Objectives (5 - 7 minutes)

1. Understand the concepts of prime and composite numbers: The teacher should make an effort to ensure that students have a clear and concise understanding of what prime and composite numbers are. This involves explaining the difference between the two and providing practical examples to illustrate the point.

2. Identify prime and composite numbers: The teacher should teach students how to identify whether a number is prime or composite. This can be done through practical activities, such as decomposing a number into its prime factors.

3. Solve problems involving prime and composite numbers: The teacher should provide students with problems that involve the use of prime and composite numbers. This will help consolidate students' understanding of the subject and allow them to apply what they have learned in a practical way.

Secondary objectives:

• Promote classroom discussion: The teacher should encourage active student participation in the class, encouraging them to ask questions and share their ideas and thoughts on the topic. This will help create a collaborative learning environment and deepen students' understanding of the subject.

• Develop problem-solving skills: By solving problems involving prime and composite numbers, students will also be developing their problem-solving skills, which are essential in mathematics and many other aspects of life.

• Stimulate critical thinking: By discussing and analyzing the concepts of prime and composite numbers, students will also be stimulating their critical thinking, which is a valuable skill for life.

Introduction (10 - 15 minutes)

1. Review of related content: The teacher should start the lesson by briefly reviewing the concepts of factors and multiples, as these concepts are closely related to prime and composite numbers. This review can be done through a quick quiz or a brief classroom discussion. The goal is to ensure that students have the necessary foundation to understand the new concepts that will be presented.

2. Problem situation: The teacher can propose two problem situations to capture students' attention and show the importance of prime and composite numbers. The first situation could be: 'Imagine you have to divide 30 balls equally into groups. How many different groups can you make?' The second situation could be: 'If you have a house with 45 light bulbs and want to turn them all on at the same time, which numbers would you have to consider to ensure that all light bulbs are turned on?'

3. Contextualization: The teacher should explain that prime and composite numbers have practical applications in the real world. For example, they are used in cryptography to ensure the security of confidential information. Additionally, the idea of prime factors is fundamental to many other areas of mathematics, such as number theory.

4. Introduction to the topic: To introduce the topic in an engaging way, the teacher can share some curiosities about prime numbers. For example, they can mention that the largest known prime number currently has over 24 million digits. Or they can talk about the Clay Mathematics Institute's Million Dollar Prize, which offers a reward to anyone who can prove the Goldbach Conjecture, which states that every even number greater than 2 can be represented as the sum of two prime numbers.

5. Capturing students' attention: The teacher can end the Introduction by sharing the following curiosity: 'Did you know that prime numbers are like the superheroes of mathematics? This is because they have a unique characteristic: they can only be divided by 1 and by themselves. This makes them very special and unique! And today we are going to learn how to identify these mathematics superheroes and also their 'opposites': composite numbers.' This analogy can help spark students' interest in the topic and facilitate the understanding of the concepts.

Development (20 - 25 minutes)

1. Explanation of concepts (10 - 12 minutes):

   • What are prime numbers? The teacher should start by explaining that prime numbers are numbers that have only 2 divisors: 1 and themselves. They cannot be formed by multiplying other numbers besides 1 and themselves. The teacher should provide examples of prime numbers, such as 2, 3, 5, 7, 11, 13, etc.
   • What are composite numbers? Next, the teacher should explain that composite numbers are all numbers that are not prime. In other words, they are numbers that have more than 2 divisors. The teacher should provide examples of composite numbers, such as 4, 6, 8, 9, 10, 12, etc.
   • Difference between prime and composite numbers: The teacher should emphasize that prime and composite numbers are different because prime numbers have only 2 divisors, while composite numbers have more than 2 divisors.

2. Identification of prime and composite numbers (5 - 7 minutes):

   • Sieve of Eratosthenes rule: The teacher should teach students the Sieve of Eratosthenes rule to identify prime numbers. This is a simple and effective technique that involves crossing out all multiples of a number, starting with the first prime number (2). The numbers that are not crossed out are prime. The teacher should demonstrate this technique on the board and ask students to try to identify prime numbers using this rule.

3. Decomposition into prime factors (5 - 6 minutes):

   • Definition of prime factors: The teacher should explain that the decomposition into prime factors is a way to express a number as the product of its prime factors. Prime factors are the prime numbers that divide the original number.
   • Process of decomposition into prime factors: The teacher should demonstrate on the board how to decompose a number into its prime factors. This can be done using the technique of successive division, dividing the number by the smallest prime number that divides it, and repeating the process until the number is 1. The teacher should provide examples and ask students to try to decompose other numbers into prime factors.

