Objectives (5 - 7 minutes)

1. Understand the Basics of Cartesian Geometry: Students will learn about the fundamental principles of Cartesian geometry, including the coordinate plane, x and y axes, and the concept of points and their locations on the plane. This will serve as the foundation for the rest of the lesson.

2. Recognize the Conic Sections: Students will learn to identify the four conic sections - the circle, the ellipse, the parabola, and the hyperbola. They will understand the basic shapes and properties of each conic section.

3. Learn the Equation of Conics: Students will learn the standard equation for each conic section. They will understand how the coefficients in the equation affect the shape, size, and orientation of the conic section.

Secondary Objective:

• Apply the Concepts: Students will apply their understanding of Cartesian geometry and the equation of conics to solve simple problems. They will learn to identify the conic section represented by an equation and sketch the conic section based on the equation.

Introduction (10 -15 minutes)

1. Review of Required Knowledge: The teacher begins the lesson by reminding students of the fundamental concepts of Cartesian Geometry that they need to understand the lesson. This includes the coordinate plane, x and y axes, and the concept of points and their locations on the plane. The teacher can use a quick quiz or a few sample questions to gauge the students' understanding and refresh their memories.

2. Problem Situations: The teacher presents two problem situations to the students that will serve as starters for the development of the theory. The first problem could be about finding the distance from a point to a line, and the second problem could be about finding the equation of a circle given its center and a point on it. The teacher stresses that these problems will be solved using the concepts learned in the lesson.

3. Real-World Applications: The teacher connects the theory to real-world applications to make it more relatable and engaging for the students. For example, the teacher can explain how the equation of a parabola is used in physics to describe the path of a projectile or how the equation of an ellipse is used in astronomy to describe the orbit of planets.

4. Topic Introduction: The teacher introduces the topic of the lesson - Cartesian Geometry: Equation of Conics - with a brief explanation. The teacher explains that conic sections are the shapes that result from the intersection of a double-napped cone with a plane. The teacher also points out that the standard equation of each conic section can be derived from its geometric properties.

5. Curiosities and Fun Facts: To grab the students' attention, the teacher shares some interesting facts and curiosities related to the topic. For example, the teacher can tell the students that the ancient Greeks discovered conic sections while studying the shapes that resulted from slicing a cone, and these shapes have been used in various fields of study, from physics and engineering to art and architecture.

Development (20 - 25 minutes)

Activity 1: Conic Art (10 - 12 minutes)

1. The teacher sets up the class into groups of 3 or 4 students and distributes a large piece of white paper to each group, along with a compass, a ruler, and a protractor.

2. The teacher instructs each group to draw a Cartesian coordinate plane on their paper and label the x and y axes.

3. The teacher then explains that each group's task is to draw the best representation of a conic section on their coordinate plane. They must decide within their group whether they want to draw a circle, an ellipse, a parabola, or a hyperbola.

4. The teacher mentions that the conic section's equation should meet specific criteria, like the size, shape, and orientation of the conic section, based on the standard equation of the conic section.

5. The teacher walks around the class, providing support and guidance as needed. They check to make sure each group understands the equation of the conic section they have chosen and how to translate this onto their coordinate plane.

6. Once completed, the students should color and label their conic section, including its equation.

7. The teacher then asks each group to present their conic section to the class, explaining their thought process and how they used the equation to create their drawing.

Activity 2: Conic Carousel (10 - 13 minutes)

1. The teacher explains that the class will now participate in a carousel activity. The teacher places large poster boards with a blank coordinate plane and a conic section labeled on each table around the classroom in a circle.

2. Each group starts at a different table, and they are given a specific amount of time (around 2 minutes) to examine the conic section and its labeled equation.

3. The teacher then calls "rotate," and each group moves to the next table clockwise.

4. The task for each group is to determine whether the labeled conic section and its equation match. If they do not, the group must correct it.

5. This process continues several times until each group has had the chance to examine and correct each conic section.

6. The teacher then leads a class discussion about the common mistakes made and the correct way to match a conic section with its equation.

