Objectives (5 - 7 minutes)

1. Understand the concept of Greatest Common Divisor (GCD): The teacher should introduce students to the definition and concept of Greatest Common Divisor, explaining how it is calculated and why it is useful in various situations.

2. Learn to calculate the GCD of two or more numbers: Students should be able to apply the concept of GCD to calculate the Greatest Common Divisor of two or more numbers. They should understand the step-by-step calculation process and be able to perform the calculations manually.

3. Solve problems involving the GCD: Students should be able to apply the concept of Greatest Common Divisor to solve practical problems. This implies identifying situations where the GCD can be useful and applying the calculation effectively to find the answer.

Secondary Objectives:

• Promote group interaction: The teacher should encourage collaboration and communication among students during the lesson, promoting discussion and joint problem solving.

• Develop critical thinking and problem-solving skills: Calculating the Greatest Common Divisor involves the application of formulas and solving complex problems. Therefore, students should be encouraged to think critically and develop their problem-solving skills.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reminding students about the concepts of divisibility and prime factors, which are fundamental to understanding the Greatest Common Divisor (GCD). The teacher can briefly review how to identify the prime factors of a number and how to use this information to determine if one number divides another. Additionally, it is important to review the concept of exact division and remainder.

2. Initial problem situations: To spark students' interest, the teacher can present two problem situations involving the Greatest Common Divisor. For example, "Alice has 12 chocolates and Bob has 15. They want to divide the chocolates equally between them. How many chocolates will each of them receive and how many will be left?" or "If I want to divide a 30-meter rope into pieces of the same size, what would be the maximum length of each piece?"

3. Contextualization of the importance of the GCD: The teacher should then explain the importance of the Greatest Common Divisor in everyday situations and in other areas of mathematics. For example, the GCD is widely used in proportion problems, fractions, and in cryptography algorithms. Additionally, the GCD is a fundamental concept in number theory, which has applications in computer science, cryptography, and many other areas.

4. Introduction to the topic: To introduce the topic in an interesting way, the teacher can share some curiosities or applications of the Greatest Common Divisor. For example, the GCD is one of the few mathematical concepts that has prehistoric origins, being used by ancient civilizations in the construction of monuments, land division, and time measurement. Another curiosity is that the Greatest Common Divisor is one of the first mathematical operations that digital computers were able to perform.

5. Capturing students' attention: The teacher can conclude the Introduction by telling a short story or anecdote related to the Greatest Common Divisor. For example, the story of the Greek mathematician Euclid, who developed Euclid's algorithm, an efficient technique for calculating the GCD, over 2000 years ago. Or the story of how the GCD is used in cryptography to ensure the security of online transactions and digital communications.

Development (20 - 25 minutes)

1. Activity "Find the GCD" (10 - 12 minutes):

   • Group formation: The teacher divides the class into groups of 4 to 5 students. Each group receives a sheet of paper and pencils to carry out the activity.

   • Activity description: The teacher explains that each group will receive a set of cards with numbers. The group's task is to organize the cards in pairs and calculate the Greatest Common Divisor of each pair. The group should record the calculations and the result on their sheet of paper. The goal is for the group to find the largest Greatest Common Divisor among all pairs of numbers.

   • Activity execution: The teacher distributes the sets of cards to each group. The students start organizing the cards in pairs and calculating the Greatest Common Divisor. The teacher circulates around the class, observing the groups' work, answering questions, and offering guidance if necessary.

   • Discussion and Conclusion: After a set time, the teacher asks each group to share the largest Greatest Common Divisor they found. The teacher records the results on the board. Then, the teacher guides a discussion about the strategies used by the groups, the patterns observed, and the difficulties encountered. The teacher highlights the importance of the Greatest Common Divisor and how it can be useful in various situations.

2. Activity "Real-World Problems" (10 - 13 minutes):

   • Introduction to the activity: The teacher presents the activity as an opportunity to apply what has been learned about the Greatest Common Divisor to real-world problems. The teacher should propose some problems, such as calculating the Greatest Common Divisor between the number of students in a school and the number of snacks available for a party, or calculating the Greatest Common Divisor between the number of feet in a room and the number of tables that can be placed so that each table has the same number of feet.

   • Group formation and distribution of problems: The teacher divides the class into groups and distributes the problems. Each group receives a different problem to solve.

   • Problem solving: Students work in their groups to solve the assigned problem. They must identify the numbers involved, calculate the Greatest Common Divisor, and interpret the result in relation to the problem.

