Lesson Plan | Traditional Methodology | Prime and Composite Numbers

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to provide students with a clear and initial understanding of the concept of prime and composite numbers, as well as introduce the criteria for divisibility. This basic understanding is crucial for students to follow the development of the lesson and actively engage in subsequent activities.

Main Objectives

1. Explain the definition and importance of prime and composite numbers.

2. Demonstrate how to identify prime and composite numbers through practical examples.

3. Introduce the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 100, and 1000.

Introduction

Duration: 10 - 15 minutes

The purpose of this stage is to provide students with a clear and initial understanding of the concept of prime and composite numbers, as well as introduce the criteria for divisibility. This basic understanding is crucial for students to follow the development of the lesson and actively engage in subsequent activities.

Context

📚 To start the lesson, explain to the students that numbers are present in all aspects of our lives, from counting objects to performing more complex mathematical operations. Tell them that today, they will learn about two special types of numbers: prime numbers and composite numbers. Emphasize that these concepts are fundamental in mathematics and that understanding the difference between them will help in various areas of mathematical knowledge.

Curiosities

🔍 An interesting fact is that prime numbers are used in cryptography, which is the basis of security in online banking transactions and the protection of sensitive information. For example, when we use credit cards on the internet, algorithms based on prime numbers help keep our information secure.

Development

Duration: 45 - 55 minutes

The purpose of this stage is to deepen students' understanding of prime and composite numbers, as well as the rules of divisibility. This understanding will allow students to identify and classify numbers independently and appropriately, strengthening their mathematical foundation and preparing them to solve more complex problems.

Covered Topics

1. 📌 Definition of Prime Numbers: Explain that a prime number is a natural number greater than 1 that can only be divided by 1 and itself. For example, the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc., are prime. 2. 📌 Definition of Composite Numbers: Detail that a composite number is a natural number greater than 1 that can be divided by 1, by itself, and by other natural numbers. For example, the numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc., are composite. 3. 📌 Divisibility Rules: Introduce the divisibility rules to help students identify whether a number is prime or composite. Explain the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 100, and 1000 clearly and with practical examples: Divisibility by 2: A number is divisible by 2 if it is even, i.e., it ends in 0, 2, 4, 6, or 8. Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Divisibility by 4: A number is divisible by 4 if the last two digits form a number divisible by 4. Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5. Divisibility by 6: A number is divisible by 6 if it is divisible by 2 and 3. Divisibility by 8: A number is divisible by 8 if the last three digits form a number divisible by 8. Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9. Divisibility by 10: A number is divisible by 10 if it ends in 0. Divisibility by 100: A number is divisible by 100 if it ends in two zeros. Divisibility by 1000: A number is divisible by 1000 if it ends in three zeros.

Classroom Questions

1. 1️⃣ Determine whether the following numbers are prime or composite: 17, 28, 31, 49, and 53. 2. 2️⃣ Using the divisibility rules, check if the numbers 144, 250, and 729 are divisible by 2, 3, 4, 5, 6, 8, 9, and 10. 3. 3️⃣ Explain why the number 1 is not considered a prime number nor a composite number.

Questions Discussion

Duration: 20 - 25 minutes

The purpose of this stage is to check and consolidate students' understanding of prime and composite numbers, as well as the criteria for divisibility. Detailed discussion of the answers enables the identification of possible doubts and their resolution, while the questions and reflections encourage critical thinking and active student engagement.

Discussion

• 1️⃣ Determine whether the following numbers are prime or composite: 17: It is a prime number because it can only be divided by 1 and itself. 28: It is a composite number because it can be divided by 1, 2, 4, 7, 14, and 28. 31: It is a prime number because it can only be divided by 1 and itself. 49: It is a composite number because it can be divided by 1, 7, and 49. 53: It is a prime number because it can only be divided by 1 and itself.

• 2️⃣ Using the divisibility rules, check if the numbers 144, 250, and 729 are divisible by 2, 3, 4, 5, 6, 8, 9, and 10: 144: Divisible by 2 (ends in 4), by 3 (1+4+4=9, which is divisible by 3), by 4 (the last two digits 44 are divisible by 4), by 6 (divisible by 2 and by 3), by 8 (the last three digits 144 are divisible by 8), by 9 (1+4+4=9, which is divisible by 9), but not by 5 (does not end in 0 or 5) and not by 10 (does not end in 0). 250: Divisible by 2 (ends in 0), by 5 (ends in 0), by 10 (ends in 0), but not by 3 (2+5+0=7, which is not divisible by 3), not by 4 (the last two digits 50 are not divisible by 4), not by 6 (not divisible by 3), not by 8 (the last three digits 250 are not divisible by 8), and not by 9 (2+5+0=7, which is not divisible by 9). 729: Divisible by 3 (7+2+9=18, which is divisible by 3), by 9 (7+2+9=18, which is divisible by 9), but not by 2 (not even), not by 4 (the last two digits 29 are not divisible by 4), not by 5 (does not end in 0 or 5), not by 6 (not divisible by 2), not by 8 (the last three digits 729 are not divisible by 8), and not by 10 (does not end in 0).

• 3️⃣ Explain why the number 1 is not considered a prime number nor a composite number: The number 1 is not considered prime because a prime number must have exactly two distinct divisors: 1 and itself. Since 1 only has one divisor (itself), it does not meet this definition. The number 1 is also not considered composite because a composite number must have more than two divisors. As 1 only has one divisor, it cannot be classified as composite.

Student Engagement

1. 🔢 Question: Why is it important to understand prime numbers for cryptography? 2. 🔢 Reflection: Think about how the divisibility rules can help simplify calculations in everyday problems. 3. 🔢 Question: If you encounter a very large number, how would you check if it is prime or composite? Discuss strategies. 4. 🔢 Reflection: How does understanding prime and composite numbers influence other areas of mathematics, such as factoring?

Conclusion

Duration: 10 - 15 minutes

The purpose of this stage is to consolidate students' learning by recapping the main points of the lesson and reinforcing the connection between theory and practice. This ensures that students understand the relevance of the concepts learned and are prepared to apply this knowledge in future contexts.

Summary

• Definition of prime numbers: natural numbers greater than 1 that can only be divided by 1 and themselves.
• Definition of composite numbers: natural numbers greater than 1 that can be divided by 1, by themselves, and by other natural numbers.
• Divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 100, and 1000, with practical examples.
• Discussion and problem-solving to identify prime and composite numbers.
• Explanation of the importance of prime numbers in cryptography and other practical applications.

The lesson connected the theory of prime and composite numbers with practice through the application of the divisibility rules to identify these numbers. Problem-solving and practical examples helped solidify the theoretical understanding of the concepts presented.

Understanding prime and composite numbers is fundamental not only for mathematics but also for computer science and cryptography, which are used to protect sensitive information. Knowing how to identify these numbers and apply the divisibility rules simplifies calculations and is a useful skill in various everyday situations.