Lesson Plan | Active Learning | Combinatorial Analysis: Combination

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 essential to direct the focus of students and the teacher toward the concrete learning goals. By clearly establishing what is expected to be achieved, this stage prepares the ground for a structured and effective approach during the lesson. The outlined objectives aim to ensure that students are capable of applying the concept of combination in various contexts, thereby consolidating previously acquired knowledge in self-study.

Main Objectives:

1. Empower students to solve problems involving the calculation of combinations, focusing on situations where the order of elements is not relevant, such as in the example of forming groups of people.

2. Develop analytical skills and logical reasoning through the practice and manipulation of formulas and concepts of combinations.

Side Objectives:

1. Encourage collaboration among students during practical activities to promote an interactive and engaging learning environment.

Introduction

Duration: (15 - 20 minutes)

The introduction serves to engage students with the content they studied previously, using problem situations that require the direct application of combinations and contextualizations that show the relevance of the subject in different areas. This approach seeks to activate students' prior knowledge, increasing curiosity and motivation to explore the topic in depth during the lesson.

Problem-Based Situations

1. Imagine an event planning company needs to choose 3 bands from a total of 8 to play at a festival. How could they calculate how many unique combinations of bands could be selected?

2. A restaurant offers a menu of 5 appetizers, 10 main courses, and 3 desserts. If a customer wishes to choose a complete meal, consisting of an appetizer, a main course, and a dessert, in how many different ways can they do this?

Contextualization

Combinatorial Analysis, specifically the study of Combinations, has practical applications in various everyday situations and professional areas such as event planning, menu organization, computer programming, and even in statistics. Interestingly, this concept also has applications in non-mathematical areas, such as genetics, when calculating different possibilities of genetic combinations in crosses, or even in literature, when analyzing structures of poems and texts. This contextualization helps to show students the relevance of the subject and how it is integrated into various facets of life and knowledge.

Development

Duration: (75 - 80 minutes)

The Development stage is designed to allow students to practically and contextually apply the combination concepts they previously studied. The suggested activities aim to consolidate theoretical understanding through the resolution of real and creative problems, stimulating logical reasoning, collaboration, and calculation skills. This practical approach helps students visualize and internalize the mathematical concept of combinations in everyday situations and professional contexts. Each activity is carefully planned to be interactive and engaging, ensuring that students can explore the theme in a thorough and fun way.

Activity Suggestions

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

Activity 1 - The Combinations Party

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of combinations in a practical and creative context, developing calculation and logical reasoning skills.

- Description: Students are invited to plan a party where they must choose the decoration, menu, and activities using the concept of combinations. Each group of students receives a list of options for each category (for example, types of decoration, dishes for the menu, and games for the activities) and must calculate how many unique combinations can be made.

- Instructions:

• Form groups of up to 5 students.

• Distribute to each group the lists of options for decoration, menu, and activities.

• Ask each group to choose a combination of items from each list for their party.

• Students must calculate the total number of possible combinations.

• Each group will present their party and explain how they arrived at their combinations calculation.

Activity 2 - The Password Mystery

> Duration: (60 - 70 minutes)

- Objective: Understand the importance of calculating combinations in password security and practice using combinations in a cybersecurity context.

- Description: In this activity, students work in groups to create secure passwords. The security of the password is determined by the number of possible combinations that an intruder would have to try to guess it. Students must use the concept of combinations to calculate the security of several passwords created by themselves.

- Instructions:

• Organize students into groups of no more than 5 people.

• Provide a set of rules for creating passwords (for example, it must contain uppercase letters, lowercase letters, numbers, and symbols).

• Ask each group to create three passwords following the rules.

• Students must calculate how many unique password combinations can be made with the given rules.

• Each group presents their passwords and the calculation of possible combinations.

Activity 3 - The Mathematical Fashion Show

> Duration: (60 - 70 minutes)

- Objective: Use the concept of combinations to solve an organization, planning, and creativity problem, in addition to reinforcing mathematical understanding of combinations.

- Description: Students, in groups, must organize a fashion show using a combination of available clothes and accessories, so that each model has a unique set. They need to calculate the number of possible combinations to ensure that each model has a unique look.

- Instructions:

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

• Present a set of clothes and accessories available for the fashion show.

• Each group must choose unique combinations of clothes and accessories for each model.

• Students must calculate the total number of possible combinations.

• Each group presents their models and explains the calculation of the combinations.

Feedback

Duration: (15 - 20 minutes)

This stage of the lesson plan is crucial for consolidating students' learning, allowing them to articulate what they have learned and how they applied the knowledge in practical situations. Group discussion helps develop communication skills and deepens understanding of the material through the exchange of ideas and experiences. Moreover, by answering the key questions, students have the opportunity to reflect on their own learning process and identify areas that may need more attention or study.

Group Discussion

To initiate the group discussion, the teacher can ask each group about the decisions made during the activity and how they applied the concept of combinations to solve the proposed problems. Additionally, they can ask to share any challenges encountered and how they overcame them. This is an opportunity for students to reflect on the learning process and share insights with each other.

Key Questions

1. What were the main challenges in applying the concept of combinations in the proposed activities?

2. How can the understanding of combinations be applied in everyday situations or other subjects?

3. Was there any specific strategy or method your group found particularly effective during the resolution of the problems?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the Conclusion stage is to reinforce students' learning, ensuring that they can consolidate and internalize the concepts discussed during the lesson. Summarizing the key points, linking theory to practice, and highlighting the relevance of the topic for real-world applications helps to conclude the lesson with a clear understanding and an expanded view of the impact of combinations in different contexts.

Summary

In the conclusion, the teacher should summarize the main points covered regarding combinations, reiterating the definition and exemplifying how they were applied in practical activities. It is essential to recap the key formulas and concepts to ensure that students have a clear understanding of the material.

Theory Connection

During the class, the teacher connected theory and practice by exploring how combinations are used in real situations, such as event planning and password security. This approach helps solidify learning, showing students the relevance of theoretical content in practical and everyday contexts.

Closing

The importance of combinations transcends the academic environment, being applicable in various areas, from financial mathematics to cybersecurity. Understanding this concept not only enriches students' mathematical knowledge but also prepares them to tackle practical and professional challenges where logical reasoning and analysis are essential.