Objectives (5 - 7 minutes)

1. Understand the concept of prime and composite numbers: Students should be able to clearly define what prime and composite numbers are. They should understand that prime numbers are those that can only be divided by 1 and themselves, while composite numbers are those that have more than two divisors.

2. Identify prime and composite numbers: Students should learn to identify whether a number is prime or composite, applying the learned concept. They should be able to apply some quick tests to determine the nature of a given number.

3. Solve problems involving prime and composite numbers: Students should be able to apply the acquired knowledge to solve problems involving prime and composite numbers. They should be able to factor a composite number into its prime factors and, conversely, multiply prime factors to obtain a composite number.

Secondary Objectives:

• Establish the importance of the theory of prime and composite numbers: Students should be able to understand the importance of these concepts in mathematics and in practical applications. They should be able to see how the theory of prime and composite numbers is used in cryptography, for example.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing basic concepts that are necessary for the understanding of the current topic. This may include the definition of divisors, multiples, and factors, as well as the decomposition of numbers into prime factors. This review can be done through direct questions to the students or through a quick quiz.

2. Presentation of problem situations: The teacher should then present two problem situations involving prime and composite numbers. For example, 'If a number can only be divided by 1 and by itself, what happens if we try to divide the number 8 by 3?' and 'How can we factor the number 24 into its prime factors?' These problem situations will serve to arouse the students' interest in the topic and to show the practical relevance of what will be learned.

3. Contextualization of the subject: The teacher should then contextualize the subject, explaining how prime and composite numbers are used in our daily lives. For example, prime numbers are used in cryptography to protect our personal information on the internet. The teacher may also mention that the decomposition of numbers into prime factors is an important tool in many areas of mathematics and science.

4. Capturing students' attention: To capture students' attention, the teacher can share some curiosities about prime numbers. For example, the fact that there is no obvious pattern for the occurrence of prime numbers, or that the largest known prime has more than 17 million digits. Another interesting curiosity is that the distribution of prime numbers follows what is called the Prime Number Theorem, a surprisingly deep result in mathematics.

5. Introduction to the topic: Finally, the teacher should introduce the topic of the lesson, explaining that students will learn to distinguish between prime and composite numbers, to factor numbers into their prime factors, and to multiply prime factors to obtain composite numbers. He should also clarify that these skills are essential for solving many mathematical problems and for understanding more advanced concepts in mathematics and science.

Development (20 - 25 minutes)

1. Presentation of the theory (10 - 12 minutes):

   1.1. Definition of prime and composite numbers: The teacher should start by explaining the concept of prime and composite numbers. He should emphasize that a prime number is one that has only two divisors: 1 and the number itself, while a composite number has more than two divisors.

   1.2. Divisibility test: The teacher should then present the divisibility test for 2, 3, 5, and 7, which are the most common tests used to identify whether a number is prime or composite. He should explain how each of these tests works and give examples of how to apply them.

   1.3. Decomposition into prime factors: The teacher should then explain how to decompose a number into its prime factors. He should start with simple examples and then move on to more complex examples. He should emphasize that the decomposition into prime factors is an essential skill in mathematics and in many areas of science.

   1.4. Multiplication of prime factors: Finally, the teacher should explain how to multiply prime factors to obtain a composite number. He should give several examples to illustrate the process and should emphasize that the ability to multiply prime factors is the inverse skill of the decomposition into prime factors.

2. Application of the theory (10 - 13 minutes):

   2.1. Practical activity - Identifying prime and composite numbers: The teacher should then move on to a practical activity in which students will have to identify whether a number is prime or composite. The teacher can distribute cards with numbers to the students and ask them to use the learned divisibility tests to identify the nature of these numbers.

   2.2. Practical activity - Factoring numbers into prime factors: The teacher should then move on to an activity in which students will have to factor numbers into prime factors. The teacher can give each student a number to factor and ask them to show their work on the board.

   2.3. Practical activity - Multiplication of prime factors: Finally, the teacher should move on to an activity in which students will have to multiply prime factors to obtain a composite number. The teacher can give each student a set of prime factors and ask them to multiply them to obtain a composite number.