   • Presentation of solutions: After a set time, each group presents its solution to the problem. The teacher guides a discussion about the different approaches used by the groups and the effectiveness of each. The teacher highlights how the Greatest Common Divisor was used to solve the problem and the relevance of the GCD in real life.

3. Activity "GCD Game" (optional, if time allows):

   • Game description: The teacher presents a simple board game in which players advance according to the Greatest Common Divisor of the numbers they land on. For example, if a player is on number 15 and the next number on the board is 9, the player advances 3 spaces (which is the Greatest Common Divisor of 15 and 9). The first player to reach the end of the board wins.

   • Group formation and game: Students organize into groups and play the game. The teacher circulates around the class, observing the game, answering questions, and offering guidance if necessary.

   • Discussion and Conclusion: After the game, the teacher leads a discussion about the game experience, the strategies used by the players, and the importance of the Greatest Common Divisor in the game. The teacher highlights how the game helped reinforce the concept of Greatest Common Divisor in a fun and engaging way.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • Sharing Experiences: The teacher asks each group to share their solutions or conclusions from the activities carried out. Each group has up to 2 minutes to present. The teacher should ensure that all groups have the opportunity to speak. During the presentations, the teacher should encourage other groups to ask questions or make comments about the solutions presented.

   • Comparison of Solutions: After the presentations, the teacher should lead a general discussion, comparing the different approaches and solutions presented. The teacher can highlight common points among the solutions, as well as differences in approach. The goal is for students to see that there are several ways to approach a problem and that different strategies can lead to the same answer.

2. Connection with Theory (2 - 3 minutes):

   • Review of Concepts: The teacher should recap the main concepts and procedures related to the Greatest Common Divisor (GCD), recalling the definition, the way to calculate it, and the importance of this concept.

   • Application of Theory: The teacher should then make the connection between theory and the practical activities carried out. For example, the teacher can explain how the activity "Find the GCD" helped reinforce the process of calculating the Greatest Common Divisor. Similarly, the teacher can highlight how the activity "Real-World Problems" allowed students to apply the concept of GCD to real situations.

3. Individual Reflection (2 - 3 minutes):

   • Reflection Moment: The teacher proposes that students reflect individually on what they learned in the lesson. The teacher can ask guiding questions, such as: "What was the most important concept you learned today?" or "What questions have not been answered yet?" The teacher should give students enough time to think about the questions and write their answers.

   • Sharing Reflections: After the reflection moment, the teacher can ask some students to share their answers with the class. This can help identify which concepts were well understood and which areas may need further review or practice.

4. Lesson Closure (1 minute):

   • Thanks and Motivation: The teacher thanks everyone for their participation and encourages them to continue exploring the concept of Greatest Common Divisor (GCD) in their daily activities. The teacher can share a final curiosity or an interesting practical application of the GCD to motivate students to continue learning about the subject.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes):

   • The teacher recaps the main points covered during the lesson, recalling the definition of the Greatest Common Divisor (GCD) and the methods to calculate it.

   • It is also important to review the strategies used during the practical activities, highlighting how students were able to apply the concept of GCD to solve real-world problems.

2. Connection between Theory and Practice (1 - 2 minutes):

   • The teacher emphasizes how the lesson combined theory with practice, allowing students to understand the concept of GCD and, at the same time, apply it in real situations.

   • The teacher can highlight how the group activities provided an opportunity for students to discuss and exchange ideas, reinforcing their understanding of the content.

3. Additional Materials (1 - 2 minutes):

   • The teacher suggests additional study materials for students who wish to deepen their knowledge of the Greatest Common Divisor. This may include textbooks, educational websites, explanatory videos, and practice exercises.

4. Practical Applications (1 minute):

   • The teacher reinforces the relevance of the Greatest Common Divisor, explaining that it is often used in various areas of mathematics, such as fractions, proportions, and cryptography algorithms.

   • Additionally, the teacher can highlight how the GCD can be useful in everyday situations, such as dividing things equally among a group of people, or when planning the arrangement of furniture in a room.

5. Lesson Closure (1 minute):

   • The teacher thanks the students for their participation and effort during the lesson. He can encourage them to continue practicing the calculation of the Greatest Common Divisor and to explore its applications.

   • The teacher can conclude by reminding students that, although mathematics may seem complex and abstract, it is present in many aspects of our daily lives, and that understanding and applying mathematical concepts can help us solve problems and make decisions more efficiently.