7. The teacher emphasizes the importance of understanding the equation of conics and its impact on the shape and position of the conic section on the coordinate plane.

Closure: (2-3 minutes)

1. The teacher wraps up the Development stage by summarizing the key concepts learned during the activities. They remind students about the standard equation of each conic section and how it affects the conic section's size, shape, and orientation on the coordinate plane.

2. The teacher also highlights the importance of understanding these concepts, as they will be used not only in their math class but also in other subjects and real-world applications.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher proposes a group discussion where each group is given up to 2 minutes to share their most significant learning point from the lesson.
   • The teacher guides the discussion to ensure that the key points of the lesson are covered. These include understanding the basics of Cartesian geometry, recognizing the conic sections, and learning the equations for each conic section.
   • The teacher also prompts the students to discuss how they applied these concepts in the activities. For instance, how they used the equation of a conic section to draw it on their coordinate plane in the Conic Art activity or how they matched the conic sections with their equations in the Conic Carousel activity.

2. Individual Reflection (3 - 4 minutes):

   • The teacher then asks the students to take a moment for individual reflection. They are asked to consider the following questions:
     1. What was the most important concept you learned today?
     2. Which questions have not yet been answered?
   • The students are encouraged to write down their reflections. This will not only help them to solidify their understanding of the topic but also provide valuable feedback for the teacher about any areas that may need further clarification or reinforcement in future lessons.

3. Sharing Reflections (2 minutes):

   • The teacher invites a few students to share their reflections with the class. This can be done either by volunteering or by the teacher randomly selecting students.
   • The teacher makes sure to address any unanswered questions and provides additional explanations or examples as needed.

4. Summarizing the Lesson (1 - 2 minutes):

   • The teacher concludes the feedback stage by summarizing the main points of the lesson and its objectives. They also remind the students about the importance of understanding Cartesian Geometry and the Equation of Conics, not only for their math class but also for its real-world applications.

This feedback stage allows the teacher to assess the students' understanding of the lesson, address any remaining questions or misconceptions, and provide closure to the lesson. It also provides the students with an opportunity to reflect on their learning and consolidate their understanding of the topic.

Conclusion (5 - 7 minutes)

1. Summary of the Lesson (2 minutes):

   • The teacher begins the conclusion by summarizing the main points of the lesson. They remind the students about the fundamental principles of Cartesian geometry, the four conic sections (circle, ellipse, parabola, and hyperbola), and their standard equations.
   • The teacher also recaps the activities that the students engaged in during the lesson, such as the Conic Art and Conic Carousel, and how these activities helped them to understand and apply the concepts.

2. Connecting Theory, Practice, and Applications (2 minutes):

   • The teacher then explains how the lesson connected theory, practice, and applications. They highlight how the theoretical understanding of Cartesian geometry and the equation of conics enabled the students to solve practical problems in the activities.
   • They also discuss how the real-world applications of the conic sections, which were touched upon in the introduction, were applied in the activities. For example, in the Conic Carousel, the students had to match the conic sections with their equations, which is a skill used in various fields such as physics and engineering.

3. Additional Learning Resources (1 - 2 minutes):

   • To further enhance the students' understanding of the topic, the teacher suggests a few additional learning resources. These could include relevant chapters from the textbook, online tutorials or videos about Cartesian geometry and conic sections, and practice problems for the students to work on at home.
   • The teacher encourages the students to explore these resources and to reach out if they have any questions or need further clarification.

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

   • The teacher concludes the lesson by discussing the relevance of the topic to everyday life. They explain that understanding Cartesian geometry and the equation of conics can help the students to visualize and understand various phenomena in the world around them.
   • For example, the teacher can mention how the shape of a car's headlight or the path of a basketball shot can be described by a conic section. They can also explain how the concept of conic sections is used in fields such as architecture, art, and physics.
   • The teacher emphasizes that learning about Cartesian geometry and the equation of conics is not just about solving math problems, but also about developing a deeper understanding of the world and its many fascinating shapes and patterns.