Objectives (5 - 10 minutes)

1. Understand the concept of translation of flat figures and the importance of this transformation for geometry.
2. Develop the ability to graphically represent the translation of flat figures.
3. Solve practical problems involving the translation of flat figures, applying the concept of translation vector.

Secondary Objectives:

1. Stimulate logical thinking and mathematical reasoning through problem-solving.
2. Foster the ability to apply acquired knowledge in practical and everyday situations.
3. Promote interaction and collaboration among students through group activities.

The teacher should start the lesson by clearly presenting these Objectives, explaining the importance of each one and how they will be achieved throughout the lesson.

Introduction (10 - 15 minutes)

1. Review of Previous Content:

   • The teacher should start the lesson by briefly reviewing the concepts of flat figures, especially squares and rectangles, and vectors, which were studied in previous classes. It is important to ensure that all students have an understanding of these concepts before moving on to the new topic. (5 minutes)

2. Problem Situations:

   • Problem 1: The teacher can present the following situation: 'Imagine you have a drawing of a square on a piece of paper and, without changing the shape of the drawing, you need to move it to a different position on the paper. How would you do that?' (3 minutes)
   • Problem 2: The teacher can propose the following situation: 'You have a rectangle drawn on a piece of paper and need to translate it so that it is exactly next to the original rectangle, without rotating it. How can you do that?' (3 minutes)

3. Contextualization:

   • The teacher should explain that the translation of flat figures is a very important operation in geometry and in many other areas, such as architecture, art, engineering, and computer graphics. For example, when designing a building, architects need to know how to translate flat figures so that they fit correctly in the structure. (2 minutes)

4. Introduction to the Topic:

   • The teacher should introduce the topic of the lesson, explaining that the translation of flat figures is an operation that allows moving a figure from one place to another without changing its size, shape, or orientation. To do this, the teacher can use the following definition: 'The translation of a figure is the operation that, while maintaining the orientation, shape, and size of the figure, moves it from one position to another on the plane.' (2 minutes)

The teacher should ensure that students are engaged during the Introduction, encouraging them to participate in discussions and think of solutions to the problem situations presented.

Development (20 - 25 minutes)

1. Theory Explanation:

   • The teacher should start the theory explanation by reinforcing the definition of translation of flat figures presented in the Introduction. Next, introduce the concept of translation vector, which is a vector that indicates the direction and distance that the figure will be translated.
   • To illustrate the concept, the teacher can use a drawing of a figure on the board and draw a translation vector next to it, explaining that the figure will be moved in the direction and distance indicated by the vector.
   • The teacher should emphasize that, in translation, the figure is moved without rotating, distorting, or changing size. To check students' understanding, the teacher can propose some examples of figures that do not undergo changes in these properties when translated, such as squares, rectangles, and circles.
   • The teacher should also clarify that, in translation, the initial point and the final point of the figure are called corresponding points. This is important for the next step, which is the graphical representation of the translation.

2. Graphical Representation of Translation:

   • The teacher should explain that the graphical representation of translation consists of drawing the original figure and the translated figure on a Cartesian plane, using the translation vector to determine the new position of the figure.
   • To demonstrate how to do this, the teacher should choose a simple and guided example, such as the translation of a square. The teacher should draw the original square at a point on the Cartesian plane and then draw the translated square from the translation vector.
   • The teacher should explain that to draw the translated figure, one simply needs to move each vertex of the original square in the direction and distance indicated by the translation vector.
   • The teacher should repeat the process with other examples of figures to ensure that students have understood the method.

3. Practical Exercises:

   • After the theory explanation and graphical representation, the teacher should propose some practical exercises for students to apply what they have learned.
   • The exercises should start with simple examples and progress to more complex ones, so that students can gradually develop their skills.
   • The exercises should include the graphical representation of translation, determining the translation vector from translated figures, and solving problems involving the translation of figures.
   • The teacher should circulate around the classroom, offering help and clarifying doubts as students work on the exercises.

