Objectives (5 - 10 minutes)

1. Define and explain the concept of permutations - Students should be able to understand that permutations refer to the arrangement of objects, where the order is important.
2. Show how permutations are applied in real-world scenarios - This would give students a practical understanding of the concept, allowing them to relate the topic to everyday experiences.
3. Understand and apply the formula for permutations - Students should be able to use the formula for permutations to solve mathematical problems.

Secondary objectives:

1. Encourage active class participation - The teacher should ask questions and encourage students to contribute to the class discussion. This will help to ensure that all students understand the topic and can apply the concepts learned.
2. Foster critical thinking - The teacher should present problems that require students to apply their understanding of permutations and to think critically about the problem-solving process.

Introduction (10 - 15 minutes)

1. Review of necessary content (2 - 3 minutes)

   • The teacher begins the lesson by reviewing the concept of factorials as a prerequisite to understanding permutations.
   • They can use a quick activity, like having the students calculate the factorial of a small number on mini whiteboards or in their notebooks.

2. Problem situations to introduce the concept (4 - 5 minutes)

   • The teacher introduces a problem: "In how many different ways can you arrange 3 books on a shelf?"
   • They then ask a more complex problem: "Now, how many different ways can you arrange 5 books on a shelf?"
   • Students can share their thoughts and reasoning before the teacher reveals the correct answers and how they are related to the concept of permutations.

3. Contextualizing the importance of the topic (1 - 2 minutes)

   • The teacher explains that permutations are not just about arranging books, but they are used in many areas, including computer science (for password combinations), business (for scheduling), and biology (for genetic variations).
   • They emphasize that understanding permutations can help students solve complex problems, enhance their critical thinking skills, and open doors to various careers.

4. Introducing the topic with captivating elements (3 - 5 minutes)

   • The teacher then shares a curiosity about permutations: "Did you know that the number of possible arrangements of a standard 52-card deck is greater than the number of atoms on Earth?"
   • They also tell a short story: "In the 18th century, a famous mathematician named Euler used permutations to solve the '36 officers problem,' which puzzled many mathematicians of his time. The problem was about arranging 36 officers, each from 6 different regiments and of 6 different ranks, in a square formation so that each row and column contains one officer of each rank and one from each regiment. Euler found that it was impossible to do so, which led to the development of a whole new area in mathematics. Today, you'll learn the basics of permutations, which is the first step to understanding complex problems like the one Euler solved!"
   • The teacher concludes the introduction by saying: "So, let's dive into the world of permutations and see how many different ways we can arrange, organize, and make decisions!"

Development

Pre-Class Activities (10 - 15 minutes)

1. Research and Read (5 - 7 minutes)

   • Students should conduct research on permutations, focusing on its definition and uses in real life.
   • They can use online resources, such as math-related websites, online textbooks, or videos to gain a comprehensive understanding of the topic.
   • As part of their research, students are to note down key ideas and questions they may want to bring up during class discussion.

2. Self-guided Learning (3 - 5 minutes)

   • After their research, students should watch an interactive video about permutations, arranged by their teacher in advance. The video should explain the concept, the formula, and examples of how to solve permutation problems.
   • Here's a suggested video: Understanding Permutations

3. Preparatory Exercise (2 - 3 minutes)

   • Students should then complete a short online quiz based on the video to ensure their understanding of the topic.
   • The quiz can be created using tools like Google Forms or Quizizz, and should be shared by the teacher before the class.

In-Class Activities (20 - 30 minutes)

1. Activity: Permutation Puzzlers (10 - 15 minutes)

   • The teacher divides the students into groups of five and hands out "Permutation Puzzler" cards to each group.
   • Each card contains a puzzle which requires the use of permutations to solve.
     • For instance, a card could pose a question like "A graphic designer has 4 colors to make a logo. How many different combinations, assuming he needs to use all 4 colors and each color can only be used once, can he make?"
   • The teacher encourages each group to collaborate and solve their puzzle, with the teacher walking around the room to provide assistance if necessary.
   • After the groups have finished, they present their puzzles and solutions to the class. The teacher guides the review of each solution, ensuring the correct usage of permutation concepts.

2. Activity: Permutations Chain Reaction (10 - 15 minutes)

   • The teacher initiates a playful activity called "Permutations Chain Reaction." In this activity, the first group starts by posing a permutations problem. The problem can be creative and relevant, with a real-life context.
   • The next group has to solve the problem before posing their own problem.
   • This chain continues until each group has had the chance to pose and solve at least one problem.
   • This activity allows the students to practice applying permutations to problem-solving and encourages creativity and teamwork in a fun, engaging manner.
   • To wrap up the activity, the teacher summarises the class discussion and provides any necessary clarification on solving permutation problems.

Conclusion (10 - 15 minutes)

1. Classwide Discussion (5 - 10 minutes)

   • The teacher opens a classwide discussion, encouraging students to share their thoughts on the topic, their understanding, and ways they see permutations used in everyday life.
   • They can address any questions brought up during the pre-class research students conducted.

