Objectives (5 - 7 minutes)

1. Recognition of prime and composite numbers: Students should be able to differentiate between prime numbers and composite numbers. They should understand that a prime number is a number greater than 1 and can only be divided by 1 and itself, while a composite number can be divided by more than two numbers.

2. Identification of factors: Students should learn to identify the factors of a number. They should be able to list all the factors of a given number and identify whether the number is prime or composite based on these factors.

3. Factoring numbers: Students should be able to factor a number into its prime factors. They should understand that every composite number can be expressed as a product of prime factors and that this expression is unique.

Secondary Objectives

• Development of logical reasoning: The study of prime and composite numbers helps in the development of students' logical reasoning, as they need to think systematically to identify factors and perform factorization.

• Application in everyday situations: The teacher should encourage students to identify everyday situations where understanding prime and composite numbers can be useful. This helps reinforce learning and demonstrate the relevance of the topic.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the concepts of factors, divisors, and multiples, which are fundamental to understanding prime and composite numbers. This can be done through direct questions to students or through practical activities involving the application of these concepts. For example, the teacher can ask students to list the factors of a number or to identify if one number is a divisor of another.

2. Problem situation 1: The teacher can propose the following problem situation: "Imagine you are a mathematician trying to discover a new type of number. You find out that this number is greater than 1 and can only be divided by 1 and itself. What can you conclude about this number? Is it prime or composite? Why?"

3. Contextualization: The teacher should explain to students that the idea of prime and composite numbers is not just an abstract concept in mathematics, but has practical applications in various areas, such as cryptography, number factorization in computer science, among others. Concrete examples can be cited to illustrate this, such as the famous RSA cryptography, which is based on the difficulty of factoring large numbers into their prime factors.

4. Curiosities: To spark students' interest, the teacher can share some curiosities about prime numbers. For example, it can be mentioned that prime numbers are infinite and that there is no predictable pattern for their distribution. The famous unsolved problem in mathematics, the Goldbach conjecture, can also be mentioned, which states that every even number greater than 2 can be expressed as the sum of two prime numbers.

5. Problem situation 2: The teacher can propose another problem situation: "You have heard of QR codes, right? They are a type of two-dimensional barcode used in many places, such as movie tickets. These codes are generated from very large prime numbers. Can you think of a reason for this? How can the property of prime numbers being difficult to factor be useful in this case?"

By the end of the Introduction, students should be curious and motivated to learn more about prime and composite numbers.

Development (20 - 25 minutes)

1. "Treasure Hunt of Prime Numbers" Activity (10 - 12 minutes):

   • Preparation: The teacher should prepare cards in advance with numbers from 1 to 100. On each card, some numbers will be prime and others will be composite. The cards will be hidden around the classroom.
   • Execution: Students, divided into groups of 4 to 5, will receive a checklist and the task of finding and classifying the numbers on the hidden cards as prime or composite. They should base their classification on the rule learned: prime numbers can only be divided by 1 and themselves, while composite numbers can be divided by more than two numbers.
   • Reflection: After the activity, students should discuss in groups the strategies used to identify prime numbers. Each group should share their strategies with the class. This activity, in addition to reinforcing the differentiation between primes and composites, promotes collaboration and teamwork.

2. "Building the Prime Factor Tree" Activity (10 - 12 minutes):

   • Preparation: The teacher should provide each group with a large sheet of paper and colored pens. The teacher should choose a large composite number (for example, 72) and factor it into its simplest form (in the example, 2 x 2 x 2 x 3 x 3).
   • Execution: Students, in their groups, should draw a tree with the composite number at the root and its prime factors as branches. Each prime factor should be represented by a different color. For example, the branches of 72 would be colored with two different colors, one for each prime factor (2 and 3).
   • Reflection: After the activity, students should reflect on how the prime factor tree helps visualize the factorization of a composite number. The teacher can ask students to identify patterns or similarities between the trees created by different groups. This activity promotes visualization and understanding of the factorization of composite numbers.

