Objectives (5 - 7 minutes)

1. Understand what a circle is and how it is defined in analytical geometry.
2. Learn how to obtain the equation of a circle from its characteristics.
3. Develop the ability to solve problems involving the circle equation.

Secondary Objectives:

• Practice the use of coordinates and mathematical formulas in problem solving.
• Promote logical and analytical thinking skills.
• Encourage teamwork and active student participation in the class.

Introduction (10 - 15 minutes)

1. Review of Previous Content:

   • The teacher should start the class by reviewing the concepts of Cartesian coordinates, distance between two points, and basic geometry formulas. This is essential for students to understand and correctly apply the concepts that will be addressed in this class.
   • Suggestion: The teacher can ask quick questions to verify if students remember the previous concepts, such as 'What are Cartesian coordinates?' or 'How can we calculate the distance between two points on a plane?'.

2. Contextualization:

   • Next, the teacher should contextualize the importance of analytical geometry, explaining that it is widely used in various areas of science and engineering, from computer graphics and physics to navigation and architecture.
   • To make the contextualization more concrete, the teacher can give examples of everyday situations where analytical geometry is used, such as locating a point on a map, modeling an object in a computer program, or calculating the orbits of planets.

3. Topic Presentation:

   • The teacher should introduce the topic of the class, explaining that students will learn how to obtain the equation of a circle from its characteristics. He should emphasize that this is a direct application of analytical geometry, and that the knowledge gained will be useful for solving problems in various areas.
   • To spark students' interest, the teacher can share a curiosity about analytical geometry, such as the fact that it was independently developed by René Descartes and Pierre de Fermat in the 17th century, and that this discovery revolutionized mathematics and physics.

4. Engaging Students' Attention:

   • To capture students' attention, the teacher can propose two problem situations: the first involving the location of a point on a circle knowing its equation, and the second involving the determination of the equation of a circle from three given points.
   • The teacher can explain that by the end of the class, students will be able to solve these problems and many others, using the concepts that will be presented.

With the Introduction, students will be prepared to start the class, with a clear understanding of what will be covered and the importance of the subject.

Development (20 - 25 minutes)

1. Activity 1 - 'Finding the Circle':

   • The teacher will divide the class into groups of up to 5 students and provide each group with a set of cards, where each card will have a set of coordinates (x, y) representing points scattered on a large sheet of paper.
   • The objective of the activity will be for students, using these cards with coordinates, to position the points on the paper according to the provided coordinates, so that they form a circle.
   • After forming the circle, students should measure the radius and diameter of the formed circle and calculate its area and perimeter.
   • At this stage, the teacher should circulate around the room, assisting groups that encounter difficulties and asking guiding questions to stimulate students' critical thinking. The idea is for them to perceive the relationship between the coordinates of the points and the characteristics of the formed circle.
   • At the end of the activity, each group should present to the class the circle they formed, the measurements they obtained, and the conclusions they reached based on the calculations.

2. Activity 2 - 'Discovering the Equation':

   • Using the same set of cards from the previous activity, the teacher will propose a new challenge: students must discover the equation of the circle they formed.
   • For this, the teacher will explain that the equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) are the coordinates of the center of the circle and r is the radius.
   • Students should then use the coordinates of the points they positioned on the paper and the measurements they obtained in the first activity to find the values of h, k, and r, and thus write the equation of the circle.
   • At this stage, the teacher should encourage students to try different strategies to solve the problem, such as using the average of the coordinates of the points to find the coordinates of the center of the circle, or applying the Pythagorean Theorem to calculate the radius from the measurements they obtained.
   • At the end of the activity, each group should present to the class the equation of the circle they discovered and explain how they arrived at that equation.