2. Summarizing the Lesson (3 - 5 minutes)

   • The teacher summarizes the key concepts learned, emphasizing the formula and use of permutations in problem-solving.
   • They highlight the importance of understanding permutations in various fields.

3. Homework Assignment (1 - 2 minutes)

   • The teacher assigns homework, which consists of a set of problems involving permutations for the students to solve independently, further cementing their understanding of the lesson. They are encouraged to use critical thinking and problem-solving skills gained in class to help solve the problems.
   • The teacher should make it clear that they are available for further doubts and questions either online or in the next face-to-face encounter.

This approach to teaching permutations should help students understand the topic fully and equip them with useful problem-solving skills. The flipped classroom methodology encourages research, independent learning, and collaboration in a fun, engaging environment. The emphasis on real-world examples and practical application helps students appreciate the relevance of permutations in various fields.

Feedback (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes)

   • The teacher opens the floor for a group discussion where each group gets the opportunity to share their solutions or conclusions from the in-class activities. Each group is given a maximum of 3 minutes to present their work.
   • The teacher ensures that the discussion is inclusive, engaging, and productive, providing constructive feedback to each group and addressing any misconceptions or errors.
   • As groups share their work, the teacher links their findings to the theoretical concepts of permutations, thereby strengthening the connection between theory and practice.

2. Assessment of Learning (2 - 3 minutes)

   • The teacher then assesses what was learned from the group activities. They do this by asking reflective questions, such as:
     • How does the activity connect with the theory of permutations?
     • Can you explain how you used the permutations formula to solve the puzzles?
   • The teacher uses this opportunity to gauge the students' understanding of permutations and their ability to apply the concept in practice.

3. Reflection (3 - 5 minutes)

   • Finally, the teacher proposes that students take a moment to reflect on what they have learned during the lesson. They can do this individually or share their thoughts with the class.
   • The teacher prompts reflection with questions such as:
     1. What was the most important concept you learned today?
     2. What questions remain unanswered about permutations?
   • Reflecting on these questions allows students to consolidate their learning and identify areas they need to revise or seek clarification on.

4. Closing (1 - 2 minutes)

   • To conclude the feedback stage, the teacher reiterates the importance of understanding and applying permutations, encouraging students to continue practicing and exploring this concept in different contexts.
   • The teacher also reminds students of their availability for any further questions or doubts that might arise while reviewing the lesson or doing homework.

This feedback stage is crucial in the flipped classroom methodology as it allows for active learning, encourages critical thinking, and fosters a deep understanding of the subject matter. By combining group discussion, assessment, and reflection, students have the opportunity to enhance their comprehension of permutations and improve their problem-solving skills.

Conclusion (10 - 15 minutes)

1. Summarizing the Lesson (3 - 5 minutes)

   • The teacher starts by summarizing the main points of the lesson. They reiterate the definition and concept of permutations and remind students of the formula for calculating permutations.
   • They also recap the practical application of permutations in real-world scenarios, emphasizing its relevance to various fields such as computer science, business, and biology.
   • They remind students of the engaging activities completed during class, such as the "Permutation Puzzlers" and "Permutations Chain Reaction," and how these activities helped in reinforcing the understanding of permutations.

2. Connecting Theory, Practice, and Applications (3 - 5 minutes)

   • The teacher then highlights how the lesson connected theory, practice, and applications. They explain how students started with understanding the theoretical concept of permutations and then moved on to apply this concept in practice through various activities.
   • They draw attention to how these activities helped students understand the practical applications of permutations, allowing them to recognize its importance in everyday life.
   • They stress that the ability to calculate permutations is not only a mathematical skill but also a problem-solving tool that can be applied in various contexts.

3. Additional Resources (2 - 3 minutes)

   • To further support students' understanding of the topic, the teacher suggests additional resources. This could include supplementary reading materials, websites for further study, and interactive online games or quizzes about permutations.
   • They suggest resources like Khan Academy, which offers comprehensive lessons on permutations, or Math Is Fun, which provides interactive permutation exercises.
   • They also recommend more advanced resources for students who wish to delve deeper into the topic, such as books or academic papers on combinatorics and permutations.

4. Relevance of the Topic (2 - 3 minutes)

   • Lastly, the teacher discusses the importance of permutations in everyday life. They provide examples of how understanding permutations can help in decision-making, organizing, and problem-solving.
   • They highlight that the knowledge of permutations is not restricted to mathematics, but extends to various fields and everyday scenarios. For instance, knowing permutations can help in planning schedules, calculating probabilities, understanding genetic variations, or even creating secure passwords in computer science.
   • They conclude by encouraging students to apply their knowledge of permutations in their daily lives and appreciate the beauty and utility of mathematics.

This conclusion stage helps students to synthesize their learning, understand the relevance of the topic to their lives, and gives them direction for further exploration of the subject. The teacher's role in this stage is to guide students in making the connections between theory and practice and to instill in them a curiosity and appreciation for the topic.