Objectives (5 - 7 minutes)

1. Understanding the Cartesian Plane: Students should be able to understand what a Cartesian plane is, how it is structured, and how it is used to represent points and distances in space. This involves identifying the x- and y-axes, interpreting coordinates, and visualizing points on the plane.

2. Calculating the Distance Between Two Points: Students should be able to calculate the distance between two points on a Cartesian plane. This requires applying the distance formula between two points, which involves the difference of coordinates and the use of the Pythagorean theorem.

3. Practical Application of the Concept: Students should be able to apply the concept of distance between points in practical situations. This may involve solving problems that require using the distance formula between two points.

Secondary Objectives:

• Development of Logical-Mathematical Thinking: Throughout the lesson, students should be encouraged to think logically and apply mathematical concepts to solve problems.

• Encouraging Participation and Collaboration: Students should be encouraged to actively participate in the lesson by sharing their ideas, doubts, and problem-solving approaches. The teacher should create a collaborative learning environment where students feel comfortable interacting and helping each other.

Introduction (8 - 10 minutes)

1. Review of Previous Content: The teacher initiates the lesson by briefly reviewing the concepts of coordinates, x- and y-axes, and the structure of the Cartesian plane. This is crucial to ensure that all students have the necessary foundation for understanding the new content. The teacher may ask students to recall these concepts, or even conduct a quick quiz to assess students' prior knowledge.

2. Problem Situation: The teacher presents two problem situations to spark students' interest and contextualize the importance of the topic.

• Situation 1: Imagine you are in a labyrinth and need to find the shortest distance between two points. How can Mathematics help us solve this problem?
• Situation 2: Suppose you are planning a trip and need to know the distance between two cities on a map. How can we use Mathematics to calculate this distance?

3. Contextualization: The teacher explains that the ability to calculate the distance between points is essential in various fields, such as navigation, engineering, architecture, computer science, and even computer games. Furthermore, Mathematics is a powerful tool for solving everyday problems, like finding the shortest route between two points in a city.

4. Introduction to the Topic: To pique students' curiosity and interest, the teacher may share some trivia or interesting applications of the topic:

• Fun Fact 1: The formula for calculating the distance between points on a Cartesian plane is a direct application of the Pythagorean theorem.
• Fun Fact 2: The idea of a Cartesian plane and the formula for calculating the distance between points were developed by the French mathematician René Descartes, who is considered one of the founders of analytic geometry.

The teacher should promote students' interaction with the Introduction by encouraging them to share their ideas, curiosities, or experiences related to the topic.

Development (20 - 25 minutes)

1. Activity 1 - Paths on the Plane (10 minutes):

• Description: The teacher poses a challenge for students to find the shortest path between two points on a Cartesian plane. To do this, the teacher creates a "labyrinth" on the Cartesian plane, with obstacles that students must avoid, and indicates two start and end points. Students must calculate the distance between these points and find the shortest path, avoiding the obstacles.
• Step-by-Step:

1. The teacher divides the class into groups of 4 to 5 students.

2. Each group receives a "labyrinth" on the Cartesian plane and a sheet of graph paper to perform the calculations.

3. Students should calculate the distance between the start and end points using the distance formula between two points.

4. Then, students should plot the shortest path on the "labyrinth", avoiding the obstacles.

5. The teacher circulates around the room, assisting groups and clarifying doubts.

6. At the end, each group presents their shortest path and the calculations performed. The teacher can then discuss the different strategies used by the groups and how the distance formula between points was applied.

7. Activity 2 - Discovering the Formula (10 minutes):

• Description: In this activity, students will discover the formula for the distance between two points on the Cartesian plane in an investigative manner.
• Step-by-Step:

1. The teacher provides students with a grid of points on the Cartesian plane, with two points highlighted.

2. The teacher guides students to measure the distance between these points using a ruler.

3. The teacher then asks students to try to relate the distance measurement to the coordinates of the points.

4. Students, with the teacher's guidance, should realize that the distance between the points can be found by applying the Pythagorean theorem to the differences of the x and y coordinates.

5. The teacher guides students to write the formula for the distance between two points: √((x2 - x1)² + (y2 - y1)²).

6. The teacher reinforces the concept by showing that the formula is a generalization of the Pythagorean theorem to the Cartesian plane.

7. Activity 3 - Real-World Problems (5 minutes):

• Description: The teacher proposes solving two real-world problems that involve calculating the distance between points on the Cartesian plane.
• Step-by-Step:

1. The teacher presents the problems, which could be:
2. Calculate the distance between two cities on a map, knowing that each unit on the x-axis represents 100 km and each unit on the y-axis represents 50 km.
3. Calculate the distance an airplane travels when flying from one point to another on a Cartesian plane, knowing that the airplane's speed is constant and the distance between the points is given in units of time.
4. Students, in their groups, should discuss and propose solutions to the problems.
5. The teacher circulates around the room, assisting groups and clarifying doubts.
6. At the end, each group presents their solutions and the teacher discusses the strategies used and the application of calculating the distance between points.

