Lesson Plan | Active Learning | Factorial

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 the learning goals of the lesson. By focusing on the notation and properties of factorials, this section aims to prepare students to apply prior knowledge in new and challenging situations. With a clear understanding of the objectives, students can better direct their efforts during practical activities in class, maximizing the effectiveness of learning.

Main Objectives:

1. Enable students to recognize and manipulate the mathematical notation of factorials, applying it in calculations involving factorial expressions.

2. Develop an understanding of the main properties of factorials, such as recursive definition and basic operations with factorials, so that students can solve problems autonomously.

Side Objectives:

1. Encourage logical reasoning and the ability to generalize mathematically through practical examples involving factorials.

Introduction

Duration: (15 - 20 minutes)

The introduction stage aims to engage students with the theme of factorials, using problem situations that stimulate the review of previously studied content and its practical application. Furthermore, the contextualization serves to show the importance and real-world applications of the factorial, increasing students' interest in the subject and demonstrating that mathematics is present in everyday situations and in other fields of knowledge.

Problem-Based Situations

1. Considering that the factorial of a number is the product of all positive integers less than or equal to that number, how would you solve the expression 4! + 3! - 2!? Suggest steps to simplify and calculate the final value.

2. Imagine that in a math competition you need to calculate the factorial of a large number such as 10 or 20. What strategies would you use to simplify this calculation and make it more efficient?

Contextualization

The factorial is a mathematical function with applications in various areas, such as statistics, combinatorics, and algorithms. For example, in probability theory, the factorial is used to calculate permutations and combinations, which are fundamental for solving arrangement and selection problems. Additionally, curiosities such as the use of factorials in calculating possible shuffling sequences of cards or in computing to measure algorithmic complexity highlight the importance and relevance of this mathematical concept.

Development

Duration: (75 - 85 minutes)

The development stage is designed to allow students to apply and deepen their knowledge of factorials in a practical and interactive environment. The proposed activities aim not only to reinforce the understanding of the concepts studied but also to develop problem-solving skills, collaboration, and critical thinking. Choosing just one activity allows for an intense and profound focus, ensuring that students can fully explore the topic at hand.

Activity Suggestions

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

Activity 1 - The Factorial Race

> Duration: (60 - 70 minutes)

- Objective: Develop quick and precise calculation skills in pressure situations, as well as solidify understanding of factorials and their properties.

- Description: In this activity, students will be divided into groups of up to 5 people and participate in a math race to solve a series of problems involving factorials. Each group will receive cards with factorial expressions and must calculate the value of each one as quickly as possible.

- Instructions:

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

• Distribute the cards with factorial expressions to each group.

• Set a time limit for each round (for example, 3 minutes).

• At the end of the time, each group must present the calculated answers.

• Score according to the accuracy and speed of the answers.

• Conduct several rounds with increasingly difficult cards.

Activity 2 - The Enigma of the Lost Factorials

> Duration: (60 - 70 minutes)

- Objective: Stimulate logical reasoning and collaboration among group members while reinforcing knowledge of factorials and their applications.

- Description: Students, in groups, will receive a series of enigmas that will lead them to discover the value of 'lost' factorials in an interactive story. Each correctly solved enigma will provide clues for solving the next one, forming a playful and challenging narrative.

- Instructions:

• Divide the students into groups of up to 5.

• Give the first enigma, which involves calculating a factorial to reveal part of the story.

• After solving the enigma, students will receive the next part of the narrative and the next calculation to make.

• Each correctly solved enigma will give the group a clue for the next.

• The group that reaches the end of the story first, with all correct calculations, wins.

Activity 3 - Constructors of Factorial Sequences

> Duration: (60 - 70 minutes)

- Objective: Develop skills of observation, deduction, and application of mathematical patterns, as well as reinforce knowledge of factorials and sequences.

- Description: In this scenario, each group of students receives a set of numbers representing the first terms of a factorial sequence. The challenge is to discover the rule governing the sequence and predict the next terms, in a board game format where each correct step advances the group on the board.

- Instructions:

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

• Give each group a 'box' containing cards with the first terms of factorial sequences.

• Students must analyze the numbers and determine the rule governing the generation of the sequence.

• Each time a group finds a correct rule, they receive additional cards with the next terms.

• The game continues until all rules are discovered and the sequences are completed.

• The first group to discover all the rules and complete all the sequences wins.

Feedback

Duration: (10 - 15 minutes)

This stage of the lesson plan aims to consolidate student learning by providing a space for them to articulate what they have learned and how they applied that knowledge. The group discussion helps to identify gaps in understanding and reinforces concepts through explanation to peers. Additionally, it allows the teacher to assess student progress and clarify any remaining doubts.

Group Discussion

After completing the activities, organize a group discussion with all students. Start the discussion with a brief review of the factorial concepts addressed, and then ask each group to share their discoveries and strategies used during the activities. Encourage students to reflect on the application of knowledge in practical situations and to discuss the difficulties encountered and how they overcame them.

Key Questions

1. What were the main properties of the factorial that you applied during the activities and how did they help in solving the problems?

2. Was there any calculation strategy that proved to be more effective than the others? Why?

3. How do you see the application of the concept of factorial in everyday situations or in other subjects?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the conclusion stage is to consolidate learning, ensuring that students have a clear and integrated view of the discussed topics. This section helps to reinforce the link between theory and practice, highlighting the relevance of the learned content in real and everyday contexts. Moreover, it allows students to reflect on the importance of the topic and how it applies in various practical situations.

Summary

To conclude, let us recap the key concepts addressed about factorials. We revisited the definition and notation, explored fundamental properties such as the recursive definition and basic operations, and applied this knowledge in various practical activities, reinforcing calculation ability and understanding of the applications of factorials.

Theory Connection

During the lesson, the connection between theory and practice was established through interactive activities that simulated real situations where knowledge of factorials is essential, such as in math competitions and solving sequence problems. This practical approach not only reinforced theoretical understanding but also demonstrated the applicability of factorials in various contexts.

Closing

The importance of studying factorials transcends the academic environment, being essential in fields such as statistics, engineering, and computer science. Understanding and manipulating factorials not only enriches students' mathematical reasoning but also prepares them to face practical challenges in their future careers and studies.