Objectives (5 - 10 minutes)

1. Understanding the Concept of Distance between Points in Analytic Geometry: The teacher should ensure that students understand the concept of distance between two points on the Cartesian plane. This includes applying the distance formula between two points and interpreting the result obtained.

2. Distance between Points Calculation Skills: Students should be able to apply the distance formula between two points to calculate the distance between specific points. This should be done both manually (with the aid of calculators, if necessary) and using geometric drawing software, such as Geogebra.

3. Applying the Concept of Distance between Points in Real-World Situations: Finally, students should be able to apply the concept of distance between points to solve problems involving real-life situations. This will help solidify their understanding of the topic and realize the relevance of analytic geometry in practical contexts.

Secondary objectives:

• Developing Logical Thinking: Solving analytic geometry problems requires well-developed logical thinking. By working through this topic, students will have the opportunity to enhance their logical reasoning skills.

• Improving Teamwork Skills: The hands-on activities suggested in this lesson plan can be carried out in groups. This will provide students with the opportunity to improve their teamwork and communication skills.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher should begin the lesson by briefly reviewing mathematical concepts that are prerequisites for understanding analytic geometry, such as the Cartesian coordinate system, points on a plane, and equations of lines. This can be done through a short classroom discussion or a review activity, such as a quiz game.

2. Presentation of Problem Situations: The teacher should then present students with one or two real-world situations that can be solved using the concept of distance between points. For example, the distance between two cities on a map, the distance between two players on a soccer field, or even the distance between two points in a video game. These problem situations will serve as motivation for studying the topic.

3. Contextualization of the Importance of the Subject: The teacher should then explain the importance of analytic geometry, and more specifically, the distance between points, in various areas of knowledge and practical life, such as architecture, engineering, navigation, geography, physics, among others. This will help spark students' interest and show the relevance of the subject.

4. Introduction of the Topic with Curiosities or Stories: To capture students' attention, the teacher can introduce the topic of analytic geometry with some curiosities or stories. For example, one could mention that analytic geometry was developed by René Descartes, a famous philosopher and mathematician of the 17th century, who created it to solve geometry problems using algebra. Another interesting curiosity is that analytic geometry is used in many animated films, such as Toy Story and Shrek, to create characters and objects in a three-dimensional space.

At the end of the Introduction, students should be motivated to learn about the distance between points in analytic geometry and understand its importance and application.

Development (20 - 25 minutes)

1. Hands-on Activity with Strings and Thumbtacks: (10 - 15 minutes)

• Materials Required: Strings of different lengths and thumbtacks.

• Preparation: The teacher should prepare the environment for the activity by drawing a Cartesian plane on the classroom floor, with a visible scale. Then, a thumbtack should be tied to one end of each string, representing a point on the Cartesian plane.

• Step-by-Step Activity:

1. Divide the class into groups of 4 or 5 students.
2. Distribute the strings and thumbtacks to each group.
3. Explain to the students that they should position the thumbtacks at specific points on the Cartesian plane and then use the strings to measure the distance between those points.
4. Suggest that the students choose some random points on the Cartesian plane and calculate the distance between them using the distance formula between two points.
5. Walk around the groups to monitor progress, clarify doubts, and provide guidance, if necessary.
6. At the end of the activity, ask each group to share their findings and conclusions with the class.

• Objective of the Activity: This activity aims to help students visualize the concept of distance between points on the Cartesian plane in a concrete and playful way. In addition, it allows for the practice of calculating the distance using the specific formula.

2. Hands-on Activity with Geogebra: (10 - 15 minutes)

• Materials Required: Computers or mobile devices with internet access and Geogebra software.

• Preparation: The teacher should ensure that all students have access to Geogebra software and know how to use it to draw on the Cartesian plane and calculate distances.

• Step-by-Step Activity:

1. Divide the class into groups of 4 or 5 students.
2. Explain to the students that they should use Geogebra software to draw some points on the Cartesian plane and calculate the distance between them.
3. Ask students to choose some points on the plane and calculate the distance between them using Geogebra's "Distance" tool.
4. Walk around the groups to monitor progress, clarify doubts, and provide guidance, if necessary.
5. At the end of the activity, ask each group to share their findings and conclusions with the class.

