Lesson plan of Spatial Geometry: Dihedrals and Trihedrals

Objectives (5-7 minutes)

• Main Objective: To introduce and deepen the concept of dihedral and triedral angles in spatial geometry. The teacher should make sure that the students understand what dihedral and trihedral angles are, how they are formed and how they can be identified in different spatial figures.

• Secondary Objectives:

  1. Develop the students' spatial observation skills. They should be able to identify dihedral and trihedral angles in different three-dimensional figures and everyday objects.

  2. Promote students' logical-mathematical reasoning skills. They should be able to understand the relationship between the angles formed by the planes that make up a dihedral or triedral angle.

  3. Encourage students' active participation in the learning process. They should be stimulated to ask questions, propose solutions, and share their ideas and understandings during the lesson.

  4. Stimulate critical thinking and problem solving. Students should be encouraged to apply the concept of dihedral and trihedral angles in problem situations in order to consolidate their understanding and improve their mathematical skills.

Introduction (10-15 minutes)

• Review of previous content: The teacher should start by reviewing the basic concepts of spatial geometry, such as points, lines and planes, and how they relate to three-dimensional space. This review is crucial for students to understand and correctly apply the concept of dihedral and trihedral angles. (3-5 minutes)

• Problem situation 1: The teacher can present the students with a cube and ask: "How many planes form each corner of this cube?" This question aims to arouse the students' curiosity and prepare them for the introduction of the concept of dihedral and trihedral angles. (2-3 minutes)

• Contextualization: The teacher should explain the importance of spatial geometry in our daily lives, highlighting how it is applied in areas such as architecture, engineering, design and even in digital and board games. Practical examples can be cited, such as the construction of bridges and buildings, the modeling of characters in 3D animations and the solving of spatial puzzles. (2-3 minutes)

• Problem situation 2: The teacher can then present a more complex problem situation: "If we have a triangular-based prism, how many triedral angles will it have and how many right angles can be formed?" This question aims to challenge students to apply what they have learned about dihedral and trihedral angles in a practical context. (3-4 minutes)

• Gain attention: Finally, the teacher can introduce the topic of the lesson in a playful and interesting way. For example, you could show a short video of a magician performing tricks with three-dimensional figures, and then ask the students: "How do you think he managed to do those tricks? What can we learn about dihedral and trihedral angles from these illusions?" This strategy aims to capture the students' attention and arouse their interest in the subject. (2-3 minutes)

Development (20 - 25 minutes)

• Activity 1: Building dihedral and trihedral angles (10 - 12 minutes)

  1. The teacher should provide students with popsicle sticks and modeling clay. Popsicle sticks will represent the edges of polyhedra, and modeling clay will be used to set the edges and form dihedral and trihedral angles.
  2. The teacher should ask the students, in groups of 3 to 4 people, to build different three-dimensional figures, such as cubes, prisms and pyramids, using the materials provided.
  3. After building the figures, students should identify and mark the planes that form the dihedral and trihedral angles in each figure.
  4. Students should record their observations and conclusions in their notebooks, drawing the constructed figures and indicating the dihedral and trihedral angles.

• Activity 2: Solving problems with dihedral and trihedral angles (10 - 12 minutes)

  1. The teacher should present the students with a series of problems involving dihedral and trihedral angles. For example: "If a cube has 12 edges, how many dihedral angles does it have?" or "If a prism has 6 faces and 9 edges, how many triedral angles does it have and how many right angles can be formed?".
  2. In their groups, students should discuss and try to solve the problems. They should apply what they have learned about forming and identifying dihedral and trihedral angles to find the correct answers.
  3. After solving the problems, the teacher should promote a discussion in class, asking different groups to present their solutions and explain their reasoning. This activity aims not only to apply the acquired knowledge, but also to develop the students' communication and argumentation skills.

• Activity 3: Exploring the application of dihedral and trihedral angles (5 - 7 minutes)

  1. The teacher should propose to the students that, in their groups, they search for objects in the school environment or in their backpacks that can be represented by three-dimensional figures. For example, a can of soda can be represented by a cylinder, a book by a parallelepiped, etc.
  2. Students should identify and mark the dihedral and trihedral angles of the selected objects, and discuss how knowledge of dihedral and trihedral angles can be useful for understanding and describing everyday objects.
  3. Students should record their observations and conclusions in their notebooks, along with drawings of the objects and indications of the dihedral and trihedral angles.