3. "Deciphering the Prime Number Enigma" Activity (5 - 8 minutes):

   • Preparation: The teacher should prepare a list of composite numbers in advance and an enigma that can only be solved if students can correctly factor the composite numbers.
   • Execution: The teacher should provide each group with a list of enigmas and the list of composite numbers. Students, in their groups, should work together to factor the composite numbers and solve the enigmas.
   • Reflection: After the activity, students should discuss in groups how the ability to factor composite numbers helped them solve the enigmas. This activity promotes the practical application of the concept of factorization of composite numbers.

By the end of the Development, students should have reinforced their understanding of prime and composite numbers, as well as the ability to factor composite numbers. Additionally, they should have had the opportunity to work in teams, discuss strategies, reflect on their learning, and apply what they learned in everyday situations.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher should invite each group to share their solutions or conclusions from the activities. Each group will have a maximum of 3 minutes to present.
   • During the presentation, the teacher should encourage other students to ask questions or make comments, thus promoting interaction between the groups.

2. Connection to Theory (2 - 3 minutes):

   • After all presentations, the teacher should summarize the main points raised by each group and connect them to the theory presented at the beginning of the lesson.
   • The teacher can reinforce the concepts of prime number, composite number, factors, and factorization, highlighting how these concepts were applied in the activities.

3. Individual Reflection (2 - 3 minutes):

   • The teacher should propose that students reflect for one minute on the following questions:
     1. What was the most important concept learned today?
     2. What questions have not been answered yet?
   • After the minute of reflection, the teacher should ask some students to share their answers with the class. The teacher should listen carefully to the answers, as they can provide valuable feedback for planning future lessons.

4. Teacher Feedback (1 minute):

   • To end the lesson, the teacher should provide brief feedback on the class performance, highlighting strengths and areas that need improvement.
   • The teacher can also give a brief glimpse of what will be covered in the next lesson to keep students engaged and motivated.

By the end of the Return, students should have had the opportunity to share their learnings, ask questions, reflect on what they learned, and receive feedback from the teacher. This helps consolidate learning and prepare students for the next lesson.

Conclusion (5 - 7 minutes)

1. Summary of Main Contents (2 - 3 minutes):

   • The teacher should recap the main points covered during the lesson. This includes the definition of prime and composite numbers, the identification of factors, and the factorization of numbers. This can be done through a brief review, emphasizing the most important concepts.
   • Additionally, the teacher should remind students of the practical activities carried out and how they helped reinforce the understanding of theoretical concepts. The teacher can highlight the importance of teamwork, discussion, and reflection during these activities.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes):

   • The teacher should explain how the lesson connected the theory of prime and composite numbers with the practice of the activities carried out. It can be discussed how the theory of factors and factorization was used to solve the enigmas and classify the numbers in the "Treasure Hunt of Prime Numbers" activity.
   • Additionally, the teacher should highlight the practical applications of prime and composite numbers, such as in cryptography and number factorization in computer science. The problem situations presented in the Introduction can be recalled and how the concepts learned in the lesson could be used to solve them.

3. Extra Study 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 math books, educational math websites, explanatory videos on YouTube, among others. The teacher can share these suggestions by email or through the school's virtual learning environment.

4. Importance of the Subject for Everyday Life (1 - 2 minutes):

   • To conclude the lesson, the teacher should emphasize the importance of knowledge about prime and composite numbers for everyday life. It can be mentioned again the practical applications, such as in cryptography and number factorization in computer science.
   • Additionally, the teacher can engage students in a reflection exercise, asking how they could use what they learned in the lesson in everyday situations. For example, they could think about how the ability to factor prime numbers could be useful for solving divisibility problems or for better understanding the numbers they encounter in their lives, such as in QR codes or product serial numbers.

By the end of the Conclusion, students should have reinforced their understanding of the concepts of prime and composite numbers, the importance of these concepts for practice and everyday life, and have received suggestions for extra study materials.