Objectives (5 - 10 minutes)

1. Main Objective: To comprehend the concept of triangles and their classifications, students will be able to identify and differentiate between equilateral, isosceles, and scalene triangles.

2. Secondary Objective: To develop problem-solving skills through the application of specific properties of each type of triangle.

3. Secondary Objective: To promote interaction and teamwork among students, encouraging discussion and exchange of ideas during the proposed activities.

Specific Objectives:

• Identify the main characteristics of each type of triangle (equal or different sides and angles).
• Apply the Pythagorean Theorem to determine whether a triangle is equilateral, isosceles, or scalene.
• Solve practical problems involving the classification of triangles.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher will start the lesson by reviewing the basic concepts of plane geometry that were previously studied, such as the definition of polygons and the sum of their internal angles. This review is crucial for understanding the concepts that will be addressed in the current lesson.

2. Problem-Solving Situations: The teacher will present two problem-solving situations to the students involving the concept of triangles and their classifications. The first could be: 'If I have a triangle with all sides equal, what would be the measures of the internal angles?' The second could be: 'If I have a triangle with two equal sides, what would be the measures of the internal angles?' The teacher will encourage students to think about these questions and share their hypotheses.

3. Contextualization: The teacher should contextualize the importance of studying triangles and their classifications, explaining that these concepts are fundamental in mathematics and various areas of science and engineering. For example, in physics, the study of triangles is essential for understanding phenomena such as wave propagation and the resultant force in a system of forces.

4. Gaining Attention: To spark students' interest, the teacher may share some curiosities about triangles. For example, they may mention that the equilateral triangle is the only regular polygon that can be inscribed in a circle, or that the sum of the internal angles of any triangle is always equal to 180 degrees. Another interesting curiosity is the famous 'Fermat's Last Theorem,' which states that there are no three different non-zero integers that satisfy the equation x^n + y^n = z^n for any value of n greater than 2. This theorem, formulated by the mathematician Pierre de Fermat in 1637, remained unproven for over 350 years until it was finally demonstrated by the British mathematician Andrew Wiles in 1995, using advanced concepts of geometry and algebra, including the study of triangles.

5. Introduction to the Topic: Finally, the teacher will introduce the topic of the lesson: 'Triangles and their Classifications.' They will explain that during the lesson, students will learn to differentiate between equilateral, isosceles, and scalene triangles, and apply the Pythagorean Theorem to determine the classification of a triangle. The teacher may show some images of different triangles and ask students if they can identify the classification of each one. This activity serves to arouse students' curiosity and encourage their participation from the beginning of the lesson.

Development (20 - 25 minutes)

1. Playful Activity - 'Building Triangles' (10 - 15 minutes)

   • Objective: Allow students to visualize and manipulate triangles of different types to better understand their characteristics.

   • Materials Needed: Ruler, compass, paper, colored pencils or pens.

   • Description: The teacher will divide the class into groups of 3 to 5 students. Each group will receive the necessary materials for the activity. The teacher will explain that each group should build three triangles: one equilateral, one isosceles, and one scalene.

   • Step by Step:

     1. The teacher will show how to use the compass to draw a circle with a certain radius.
     2. Next, the teacher will show how to use the ruler to draw three line segments of the same length, connected in the form of a triangle (equilateral triangle).
     3. The teacher will show how to use the ruler to draw a line segment and, from one of its ends, draw two other line segments of the same length (isosceles triangle).
     4. Finally, the teacher will show how to use the ruler to draw three line segments of different lengths, connected in the form of a triangle (scalene triangle).
     5. After the demonstration, students in their respective groups will try to repeat the steps to build their own triangles.
     6. Once built, each group will present their triangles to the class, explaining why they believe each triangle belongs to a certain classification. The teacher will correct and clarify any confusion or errors that may have occurred.

2. Practical Activity - 'Classifying Triangles' (10 - 15 minutes)

   • Objective: Apply the concepts learned to classify triangles based on their measurements.

   • Materials Needed: Printed activity sheets with triangles of different measurements, pencil or pen.

   • Description: The teacher will give each group an activity sheet with various triangles of different measurements. Each group should classify the triangles as equilateral, isosceles, or scalene.

