Lesson Plan | Active Learning | Combinatorial Analysis: Circular Permutation

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The objectives stage is crucial for establishing the learning goals of the lesson. By clearly defining what is expected of students, they can better direct their efforts and focus on the specific skills that will be applied and developed during the activities. Furthermore, this section serves to align the expectations of both the students and the teacher, ensuring that everyone is aware of the desired learning outcomes.

Main Objectives:

1. Develop the ability to solve combinatorial analysis problems involving circular permutations, such as arranging people around a table.

2. Enable students to apply specific formulas for calculating circular permutations and to understand the logic behind these formulas.

Side Objectives:

1. Encourage critical thinking and a systematic approach to solving mathematical problems.
2. Promote collaboration and discussion among students during practical activities.

Introduction

Duration: (15 - 20 minutes)

The introduction stage serves to engage students and revive prior knowledge about circular permutations, using problem situations that stimulate reflection and the practical application of the concept. The contextualization seeks to show the relevance of the topic in the real world, encouraging students to perceive mathematics not just as a theoretical discipline, but as a tool applicable in various everyday situations.

Problem-Based Situations

1. Imagine that five friends are playing a card game around a round table. How could one calculate how many different ways these friends can sit, considering that the positions are identified?

2. Consider a situation where seven people are organizing an event and need to decide how they will be arranged around a circular table to discuss preparations. How many distinct ways are there for these people to be seated, knowing that the order matters and the table is circular?

Contextualization

Circular permutation is not just a mathematical concept, but also a useful tool in everyday situations, such as organizing events, games, or even in cryptography. For example, when solving a puzzle, the order of letters may be important and represented in a circular form, where different arrangements can lead to different meanings. Additionally, understanding circular permutation can help develop analytical and planning skills, which are fundamental in various professional and academic fields.

Development

Duration: (65 - 75 minutes)

The development stage is designed to allow students to apply the concepts of circular permutation in a practical and engaging manner. Through group activities, students can explore fictional scenarios that simulate real situations where circular permutation is crucial, such as in event arrangements or meetings. This approach not only reinforces theoretical learning but also promotes teamwork, problem-solving, and critical thinking skills.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - Mathematical Banquet

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of circular permutation in practice, developing logical reasoning skills and teamwork.

- Description: In this activity, students will be challenged to organize a mathematical 'banquet,' where they must seat a specific number of guests around a circular table. Each group will be tasked with seating 8 guests (identified by name cards) in 8 chairs arranged in a circle. The challenge is to determine how many distinct ways the guests can be seated, considering two arrangements are considered identical if the guests are in corresponding positions at the table, even if rotated.

- Instructions:

• Divide the class into groups of up to 5 students.

• Distribute name cards for the guests to each group.

• Guide the students to draw a representation of the circular table and identify the chairs.

• Ask each group to list the possible circular permutations of the guests.

• The groups must present their solutions and explain how they arrived at the result.

Activity 2 - Surprise Birthday Party

> Duration: (60 - 70 minutes)

- Objective: Solve a practical problem involving circular permutation, stimulating creativity and the ability to tackle complex challenges.

- Description: Students will plan a surprise birthday party for 10 people. The party venue has a round table with 10 chairs. Each student will be represented by a fictional character, and the challenge is to organize these characters at the table so that the birthday person is exactly between two specific people, not considering the order of the other people at the table.

- Instructions:

• Form groups of up to 5 students.

• Assign each group a specific configuration for the birthday person and the people who should be next to them.

• The students must determine how many distinct ways they can arrange the other 7 people around the table.

• Each group must present their solution and justify the reasoning used.

Activity 3 - The Royal Chess Tournament

> Duration: (60 - 70 minutes)

- Objective: Develop combinatorial analysis skills in a playful and creative context, promoting critical thinking and collaboration.

- Description: Imagine a chess tournament where 6 kings from different kingdoms must sit around a circular table to discuss an alliance. Each group of students will receive the descriptions and preferences of the kings (e.g., 'King A does not want to sit next to King B') and must determine how many different ways there are to seat them, respecting the given restrictions.

- Instructions:

• Divide the class into groups of no more than 5 students.

• Provide each group with the preferences and restrictions of the kings.

• Guide the students to use circular permutation strategies to solve the problem.

• Each group must present their final solution, explaining the resolution process.

Feedback

Duration: (15 - 20 minutes)

The purpose of this stage is to consolidate the learning acquired during the practical activities, allowing students to articulate and reflect on the strategies used and the challenges faced. The group discussion helps reinforce understanding of the concepts of circular permutation, in addition to promoting communication and collaboration skills. This collective feedback also provides the teacher with insights into student understanding and the effectiveness of the activities, guiding possible future revisions in the teaching of the topic.

Group Discussion

At the end of the activities, gather all students for a group discussion. Start the discussion with a brief introduction: 'Now that everyone has had the opportunity to explore circular permutation in different contexts, let’s share our discoveries and challenges. Each group will have the chance to present a summary of what they discussed and the strategies they used. Let’s take this moment to learn from each other and see how different approaches can lead to the same results.'

Key Questions

1. What were the biggest challenges your group faced when applying the concept of circular permutation during the activities?

2. How can understanding circular permutations be useful in everyday situations or in other subjects?

3. Was there any particular strategy or method that your group found especially effective while solving the problems?

Conclusion

Duration: (5 - 10 minutes)

The purpose of this stage is to ensure that students have understood the essential concepts of the lesson and can relate them to real and theoretical situations. The summary helps consolidate learning, while the discussion about the connection between theory and practice reinforces the importance of the topic. This moment also serves to clarify remaining doubts and ensure that students leave the lesson with a clear and applicable understanding of the studied topic.

Summary

In the final stage, the teacher should summarize the main concepts covered, highlighting the formulas and methods used to solve circular permutations. It is important to revisit the simulated practical situations, such as organizing events and seating people at circular tables, to consolidate learning.

Theory Connection

Today's lesson connected combinatorial analysis theory with practical applications through interactive activities, allowing students not only to understand mathematical concepts but also to visualize and apply these concepts in everyday contexts, reinforcing the importance and utility of mathematics.

Closing

Finally, it is crucial to emphasize the relevance of studying circular permutations. This knowledge not only enriches the students' mathematical understanding but also has practical applications in various situations, such as event planning and information technology, where permutation algorithms are essential.