Objectives (5-7 minutes)

1. Understanding the Pythagorean theorem: The teacher should make sure that the students understand the concept and how to apply the Pythagorean theorem. This can be done by explaining the theorem formula, as well as by giving practical examples.

2. Identifying right triangles: Students must be able to identify right triangles in different contexts, either through figures or in real-world situations. This is fundamental for applying the Pythagorean theorem.

3. Solving problems using the Pythagorean theorem: The final goal of the class is for students to be able to apply the Pythagorean theorem to solve mathematical problems. The teacher should provide a variety of scenarios for students to practice applying the theorem.

Secondary objectives:

• Promoting critical thinking: By solving problems, students are encouraged to think critically and develop strategies for the solution. The teacher should encourage this approach and provide guidance when necessary.

• Stimulating active participation: The teacher should create a classroom environment where students feel comfortable actively participating in discussions and activities. This can be done by implementing participative teaching strategies, such as using questioning and group discussions.

Introduction (10-15 minutes)

1. Review of previous concepts: The teacher should start the class by briefly reviewing the concepts of basic geometry that are necessary for understanding the Pythagorean theorem. This includes the definition of a triangle and the properties of a right triangle.

2. Problem situation (1): Present the following situation: “Imagine that you are building a ladder that needs to reach a window in a building. You need to know the length of the ladder to make sure it’s safe. However, you only know the height of the window and the distance from the base of the wall to the window. How can you use this information to determine the length of the ladder?”

3. Providing context for the importance of the Pythagorean theorem: The teacher should then explain that the Pythagorean theorem is a tool widely used in many fields, including architecture, engineering, physics, and even in computer games. It is one of the fundamental tools for solving problems involving measurements and distances.

4. Curiosities (1): The teacher can share some curiosities about the Pythagorean theorem to arouse the students’ interest. For example, the discovery of the theorem is attributed to Pythagoras, a Greek mathematician from the 6th century BC, but there is evidence that ancient civilizations in Mesopotamia and Egypt already knew and used the theorem before that.

5. Providing context for applying the Pythagorean theorem (2): The teacher should then present a few more real-world situations that involve applying the theorem. For example, determining the distance between two points on a map, building ramps in parking lots, and figuring out the shortest path between two points in a maze.

6. Problem situation (2): The teacher can then present another situation: “Imagine that you are in a field and want to determine the distance between two trees. You can measure the distance between the two trees and the distance from one tree to the base of the other. How can you use the Pythagorean theorem to solve this problem?”

This introduction will prepare students to understand the Pythagorean theorem and how to apply it to real problems.

Development (20-25 minutes)

1. Hands-on activity: Building right triangles (10-12 minutes):

   • The teacher should divide the class into groups of 3 to 4 students. Each group will receive cardboard, a ruler, a compass, and scissors.
   • The teacher should instruct the students to use the materials provided to build right triangles. They should start by drawing a line segment on the cardboard. At one end, they should use the compass to mark a point. From this point, they should use the ruler and compass to draw an arc. The point where the arc intersects the line is the vertex of the triangle. They should repeat the process to draw the other two sides of the triangle.
   • After building the triangle, students should measure the lengths of the sides and check if they fit into the Pythagorean theorem formula (a² + b² = c²).
   • The teacher should circulate around the room, observing the students’ progress, and helping them when necessary. They should also encourage discussion among group members and joint problem-solving.

2. Problem-solving activity (10-13 minutes):

   • After the hands-on activity, the teacher should provide students with a series of problems to solve using the Pythagorean theorem. The problems should vary in difficulty and application, so that students can practice different aspects of the theorem.
   • Students should work in their groups to solve the problems. The teacher should encourage discussion and collaboration among group members.
   • The teacher should circulate around the room, offering guidance and clarifying doubts when necessary. They should also observe and assess students’ problem-solving process, rather than just focusing on the correct answer.

