Objectives (5 - 10 minutes)

1. Introduce and develop the concept of spatial geometry, focusing on deformations in projections, which is the main theme of the lesson.

2. Present and explain the sphere, cylinder, and cone as three-dimensional geometric shapes and study their properties regarding deformations in projections.

3. Develop students' ability to visualize and interpret the transformations that occur when a three-dimensional object is projected onto a plane, thus strengthening spatial reasoning.

   Secondary objectives:

   • Stimulate active participation of students, encouraging questioning and discussion about the presented concepts.

   • Develop the ability to solve practical problems involving deformations in projections, applying the studied theoretical concepts.

   • Promote the appreciation of mathematics as a powerful and fascinating tool to understand and describe the world around us, especially in the context of spatial geometry.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher should start the lesson by quickly reviewing the concepts of plane geometry, especially two-dimensional shapes like the circle, square, and triangle. This review is important so that students have a solid starting point for understanding spatial geometry. (3 - 5 minutes)

2. Problem Situation 1: Present the following situation: "Imagine you are in a room with a large sphere, a cylinder, and a cone. Now, imagine you have a very strong lamp in your hands and are trying to project the shadow of these objects on a wall. What would be the shadow of each object?" This problem situation will serve as a hook for the introduction to the theory of deformations in projections. (3 - 5 minutes)

3. Problem Situation 2: Present a second situation: "Now, imagine you are looking at a coffee cup, which is similar to a cone. If you look at it from above, you will see a circle. If you look at it from the side, you will see an ellipse. Why does this happen?" This situation will help illustrate how projections can alter the shape of a three-dimensional object. (3 - 5 minutes)

4. Contextualization: The teacher should then contextualize the importance of deformations in projections, explaining how they are applied in various areas, such as architecture and interior design, where they are used to project the final appearance of an object, like a building, from a three-dimensional model. (2 - 3 minutes)

5. Introduction of the Topic: To spark students' interest, the teacher can introduce the topic by sharing some curiosities about deformations in projections. For example, they can mention that the concept of projection is fundamental in creating 3D animated movies, where animators virtually project characters and scenarios onto a flat screen. Another interesting curiosity is that the projection of a three-dimensional object onto a plane can create an image that is smaller, equal, or larger than the real object, depending on the distance between the object and the projection plane. (2 - 4 minutes)

Development (20 - 25 minutes)

1. Theory of Spatial Geometry (10 - 12 minutes):

   1.1. Introduction to Spatial Geometry: The teacher should start by explaining what spatial geometry is, which is the part of mathematics that studies the shapes and properties of three-dimensional space. It should be emphasized that, unlike plane geometry, which deals with two-dimensional shapes, spatial geometry deals with three-dimensional shapes, such as the sphere, cylinder, and cone. (2 - 3 minutes)

   1.2. Three-Dimensional Geometric Shapes: The teacher should then present and describe the main characteristics of each of the three-dimensional geometric shapes that will be studied in the lesson - the sphere, cylinder, and cone. For each shape, the teacher should explain how it is defined, what its main parts are, and what its unique properties are. (4 - 5 minutes)

   1.3. Deformations in Projections: Next, the teacher should introduce the concept of deformations in projections. It should be explained that when a three-dimensional object is projected onto a plane, the resulting image may be different from the real object, due to deformations that occur during the projection. The teacher should illustrate this idea with simple examples, such as the projection of a cube onto a piece of paper. (4 - 5 minutes)

2. Applications and Examples (5 - 7 minutes):

   2.1. Example 1 - Projection of a Sphere: The teacher should now present a practical example of deformation in projection. They should show how the projection of a sphere onto a plane can result in an image that is a circle with a radius smaller than that of the sphere. The teacher should explain why this deformation occurs, emphasizing the idea that the projection of a point on a sphere to a plane is always a point, and that this projection preserves the distance between points, but does not preserve the area or volume. (2 - 3 minutes)

   2.2. Example 2 - Projection of a Cylinder: The teacher should then show how the projection of a cylinder onto a plane can result in an image that is a rectangle. They should explain that, in this case, the deformation occurs because the projection of a line parallel to the cylinder's axis is a straight line, while the projection of a line perpendicular to the axis is a line segment. (2 - 3 minutes)

   2.3. Example 3 - Projection of a Cone: Finally, the teacher should illustrate the projection of a cone onto a plane, showing that the projection of a circle from the cone's base is an ellipse, and that the projection of a line passing through the cone's vertex and base is a straight line. The teacher should emphasize that, in this case, the deformation is more complex, as the resulting shape is a combination of a straight line and a circle. (1 - 2 minutes)

3. Practical Activity (5 - 6 minutes):

   3.1. Deformations in Projections (3 - 4 minutes): The teacher should propose a practical activity in which students must draw the projection of a three-dimensional object onto a plane. Students can choose between the sphere, cylinder, and cone. The teacher should provide models of each of these objects and sheets of paper for the activity. Students should be guided to draw the three-dimensional object on one side of the sheet and the projection on the other side, and to observe the deformations that occur during the projection. The teacher should walk around the room, observing the students' work and providing help when needed. (2 - 3 minutes)

