Objectives (5 - 7 minutes)

1. Understanding the concept of triangle similarity: Students should be able to clearly understand and articulate what it means for two triangles to be similar. They should be able to identify the properties that define triangle similarity.

2. Identification of similar triangles: Students must be able to identify which triangles out of a set are similar and which are not. They should be able to justify their answers, referring to the properties of triangle similarity.

3. Application of similarity ratios: Students must be able to apply the concept of triangle similarity to determine the similarity ratios between the corresponding measurements of similar triangles. This includes the ability to create proportions and solve simple equations.

Secondary objectives:

• Development of logical-mathematical thinking: In addition to acquiring specific triangle similarity skills, students should develop their mathematical thinking, including the ability to think logically, solve problems, and justify their answers.

• Application of mathematics in real-world situations: Students should be able to apply the concept of triangle similarity in practical situations, such as determining inaccessible heights or distances.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should begin the lesson by briefly reviewing the concepts of proportion and ratio, which were covered in previous lessons. This will serve as a basis for understanding the concept of triangle similarity. The teacher can suggest that students recall some situations in which proportion and ratio are used in their daily lives (for example, when cooking, mixing chemicals, etc.).

2. Problem situations: The teacher should present two problem situations involving triangle similarity, but without going into detail about how to solve them. The first could be determining the height of a building, using the building's shadow and the shadow of a pole. The second could be determining the height of a tree, using the tree's shadow and the student's shadow. These problem situations will serve to arouse students' interest and show the applicability of the concept that will be studied.

3. Contextualization: The teacher should explain that triangle similarity is a fundamental concept in mathematics and many other areas, such as physics, engineering, architecture, cartography, among others. For example, in photography and cinema, triangle similarity is used to determine the focal length of lenses and the scale of objects. In medicine, triangle similarity is used to determine a person's height from an X-ray image.

4. Introduction to the topic: To gain students' attention, the teacher can share some interesting facts and applications of triangle similarity. For example, he or she might mention that triangle similarity is used in Renaissance paintings to create the illusion of depth. Another interesting fact is that triangle similarity is used to calculate the height of large monuments, such as the Great Pyramid of Giza in Egypt, which has a height of 146.6 meters and a base of 230.4 meters, forming a right triangle.

Development (20 - 25 minutes)

1. Activity "Building Similar Triangles" (10 - 12 minutes): This hands-on activity will allow students to explore the concept of triangle similarity in a tangible way. Students will be divided into groups of three and given a set of sticks of varying sizes. Each group will have to build two triangles with their sticks, making sure that the corresponding sides of the two triangles are proportional. They will then measure the sides of the triangles and calculate the similarity ratios. The goal is for students to see how the similarity of triangles is determined by the proportions between the corresponding sides and how the similarity ratios are calculated.

   1. The teacher should provide the sticks of various sizes and the rulers to measure the sides of the triangles.

   2. The teacher should circulate around the room, guiding groups as needed and clarifying any doubts.

   3. At the end of the activity, each group should present their results to the class, explaining how they determined that the triangles were similar and what similarity ratios they found.

2. Activity "Application of Triangle Similarity" (10 - 12 minutes): In this activity, students will be challenged to apply what they have learned about triangle similarity to solve real-world problems. The teacher should present students with two different problem situations involving triangle similarity. For example, determining the height of a building from its shadow and the shadow of a pole (as in the Introduction) and determining the height of a tree from its shadow and the student's shadow (as in the problem-situation activity).

   1. The teacher should guide students in identifying the similar triangles in each situation and determining the similarity ratios.

   2. Students should then apply the similarity ratios to solve the problem situations and determine the heights of the objects.

   3. The teacher should circulate around the room, guiding students as needed and clarifying any doubts.

   4. At the end of the activity, each group should present their solutions to the class, explaining how they used triangle similarity to solve the problem situations.

3. Discussion and Reflection (3 - 5 minutes): After the conclusion of the activities, the teacher should open up the floor for discussion and reflection. Students should be encouraged to share their experiences, questions, and discoveries. The teacher should guide the discussion, asking questions that encourage students to think more deeply about the concept of triangle similarity and its applications. For example, the teacher could ask: "How do you think triangle similarity could be used in other real-world situations?" or "What were some of the challenges you faced trying to determine the similar triangles and similarity ratios in the activities?"

   1. The teacher should encourage all students to participate in the discussion and ensure that all questions are answered.

   2. The teacher should take this time to clarify any misunderstandings and reinforce the important concepts.

Note: The suggested activities are just suggestions. The teacher can adapt them according to the needs and resources available in the classroom.

