Objectives (5 - 10 minutes)

1. Familiarize students with the concept of symmetry: The teacher will present the concept of symmetry in a simple and concrete way, relating it to everyday objects and images. The goal is for students to understand what symmetry is and be able to identify it in different contexts.

2. Introduce the use of the Cartesian plane: The teacher will explain what the Cartesian plane is, using visual and practical examples. The goal is for students to understand how the Cartesian plane is used to represent the position of points in space.

3. Spark students' interest in the subject: During the topic introduction, the teacher will present problem situations and curiosities involving symmetry and the use of the Cartesian plane. The goal is to spark students' curiosity and interest in the subject, encouraging active participation and engagement with the lesson.

Introduction (10 - 15 minutes)

1. Review of concepts: The teacher will review with students the concepts of plane geometry already studied, such as basic geometric shapes (circle, square, rectangle, triangle), straight lines, curved lines, and points. This is important so that students can understand symmetry and the Cartesian plane more effectively.

2. Problem situations: The teacher will propose two problem situations involving symmetry and the use of the Cartesian plane. The first situation could be: 'If you draw a vertical straight line on a paper, and then draw a point to the left of this line and another point to the right, are these points symmetric with respect to the line? How can we represent this symmetry on the Cartesian plane?' The second problem situation could be: 'If you draw a point on a paper, and then draw another point at a certain distance and in a certain direction from this first point, how can we represent the position of this second point on the Cartesian plane?'

3. Contextualization: The teacher will contextualize the importance of symmetry and the Cartesian plane in everyday life. For example, explaining how the Cartesian plane is used in maps to represent the location of places, or how symmetry is used in interior design to create balanced and pleasant environments. The teacher can also show images of famous buildings that exhibit symmetry, such as Notre-Dame Cathedral, and explain how symmetry is used in architecture.

4. Introduction to the topic: The teacher will introduce the topic of the lesson, explaining that they will learn about symmetry and how to represent it on the Cartesian plane. To gain students' interest, the teacher can share some curiosities, such as the fact that symmetry is a fundamental characteristic in nature, or that symmetry is used in cryptography to ensure message security. The teacher can also show images of famous art pieces that exhibit symmetry, such as church stained glass windows, and explain how symmetry is used in art to create visual effects and convey meaning.

Development (20 - 25 minutes)

1. Symmetry Theory (10 - 12 minutes)

   • Definition of Symmetry: The teacher will explain that symmetry is a property that some objects have when they can be divided into parts that reflect or repeat themselves. Through practical examples, like a mirror, a butterfly, or a heart cut in half, the teacher will show that half of an object is equal to its other half when reflected.

   • Types of Symmetry: Next, the teacher will explain that there are two types of symmetry: axial symmetry and radial symmetry. Axial symmetry is the symmetry that occurs in relation to a line, like in a mirror. Radial symmetry occurs when the elements of a figure are arranged around a center. The teacher will illustrate both types of symmetry with visual examples.

   • Symmetry in the Cartesian Plane: The teacher will explain that when studying symmetry in the Cartesian plane, we are interested in axial symmetry. The teacher will show how the line y = 0 divides the Cartesian plane into two symmetrical parts, each containing points that are symmetric with respect to this line.

2. Introduction to the Cartesian Plane (10 - 12 minutes)

   • Definition of the Cartesian Plane: The teacher will explain that the Cartesian plane is a mathematical tool that allows us to numerically represent the location of points in a two-dimensional plane. It is formed by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), intersecting at a point called the origin.

   • Using the Cartesian Plane: The teacher will show how to use the Cartesian plane to represent points. They will choose a point (x, y) and represent it on the Cartesian plane, counting x steps to the right or left from the origin, and y steps up or down from the origin.

   • Symmetry in the Cartesian Plane: The teacher will explain how symmetry in the Cartesian plane can be represented. They will draw a point on the Cartesian plane and then draw a symmetric point with respect to the y-axis = 0. They will repeat the process, but this time drawing a symmetric point with respect to the x-axis = 0. This way, students will be able to visualize symmetry in the Cartesian plane.

3. Practical Activity (5 - 10 minutes)

   • The teacher will propose a practical activity in which students will have to draw points on the Cartesian plane according to the given coordinates. Some of the provided coordinates will be symmetric with respect to the x-axis = 0 or the y-axis = 0, and students will have to identify the symmetry and draw the corresponding point.

   • To make the activity more interesting, the teacher can turn it into a game. Students can form teams and each team will have to draw the points as quickly as possible on the Cartesian plane. The team that finishes first and correctly will be the winner. This will encourage collaboration and healthy competition among students.

