifying doubts as needed. After the completion of the activity, the teacher asks one representative from each group to share their answers with the class. The teacher then discusses how the necklace activity relates to circular permutations and how the concept can be applied to real-world situations.

   Materials Required: Set of beads of different colors and string for each group, sheets of paper, and pencils.

3. Group Discussion Activity: "Applying Circular Permutations" (5 min): To conclude the Development stage, the teacher facilitates a group discussion. Students are invited to share their thoughts on how circular permutations can be applied to other everyday situations. The teacher guides the discussion, asking questions to stimulate critical thinking and real-world connections. For example, the teacher could ask: "Can you think of other situations where circular permutations could be applied?" or "How does circular permutation relate to the problems we solved?" The teacher should encourage all students to participate and value all answers and contributions.

   Materials Required: None

Return (10 - 15 min)

1. Group Discussion (5 - 7 min): The teacher facilitates a general discussion of the solutions or conclusions reached by each group in the "The Carousel" and "The Necklace" activities. To do this, the teacher may ask a representative from each group to share the solution they found or the conclusion they reached with the class. The goal is to encourage the exchange of ideas and collaborative learning, allowing students to learn from each other. During the group presentations, the teacher should encourage questions and comments from other students, promoting active participation from the entire class.

2. Connecting to Theory (3 - 5 min): After the group presentations, the teacher connects the hands-on activities to the theory of circular permutations. They can highlight how circular permutations are applied in the situations presented by the activities and how students were able to solve the problems by applying the concept. The teacher should reinforce the most important points of the theory and clarify any doubts that may have arisen during the activities.

3. Individual Reflection (2 - 3 min): To wrap up the lesson, the teacher asks students to reflect individually on what they have learned. They can ask questions such as: "What was the most important concept you learned today?" and "What questions do you still have?" Students should jot down their answers on a piece of paper or in a notebook. This step is important for students to consolidate their learning and identify any gaps in their understanding, which can be addressed in future lessons.

4. Teacher Feedback (1 - 2 min): The teacher then gathers the attention of the class and shares their insights on the lesson. They can commend the students' effort and participation, highlight positive learning outcomes, and offer suggestions for improvement. The teacher should also encourage students to continue practicing the concept of circular permutations outside of class, through exercises or by applying the concept to everyday situations.

   Materials Required: Sheets of paper or notebook for students to write down their reflections.

Conclusion (5 - 7 min)

1. Summary and Recap (2 - 3 min): The teacher begins the Conclusion by recapping the main points covered during the lesson. They should reiterate the concept of circular permutations, the difference between circular and linear permutations, and the practical applications of the concept. This will help solidify the students' learning and remind them of the key concepts to retain.

2. Connecting Theory, Practice, and Applications (1 - 2 min): The teacher then emphasizes how the lesson connected the theory of circular permutations with practice through the "The Carousel" and "The Necklace" activities. They should also highlight how the concept of circular permutations can be applied to real-world situations, as mentioned during the Introduction. This will demonstrate to students the importance and relevance of what they have learned.

3. Supplementary Materials (1 min): The teacher can then suggest supplementary materials for students who wish to further their understanding of circular permutations. These materials could include additional exercises, explanatory videos, mathematics websites, or reference books. The teacher should encourage students to explore these resources on their own as part of the self-directed learning process.

4. Importance of the Topic (1 - 2 min): Finally, the teacher summarizes the importance of the topic covered in the lesson. They can reinforce how circular permutations are a fundamental concept in mathematics, with applications in various fields, including computer science, gaming, cryptography, and more. The teacher can also highlight how the ability to solve circular permutation problems can help students develop valuable skills, such as logical thinking, problem-solving, and collaboration.

   Materials Required: None