Review (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes): The teacher should call all students' attention to the front of the room and ask each group to share their solutions or conclusions from the activities performed. Each group should have a maximum of 3 minutes to present. The goal is for students to learn from the different approaches and solutions presented by their classmates. During the presentations, the teacher should encourage other students to ask questions and make comments.

   1. The teacher should make sure that all groups have the opportunity to present and that the time is respected.

   2. The teacher should intervene, if necessary, to clarify misunderstood points or correct conceptual errors.

2. Connection to the Theory (2 - 3 minutes): After the presentations, the teacher should revisit the theoretical concepts discussed in the Introduction of the lesson and make the connection to the practical activities performed. The teacher should highlight how the activities illustrate and reinforce the theoretical concepts. For example, the teacher could observe how the activity "Building Similar Triangles" demonstrates that the similarity of triangles is determined by the proportions between the corresponding sides and how the activity "Application of Triangle Similarity" shows how triangle similarity can be used to solve real-world problems.

   1. The teacher should emphasize the main points and clarify any misunderstandings that may have arisen during the activities.

3. Individual Reflection (2 - 3 minutes): To conclude the lesson, the teacher should ask the students to briefly reflect on what they have learned. The teacher should ask the following questions:

   1. "What was the most important concept you learned today?"

   2. "What questions still remain unanswered?"

   3. "How can you apply what you learned today to real-world situations?"

   4. "What did you think of the hands-on activities? Did they help you better understand the concept of triangle similarity?"

   5. "What steps would you follow to solve problems similar to the ones you faced in the activities?"

   6. "What aspects do you think need more practice or study?"

   7. "What strategies did you use to solve the problem situations? Were they effective?"

   8. "Do you have any suggestions for improving the activities or the teaching of this topic?"

   9. "Would you like to share any interesting facts or applications of the concept of triangle similarity?"

   10. "Would you like to ask for help or clarification on any aspect of the topic?"

   11. "Are there any questions or concerns you have about the topic that were not addressed today?"

   12. "Do you feel prepared to solve problems involving triangle similarity independently?"

   13. "What do you think would be helpful to review or study more about triangle similarity?"

   The teacher should allow a moment of silence for students to reflect on the questions. Then, the teacher can ask some students to share their answers with the class, if they feel comfortable. The teacher should respect students' privacy and not force anyone to speak if they do not want to.

   1. The teacher should encourage all students to participate in the reflection and ensure that all questions are answered.

   2. The teacher should take this time to clarify any misunderstandings and reinforce the important concepts.

Note: The Review is a crucial part of the lesson, as it allows the teacher to assess students' understanding of the topic and identify any gaps or misunderstandings that need to be addressed. Furthermore, the Review helps students to consolidate what they have learned and to reflect on the learning process. The teacher should be open to feedback and suggestions from students and use this information to improve their future lessons.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes): The teacher should make a brief summary of the main points covered during the lesson. This could include the definition of triangle similarity, the identification of similar triangles, the application of similarity ratios, and the real-world situations where triangle similarity can be applied. The teacher should emphasize the key concepts and problem-solving strategies that were discussed.

   1. The teacher should make sure that all students understood the main concepts and clarify any doubts that may have arisen.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should explain how the lesson connected the theoretical concepts of triangle similarity with the hands-on activities performed by the students and the real-world applications discussed. The teacher should emphasize how understanding the theory is crucial for solving practical problems and applying mathematics in real situations.

   1. The teacher should reinforce the importance of connecting theory, practice, and applications for meaningful and lasting learning.

3. Supplementary Materials (1 minute): The teacher should suggest additional study materials for students who wish to deepen their knowledge of triangle similarity. This could include math books, educational websites, explanatory videos, interactive games, and learning apps. For example, the teacher could suggest using a mobile phone app that allows students to explore triangle similarity in an interactive and fun way.

4. Importance of the Topic (1 - 2 minutes): Finally, the teacher should emphasize the importance of triangle similarity for everyday life and for understanding other mathematical concepts. The teacher should mention examples of how triangle similarity is used in various areas, such as physics, engineering, architecture, cartography, among others. The teacher should also highlight that the ability to identify and work with similar triangles is a fundamental skill in geometry and is often tested on standardized exams.

   1. The teacher should encourage students to apply what they learned about triangle similarity in their everyday lives and in other disciplines.

Note: The Conclusion is an essential part of the lesson, as it allows the teacher to reinforce the key concepts, connect the theory with practice and applications, and motivate students to continue learning. The teacher should make sure that all students understood the main points of the lesson and are ready to move on to the next topic. The teacher should also be available to clarify any doubts that students may have after the lesson.