Lesson Plan | Traditional Methodology | Combinatorial Analysis: Permutation with Repetition

Objectives

Duration: 10 - 15 minutes

The purpose of this stage is to provide students with a clear understanding of what will be covered in the class, establishing a reference point for the content and skills that will be developed. This will help students focus on the key concepts and understand the relevance of the topic for solving specific problems.

Main Objectives

1. Understand the concept of permutation with repetition and its applications.

2. Learn the formula to calculate permutations with repeated elements.

3. Solve practical problems involving permutations of words with repeated letters, such as 'BANANA'.

Introduction

Duration: 10 - 15 minutes

The purpose of this stage is to provide students with a clear understanding of what will be covered in the class, establishing a reference point for the content and skills that will be developed. This will help students focus on the key concepts and understand the relevance of the topic for solving specific problems.

Context

To start the class on Combinatorial Analysis focusing on Permutation with Repetition, begin by establishing the relevance of the topic. Explain that in mathematics, combinatorial analysis is a powerful tool for counting and organizing elements of a set. Permutations are one of the main parts of this analysis. When some elements repeat, permutations with repetition come into play. Give a practical example, such as organizing letters in a word where there is repetition, like 'BANANA'.

Curiosities

Did you know that the concept of permutation with repetition is used in various fields of knowledge? For example, in cryptography, to generate secure password combinations, and in biology, to study the different ways of combining nucleotides in DNA. Additionally, in everyday life, we can think about how to organize different items, such as books on a shelf or clothes in a suitcase, considering identical items.

Development

Duration: 40 - 50 minutes

The purpose of this stage is to deepen students' understanding of permutation with repetition through detailed content exposure and solving practical problems. This will allow students to consolidate theoretical knowledge with practice and develop skills to apply the permutation formula with repeated elements in different contexts.

Covered Topics

1. Concept of Permutation with Repetition: Explain that permutation with repetition occurs when we have to permute elements where some are identical. Emphasize that the formula for calculating permutation with repetition is P = n! / (n1! * n2! * ... * nk!), where n is the total number of elements and n1, n2, ..., nk are the repetitions of each element. 2. Formula and Application: Detail the formula P = n! / (n1! * n2! * ... * nk!) and show how to apply it. Exemplify with the word 'BANANA', where we have 6 letters in total (n = 6), with 3 repetitions of 'A', 2 of 'N' and 1 of 'B'. The formula would be: P = 6! / (3! * 2! * 1!) 3. Practical Examples: Provide practical examples for students to solve together. Use words like 'MASSA', 'LIVRO' and 'COCADA' so that students can see different cases of permutations with varied repetitions.

Classroom Questions

1. Calculate the number of distinct permutations of the word 'MASSA'. 2. How many distinct permutations can be formed with the word 'LIVRO'? 3. Determine the number of distinct permutations of the word 'COCADA'.

Questions Discussion

Duration: 20 - 25 minutes

The purpose of this stage is to review and consolidate the knowledge acquired by students during the class. By discussing the solutions to the questions and engaging students with reflective questions, we ensure they deeply understand the concept of permutation with repetition and know how to apply it in different contexts.

Discussion

• Word 'MASSA': To calculate the number of distinct permutations of the word 'MASSA', we will use the formula P = n! / (n1! * n2! * ... * nk!). We have 5 letters in total (n = 5), with 2 repetitions of 'S' and 2 of 'A'. The formula would be: P = 5! / (2! * 2!) = 120 / (2 * 2) = 120 / 4 = 30. Therefore, there are 30 distinct permutations for the word 'MASSA'.

• Word 'LIVRO': To calculate the number of distinct permutations of the word 'LIVRO', we use the same formula. We have 5 letters in total (n = 5) and there are no repetitions. The formula would be: P = 5! / (1! * 1! * 1! * 1! * 1!) = 120 / 1 = 120. Therefore, there are 120 distinct permutations for the word 'LIVRO'.

• Word 'COCADA': To calculate the number of distinct permutations of the word 'COCADA', again we use the formula. We have 6 letters in total (n = 6), with 2 repetitions of 'C' and 2 of 'A'. The formula would be: P = 6! / (2! * 2!) = 720 / (2 * 2) = 720 / 4 = 180. Therefore, there are 180 distinct permutations for the word 'COCADA'.

Student Engagement

1. Why is it important to consider repetitions when calculating permutations? 2. How can we apply the concept of permutation with repetition in other areas besides words? 3. What is the difference between simple permutation and permutation with repetition? 4. Explain a practical example where you could use permutation with repetition in your daily life. 5. How does the formula for permutation with repetition change if we have more groups of repeated elements?

Conclusion

Duration: 10 - 15 minutes

The purpose of this stage is to consolidate the knowledge acquired by students during the class, recapping the main points covered. By connecting theory with practice and discussing the relevance of the content, it ensures that students understand the importance of the topic and know how to apply it in different contexts.

Summary

• Understanding the concept of permutation with repetition and its applications.
• Learning the formula to calculate permutations with repeated elements: P = n! / (n1! * n2! * ... * nk!).
• Practical resolution of problems involving permutations of words with repeated letters, such as 'BANANA', 'MASSA', 'LIVRO' and 'COCADA'.

The class connected theory with practice through the detailed explanation of the concept of permutation with repetition and the corresponding formula, followed by the guided resolution of practical problems. This allowed students to see the direct application of theory in concrete examples, facilitating understanding and memorization of the content.

Understanding permutations with repetition is fundamental not only for solving mathematical problems but also in various areas such as cryptography, biology, and organization of items in everyday life. Knowing how to calculate permutations with repeated elements allows for better organization and understanding of patterns, which is useful in various everyday situations.