Lesson plan of Triangles: Sum of Angles

Objectives (5 - 7 minutes)

Main Objectives

1. Understand the concept of the sum of the interior angles of a triangle and be able to apply it in different situations.
2. Apply the concept of the sum of the interior angles of a triangle to solve practical problems.
3. Develop logical and critical reasoning skills when dealing with the properties of triangles.

Secondary Objectives

1. Foster teamwork and collaboration among students through hands-on activities and discussions.
2. Promote the use of digital technologies (if available) for interactive exploration of the content.
3. Stimulate mathematical thinking and complex problem solving.

Introduction (10 - 12 minutes)

1. Review of previous concepts: The teacher begins the class by recapping the concepts studied previously that are fundamental to understanding the current topic. In this case, the teacher should briefly review what angles are, how they are measured, and the different types of angles (acute, right, obtuse). In addition, you should recall the basic properties of triangles (the sum of the interior angles is always 180 degrees).

2. Problem situation: To arouse students' interest, the teacher presents two problem situations. The first is the construction of a triangle whose interior angles add up to 180 degrees, but with sides of different sizes. The second is the construction of a triangle with sides of the same size, but whose interior angles do not add up to 180 degrees.

3. Contextualization: The teacher then contextualizes the importance of the topic, explaining how the sum of the interior angles of a triangle is a fundamental property for geometry and has practical applications in various areas, such as architecture, engineering, physics, and cartography.

4. Introduction to the topic: To introduce the topic in an engaging way, the teacher can share curiosities such as the origin of the study of triangles (in ancient Greek mathematics) and the relevance of this geometric figure in nature (for example, in the formation of crystals and in the structure of molecules).

5. Attention grabber: To capture the students' attention, the teacher can present two situations involving the sum of the interior angles of a triangle in an unusual way. The first is the famous "Gauss's Theorem", which states that the sum of the interior angles of a triangle is equal to 180 degrees, even if the triangle is drawn on a curved surface, such as the Earth's surface. The second is the "Triangle Sine Theorem", which is used in air navigation and states that the sum of the sines of the angles of a triangle is always equal to 2.

Development (20 - 25 minutes)

1. Activity 1 - "Building Triangles": (10 - 15 minutes)

   • The teacher will divide the class into groups of 3 to 4 students.
   • Each group will receive a set of sides (matchsticks, straws, etc.) of different lengths and the task of assembling triangles.
   • The challenge is to assemble triangles so that the sum of the interior angles is always 180 degrees.
   • The teacher will circulate around the room, observing and guiding the groups as needed.
   • At the end of the activity, each group will present its triangle and explain how it came to the conclusion that the sum of the interior angles is 180 degrees.

2. Activity 2 - "Exploring with Dynamic Geometry": (10 - 15 minutes)

   • The teacher will use dynamic geometry software (e.g., GeoGebra) designed on the blackboard or on a computer connected to a projector.
   • The teacher will ask students to draw any triangle in the software and measure the interior angles.
   • Students will then drag the vertices of the triangle to see how the interior angles change, but always adding up to 180 degrees.
   • Students will also be able to explore what happens when the triangle is drawn so that it is not planar, that is, when the vertices are not all on the same plane.
   • This activity will allow students to visualize and manipulate triangles interactively, which will facilitate understanding of the concept of the sum of the interior angles.

3. Activity 3 - "Application Problems": (5 - 10 minutes)

   • To consolidate understanding of the concept, the teacher will provide the groups with a series of application problems that involve the sum of the interior angles of a triangle.
   • Problems can include real-world situations (e.g., determining the location of the sun from the shadows cast by three poles) and fictitious situations (e.g., solving a puzzle where the sum of the interior angles of a triangle is a clue).
   • Students will have to discuss in groups and present the solutions to the class, explaining how they applied the concept of the sum of the interior angles to arrive at the answer.

These activities promote interaction between students, teamwork, active exploration of the content, and practical application of the concept studied. In addition, by using digital technologies and playful approaches, the teacher makes the class more engaging and effective.

Feedback (8 - 10 minutes)

1. Group Discussion: (3 - 4 minutes)

   • The teacher will ask each group to share their conclusions and solutions found during the hands-on activities.
   • Each group will have up to 3 minutes to present their findings to the class.
   • During the presentations, the teacher will reinforce key points, clarify doubts, and correct possible misconceptions.
   • The teacher will also encourage other students to ask questions and offer constructive feedback.

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

   • After the group presentations, the teacher will summarize the main ideas discussed, connecting them to the theory presented in the Introduction of the class.
   • The teacher will highlight how the hands-on activities helped to illustrate and consolidate the concept of the sum of the interior angles of a triangle.
   • The teacher may also propose a new challenge, such as: "How could you prove that the sum of the interior angles of an n-sided polygon is equal to (n-2) x 180 degrees?" This question, in addition to deepening the understanding of angle sums, will introduce the concept of polygons and their properties.

3. Individual Reflection: (2 - 3 minutes)

   • The teacher will conclude the class by proposing that students reflect individually on what they have learned.
   • The teacher will ask questions such as "What was the most important concept learned today?" and "What questions have not yet been answered?".
   • Students will have one minute to think about their answers. Then they can share their reflections with the class if they wish.
   • This reflection stage is crucial for students to consolidate what they have learned, identify possible gaps in their understanding, and prepare for upcoming lessons.

This Feedback moment is essential for assessing the effectiveness of teaching and learning, for clarifying any remaining doubts, and for stimulating students' reflection and metacognition. In addition, by valuing the ideas and contributions of each student, the teacher reinforces the importance of respect, active listening, and collaboration in the classroom.

Conclusion (5 - 7 minutes)

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

   • The teacher begins the Conclusion by recalling the key points of the lesson, highlighting the concept of the sum of the interior angles of a triangle and how it applies in different situations.

   • The teacher may, for example, emphasize that, regardless of the shape or size of the triangle, the sum of its interior angles will always be 180 degrees.

   • In addition, the teacher may reinforce the importance of logical reasoning and problem-solving skills in mathematics, skills that were developed during the hands-on activities of the lesson.

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

   • The teacher will then explain how the lesson connected theory (the concept of the sum of interior angles), practice (the activities of building triangles and exploring with dynamic geometry), and applications (the application problems).

   • The teacher may, for example, mention that, by constructing and manipulating triangles, students were able to visualize and experience in practice what the sum of the interior angles means.

   • In addition, the teacher may point out that the application problems allowed students to understand how the sum of the interior angles can be useful in solving real-world problems.

3. Extra Materials and Independent Study: (1 - 2 minutes)

   • The teacher will suggest additional materials for students who wish to deepen their knowledge of the sum of the interior angles of a triangle.

   • These materials may include explanatory videos, interactive math websites, educational games, books, and scientific articles.

   • The teacher may, for example, suggest that students watch a YouTube video explaining the formal proof of the sum of the interior angles theorem of a triangle, or that they play a digital game that challenges them to build triangles with certain characteristics (such as the sum of the interior angles being a specific value).

   • In addition, the teacher may propose a self-study activity for home, such as solving a set of math problems involving the sum of the interior angles of triangles.

4. Importance of the Topic in Everyday Life: (1 minute)

   • Finally, the teacher will reinforce the importance of the topic covered for everyday life.

   • The teacher may, for example, mention that understanding the sum of the interior angles of a triangle is essential for various areas of life, such as architecture (for the construction of safe and efficient structures), navigation (for orientation on maps and compasses), and art (for the composition of images and scenaries).

   • In addition, the teacher may emphasize that the development of logical reasoning and problem-solving skills are valuable skills not only in mathematics, but in all areas of knowledge and life.