   • Step by Step:

     1. The teacher will explain that to classify a triangle, students must check if all sides are equal (equilateral triangle), if two sides are equal (isosceles triangle), or if all sides are different (scalene triangle).
     2. Students in their respective groups will work together to classify the triangles on the activity sheet.
     3. After completing the activity, the teacher will ask a representative from each group to share their answers and explain how they arrived at each classification. The teacher will correct and clarify any confusion or errors that may have occurred.

3. Group Research Activity - 'Triangles in the World' (5 - 10 minutes)

   • Objective: Connect the learned content with the real world, encouraging students to recognize the presence and importance of triangles in their daily lives.

   • Materials Needed: Electronic devices with internet access.

   • Description: The teacher will ask each group to research and present an example of how triangles are used in some aspect of the real world. It could be in architecture, art, nature, technology, etc.

   • Step by Step:

     1. The teacher will provide some examples to start the activity, such as the use of triangles in bridge construction, origami art, crystal formation, among others.
     2. Students in their respective groups will choose an example to research further.
     3. Each group will present their findings to the class, explaining how triangles are used in the chosen example. The teacher will encourage the participation of all students, asking questions and promoting discussion.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes)

   • Objective: Promote the exchange of experiences and knowledge among students, encouraging the expression of opinions and respect for different perspectives.
   • Description: The teacher will gather the class in a discussion circle and ask each group to share their solutions or conclusions from the activities carried out. Each group will have a maximum of 3 minutes to present their findings. During the presentations, the teacher should ask questions to stimulate critical thinking and reflection among students. The teacher should also ensure that all students are actively participating in the discussion, encouraging those who are more timid to share their ideas.

2. Connection with Theory (3 - 5 minutes)

   • Objective: Verify if students were able to apply theoretical concepts in practice, identifying difficulties and points to be reinforced.
   • Description: The teacher will provide a brief summary of the activities carried out, highlighting the main connections with the theory. Then, the teacher will ask students to reflect on what they have learned and share their insights. The teacher should ask questions like: 'What was the most important concept you learned today?', 'What questions have not been answered yet?' and 'What would you do differently if you had to repeat the activities?'. The goal is to make students think critically about what they have learned and how they could improve their performance.

3. Final Feedback (2 - 3 minutes)

   • Objective: Evaluate the effectiveness of the lesson and identify possible improvements for future lessons.
   • Description: The teacher will ask students to give final feedback on the lesson. Students can express their opinions, ask questions, or suggest improvements. The teacher should take note of the students' feedback and use this information to plan future lessons. The teacher should also take this opportunity to clarify any doubts students may have about the lesson content.

Conclusion (5 - 10 minutes)

1. Content Summary (2 - 3 minutes)

   • Objective: Recap the main points learned during the lesson and reinforce key concepts.
   • Description: The teacher will summarize the main concepts, recalling the definitions of equilateral, isosceles, and scalene triangles, as well as reinforcing the application of the Pythagorean Theorem to determine the classification of a triangle. The teacher may use diagrams and examples to illustrate each concept and ensure that all students understand clearly.

2. Theory-Practice Connection (1 - 2 minutes)

   • Objective: Explain how the lesson connected theory with practice, highlighting the relevance of the studied concepts.
   • Description: The teacher will reinforce how the activities carried out during the lesson allowed students to apply the theoretical concepts learned in practice. The teacher may use examples from the activities, such as building triangles and classifying them, to illustrate this connection and emphasize the importance of the acquired knowledge.

3. Extra Materials (1 - 2 minutes)

   • Objective: Suggest additional materials for students to deepen their studies on the topic.
   • Description: The teacher may suggest some extra materials for students who wish to deepen their knowledge of triangles. These materials may include books, websites, videos, and math apps that address the topic interactively and playfully. The teacher may also indicate some exercise questions for students to practice at home.

4. Applications in Daily Life (1 - 2 minutes)

   • Objective: Demonstrate the importance of triangles and their classifications in everyday life.
   • Description: To conclude, the teacher will highlight some practical applications of triangles and their classifications in daily life. For example, the teacher may mention how knowledge about triangles is essential in building structures, solving physics and engineering problems, art and design, among others. This connection to the real world will help students realize the relevance of what they have learned and motivate them to continue exploring the topic.