• Objective of the Activity: This activity aims to familiarize students with the use of geometric drawing software, such as Geogebra, for solving analytical geometry problems. In addition, it allows for the exploration of the concept of distance between points in an interactive and visually appealing way.

Feedback (10 - 15 minutes)

1. Group Discussion: (5 - 7 minutes)

• Preparation: The teacher should prepare the group discussion by dividing the class into groups of 4 or 5 students and assigning a leader to each group.

• Step-by-Step Activity:

1. Each group leader should share with the class the conclusions that their group reached during the hands-on activities.
2. The teacher should encourage participation from all students in the discussion, asking open-ended questions and requesting that students justify their answers based on what they have learned.
3. During the discussion, the teacher should clarify any misunderstandings, reinforce important concepts, and provide constructive feedback.

• Objective of the Activity: The group discussion allows students to share their ideas and findings, learn from each other, and develop their communication and argumentation skills.

2. Connection to Theory: (3 - 5 minutes)

• Step-by-Step Activity:

1. After the group discussion, the teacher should briefly review the theoretical concepts covered in the lesson, reinforcing the distance formula between two points, the use of Cartesian coordinates, and the interpretation of the result obtained.
2. Then, the teacher should connect the theory with the hands-on activities, explaining how the activities illustrate and apply the theoretical concepts.
3. The teacher should also highlight the skills that students have developed during the lesson, such as the ability to work in teams, solve problems, think critically, and use technological tools.

• Objective of the Activity: Connecting to theory helps to consolidate students' understanding of the concepts covered in class and realize the relevance and application of these concepts in different contexts.

3. Final Reflection: (2 - 3 minutes)

• Step-by-Step Activity:

1. To close the lesson, the teacher should ask students to reflect silently for a minute on what they have learned.
2. Then, the teacher should ask some reflection questions, such as: "What was the most important concept you learned today?" and "What questions have not yet been answered?"
3. Students should have the opportunity to share their answers with the class, if they wish. The teacher should encourage students to be honest in their answers and to use them as a basis for planning the next lessons.

• Objective of the Activity: The final reflection helps students to consolidate what they have learned, identify any gaps in their understanding, and prepare for future learning. In addition, it provides the teacher with valuable feedback on the effectiveness of the lesson and on the students' learning needs.

Conclusion (5 - 10 minutes)

1. Summary of the Content: (2 - 3 minutes)

• The teacher should begin the Conclusion by recapping the main points that were covered during the lesson. This includes the concept of distance between points in analytic geometry, the distance formula between two points and how to apply it, and the application of this concept in real-world situations.
• The teacher can reference the hands-on activities that were carried out, highlighting students' findings and delving deeper into the connection between theory and practice.

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

• The teacher should then reinforce the importance of the connection between theory, practice, and applications. He should explain how the hands-on activities helped to illustrate and apply the theoretical concepts, and how the understanding of these concepts can be useful in solving real-world problems.

3. Complementary Materials and Future Studies: (1 - 2 minutes)

• The teacher should suggest complementary materials for students who wish to deepen their knowledge of the topic. This may include additional readings, explanatory videos, interactive mathematics websites, and practice exercises.
• The teacher may also suggest related topics that students can explore in future studies, such as the equation of a line, the circle, and the ellipse, which are closely linked to the concept of distance between points in analytic geometry.

4. Relevance of the Topic to Daily Life: (1 - 2 minutes)

• Finally, the teacher should emphasize the importance of the topic to daily life. He can cite examples of how analytic geometry, and more specifically, the concept of distance between points, is applied in different areas, such as architecture, engineering, navigation, geography, and physics.
• The teacher should emphasize that, by understanding and applying these concepts, students are developing skills that can be useful in many aspects of their lives, from plotting routes on GPS to designing buildings and bridges.