3. Discussion and clarification of doubts (5 - 7 minutes):

   3.1. Group discussion: The teacher should conclude the theoretical part of the lesson with a group discussion about the concepts learned. Students should be encouraged to ask questions and share their own strategies for identifying prime and composite numbers, factoring numbers, and multiplying prime factors.

   3.2. Clarification of doubts: The teacher should then dedicate time to clarify any doubts that students may have. He should reinforce the most difficult concepts and answer any questions that students may have.

   3.3. Final reflection: Finally, the teacher should ask students to reflect on what they have learned in the lesson. He should ask students what the most important concepts they learned were and what questions have not been answered yet. This will allow the teacher to assess the effectiveness of the lesson and to plan future lessons according to the students' needs.

Return (10 - 12 minutes)

1. Review of concepts (3 - 5 minutes):

   1.1. Group discussion: The teacher should start the review of concepts by discussing with the class what was learned. He should ask students about the difference between prime and composite numbers and request examples of each.

   1.2. Practical application: The teacher should then ask students to share how they used the acquired knowledge during the practical activities. For example, how they used the divisibility tests to identify prime and composite numbers, or how they used the factorization of numbers into prime factors to solve problems.

   1.3. Theory review: Next, the teacher should briefly review the theory, reinforcing the definition of prime and composite numbers, the divisibility test, and the decomposition and multiplication of prime factors. He should take this opportunity to clarify any remaining doubts that students may have.

2. Connection with practice (3 - 5 minutes):

   2.1. Group discussion: The teacher should then discuss with the class how the theory relates to practice. He should ask students to reflect on how the theory was applied during the practical activities and how practice helped solidify the understanding of the theory.

   2.2. Practical examples: The teacher should then present some additional practical examples of how the theory of prime and composite numbers is used in the real world. For example, he can mention how public key cryptography uses the factorization of large numbers into prime factors, or how number factorization is used in many areas of mathematics and science to solve complex problems.

3. Final reflection (2 - 3 minutes):

   3.1. Reflective questions: Finally, the teacher should ask students to silently reflect on the lesson. He should ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.

   3.2. Teacher feedback: The teacher should then end the lesson by providing feedback to the students. He should praise the students' efforts and highlight the class's strengths. He should also provide guidance on how students can continue to learn about prime and composite numbers, whether through additional readings, practices, or online reviews.

   3.3. Encouragement: Finally, the teacher should encourage students to continue practicing what they have learned and to seek help if they have doubts. He should emphasize that practice is the key to mastering the concepts of prime and composite numbers and that all students are capable of success if they persist in their studies.

Conclusion (5 - 7 minutes)

1. Summary of contents (2 - 3 minutes):

   • The teacher should start the Conclusion by summarizing the main points discussed during the lesson. He should reaffirm the difference between prime numbers (which can only be divided by 1 and themselves) and composite numbers (which have more than two divisors).
   • The teacher should recall the divisibility tests for 2, 3, 5, and 7, which are the most common ones to determine the nature of a number, as well as the importance of decomposition into prime factors and the multiplication of prime factors to solve problems involving prime and composite numbers.

2. Theory-practice connection (1 - 2 minutes):

   • The teacher should highlight how the lesson connected the theory of prime and composite numbers with practice. He can mention the practical activities carried out in the classroom, where students applied the learned concepts to identify prime and composite numbers, to factor numbers into prime factors, and to multiply prime factors to obtain composite numbers.
   • The teacher should emphasize that practice is essential to consolidate theoretical understanding and that students should continue to practice these skills outside the classroom.

3. Extra materials (1 minute):

   • The teacher should suggest additional materials for students who wish to deepen their knowledge of prime and composite numbers. This may include math books, educational websites, online videos, math games, and practical exercises.

4. Application in daily life (1 - 2 minutes):

   • Finally, the teacher should emphasize the importance of prime and composite numbers in our daily lives. He can mention examples of how the theory of prime numbers is used in public key cryptography, which is widely used to protect information online.
   • The teacher can also mention that the ability to factor numbers into prime factors is used in many areas of mathematics and science to solve complex problems.
   • The teacher should encourage students to realize the presence and usefulness of prime and composite numbers in their daily lives, which can help motivate them to continue studying and practicing these concepts.