4. Problem solving (5 - 7 minutes):

   • Problems of identifying prime and composite numbers: The teacher should provide students with a series of problems that involve identifying prime and composite numbers. This can be done through multiple-choice questions or free-form resolution problems. The teacher should ensure that students understand the problem and know how to identify whether a number is prime or composite before they start solving.
   • Problems of decomposition into prime factors: The teacher should provide students with problems that require the decomposition of a number into its prime factors. This will help consolidate students' understanding of the technique and allow them to practice it. The teacher should guide students during the problem-solving process, providing tips and feedback as needed.

5. Practical activity (optional): If time allows, the teacher can propose a practical activity where students will have to identify and decompose a series of numbers into prime factors. This will help reinforce the concepts learned in a fun and engaging way.

Return (8 - 10 minutes)

1. Group discussion (3 - 4 minutes):

   • The teacher should start a group discussion, asking students to share their answers and solutions to the problems proposed during the lesson. This will allow students to learn from each other and see different approaches to problem solving.
   • The teacher should encourage students to explain their reasoning and justify their answers, thus promoting a deep understanding of the concepts and the application of critical thinking skills.

2. Connection with theory (2 - 3 minutes):

   • The teacher should then make the connection between the group discussion and the theory presented at the beginning of the lesson. This can be done by highlighting how the identification and decomposition of prime and composite numbers are applied in problem solving.
   • The teacher should emphasize that theory and practice are intrinsically linked and that a deep understanding of theory is essential for the effective application of practice.

3. Individual reflection (2 - 3 minutes):

   • The teacher should ask students to reflect individually on what they have learned in the lesson. This can be done through questions such as: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • The teacher should allow a moment of silence for students to reflect, and then encourage those who feel comfortable to share their reflections with the class.
   • The teacher should listen carefully to students' responses, addressing any unanswered questions and reinforcing important concepts that were learned.

4. Feedback (1 minute):

   • Finally, the teacher should ask for feedback from students about the lesson. This can be done through a quick oral survey, where students can express what they liked about the lesson and what they think could be improved.
   • The teacher should take into account students' feedback when planning future lessons, ensuring that students' needs and interests are met.

5. Lesson closure (1 minute):

   • To conclude the lesson, the teacher should summarize the main points discussed and reinforce the key concepts. The teacher can also make a brief announcement about what will be covered in the next lesson, so that students can prepare accordingly.
   • The teacher should thank the students for their active participation and effort during the lesson, and encourage them to continue studying and practicing the concepts learned.

Conclusion (5 - 7 minutes)

1. Summary of key concepts (2 - 3 minutes):

   • The teacher should summarize the main points discussed during the lesson, reiterating the definition of prime and composite numbers, the difference between them, and how to identify them.
   • The teacher should also recap the technique of decomposition into prime factors, highlighting its usefulness in mathematics and other areas, such as cryptography.
   • The teacher should remind students of the importance of understanding and applying these concepts, as they are fundamental in many aspects of mathematics and everyday life.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   • The teacher should reaffirm how the lesson connected theory (concepts of prime and composite numbers and decomposition into prime factors) with practice (problem solving) and real-world applications (such as cryptography).
   • The teacher should emphasize that a deep understanding of theory is essential for the effective application of practice, and that real-world applications help make the concepts more tangible and relevant to students.

3. Extra materials (1 minute):

   • The teacher should suggest extra materials for students who wish to deepen their knowledge of prime and composite numbers. This may include books, websites, videos, and educational games that address the topic in different and engaging ways.
   • The teacher should emphasize that independent exploration of these materials can help students consolidate what they have learned in the lesson and expand their understanding.

4. Importance of the topic (1 minute):

   • Finally, the teacher should highlight the importance of prime and composite numbers in various fields, from mathematics and science to technology and information security.
   • The teacher should emphasize that by understanding and being able to work with prime and composite numbers, students are developing crucial mathematical skills, such as critical thinking, problem solving, and the application of concepts.
   • The teacher should encourage students to see mathematics as a powerful and versatile tool that can be applied in many aspects of their lives, and to continue exploring and learning more about the subject.