Objectives (5 - 10 minutes)

1. Understand the concept of distance between two points on the Cartesian plane: Students should be able to comprehend what the distance between two points on a Cartesian plane is, recognizing the importance of this concept for solving mathematical problems.

2. Apply the formula for the distance between two points: Once the concept has been understood, students should be able to apply the formula for the distance between two points to calculate the distance between any two points on a Cartesian plane.

3. Solve problems using the distance between points: In addition to calculating the distance between two points, students should be able to use this skill to solve practical problems, developing their critical thinking and problems solving skills.

Secondary Objectives:

• Review previous concepts: During the lesson, students will have the opportunity to review previous mathematical concepts, such as the use of coordinates on the Cartesian plane and the application of mathematical formulas.

• Promote logical-mathematical thinking: By solving problems involving the distance between points, students will be encouraged to think logically and analytically, enhancing their mathematical thinking skills.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the concepts of Cartesian plane, coordinates, and mathematical formulas that were learned in previous classes. This can be done through a quick interactive review, where students are encouraged to participate, recalling and explaining the concepts. (3 - 5 minutes)

2. Problem situation: The teacher should propose two problem situations that involve calculating the distance between two points on the Cartesian plane. For example, "Imagine you have to draw a straight line from point A to point B on a piece of paper. How can you measure the distance between these two points?" or "Suppose you are in a maze and need to find the shortest distance between the entrance and the exit. How can you use the Cartesian plane and the formula for the distance between two points to solve this problem?" These problem situations should arouse students' interest and encourage them to think about the importance of the subject that will be addressed in the lesson. (2 - 3 minutes)

3. Contextualization: The teacher should explain the importance of calculating the distance between two points on the Cartesian plane, showing how this is applied in everyday situations and in other disciplines. For example, in physics, the distance between two points is used to calculate the speed of an object over time; in geography, the distance between two points is used to calculate the shortest route between two places; in computer programming, the distance between two points is used to solve algorithm problems. (3 - 5 minutes)

4. Introduction to the topic: To arouse students' interest, the teacher can introduce the topic of distance between points with some curiosities or stories. For example, one can talk about how the Greek mathematician Euclid was the first to develop a formula for calculating the distance between two points on a plane over 2000 years ago; or one can talk about how the distance between points is used in navigation, both on land and at sea, to plot routes. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Theory: Distance between two points on the Cartesian plane (8 - 10 minutes)

   1.1. Definition: The teacher should start by explaining what the distance between two points on a Cartesian plane is. It should be emphasized that the distance between two points is always a positive measure, regardless of the points' location on the plane.

   1.2. Distance formula: The teacher should present the formula for calculating the distance between two points, which is the square root of the sum of the squares of the differences between the x and y coordinates of the two points. It should be explained that this formula is a direct application of the Pythagorean theorem.

   1.3. Visual example: The teacher should draw a Cartesian plane on the board and use arrows to represent the distance between two points, demonstrating how the distance formula is applied in practice.

2. Example exercises (10 - 12 minutes)

   2.1. Exercise 1: The teacher should propose a simple exercise where students must calculate the distance between two points on the Cartesian plane using the presented formula. The points should be chosen so that the distance is not an integer, so that students can see the practical application of the formula. The teacher should solve the exercise on the board, step by step, explaining each stage.

   2.2. Exercise 2: The teacher should propose a second, slightly more challenging exercise where students must calculate the distance between two points, but this time the points are given in the form of an algebraic expression. The goal of this exercise is to show students that the distance formula can be applied to points that are not integers.

   2.3. Discussion: After solving the exercises, the teacher should ask students to share their solutions and explain how they arrived at them. This will allow students to learn from each other and help reinforce the concepts learned.

3. Practical application (2 - 3 minutes)

   3.1. Practical problem: The teacher should propose a practical problem that involves calculating the distance between two points on the Cartesian plane. For example, the teacher may ask students to calculate the distance between two points on a map, or to calculate the distance between two points in a maze. The goal of this problem is to show students how the mathematics they are learning can be applied to real-world situations.

   3.2. Problem solving: The teacher should solve the problem on the board, step by step, explaining each stage. It should be emphasized that problem solving involves applying the theoretical concepts learned in a creative and flexible way.

