Lesson Plan | Lesson Plan Tradisional | Combinatorial Analysis: Permutation with Repetition

Objectives

Duration: 10 - 15 minutes

This stage aims to give students a clear picture of what they'll learn in the lesson, setting a reference point for the content and skills to be acquired. This approach helps students focus on the key concepts and see how the topic is relevant for addressing specific problems.

Objectives Utama:

1. Grasp the concept of permutation with repetition and explore its applications.

2. Learn the formula for calculating permutations when elements are repeated.

3. Tackle practical examples involving permutations of words with duplicate letters, like 'BANANA'.

Introduction

Duration: 10 - 15 minutes

This stage seeks to provide students with a clear understanding of the lesson's coverage, establishing a reference for the skills and content to be developed. This helps students zero in on key concepts while recognizing the topic's relevance for solving specific challenges.

Did you know?

Did you know that the idea of permutation with repetition is useful in various fields? For example, in cryptography, it helps generate secure password combinations, and in biology, it assists in analyzing different ways of combining nucleotides in DNA. Also, think about how we might organize our belongings, like books on a shelf or clothes in a suitcase, considering identical items.

Contextualization

To kick off the lesson on Combinatorial Analysis with a focus on Permutation with Repetition, start by highlighting the importance of this topic. Explain that in mathematics, combinatorial analysis is a vital tool for counting and organizing elements within a set. Permutations play a significant role in this analysis. When certain elements are repeated, we often need to consider permutations with repetition. Use a relatable example, such as arranging letters in a word that contains duplicates, like 'BANANA'.

Concepts

Duration: 40 - 50 minutes

This stage aims to deepen students' understanding of the topic by exposing them to detailed content and practical problem-solving. It reinforces theoretical knowledge through practice and cultivates skills to apply the permutation formula with repeated elements across various contexts.

Relevant Topics

1. Concept of Permutation with Repetition: Clarify that permutation with repetition occurs when we need to permute elements that include identical ones. Highlight that the formula for calculating the permutation is P = n! / (n1! * n2! * ... * nk!), where n is the total number of elements and n1, n2, ..., nk represent the repetitions of each element.

2. Formula and Application: Elaborate on the formula P = n! / (n1! * n2! * ... * nk!) and demonstrate how to apply it. Use 'BANANA' as an example, with a total of 6 letters (n = 6), comprising 3 repetitions of 'A', 2 of 'N', and 1 of 'B'. The formula thus becomes: P = 6! / (3! * 2! * 1!)

3. Practical Examples: Present students with practical examples to solve collaboratively. Use words like 'MASSA', 'LIVRO', and 'COCADA' to illustrate different cases of permutations with diverse repetitions.

To Reinforce Learning

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'.

Feedback

Duration: 20 - 25 minutes

This stage is intended to review and consolidate the knowledge gained by students during the lesson. By discussing the answers to the questions and engaging students with reflective queries, we ensure they have a profound understanding of permutation with repetition and can apply it across various scenarios.

Diskusi Concepts

1. Word 'MASSA': To find the number of distinct permutations for the word 'MASSA', we will utilize the formula P = n! / (n1! * n2! * ... * nk!). There are 5 letters in total (n = 5), with 2 repetitions of 'S' and 2 of 'A'. Therefore, the formula is: P = 5! / (2! * 2!) = 120 / (2 * 2) = 120 / 4 = 30. So, we have 30 distinct permutations for 'MASSA'. 2. Word 'LIVRO': For 'LIVRO', using the same formula, we have 5 letters in total (n = 5) with no repetitions. Hence, the formula becomes: P = 5! / (1! * 1! * 1! * 1! * 1!) = 120 / 1 = 120. Thus, there are 120 distinct permutations for 'LIVRO'. 3. Word 'COCADA': Again applying the formula for 'COCADA', we find 6 letters in total (n = 6), with 2 repetitions of 'C' and 2 of 'A'. This gives us: P = 6! / (2! * 2!) = 720 / (2 * 2) = 720 / 4 = 180. Hence, 180 distinct permutations exist for 'COCADA'.

Engaging Students

1. Why is it important to consider repetitions when calculating permutations? 2. How can we apply the concept of permutation with repetition in other fields apart from words? 3. How does simple permutation differ from permutation with repetition? 4. Can you think of a practical example in daily life where you would apply permutation with repetition? 5. How does the formula for permutation with repetition alter if we have more groups of repeated elements?

Conclusion

Duration: 10 - 15 minutes

The aim of this stage is to reinforce the knowledge gained by students during the lesson, summarizing the main aspects discussed. By linking theory with practice and emphasizing the content's relevance, we ensure that students appreciate the importance of the topic and know how to apply it in various scenarios.

Summary

['Comprehending the concept of permutation with repetition and its applications.', 'Learning the formula for calculating permutations with repeated elements: P = n! / (n1! * n2! * ... * nk!).', "Practicing problem-solving with words that contain repeating letters, such as 'BANANA', 'MASSA', 'LIVRO', and 'COCADA'."]

Connection

The lesson effectively connected theory with practice through an in-depth explanation of the permutation concept and the related formula, followed by the guided resolution of practical examples. This approach allowed students to witness the direct application of theory in real scenarios, enhancing understanding and retention of the content.

Theme Relevance

Understanding permutations with repetition is vital not just for solving mathematical problems but also has applications in fields like cryptography, biology, and organizing everyday items. Knowing how to calculate permutations with repeated elements aids in better organization and comprehension of patterns, proving useful in numerous daily situations.