3. Applying the Pythagorean theorem (5-8 minutes):

   • Finally, the teacher should propose an activity to apply the Pythagorean theorem in a real-world context. For example, the students could be challenged to calculate the distance between two points on a map, the height of an inaccessible object, or the distance a cyclist will travel while climbing a hill.
   • Students should work in their groups to solve the activity. The teacher should circulate around the room, offering guidance and clarifying doubts when necessary.
   • At the end of the activity, each group should present their solution to the rest of the class. The teacher should facilitate a discussion about the solutions, highlighting the different methods of solving and reinforcing the concept of the Pythagorean theorem.

This stage of the lesson plan allows students to experiment with the Pythagorean theorem in action, reinforcing their understanding of the concept and its application. Furthermore, carrying out the activities in groups fosters collaboration and discussion, which is essential for the development of critical thinking.

Feedback (8-10 minutes)

1. Group discussion (3-4 minutes): The teacher should gather all the students and promote a group discussion about the solutions found by each team during the activities. Each group should share their findings and the process they used to get to them. The teacher should encourage questions and comments from the other students, thus promoting the exchange of ideas and collective reflection.

2. Connection with the theory (2-3 minutes): After the group discussion, the teacher should go back to the theoretical concepts presented at the beginning of the class and connect them to the practical activities carried out. For example, the teacher can ask: “How did the activity of building right triangles help us better understand the Pythagorean theorem?” or “Which aspects of the theorem became clearer to you after solving the problems?”.

3. Individual reflection (2-3 minutes): The teacher should propose that the students reflect individually on what they learned during the class. They should think about the following questions:

   1. What was the most important concept you learned today?
   2. Which questions have not been answered yet?
   3. How can you apply what you learned today to everyday situations or other disciplines?

   The teacher should give the students a minute to think about each question. After the reflection time, the students can share their answers, if they wish to do so. The teacher should listen carefully to the students’ answers, and address any questions or concerns that might arise.

4. Feedback and final orientations (1 minute): Finally, the teacher should give feedback to the students about their performance during the class, and give orientations for the next class or activity. The teacher can, for example, compliment the collaboration and critical thinking demonstrated by the students, and suggest that they continue practicing how to apply the Pythagorean theorem at home.

This Feedback stage is crucial to consolidate the students’ learning and to promote reflection about the learning process. Furthermore, it allows the teacher to assess the effectiveness of their teaching approach and make necessary adjustments for future classes.

Conclusion (5-7 minutes)

1. Summary and recapitulation (1-2 minutes): The teacher should start the Conclusion of the class by briefly summarizing the topics discussed and the activities carried out. They should remind the students of the definition of the Pythagorean theorem and the importance of right triangles in applying the theorem. The teacher should also highlight the critical thinking and problem-solving skills that were developed during the class.

2. Connection between theory, practice and applications (1-2 minutes): Next, the teacher should explain how the class connected the theory of the Pythagorean theorem to the practice of identifying and building right triangles, as well as to applying the theorem to solve real-world problems. The teacher can, for example, revisit the problem situations presented in the Introduction and show how the students were able to apply the theorem to solve them.

3. Extra materials (1-2 minutes): The teacher should then suggest some extra materials for the students who want to further their understanding of the Pythagorean theorem. This could include online videos, interactive math websites, textbooks, and additional exercises. For example, the teacher could recommend a video that explains the theorem in a visual and intuitive way, or a website where students can solve Pythagorean theorem problems in a game-like context.

4. Everyday applications (1 minute): Finally, the teacher should emphasize the relevance of the Pythagorean theorem in everyday life. They could, for example, mention that the theorem is widely used in fields as diverse as architecture, engineering, physics, and even in computer games. The teacher could also ask the students to think about other everyday situations where the theorem could be applied, thus reinforcing the practical usefulness of what was learned.

The Conclusion of the class allows students to revisit and consolidate what they have learned, and also encourages them to continue exploring the subject on their own. Furthermore, by highlighting the practical application of the Pythagorean theorem, the teacher helps students see math not just as a set of abstract rules and formulas, but as a useful and powerful tool that can be applied to many aspects of the world around them.