4. Theory review (3 - 5 minutes)

   4.1. Review questions: The teacher should ask review questions to verify if students understood the concepts presented. These questions should include, for example: "What is the distance between two points on the Cartesian plane?", "What is the formula for calculating the distance between two points?", "How do you calculate the distance between two points when the points are given in the form of an algebraic expression?".

   4.2. Discussion: The teacher should promote a discussion about the concepts learned, encouraging students to ask questions and share their doubts and difficulties. This will allow the teacher to identify any areas that may need reinforcement or clarification.

Return (10 - 15 minutes)

1. Review and Reflection (5 - 7 minutes)

   1.1. Group Discussion: The teacher should start a group discussion, asking students to share their reflections on the lesson. This may include questions such as: "What was the most important concept you learned today?", "What questions have not been answered yet?", "How can you apply what you learned today in everyday situations?".

   1.2. Connection to the Real World: The teacher should encourage students to make connections between what they learned in the lesson and real-world situations. For example, students may be encouraged to think about how calculating the distance between points is used in GPS to plot routes, or how it is used in architecture to measure the distance between two points in a project.

   1.3. Self-assessment: The teacher may ask students to assess their own understanding of the topic, using a scale of 1 to 5, where 1 means "I didn't understand anything" and 5 means "I understood everything and can teach others". This will help students reflect on their own learning and identify any areas that may need reinforcement.

2. Questions and Answers (3 - 5 minutes)

   2.1. Clarifying Doubts: The teacher should open a question and answer session, where students can ask questions about the lesson topic. The goal is to clarify any doubts students may have and ensure that everyone has a clear understanding of the topic.

   2.2. Student Feedback: The teacher should ask for feedback from students about the lesson, asking what they liked and what they found most challenging. This will help the teacher adjust future lessons to meet the needs and interests of the students.

3. Individual Reflection (2 - 3 minutes)

   3.1. Reflection Moment: The teacher should suggest that students take a one-minute break to reflect silently on what they learned in the lesson. During this time, students should think about the following questions: "What was the most important concept I learned today?" and "What questions have not been answered yet?".

   3.2. Sharing Reflections: After the minute of reflection, the teacher should ask students to briefly share their answers. This will allow the teacher to assess the effectiveness of the lesson and identify any areas that may need reinforcement in future lessons.

   3.3. Lesson Closure: The teacher should end the lesson by thanking the students for their participation and encouraging them to continue practicing what they learned. The teacher should also remind students about the importance of continuous study and practice for effective learning.

Conclusion (5 - 10 minutes)

1. Lesson Summary (2 - 3 minutes)
   • The teacher should recap the main points of the lesson, recalling the definition of distance between two points on the Cartesian plane and the formula to calculate it.
   • It should be emphasized that the distance between two points is always a positive measure, regardless of the points' location on the plane, and that the distance formula is a direct application of the Pythagorean theorem.
   • The teacher should highlight the example exercises and practical application that were carried out during the lesson, reinforcing the importance of calculating the distance between points for solving real-world problems.
2. Connection between Theory and Practice (1 - 2 minutes)
   • The teacher should explain how the lesson connected theory, practice, and application. For example, students were introduced to the theory of distance between points and the formula to calculate it, had the opportunity to practice this skill through example exercises, and then applied what they learned to solve a practical problem.
   • The teacher should emphasize that the connection between theory and practice is essential for effective learning, as it allows students to understand the relevance and applicability of the concepts they are learning.
3. Additional Materials (1 - 2 minutes)
   • The teacher should suggest additional study materials for students who wish to deepen their knowledge of calculating the distance between points on the Cartesian plane. These materials may include textbooks, math websites, educational videos, and math learning apps.
   • The teacher should encourage students to use these resources to review the lesson content, practice the skill of calculating the distance between points, and explore additional applications of this concept.
4. Importance of the Topic (1 - 2 minutes)
   • Finally, the teacher should summarize the importance of calculating the distance between points on the Cartesian plane. It should be emphasized that this skill is fundamental for solving mathematical problems and is widely applied in various areas, including physics, geography, engineering, architecture, and computer programming.
   • The teacher should encourage students to recognize the relevance and usefulness of mathematics in their daily lives, not only as a school subject, but as a powerful tool for understanding and solving real-world problems.