4. Discussion and Clarification of Doubts:

   • After the exercises, the teacher should promote a class discussion about the difficulties encountered and the strategies used to solve the problems.
   • The teacher should clarify any remaining doubts and provide feedback to students on their efforts.
   • The teacher should encourage students to ask questions and express their opinions, so they can learn from each other and develop a deeper understanding of the topic.

The teacher should ensure that students are engaged and active during the Development, offering opportunities for participation, discussion, and clarification of doubts.

Return (10 - 15 minutes)

1. Review of Key Concepts:

   • The teacher should start the Return stage by reviewing the key concepts covered during the lesson. This includes the concept of translation of flat figures, the translation vector, and the graphical representation of translation.
   • The teacher should ask students to summarize these concepts in their own words to verify if they have been understood correctly. If necessary, the teacher should clarify any remaining doubts.
   • The teacher should also ask students to identify the importance of these concepts for geometry and other areas of knowledge, such as architecture, art, engineering, and computer graphics.

2. Connection between Theory, Practice, and Applications:

   • The teacher should explain how the lesson connected theory, practice, and applications of the presented concepts. This can be done by highlighting how practical exercises allowed students to apply theory and how real-world applications were discussed during the lesson.
   • The teacher should ask students how they see the connection between theory, practice, and applications, encouraging them to reflect on what they have learned.

3. Reflection on Learning:

   • The teacher should propose that students reflect for a minute on the following questions: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • After the minute of reflection, the teacher should ask some students to share their answers with the class. The teacher should listen carefully to students' answers and, if necessary, clarify any remaining doubts.
   • The teacher should encourage students to continue reflecting on what they have learned after the lesson and to seek answers to unanswered questions.

4. Feedback and Evaluation:

   • Finally, the teacher should provide feedback to students on their performance during the lesson. The teacher can highlight students' strengths, as well as areas that need more practice or study.
   • The teacher should also assess students' progress in relation to the learning objectives of the lesson and plan the next teaching steps based on this assessment.

The teacher should ensure that students have a clear understanding of key concepts and feel confident in their ability to apply these concepts. The teacher should also encourage students to continue exploring the topic after the lesson and to seek answers to unanswered questions.

Conclusion (5 - 10 minutes)

1. Summary of Key Contents:

   • The teacher should start the Conclusion by summarizing the main points covered during the lesson. This includes the concept of translation of flat figures, the translation vector, and the graphical representation of translation.
   • The teacher should reinforce the importance of these concepts, highlighting how they are fundamental for understanding and solving problems involving the translation of flat figures.
   • The teacher should also revisit the problem situations presented in the Introduction and explain how the concepts learned in the lesson can be applied to solve these situations.

2. Connection between Theory and Practice:

   • The teacher should explain how the lesson connected theory and practice, emphasizing how theory was applied in solving practical exercises.
   • The teacher should also reinforce that practice is essential for understanding and applying theoretical concepts, and that problem-solving is an effective way to develop these skills.

3. Supplementary Materials:

   • The teacher should suggest some complementary study materials for students who wish to deepen their knowledge on the topic. These materials may include textbooks, educational websites, videos, and online games that address the concept of translation of flat figures.
   • The teacher should explain that independent study is an important part of the learning process and that these materials can help students consolidate what they have learned in the lesson.

4. Relevance of the Topic:

   • Finally, the teacher should emphasize the relevance of the topic for daily life and for other areas of knowledge. The teacher can cite examples of real situations involving the translation of flat figures, such as drawing house plans, creating computer animations, and solving engineering problems.
   • The teacher should also emphasize that the ability to understand and apply the concept of translation of flat figures can be useful in various situations, contributing to the development of logical thinking, creativity, and problem-solving skills.

The teacher should conclude the lesson by reinforcing the importance of the topic, thanking students for their participation, and encouraging them to continue exploring the topic after the lesson.