Lesson Plan | Traditional Methodology | Divisibility Criteria: Review

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to introduce students to the main divisibility criteria and provide a solid foundation so that they can identify and apply these criteria in various mathematical contexts. Understanding these criteria is fundamental for solving problems that involve divisibility, factorization, and simplification of fractions, among other mathematical topics.

Main Objectives

1. Know and describe the main divisibility criteria for 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.

2. Check when a number is divisible by one of these numbers using the learned criteria.

3. Develop the ability to apply these criteria in different mathematical situations.

Introduction

Duration: (10 - 15 minutes)

 Purpose: The purpose of this stage is to introduce students to the main divisibility criteria and provide a solid foundation so that they can identify and apply these criteria in various mathematical contexts. Understanding these criteria is fundamental for solving problems that involve divisibility, factorization, and simplification of fractions, among other mathematical topics.

Context

 Context: Start the class by explaining that divisibility is a fundamental concept in Mathematics that facilitates solving complex problems. Mention that it is present in various areas, such as factorization, simplification of fractions, and even cryptography. Emphasize that knowing the divisibility criteria can simplify many calculations and make problem-solving more efficient.

Curiosities

 Curiosity: Did you know that divisibility criteria are used in cryptographic algorithms to ensure data security? Additionally, in everyday life, people unconsciously use these criteria when splitting bills, making measurements, or checking if a number is prime.

Development

Duration: (65 - 70 minutes)

The purpose of this stage is to provide students with a detailed and practical understanding of the divisibility criteria, allowing them to apply these criteria in various mathematical situations. This stage aims to consolidate theoretical knowledge through practical examples and solved exercises, ensuring that students can confidently identify the divisibility of numbers.

Covered Topics

1. Divisibility Criterion for 2: Explain that a number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). 2. Divisibility Criterion for 3: Detail that a number is divisible by 3 if the sum of its digits is divisible by 3. 3. Divisibility Criterion for 4: Inform that a number is divisible by 4 if its last two digits form a number divisible by 4. 4. Divisibility Criterion for 5: Explain that a number is divisible by 5 if its last digit is 0 or 5. 5. Divisibility Criterion for 6: Detail that a number is divisible by 6 if it is divisible by both 2 and 3 at the same time. 6. Divisibility Criterion for 7: Introduce that a number is divisible by 7 if, by doubling the last digit and subtracting it from the rest of the number, the result is divisible by 7. 7. Divisibility Criterion for 8: Inform that a number is divisible by 8 if its last three digits form a number divisible by 8. 8. Divisibility Criterion for 9: Explain that a number is divisible by 9 if the sum of its digits is divisible by 9. 9. Divisibility Criterion for 10: Detail that a number is divisible by 10 if its last digit is 0. 10. Divisibility Criterion for 11: Introduce that a number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is a multiple of 11.

Classroom Questions

1. Check if the number 273 is divisible by 3 and by 9. 2. Determine if the number 1,024 is divisible by 4 and by 8. 3. Assess if the number 3,630 is divisible by 2, 5, and 11.

Questions Discussion

Duration: (15 - 20 minutes)

The purpose of this stage is to consolidate the knowledge acquired during the lesson, promoting an active and engaging discussion about the divisibility criteria. By reviewing and discussing answers, students have the opportunity to clarify doubts, reinforce concepts, and understand the practical application of the criteria in different mathematical contexts.

Discussion

•  Question 1: Check if the number 273 is divisible by 3 and by 9.

• To check divisibility by 3, sum the digits of 273: 2 + 7 + 3 = 12. Since 12 is divisible by 3, 273 is also. To check divisibility by 9, sum the digits again: 2 + 7 + 3 = 12. Since 12 is not divisible by 9, 273 is not divisible by 9.

•  Question 2: Determine if the number 1,024 is divisible by 4 and by 8.

• To check divisibility by 4, look at the last two digits: 24. Since 24 is divisible by 4, 1,024 is also. To check divisibility by 8, look at the last three digits: 024. Since 24 is divisible by 8, 1,024 is also.

•  Question 3: Assess if the number 3,630 is divisible by 2, 5, and 11.

• To check divisibility by 2, see if the last digit is even: 0. Since it is even, 3,630 is divisible by 2. To check divisibility by 5, see if the last digit is 0 or 5: 0. Therefore, 3,630 is divisible by 5. To check divisibility by 11, sum and subtract the digits alternately: 3 - 6 + 3 - 0 = 0. Since 0 is a multiple of 11, 3,630 is divisible by 11.

Student Engagement

1. ❓ Question 1: How can the sum of a number's digits help us determine its divisibility by 3 and by 9? 2. ❓ Question 2: Why is it important to look at the last two or three digits of a number when checking divisibility by 4 and by 8? 3. ❓ Question 3: How does the alternating difference between the digits of a number help us check its divisibility by 11? 4.  Reflection: Think of everyday situations where these divisibility criteria could be useful. Share examples with the class.

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to recap and consolidate the main points addressed during the lesson, ensuring that students have a clear and comprehensive view of the divisibility criteria. This final review aims to reinforce acquired knowledge and highlight the importance and practical application of divisibility criteria in everyday life and in more complex mathematical contexts.

Summary

• Divisibility criterion for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
• Divisibility criterion for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility criterion for 4: A number is divisible by 4 if its last two digits form a number divisible by 4.
• Divisibility criterion for 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility criterion for 6: A number is divisible by 6 if it is divisible by both 2 and 3 at the same time.
• Divisibility criterion for 7: A number is divisible by 7 if, by doubling the last digit and subtracting it from the rest of the number, the result is divisible by 7.
• Divisibility criterion for 8: A number is divisible by 8 if its last three digits form a number divisible by 8.
• Divisibility criterion for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility criterion for 10: A number is divisible by 10 if its last digit is 0.
• Divisibility criterion for 11: A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is a multiple of 11.

The lesson connected the theory of divisibility criteria with practice through detailed examples and the resolution of mathematical problems. Each criterion was clearly explained and followed by a practical application, allowing students to visualize how to apply the divisibility rules in different mathematical situations.

Understanding divisibility criteria is essential for everyday life, as it facilitates calculations, fraction simplification, and problem-solving in factorization. Moreover, these criteria are used in areas such as cryptography, which is fundamental for data security in digital transactions. Knowing these criteria makes learning mathematics more efficient and applicable in various everyday situations.