At the end of this stage, students should be able to understand the concept of symmetry and how it can be represented on the Cartesian plane. They should also be able to represent points on the Cartesian plane and identify the symmetry of the points with respect to the axes.

Feedback (10 - 15 minutes)

1. Group Discussion (5 - 6 minutes)

   • After the completion of the practical activity, the teacher should gather all students in a large group to discuss the solutions found. The teacher should ask each team to share the points they drew and explain why these points are symmetric with respect to the axes. This allows students to learn from each other and reinforce what they learned during the lesson.
   • The teacher should encourage students to ask questions and give feedback to each other. The teacher should facilitate the discussion, clarifying doubts and reinforcing concepts as needed. The goal is for students to feel comfortable sharing their ideas and learning from each other.

2. Connection to Theory (3 - 4 minutes)

   • After the group discussion, the teacher should recap the main theoretical points covered during the lesson. The teacher should ask students how they used the theory of symmetry and the Cartesian plane to solve the practical activity. This helps reinforce the connection between theory and practice, and consolidate students' learning.
   • The teacher should provide additional examples, if necessary, to illustrate how the theory can be applied in different contexts. For example, the teacher can show how symmetry and the Cartesian plane are used in graphic design and in creating video games.

3. Final Reflection (2 - 3 minutes)

   • The teacher should propose that students reflect on what they learned in the lesson. For this, the teacher can ask two simple questions: 'What was the most interesting part of today's lesson?' and 'What did you learn today that you can use in other situations?' These questions encourage students to think critically about what they learned and consider the relevance of the lesson content.
   • The teacher should give a minute for students to reflect silently on these questions. Then, the teacher can invite some students to share their answers with the rest of the class. The teacher should praise students' answers and reinforce the importance of what they learned.

At the end of this stage, students should have a clear understanding of the concepts of symmetry and the Cartesian plane, and should be able to apply these concepts to solve practical problems. Additionally, students should feel confident in their mathematical skills and motivated to continue learning.

Conclusion (5 - 10 minutes)

1. Summary and Recap (2 - 3 minutes)

   • The teacher should start the conclusion by giving a brief summary of the main points covered in the lesson. They should remind students about the definition of symmetry, the two types of symmetry (axial and radial symmetry), and how symmetry can be represented on the Cartesian plane. The teacher should also recap what the Cartesian plane is and how points are represented on it.
   • The teacher should emphasize that symmetry and the Cartesian plane are mathematical tools that help us understand and represent the world around us. They can give examples of how symmetry and the Cartesian plane are used in different areas, such as art, architecture, design, science, and technology.

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

   • The teacher should explain that during the lesson, students had the opportunity to learn about symmetry and the Cartesian plane in a theoretical and practical way. They should reinforce that the theory of symmetry and the Cartesian plane was presented clearly and simply, and that students had the opportunity to apply this theory in the practical activity.
   • The teacher should highlight that the practical activity allowed students to see symmetry and the Cartesian plane in action, and that this practical experience helped solidify students' theoretical understanding of these concepts.

3. Extra Materials and Reinforcement Exercises (1 - 2 minutes)

   • The teacher should suggest some extra materials for students who want to deepen their knowledge of symmetry and the Cartesian plane. These materials may include books, educational websites, online games, and educational videos. The teacher can even suggest that students look for examples of symmetry in the world around them, whether at home, school, or in the community.
   • The teacher should also provide some reinforcement exercises for students to practice at home. These exercises should vary in difficulty, allowing students of different skill levels to challenge themselves. The exercises should be designed to encourage students to apply what they have learned about symmetry and the Cartesian plane in a creative and critical way.

4. Importance of the Subject (1 - 2 minutes)

   • Finally, the teacher should highlight the importance of symmetry and the Cartesian plane in everyday life. They can explain that symmetry is a fundamental property of the natural world and is used in many areas, from art and architecture to science and technology. The teacher can give specific examples of how symmetry is used in daily life, such as in maps, logos, clothing patterns, interior design, and even in the composition of photographs and paintings.
   • The teacher should also explain that the Cartesian plane is a tool used in many areas of mathematics, science, and technology. They can give examples of how the Cartesian plane is used to represent data in graphs, to solve geometry problems, for navigation, for modeling in computer games, and for many other practical applications.

At the end of this stage, students should have a clear understanding of the importance of symmetry and the Cartesian plane, and should be motivated to continue learning and exploring these concepts in a creative and critical way.