Return (8 - 10 minutes)

• Group discussion (3 - 4 minutes):

  1. The teacher should gather all the students and promote a group discussion on the solutions found by each team in the problem-solving activity. At this moment, the teacher can ask questions to stimulate the students' critical thinking and deepen their understanding of the subject. For example: "Why did you choose this strategy to solve the problem?" or "How do you know that this is the correct answer?".
  2. The teacher should emphasize the importance of justifying the answers and of using logical and mathematical arguments to solve the problems. This discussion aims not only to clarify doubts and correct possible errors, but also to strengthen collaborative learning and improve the students' communication and argumentation skills.

• Connection with the theory (2 - 3 minutes):

  1. The teacher should then make the connection between the practical activities performed by the students and the theory presented in the introduction of the lesson. The teacher can, for example, ask: "How does what we did with the sticks and modeling clay relate to the concept of dihedral and trihedral angles that we discussed earlier?" or "How do the problems we solved and the objects we analyzed help us to better understand what dihedral and trihedral angles are and how they are formed?".
  2. The teacher should reinforce that theory and practice are inseparable parts of the learning process, and that a deep understanding of a mathematical concept requires not only an understanding of its definitions and properties, but also the ability to apply it to concrete situations and to solve related problems.

• Final reflection (3 - 4 minutes):

  1. To conclude the lesson, the teacher should propose that the students reflect for a minute on the following questions: "What was the most important concept we learned today?" and "What questions have not yet been answered?". This reflection aims to consolidate learning and identify possible gaps in the students' understanding, which can be addressed in future lessons.
  2. After the minute of reflection, the teacher should invite some students to share their answers with the class. The teacher should listen attentively to the students' answers, value everyone's contributions and clarify any misunderstandings or lingering doubts.

This Return is a fundamental step in the lesson plan, as it allows the teacher to evaluate the effectiveness of the teaching, correct possible errors or misunderstandings, and adjust the planning of future lessons according to the needs and interests of the students. In addition, by promoting students' reflection and self-assessment, the teacher is encouraging the development of metacognitive skills, which are essential for autonomous and lasting learning.

Conclusion (5 - 7 minutes)

• Summary of the content (2 - 3 minutes): The teacher should summarize the main points discussed during the lesson, reinforcing the concepts of dihedral and trihedral angles, and how they are formed. The teacher should remind students that a dihedral angle is formed by two planes that meet at an edge, and a trihedral angle is formed by three planes that meet at an edge. The teacher should also emphasize the importance of being able to identify and describe dihedral and trihedral angles in different three-dimensional figures and objects.

• Theory-practice connection (1 - 2 minutes): The teacher should explain how the class connected theory, practice and the real world. The teacher should point out that, although the class has focused mainly on the theory and practice of dihedral and trihedral angles, these concepts have practical applications in many areas of everyday life, such as architecture, engineering, design and even in digital and board games. The teacher may recall the practical activities carried out by the students, such as the construction of three-dimensional figures and the analysis of everyday objects, and how these activities helped to illustrate the usefulness of dihedral and triedral angles.

• Extra materials (1 - 2 minutes): The teacher should suggest some extra materials for students who wish to deepen their understanding of the subject. These materials may include math books, educational websites, YouTube videos and geometry apps. For example, the teacher could suggest using an augmented reality app that allows students to visualize and interact with three-dimensional figures in real space, which can help consolidate understanding of dihedral and trihedral angles.

• Importance of the subject (1 minute): Finally, the teacher should emphasize the importance of the subject for the students' lives. In addition to the practical applications already mentioned, the teacher can point out that the study of three-dimensional figures, such as dihedral and trihedral angles, helps to develop important cognitive skills, such as spatial perception, logical thinking and problem solving. The teacher can also highlight that mathematics, in general, is a fundamental discipline for many careers and areas of study, and that mastering concepts such as dihedral and trihedral angles can open many doors in the future.