Objectives (5 - 7 minutes)

1. Understand the concept of irrational numbers: The teacher will introduce the concept of irrational numbers, explaining that they are numbers that cannot be expressed as a simple fraction and that have infinite non-periodic decimal places. Students should be able to understand and identify irrational numbers.
2. Identify irrational numbers on the number line: After introducing the concept, the teacher will move on to the representation of irrational numbers on the number line. Students should be able to locate irrational numbers on the number line.
3. Solve problems involving irrational numbers on the number line: The teacher will propose problem situations involving the location and comparison of irrational numbers on the number line. Students should be able to solve these problems, applying the knowledge acquired.

Secondary objectives:

• Establish the relationship between irrational and rational numbers: Students should be able to understand that irrational and rational numbers are complementary, that is, together they form the set of real numbers.
• Develop logical and analytical reasoning skills: Through solving problems involving irrational numbers on the number line, students will develop logical and analytical reasoning skills.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher begins the class by reviewing the concepts of rational numbers and their representation on the number line. It is important that students are clear about what rational numbers are and how they are represented on the number line, as these concepts will be essential for understanding irrational numbers. (3 - 5 minutes)

2. Problem Situation 1: The teacher presents the following question: "If a student divided a pizza into 6 equal pieces and ate 3 pieces, what fraction represents the amount of pizza he ate?" The students should answer that he ate half of the pizza, or 1/2. The teacher then asks: "What if he had eaten one more piece, what fraction would represent the amount he ate?" The students should realize that, in this case, the amount cannot be represented by a simple fraction, because it was not possible to divide the pizza into a whole number of equal pieces. (3 - 5 minutes)

3. Contextualization: The teacher explains that the situation presented in the previous question is an example of an irrational number, as it cannot be represented by a simple fraction. He then presents other everyday situations in which irrational numbers are present, such as calculating the diagonal of a square with side 1. (2 - 3 minutes)

4. Topic Introduction: The teacher then introduces the topic of the class, explaining that irrational numbers are numbers that cannot be represented by simple fractions and that have infinite non-periodic decimal places. He also explains that, like rational numbers, irrational numbers can be represented on the number line. (2 - 3 minutes)

5. Problem Situation 2: To end the Introduction, the teacher proposes the following question: "If the number line represents the amount of pizza the student ate, where do you think the number that represents the amount of pizza the student ate (1/2 + 1/6 + 1/6) is located? And if he had eaten one more piece, where do you think that number would be located?" These questions serve as a hook for explaining the content. (2 - 3 minutes)

Development (20 - 25 minutes)

1. "Pi Pizza" Activity (10 - 12 minutes): The teacher divides the class into groups of up to 5 students. Each group receives a cardboard circle that represents a pizza. The teacher instructs the students to divide the pizza into 8 equal pieces and to paint 3 of those pieces one color and 2 another color. Then the students must calculate the fraction that represents the amount of pizza of each color. The teacher circulates around the room, assisting the groups and clarifying any doubts. After the activity is completed, the teacher guides the students to compare the fractions and observe that the amount of pizza cannot be represented by a simple fraction. Next, the teacher introduces the concept of irrational number, explaining that the amount of pizza that could not be represented by a simple fraction is an example of an irrational number. The students should then locate the fraction that represents the amount of pizza in the irrational number on the number line, which can be drawn on the cardboard.

2. "Irrational Numbers Treasure Hunt" Activity (10 - 12 minutes): The teacher scatters cards with irrational numbers written on them around the classroom. Each group receives a sheet with a number line drawn on it and the task of locating the irrational numbers on the number line. The teacher instructs the students to walk around the room, looking for the cards and locating the numbers on the number line. The students should record their discoveries on the sheet. At the end of the activity, the teacher reviews the students' answers, clarifies any doubts, and reinforces the concept of irrational number.

3. "Irrational Numbers Puzzle" Activity (optional, 5 - 7 minutes): If there is time, the teacher can propose the "Irrational Numbers Puzzle" activity. To do this, the teacher prints an image of a puzzle and glues cards with irrational numbers written on them onto different pieces of the puzzle. The students, in their groups, must assemble the puzzle, locating the irrational numbers on the number line. This activity is a fun and entertaining way to review the concept of irrational numbers and their representation on the number line.

These fun and contextualized activities aim to make learning about irrational numbers more meaningful and enjoyable for students, providing a better understanding of the content.

Debrief (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher invites each group to share their solutions or conclusions from the "Pi Pizza" and "Irrational Numbers Treasure Hunt" activities. Each group has up to 3 minutes to present. During the presentations, the teacher should encourage the other students to ask questions and make comments, thus promoting a rich and productive discussion. The teacher should take this opportunity to reinforce key concepts, correct any misconceptions, and clarify doubts.

2. Connection with Theory (2 - 3 minutes): After the presentations, the teacher makes the connection between the activities carried out and the theory covered in class. He emphasizes the importance of understanding the concept of irrational number and its representation on the number line, and how it is present in everyday situations. For example, the teacher can return to the pizza situation and reinforce that the amount of pizza that could not be represented by a simple fraction is an irrational number.

3. Individual Reflection (1 - 2 minutes): The teacher then suggests that the students do an individual reflection on what they learned in class. He asks the following questions to guide the students' reflection:

   1. What was the most important concept learned today?
   2. What questions have not yet been answered?
   3. How can you apply what you learned today to situations in your everyday life?

4. Sharing Reflections (2 - 3 minutes): The teacher invites some students to share their answers to the reflection questions with the class. The teacher should emphasize that all answers are valid and that the purpose is only to promote reflection on learning. At the end of this stage, the teacher reinforces the importance of studying irrational numbers and the number line for understanding and using mathematical concepts in various everyday situations.

Conclusion (5 - 7 minutes)

1. Content Summary (2 - 3 minutes): The teacher gives a brief summary of the main points covered during the class. He reinforces the concept of irrational numbers, the difference between rational and irrational numbers, and their representation on the number line. The teacher also recalls the hands-on activities carried out, highlighting the students' main discoveries and learning experiences.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher explains how the class connected theory, practice, and applications. He points out that the lesson began with a review of the theory, followed by practical activities that allowed students to visualize and manipulate irrational numbers on the number line. In addition, the teacher emphasizes that the proposed activities were based on real-life situations, which allowed students to perceive the presence and importance of irrational numbers in their lives.

3. Supplementary Materials (1 - 2 minutes): The teacher suggests some supplementary materials for students who wish to further their knowledge of irrational numbers. He can indicate books, websites, videos, and online games that approach the subject in a fun and educational way. The teacher can also suggest extra exercises for students to practice at home.

4. Importance of the Content (1 minute): Finally, the teacher emphasizes the importance of the content learned for everyday life and for other subjects. He explains that understanding irrational numbers and the number line is essential for solving many mathematical problems, in addition to being important knowledge for various fields, such as physics, engineering, economics, among others. The teacher ends the class by reinforcing the importance of study and practice for effective learning of mathematics.