3. Activity 3 - 'Real Problems':

   • To conclude the Development stage, the teacher will present students with some real problems involving the equation of the circle, such as determining the trajectory of a satellite in orbit, calculating the perimeter of a race track, or modeling the shape of a bicycle wheel.
   • Students, then, in their groups, should choose one of the problems, analyze it, and propose a solution using the concepts they learned.
   • At this stage, the teacher should circulate around the room, assisting groups that encounter difficulties and asking questions to stimulate students' critical thinking.
   • At the end of the activity, each group should present to the class the problem they chose, the solution they proposed, and the conclusions they reached. The teacher should summarize the presentations, highlighting the main points and the importance of the circle equation in solving these problems.

With these activities, students will have the opportunity to apply the concepts they learned in a practical and contextualized manner, which will facilitate understanding and retention of the content. In addition, group activities will promote collaboration and communication among students, essential skills for the integral development of the student.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher should propose a group discussion, where each group will have up to 3 minutes to share the solutions or conclusions they reached during the activities.
   • The objective is for students to learn from each other, listening to different points of view and strategies used to solve the proposed problems.
   • During the presentations, the teacher should encourage the participation of all students, asking questions, requesting clarifications, and highlighting the successes and difficulties of each group.

2. Comparison with Theory (2 - 3 minutes):

   • After the presentations, the teacher should make a connection between the activities carried out and the theoretical concepts presented in the Introduction of the class.
   • For example, the teacher can ask students how they used the formulas and concepts of analytical geometry to solve the problems, or how they perceived the relationship between the coordinates of the points and the characteristics of the circle.
   • The objective is for students to see the practical application of the theory, which will help consolidate learning and understand the relevance of the subject.

3. Individual Reflection (2 - 3 minutes):

   • Finally, the teacher should propose that students make a brief individual reflection on what they learned in the class.
   • For this, the teacher can ask questions such as: 'What was the most important concept you learned today?', 'What questions have not been answered yet?' or 'How can you apply what you learned today in other situations?'.
   • Students will have a minute to think about these questions and then will be invited to share their answers with the class.
   • The teacher should listen attentively to the students' answers, take note of the questions that have not been understood, and highlight the practical applications of the content to reinforce the relevance of the subject.

With this Return, students will have the opportunity to reflect on what they learned, share their discoveries and difficulties, and make connections between theory and practice. In addition, the teacher will be able to assess students' understanding, identify possible gaps in learning, and plan the next classes according to the needs of the class.

Conclusion (5 - 7 minutes)

1. Recapitulation (2 - 3 minutes):

   • The teacher should start the Conclusion by recalling the main points covered during the class. He should recap the definition of a circle, how to determine its equation from its characteristics, and how to solve problems involving the circle equation.
   • For this, the teacher can give a brief summary of the concepts, emphasizing the practical application of each one. For example, he can recall that the equation of a circle allows calculating its position and size from a set of coordinates, which has various applications in science and technology.

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

   • Next, the teacher should highlight how the class connected theory, practice, and application. He can emphasize how group activities allowed students to apply theoretical concepts in a practical and contextualized way, and how the proposed problems reflected real situations where analytical geometry is used.
   • The teacher can also mention how group discussion and individual reflection allowed students to perceive the importance and relevance of the subject, stimulating deeper and more meaningful learning.

3. Extra Materials (1 minute):

   • The teacher should mention some additional resources that students can use to deepen their understanding of the subject. He can suggest reading math books or websites, doing online exercises, or watching explanatory videos.
   • For example, the teacher can suggest that students watch a video showing how the equation of a circle is used to model the trajectory of a satellite in orbit, or that they solve a set of online exercises to practice applying the circle equation in different contexts.

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

   • Finally, the teacher should emphasize the importance of the subject for daily life and other disciplines. He can mention, for example, how analytical geometry is used in fields such as physics, engineering, architecture, and computing.
   • The teacher can also highlight that the development of logical and analytical thinking skills, which are essential for problem solving in mathematics, is a competence that has applications in various areas of life.

With this Conclusion, students will have a clear and broad view of the subject, understand its relevance, and have resources to continue learning and practicing. In addition, the teacher will be able to assess the achievement of the lesson Objectives and plan the next classes according to the needs and level of understanding of the class.