Lesson Plan | Active Learning | Combinatorial Analysis: Simple Permutation

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The Objectives stage aims to clearly establish what is expected for students to learn and be capable of doing by the end of the lesson. By outlining the main objectives, the teacher can direct teaching and learning efforts towards the specific skills of calculating and applying permutations, providing a solid foundation for students to explore and understand the complexities of the topic.

Main Objectives:

1. Enable students to calculate the number of permutations of elements in different sets, such as numbers, letters, and objects.

2. Develop the ability to solve practical problems involving permutations, such as calculating the number of permutations of letters in specific words.

Side Objectives:

1. Encourage the application of logical and mathematical reasoning to solve combinatorial analysis problems.

Introduction

Duration: (15 - 20 minutes)

The Introduction stage is designed to engage students and revisit previously studied concepts to prepare them for practical activities in the classroom. By presenting problem situations, the teacher encourages the application of theoretical knowledge in scenarios that spark curiosity and challenge. The contextualization of the topic with real-world examples and curiosities aims to show the relevance and applicability of permutations in everyday life, increasing students' interest and motivation.

Problem-Based Situations

1. Consider the word 'CASA'. How many different ways can we rearrange the letters to form new words? What if we want to create new 4-letter words, with or without repetition, how many possibilities are there?

2. If a person has 3 shirts, 2 pants, and 2 pairs of shoes, how many different ways can they dress, ensuring that the clothing pieces match and there are no repeats?

Contextualization

Combinatorial analysis, specifically permutations, has practical applications in various fields, including password security, event organization, scheduling, and even genetics for studying gene combinations. For example, understanding permutations can help create more efficient algorithms for security systems or process optimization. Additionally, curiosities such as the origin of card games and the history of the first mathematical treaties on the subject can enrich students' understanding.

Development

Duration: (75 - 80 minutes)

The Development stage is designed to allow students to apply the concepts of permutations in a practical and interactive way. Encouraging teamwork and logical reasoning, this section seeks to consolidate theoretical knowledge in situations that simulate real and everyday problems. Through the proposed activities, students can explore creativity, enhance their communication and collaboration skills, while deepening their mathematical understanding.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - The Secret Words Challenge

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of letter permutations in words to create new words and develop collaborative and quick reasoning skills.

- Description: In this activity, students will be divided into groups of up to 5 people, and each group will receive a list of five words. The challenge is to rearrange the letters of each word to form the largest number of new words possible. Each new word must be written on a separate card. At the end, the groups will swap cards and check the correctness of the formed words.

- Instructions:

• Divide the class into groups of up to 5 students.

• Distribute a list of five words to each group.

• Groups will have 30 minutes to rearrange the letters of each word and form the largest number of new words possible.

• Each new word must be written on a separate card.

• After the allotted time, groups swap cards with other groups.

• Each group checks the list of new words received and determines which ones are correct.

• Score each group based on the number of correct words.

Activity 2 - Number Sequence Creators

> Duration: (60 - 70 minutes)

- Objective: Understand and apply numerical permutations in a playful manner, developing teamwork and logical reasoning skills.

- Description: Students, in groups, will receive cards with numerical digits and a set of rules. The objective is to create the largest numerical sequence possible following the established permutation rules. The rules may include forming even or odd sequences, increasing or decreasing sequences, among other variations. Each correct sequence completed by the group will be scored.

- Instructions:

• Organize students into groups of up to 5 people.

• Distribute cards with numerical digits and a set of permutation rules to each group.

• Set a time limit of 40 minutes for groups to form numerical sequences within the established rules.

• Each correct completed sequence is worth points.

• At the end of the time, each group presents their sequences and explains the rules followed.

• Score the groups based on the number of correct sequences and the complexity of the rules followed.

Activity 3 - Dressing Mathematics

> Duration: (60 - 70 minutes)

- Objective: Apply permutation concepts in a playful and practical context, promoting creativity and organizational skills.

- Description: In this scenario, each group of students will receive cards representing different clothing items and accessories. They must create the largest number of possible clothing combinations, ensuring that each combination is complete and without repetitions. Each correct combination generated by the group will be photographed, and at the end, the best and most creative combinations will be awarded.

- Instructions:

• Divide students into groups of up to 5 participants.

• Give each group a set of cards representing clothing items and accessories.

• Set a time of 40 minutes for groups to create the largest number of possible clothing combinations.

• Each complete combination without repetitions must be photographed.

• At the end, each group presents their photos and explains the combinations created.

• Award groups based on the originality and number of correct combinations.

Feedback

Duration: (10 - 15 minutes)

The purpose of this stage is to consolidate the learning acquired during the practical activities, allowing students to articulate and reflect on what they learned. The group discussion helps reinforce the understanding of concepts by hearing different perspectives and approaches from peers. Additionally, this stage serves to assess students' understanding and clarify any remaining doubts, ensuring that all students have a clear and comprehensive understanding of the topic.

Group Discussion

To start the group discussion, the teacher can ask each group to share their most interesting findings and the challenges faced during the activities. It is important for each student to have the opportunity to express their ideas and listen to their peers. The teacher can guide the discussion, encouraging students to relate mathematical permutations to everyday situations and explore possible practical applications of the addressed concepts.

Key Questions

1. What were the most effective strategies your group used to solve the permutation problems?

2. How can the understanding of permutations be applied in real situations outside the classroom?

3. Was there any permutation concept that still seems confusing or challenging? How can we clarify that?

Conclusion

Duration: (5 - 10 minutes)

The purpose of this stage is to ensure that students have a clear and consolidated understanding of the studied permutation concepts, as well as recognizing their applicability and relevance in the real world. Through the summary and connection between theory and practice, this section aims to reinforce learning, preparing students for future applications of the acquired knowledge and stimulating a critical and reflective view on mathematics.

Summary

In the Conclusion stage, the teacher will summarize the main points discussed about simple permutations, highlighting the formulas and calculation methods. The main practical activities will be recapped, such as the Secret Words Challenge, Number Sequence Creators, and the 'Dressing Mathematics' game, to reinforce the applicability of the concepts in various contexts.

Theory Connection

The teacher will highlight how the practical activities conducted in the classroom connect the theory of permutations with real-world applications. It will be emphasized how combinatorial analysis is not only a mathematical tool but an essential skill in everyday situations, such as password security and event organization, promoting a deeper and more meaningful understanding of the content.

Closing

Finally, the importance of studying permutations in everyday life and in various professional fields, such as information technology, logistics, and genetics, will be discussed. This topic will serve to reinforce the relevance of mathematical learning in solving practical problems and promoting logical and critical thinking.