   3.2. Discussion and Reflection (2 - 3 minutes): After the conclusion of the activity, the teacher should promote a classroom discussion so that students can share their observations and reflections. The teacher should encourage students to explain why the deformations occurred the way they did, and to discuss how deformations in projections are used in real situations. (2 - 3 minutes)

Return (10 - 15 minutes)

1. Review and Reflection (5 - 7 minutes):

   1.1. Group Discussion: The teacher should facilitate a group discussion, where students will have the opportunity to share their solutions and conclusions from the practical activity. Each group should present the three-dimensional object they chose and the projection they drew, explaining the deformations they observed. The teacher should moderate the discussion, asking questions to stimulate critical thinking and deepen students' understanding of the topic. (3 - 4 minutes)

   1.2. Connection to Theory: After each group's presentation, the teacher should make a brief connection to the theory, highlighting how the deformations observed during the practical activity are related to the concepts of spatial geometry and deformations in projections that were discussed in the first part of the lesson. The goal is to help students see the relevance and practical application of the theoretical concepts. (2 - 3 minutes)

2. Learning Verification (3 - 5 minutes):

   2.1. Questions and Answers: The teacher should then ask a series of questions to verify students' understanding of the topic. The questions should cover the main points discussed in the lesson, including the definition of spatial geometry, the characteristics and properties of the sphere, cylinder, and cone, and the deformations that occur during a projection. The teacher should give each student the opportunity to answer at least one question, ensuring that everyone is involved and actively participating. (2 - 3 minutes)

   2.2. Individual Feedback: After each student's answer, the teacher should provide individual feedback, praising correct answers and correcting incorrect ones. The teacher should also encourage students to continue studying the topic, highlighting the importance of practice and deepening mathematical concepts. (1 - 2 minutes)

3. Final Reflection (2 - 3 minutes):

   3.1. Importance of the Topic: To conclude the lesson, the teacher should propose that students reflect for a minute on the importance of the topic studied. The teacher should ask questions like: "How can deformations in projections be applied in real situations?" and "Why is it important to understand spatial geometry and deformations in projections?" After a minute of reflection, the teacher should open the discussion, giving each student the opportunity to share their reflections. (1 - 2 minutes)

   3.2. Final Feedback: The teacher should then provide final feedback, reinforcing the most important points of the lesson and highlighting the progress and difficulties observed. The teacher should also encourage students to continue studying the topic and to seek help whenever necessary. (1 - 2 minutes)

Conclusion (5 - 10 minutes)

1. Summary of Contents (2 - 3 minutes):

   1.1. Spatial Geometry: The teacher should briefly review the concept of spatial geometry, emphasizing that this is the area of mathematics that studies the shapes and properties of three-dimensional space. It should be emphasized that, unlike plane geometry, which deals with two-dimensional shapes, spatial geometry deals with three-dimensional shapes, such as the sphere, cylinder, and cone.

   1.2. Deformations in Projections: Next, the teacher should recall the concept of deformations in projections, explaining that when a three-dimensional object is projected onto a plane, the resulting image may be different from the real object, due to deformations that occur during the projection.

   1.3. Three-Dimensional Geometric Shapes: The teacher should then recap the main characteristics of each of the three-dimensional geometric shapes studied in the lesson - the sphere, cylinder, and cone. The differences between them should be highlighted and how these differences affect the deformations in projections.

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

   2.1. Practical Activity: The teacher should emphasize how the practical activity carried out during the lesson allowed students to apply the theoretical concepts of deformations in projections in a concrete and visual way. It should be highlighted that this activity provided students with the opportunity to explore and better understand deformations in projections through observation and drawing.

   2.2. Group Discussion: The teacher should also mention how the group discussion allowed students to share their observations and reflections, thus strengthening the collective understanding of the topic.

3. Additional Materials (1 - 2 minutes):

   3.1. Reading Recommendations: The teacher should suggest some reading materials for students who wish to deepen their knowledge of spatial geometry and deformations in projections. These materials may include textbooks, scientific articles, math websites, and explanatory videos.

   3.2. Additional Exercises: The teacher should also suggest some additional exercises for students to practice at home. These exercises should involve solving practical problems that require the application of spatial geometry and deformations in projections concepts.

4. Practical Application (1 - 2 minutes):

   4.1. Relevance of the Topic: To conclude the lesson, the teacher should emphasize the importance of spatial geometry and deformations in projections in everyday life. For example, how these concepts are used in areas such as architecture, interior design, 3D animation, and computer science.

   4.2. Encouragement for Continuous Study: The teacher should encourage students to continue studying the topic, emphasizing that mathematics is a discipline that requires practice and dedication. It should be reinforced that continuous study is the key to success in mathematics and other areas of knowledge.