Lesson plan of Triangles: Congruence

Objectives (5 - 7 minutes)

1. Understand the concept of triangle congruence and identify the characteristics that make two triangles congruent.
2. Develop the ability to apply the congruence criteria (side-side-side, angle-angle-side, etc.) to determine if two or more triangles are congruent.
3. Practice applying triangle congruence criteria to solve problems, reinforcing understanding of the concept and problem-solving skills.

Secondary Objectives:

• Foster logical thinking and mathematical reasoning skills by justifying the reasons for triangle congruence or non-congruence.
• Promote student interaction through group activities, strengthening cooperative work and oral communication.
• Develop abstraction skills by applying the concept of triangle congruence in different contexts.

The teacher should frame these Objectives at the beginning of class, making it clear to students what they are expected to learn and be able to do by the end of class. Doing so helps focus students and increases the effectiveness of learning.

Introduction (10 - 15 minutes)

1. Review of Prior Knowledge: The teacher begins by reviewing the concept of triangles, their properties, and their classifications, especially the different types of triangles (equilateral, isosceles, and scalene) and the sum of the interior angles of a triangle. This is essential for students to understand and apply the criteria for triangle congruence. For example, the teacher can propose a quick quiz or Q&A game to assess students' prior knowledge and activate their memories about the topic.

2. Initial Problem Situations: Next, the teacher presents two problem situations that involve the concept of triangle congruence, but cannot yet be solved with the students' current knowledge. For example, the teacher can present the problem of constructing a triangle congruent to another, or the challenge of determining if two triangles are congruent just by looking at them. These problem situations serve to pique students' interest and contextualize the importance of the topic to be studied.

3. Contextualization: The teacher then contextualizes the importance of studying triangle congruence, explaining that it is a fundamental concept in Geometry, with applications in various fields, such as Engineering (for the construction of structures), Architecture (for the design of plans and projects), Physics (for the study of motion and forces), and even Biology (for the study of shapes and structures in living beings). Real-world examples can be used to illustrate these applications.

4. Grabbing Students' Attention: Finally, the teacher presents some curiosities or interesting facts about the topic to arouse students' curiosity and demonstrate the relevance of the subject. For example, the teacher can mention that the study of triangle congruence dates back to Ancient Greece, being one of the first topics of Geometry studied, or that there are different ways to prove the congruence of triangles, such as using folding and translations. Alternatively, the teacher can tell a story related to the topic, such as that of the German mathematician Carl Friedrich Gauss, who is said to have proven the congruence of triangles using only the idea of similarity.

Development (20 - 25 minutes)

1. Activity "Constructing Congruent Triangles" (10 - 12 minutes):

   • The teacher divides the class into groups of three or four students. Each group receives a set of cards with line segments of different lengths and a protractor.
   • The activity consists of using the line segments to construct triangles on a sheet of paper, following the teacher's instructions. For example, the instructions could be "construct an equilateral triangle" or "construct an isosceles triangle with an angle of 45 degrees".
   • After constructing each triangle, students verify if the constructed triangles are congruent, using the protractor to measure the angles and the line segments to measure the sides. They record their observations in a notebook.
   • At the end of the activity, each group presents the triangles they constructed and explains why they believe they are (or are not) congruent.

2. Activity "Discovering the Congruence Criteria" (10 - 12 minutes):

   • Still in groups, students receive a set of cards with different statements about the triangles they constructed in the previous activity. For example, "if two sides of one triangle are congruent to two sides of another triangle, and the angle between those sides is the same, then the triangles are congruent".
   • The students' task is to match each statement with the triangles they constructed and verified the congruence of in the previous activity. They justify their matches by arguing from the properties of the triangles and the congruence criteria.
   • After completing the activity, each group presents their matches and justifications to the class. The teacher mediates the discussion, clarifying doubts and correcting possible misunderstandings.

3. Activity "Applying the Congruence Criteria" (5 minutes, if time permits):

   • If time is available, the teacher can propose a third activity, which is more challenging. Students receive a set of triangles drawn on a sheet of paper, and the task is to identify which pairs of triangles are congruent, applying the congruence criteria they discovered in the previous activity.
   • This activity serves to consolidate learning and assess students' autonomy in applying the congruence criteria. The teacher circulates around the room, guiding students who encounter difficulties and asking questions that lead them to reflect on the process of identifying congruence.