The teacher should promote interaction and collaboration among students during the activities, encouraging them to discuss their ideas, help each other, and think logically. Additionally, the teacher should take the opportunity to assess students' understanding and clarify any doubts that may arise.

Feedback (10 - 12 minutes)

1. Group Discussion (5 minutes):

• Description: After the Conclusion of the activities, the teacher invites each group to share their solutions or conclusions with the class. Each group has up to 3 minutes to present.
• Step-by-Step:

1. The teacher calls the first group to present and asks them to explain the strategy they used to solve the problem or activity.

2. The other groups are encouraged to ask questions or make comments after each presentation.

3. The teacher moderates the discussion, clarifying any misunderstandings and reinforcing the key concepts.

4. The process repeats for the other groups until all have presented.

5. Theory Connection (3 minutes):

• Description: The teacher takes this moment to connect the practical activities to the theory presented at the beginning of the lesson.
• Step-by-Step:

1. The teacher highlights how the distance formula between two points was used to solve the proposed practical problems.

2. The teacher reinforces the importance of understanding and correctly applying the distance formula between two points, and how it can be useful in everyday situations.

3. Individual Reflection (2 - 3 minutes):

• Description: The teacher proposes that students reflect individually on what they learned in the lesson.
• Step-by-Step:

1. The teacher suggests some questions to guide students' reflection:

2. What was the most important concept you learned today?

3. Which questions have not yet been answered?

4. Students have one minute to think about these questions.

5. Then, the teacher invites some students to share their answers with the class. The goal is to promote reflection and consolidation of learning.

6. Feedback and Closure (2 minutes):

• Description: The teacher concludes the lesson by asking for quick feedback from students about the lesson and clarifying any doubts that may remain.
• Step-by-Step:

1. The teacher asks students if they have any more questions or difficulties regarding the lesson topic.
2. The teacher thanks everyone for their participation, reinforces the importance of the content learned, and encourages students to continue practicing.

The Feedback is a crucial part of the lesson plan, as it allows the teacher to assess students' understanding, reinforce key concepts, clarify doubts, and promote reflection on learning. Also, students' feedback is valuable for the teacher to improve their teaching practices.

Conclusion (5 - 7 minutes)

1. Summary of the Content (2 minutes):

• Description: The teacher should recap the main points of the lesson, recalling the definition of the Cartesian plane, the distance formula between two points, and the practical application of these concepts.
• Step-by-Step:

1. The teacher begins by recalling the structure of the Cartesian plane, with the x- and y-axes, and the interpretation of coordinates.

2. Next, the teacher recaps the distance formula between two points and how it is applied to calculate the distance between two points on the Cartesian plane.

3. The teacher concludes the summary by reinforcing the importance of calculating the distance between points and how this concept can be applied in everyday situations.

4. Connection between Theory, Practice, and Applications (1 minute):

• Description: The teacher should reinforce how the lesson connected the theory, practice, and applications of calculating the distance between points on the Cartesian plane.
• Step-by-Step:

1. The teacher highlights how the theory was presented at the beginning of the lesson, with the definition of the Cartesian plane and the distance formula between two points.

2. Next, the teacher mentions the practical activities performed, which allowed students to apply the theory to solve problems.

3. Finally, the teacher reinforces the practical applications of calculating the distance between points, showing how this concept can be useful in various real-world situations.

4. Extra Materials (1 - 2 minutes):

• Description: The teacher should suggest extra materials for students who wish to deepen their knowledge of calculating the distance between points on the Cartesian plane.
• Step-by-Step:

1. The teacher may suggest Mathematics books that cover the topic in more detail.

2. Additionally, the teacher may indicate websites, videos, and Mathematics apps that offer explanations and exercises on calculating the distance between points.

3. Relevance of the Subject (1 - 2 minutes):

• Description: Finally, the teacher should emphasize the importance of calculating the distance between points on the Cartesian plane to everyday life and other disciplines.
• Step-by-Step:

1. The teacher may mention some practical applications of calculating the distance between points, such as in navigation, engineering, architecture, and computer science.
2. Additionally, the teacher may highlight how the ability to think logically and solve mathematical problems can be useful in various areas of life.

The Conclusion is an important step in the lesson plan, as it allows the teacher to consolidate learning, reinforce the connection between theory, practice, and applications, and motivate students to continue studying the subject. Furthermore, by emphasizing the relevance of the topic, the teacher helps students realize the importance of Mathematics to everyday life and other disciplines.