Lesson Plan | Active Learning | Permutations

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 is crucial to establish a clear direction for the lesson. By defining specific objectives, the teacher guides students on what will be focused on and what is expected of them by the end of the session. This clarity aids in the mental preparation of students and the assessment of their own progress during practical activities. Furthermore, it ensures that both the teacher and the students are aligned with the learning expectations.

Main Objectives:

1. Empower students to solve problems involving the multiplicative principle, applying it in the context of permutations.

2. Develop the ability to work with different types of permutations, such as letters, numbers, and arrangements of people in lines, through practical exercises.

Side Objectives:

1. Encourage critical thinking and problem-solving through the application of mathematical concepts in everyday situations.

Introduction

Duration: (15 - 20 minutes)

The Introduction stage serves to engage students and connect the content they studied previously with real and interesting situations. The proposed problem situations stimulate the immediate application of knowledge, while the contextualization seeks to show the relevance of studying permutations, increasing students' interest and curiosity. This interactive start prepares the ground for a practical and engaging lesson.

Problem-Based Situations

1. Imagine you have 5 different books and need to organize them on a shelf. How many different ways are there to organize these books?

2. If you have 4 types of ice cream and want to make a cone with 3 different scoops, how many ways can you choose the flavors?

Contextualization

Permutations are present in various everyday situations, from organizing objects on a shelf to forming teams in sports competitions. Knowing how to calculate and understand permutations not only helps in mathematical activities but also develops organization and logical reasoning skills that are useful in many other areas. Additionally, the history of permutations dates back to ancient times when they were used to solve problems of combinations of elements in religious and philosophical texts.

Development

Duration: (75 - 85 minutes)

The Development stage aims to put students' prior knowledge of permutations into practice, allowing them to explore the content more deeply and interactively. Through the proposed activities, students apply the multiplicative principle in diverse contexts, which not only reinforces learning but also stimulates skills such as critical thinking, creativity, and teamwork. This section is designed to be the central and most extensive part of the lesson, providing complete immersion in the theme.

Activity Suggestions

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

Activity 1 - Anagram Festival

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of permutations in a word context, developing skills in creativity and teamwork.

- Description: In this activity, students will work in groups to create the largest number of possible anagrams using the letters of a given word or phrase. They must record all possible anagrams without repeating combinations.

- Instructions:

• Form groups of up to 5 students.

• Each group will receive a word or phrase to create anagrams.

• Groups will have 15 minutes to list all possible anagrams, noting them on a piece of paper.

• At the end of the time, each group will present their anagrams to the class.

• The class will vote on the group that presented the greatest number of creative anagrams without repetitions.

Activity 2 - Ice Cream Mathematical Challenge

> Duration: (60 - 70 minutes)

- Objective: Use permutations with repetitions to solve a practical problem, promoting the use of the multiplicative principle.

- Description: Students, in groups, must solve a problem involving calculating how many different ways a customer can choose 3 scoops of ice cream from 6 available flavors, allowing for flavor repetitions.

- Instructions:

• Form groups of up to 5 students.

• Explain the problem: a customer can choose 3 scoops of ice cream from 6 available flavors, and can repeat flavors.

• Groups must calculate how many different combinations of choices the customer can make.

• Each group will present their solution and the method used.

• The teacher will discuss the different methods and solutions with the class.

Activity 3 - Book Arrangement Olympics

> Duration: (60 - 70 minutes)

- Objective: Understand and apply the concept of permutations with special conditions, promoting logical and mathematical reasoning skills.

- Description: This challenge asks students, in groups, to organize a series of books on a shelf in a way that meets certain specific conditions, such as keeping two books always together. Students must calculate the number of possible arrangements.

- Instructions:

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

• Provide each group with a list of arrangement conditions for the books (e.g., books A and B must always be together, C and D cannot be together).

• Groups must calculate how many possible arrangements satisfy the conditions.

• Each group will present their arrangement and the total number of possibilities calculated.

• Class discussion about the different strategies used by the groups.

Feedback

Duration: (10 - 15 minutes)

The purpose of this stage is to consolidate the knowledge acquired by students, allowing them to reflect on the learning process and application of permutation concepts. The group discussion helps identify gaps in understanding and strengthens students' ability to explain and justify their answers and methods, promoting a deeper understanding of the content. Additionally, this stage facilitates the transition to the conclusion, where key points are reinforced and linked to future learning.

Group Discussion

To start the group discussion, the teacher should ask each group to share their findings and challenges encountered during the activities. It is important for the teacher to circulate the room while the groups speak to ensure everyone has the opportunity to participate and provide additional support if necessary. It may be helpful to set a timer to ensure each group has equal time to speak. Encourage students to discuss not only the answers but also the reasoning processes and strategies used.

Key Questions

1. What were the biggest challenges in applying the multiplicative principle in the activities?

2. Was there any situation where you had to modify the initial approach? How did this influence the outcome?

3. How can the concept of permutations be applied in everyday situations or other subjects?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the Conclusion stage is to reinforce and solidify the knowledge acquired by students during the lesson. By summarizing key points, the teacher helps students consolidate their learning, ensuring that important concepts are retained. Moreover, by highlighting the bridge between theory and practice, and the applicability of the topic in everyday life, students are encouraged to value and apply mathematical knowledge in various contexts.

Summary

In the conclusion, the teacher should summarize the main concepts discussed about permutations, including the multiplicative principle and its application in different contexts, such as anagrams, object arrangements, and events. It is important to recap the formulas and calculation methods, as well as revisit the examples worked on during the practical activities.

Theory Connection

Today's lesson successfully established a solid connection between the theory of permutation concepts and their practical application. The exercises carried out helped demonstrate how mathematical knowledge can be used to solve real-life problems, reinforcing the importance of mathematics as an essential tool in various real situations.

Closing

Finally, it is crucial to emphasize the relevance of permutations in everyday life, such as in organizing events, planning routines, and even solving word games. Understanding and applying these concepts not only enriches students' mathematical reasoning but also develops organizational skills and logical thinking that are essential in many areas of life.