Throughout the Development of the activities, the teacher circulates around the room, observing and guiding the groups, clarifying doubts, correcting errors, and valuing the different resolution strategies. In addition, the teacher promotes discussion among the groups, asking about the strategies used by them, encouraging them to justify their answers and to argue for or against the identified congruencies. This helps promote interaction between students and develop their critical thinking and mathematical reasoning skills.

Debrief (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher organizes a group discussion with all students. Each group has a maximum of 3 minutes to present the solutions or conclusions found during the activities.
   • During the presentations, students are encouraged to explain their resolution strategies, the difficulties encountered and how they overcame them, and to justify their answers or conclusions.
   • The teacher mediates the discussion, asking questions that promote students' reflection, clarifying doubts, correcting misunderstandings, and highlighting the most relevant or interesting points presented by the groups.

2. Connecting to Theory (2 - 3 minutes):

   • After the discussion, the teacher connects the practical activities carried out to the theory presented in the Introduction.
   • For example, the teacher can highlight how the congruence criteria that students discovered in the "Discovering the Congruence Criteria" activity relate to the definitions and theorems presented at the beginning of the lesson.
   • In addition, the teacher can reinforce the importance of understanding and correctly applying the triangle congruence criteria, remembering that this is a fundamental skill for solving problems in Geometry.

3. Final Reflection (3 - 4 minutes):

   • To conclude the lesson, the teacher asks students to reflect individually on what they have learned.
   • The teacher can ask questions such as: "What was the most important concept you learned today?", "What questions have not yet been answered?", and "How can you apply what you learned today in everyday situations or in other subjects?".
   • Students write down their answers in a notebook or on a sheet of paper.
   • The teacher may ask some students to share their reflections with the class, but it is important to respect students' privacy and not pressure them to share their reflections if they do not feel comfortable doing so.

Throughout the Debrief process, the teacher pays attention to students' reactions and participation, observing whether they have demonstrated an understanding of the concept of triangle congruence and whether they have been able to apply the congruence criteria independently. The teacher notes the main difficulties and doubts observed to plan future interventions and adjust the pace and depth of instruction according to the needs of the class.

Conclusion (5 - 7 minutes)

1. Summary and Recap (2 - 3 minutes):

   • The teacher begins the lesson Conclusion by summarizing the main points covered during the lesson.
   • This includes the definition of triangle congruence, the different congruence criteria (side-side-side, angle-angle-side, etc.), and how to apply them to determine if two triangles are congruent.
   • In addition, the teacher reviews the practical activities carried out, highlighting the main observations and conclusions made by the students.
   • The teacher can also use this time to briefly review the answers to the problem situations presented in the Introduction, showing how the application of the congruence criteria allowed them to reach a solution.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes):

   • Next, the teacher emphasizes the relationship between theory and practice, showing how the understanding of the theoretical concepts of triangle congruence enabled the students to carry out the practical activities.
   • The teacher can reinforce that practice is essential to consolidate learning, but that theory is the necessary foundation for the correct understanding and application of concepts.
   • In addition, the teacher reiterates the practical applications of the concept of triangle congruence, showing how it can be useful in various areas of life and other disciplines.

3. Supplementary Materials (1 minute):

   • The teacher can suggest supplementary materials for students who wish to deepen their knowledge of triangle congruence.
   • These materials can include math books, educational websites, explanatory videos, math games, and more.
   • For example, the teacher can indicate a YouTube video that presents different ways to prove the congruence of triangles, or an interactive website where students can explore triangle congruence through virtual activities.

4. Importance of the Topic (1 - 2 minutes):

   • Finally, the teacher emphasizes the importance of the topic presented for everyday life and other disciplines.
   • For example, the teacher can mention that the ability to identify and construct congruent triangles can be useful in everyday situations, such as when assembling furniture (where it is necessary to follow an assembly diagram) or when drawing an architectural project.
   • In addition, the teacher can point out that triangle congruence is a fundamental concept for the study of other geometry topics, such as similarity, areas, and volumes, and for solving mathematical problems in general.
   • The teacher ends the lesson by encouraging students to continue exploring and applying the concept of triangle congruence in their lives and studies, and reinforcing that the teacher will be available to help them in case of doubts or difficulties.