Objectives (5 - 7 minutes)

1. Understanding the Pythagorean Theorem: The student should be able to understand the theorem in its entirety, perceiving the relationship between the lengths of the sides of a right triangle.
2. Applying the Pythagorean Theorem: The student should be able to apply the theorem in different contexts, solving practical problems involving the determination of a side of a right triangle when the other two are known.
3. Solving problems involving the Pythagorean Theorem: The student should be able to solve exercises and problems that require the application of the theorem, developing critical thinking skills and problems solving.

Secondary objectives:

• Connecting the Pythagorean Theorem to everyday life: The teacher should help students realize how the theorem is applied in everyday situations, such as in building construction and distance determination.
• Promoting group discussions: The teacher should encourage active student participation, promoting group discussions on the application of the theorem and problem-solving.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher should start the lesson by reviewing the concepts of triangles and, in particular, the right triangle. It is important for students to remember the definition of a right triangle and the characteristics of its angles. (3 - 5 minutes)

2. Problem Situations: The teacher should present two problem situations that lead students to think about the Pythagorean Theorem. For example, one can question how it is possible to determine the length of the hypotenuse of a right triangle if we know the lengths of its other two sides. Another situation may involve determining the length of one of the legs, if we know the length of the hypotenuse and the other leg. These questions should be left open for students to reflect on. (5 - 7 minutes)

3. Contextualization: The teacher should then explain that the Pythagorean Theorem is a very important mathematical tool widely used in various areas of knowledge, such as physics and engineering, as it allows determining the distance between two points on a plane. In addition, it can be mentioned that the theorem is applied in building construction and transportation engineering to calculate the height of an inaccessible object, such as a building, from its shadow and the shadow of an object whose height is known. (2 - 3 minutes)

4. Capturing Students' Attention: To captivate students' attention, the teacher can share some curiosities about the Pythagorean Theorem. For example, it can be mentioned that the theorem was not discovered by Pythagoras, but by Babylonian mathematicians about 1000 years before Pythagoras. In addition, it can be mentioned that the theorem has several demonstrations, some of which are very simple and elegant, while others are very complex. The teacher can also show a video or an animation that illustrates the theorem in a visually appealing way. (3 - 5 minutes)

Development (20 - 25 minutes)

1. Theory and Explanation of the Pythagorean Theorem (10 - 12 minutes): The teacher should explain the theorem clearly and concisely, showing the mathematical formula and its application. The teacher can use the board to draw a right triangle and highlight the three sides: hypotenuse, opposite leg, and adjacent leg. The explanation should include:

   • Definition of the Pythagorean Theorem: "In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs."
   • Presentation of the formula: a² + b² = c², where a and b are the legs and c is the hypotenuse.
   • Demonstration of the application of the formula in an example: "If we have a right triangle with legs measuring 3 and 4 units, we can determine the length of the hypotenuse using the formula: 3² + 4² = 9 + 16 = 25. Therefore, the hypotenuse has a length of 5 units."

2. Solving Examples (5 - 7 minutes): The teacher should present examples of problems involving the application of the Pythagorean Theorem. These examples should vary in difficulty and complexity, allowing students to practice applying the theorem. The teacher should guide students in solving these examples, explaining each step of the process. The teacher should also encourage students to participate actively, answering questions and proposing solutions.

3. Guided Practice (5 - 6 minutes): After solving the examples, the teacher should propose a series of exercises for students to practice applying the Pythagorean Theorem. These exercises should be solved with the teacher's guidance, who should clarify doubts and correct errors. The teacher should select exercises that are suitable for the students' level of skill and knowledge, but that also challenge them to think and apply the theorem creatively.

4. Discussion and Reflection (3 - 5 minutes): After guided practice, the teacher should promote a group discussion about what the students have learned. The teacher should ask students which concepts were the most difficult to understand and what strategies they used to solve the exercises. The teacher should also encourage students to ask questions and express their ideas and opinions. This discussion serves to consolidate learning and to identify possible difficulties that students may have with the theorem.

Return (8 - 10 minutes)

1. Review and Connection to the Real World (3 - 4 minutes): The teacher should review the concepts and skills learned during the lesson. This may include reciting the theorem, formulating the formula, and explaining how to apply it to solve problems. The teacher should also reinforce the importance of the Pythagorean Theorem in everyday life, mentioning practical examples of its application, such as in building construction, transportation engineering, and distance determination. In addition, the teacher can ask students to think of other examples of real-life situations where the theorem can be applied.

2. Learning Verification (2 - 3 minutes): The teacher should then verify if the learning objectives were achieved. This can be done through oral or written questions that should assess students' understanding of the theorem and their ability to apply it to solve problems. The teacher should ensure that all students have the opportunity to participate and demonstrate their understanding.

3. Reflection on Learning (3 - 4 minutes): The teacher should promote a reflection on the learning process. To do this, the following questions can be asked:

   • What was the most important concept learned today?
   • What questions have not been answered yet?
   • How can we apply what we learned today in other situations?
   • Which strategies were most useful for solving the exercises?

   The teacher should encourage students to think about their answers and share them with the class. This reflection serves to consolidate learning and to identify possible gaps in students' understanding, which can be addressed in future lessons.

4. Feedback and Closure (1 minute): Finally, the teacher should ask students for feedback on the lesson, asking what they liked the most and what could be improved. The teacher should also end the lesson by reinforcing the importance of the Pythagorean Theorem and encouraging students to continue practicing to improve their skills.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should recap the main points covered during the lesson, highlighting the definition of the Pythagorean Theorem, the associated formula (a² + b² = c²), and the practical application of the theorem in geometry problems and everyday situations. It is important for students to be able to recap these concepts to reinforce their understanding.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should emphasize how the lesson provided a clear connection between the theory of the Pythagorean Theorem and its practical application. The examples and exercises solved during the lesson helped students understand how to apply the theorem to solve problems. In addition, group discussions and the contextualization of the theorem with everyday life helped reinforce the relevance of the theorem in practice.

3. Extra Materials (1 - 2 minutes): The teacher should suggest extra materials for students to deepen their understanding of the Pythagorean Theorem. These materials may include explanatory videos, interactive websites, math games, and apps. The teacher can also recommend additional readings and exercises for students to practice what they have learned.

4. Importance of the Pythagorean Theorem (1 minute): Finally, the teacher should emphasize the importance of the Pythagorean Theorem for mathematics and everyday life. In addition to being one of the most fundamental theorems in geometry, the Pythagorean Theorem has practical applications in various areas, such as physics, engineering, and architecture. The teacher should encourage students to perceive the presence and importance of the theorem in their daily lives, helping them develop an appreciation for mathematics and its applications.