Objectives (5 - 7 minutes)

1. Introduce Conic Sections: The teacher will explain what conic sections are and where they are used in real life. This will help the students understand the relevance and application of the topic.

2. Understand the Equations of Conic Sections: The students will learn the basic equations of different conic sections, including circles, ellipses, parabolas, and hyperbolas.

3. Identify Conic Sections: The students will learn how to identify the type of conic section based on the given equation.

Secondary Objectives:

• Familiarize with Cartesian Plane: The teacher will refresh the students' knowledge about Cartesian Plane as it is crucial for the understanding of conic sections.
• Sharpen algebraic skills: As the students work through the equations of conic sections, they will also be enhancing their algebraic skills.

Introduction (10 - 15 minutes)

1. Revision of Necessary Concepts: The teacher will start the lesson by revisiting the concept of the Cartesian Plane, highlighting the importance of 'x' and 'y' coordinates. The students will be reminded of the concepts of 'x' and 'y' intercepts and the terminology used in equations, such as the coefficients and constants. This revision will allow the students to understand the notation and terminology used in the equations of conic sections.

2. Problem Situations: The teacher will present two problem situations to the students. The first one will involve a real-world situation where an object is projected and forms a parabolic trajectory. The second problem will involve a situation where students have to determine the shortest distance between two points, leading to the concept of a straight line, which is a degenerate form of conic sections.

3. Real-World Applications: The teacher will contextualize the importance of conic sections by explaining their frequent use in the fields of astronomy, physics, engineering, and architecture. For example, the orbits of planets being elliptical, the path of a projected object being parabolic, the shape of satellite dishes and mirrors of telescopes being parabolic for optimal signal reception and reflection of light, respectively.

4. Topic Introduction: The teacher will then introduce the concept of conic sections. They will explain that conic sections are curves obtained as the intersection of a cone with a plane. They will also mention that the angle of intersection with the plane determines the type of conic section formed: circle, ellipse, parabola, or hyperbola.

5. Interesting Facts: To grab the students' attention, the teacher will share a few interesting facts:

   • The teacher can mention that the word 'parabola' means 'comparison' or 'relation' in Greek, reflecting the fact that the curve compares the squares of distances.
   • Another interesting fact could be that the whispering gallery in the dome of St. Paul's Cathedral in London is an example of an elliptical echo, a property of ellipses.

By the end of this introduction, the students should have a clear understanding of the relevance and practical applications of conic sections, sparking their curiosity and interest in learning more about the topic.

Development

Pre-Class Activities (15 - 20 minutes)

1. Instructional Videos: Students will be asked to watch a couple of selected educational videos that give a basic introduction to the conic sections and their formulas. This will ensure they have background knowledge and a general concept of what will be discussed in class.

2. Reading Material: Students will be advised to go through some reading material explaining the applicability of conic sections in various scientific fields - astronomy, engineering, physics, and technology. This reading will foster a better appreciation for the subject matter and the historical development of these geometric concepts.

3. Assignment: Students will be tasked to write a brief reaction paper based on the videos and reading material. This paper will help the teacher assess the students' understanding of the concepts and identify points of interest or confusion, which can be addressed during the class.

In-Class Activities (20 - 30 minutes)

For in-class activities, the focus will be hands-on application of the pre-class learned information.

Activity 1:

1. The teacher will divide the class into groups and assign each group one type of conic section. They will be responsible for exploring their assigned conic section (circle, ellipse, parabola, or hyperbola).

2. Each group will receive equal-sized rectangular sheets of paper and a pair of scissors. They will be instructed to fold the paper in a specific way to form a cone.

3. Using the scissors, they are instructed to cut the paper cone at different angles, producing different conic sections. The students will see the actual shapes formed - a circle, ellipse, parabola, or hyperbola based on the angle of the cut.

4. Each group will then be required to sketch their respective conics on a Cartesian plane, discussing its basic properties and characteristics.

5. Finally, each group will present their findings to the class, encouraging discussion and comparison among the different conics.

Activity 2:

1. The teacher will present a riddle-based game in class, employing augmented reality technology. The students will need to use the principles of the conics to solve the riddles.

2. The teacher will divide the class into teams, where each team receives different riddles or problems associated with real-life situations that can be solved using the principles of conic sections.

3. Teams will be given the opportunity to discuss and utilize their knowledge of conic sections' equations to solve these riddles. They will use the AR tools to visualize the conic sections, making the learning process interactive and vivid.

4. The first team to solve all of their riddles accurately wins the game.

At the end of these activities, the students would have gained an interactive, hands-on experience of conic sections and their equations, fostering long-lasting understanding and interest in the subject.

Feedback (10 - 15 minutes)

1. Reflection and Discussion: After the classroom activities have been completed, the teacher will facilitate a group discussion. Each group will have up to 3 minutes to present their solutions and conclusions from their assigned conic section. This will encourage the students to articulate their understanding, thus reinforcing their learning.

2. Connection to Theory: The teacher will then take a moment to highlight the connection between the activities and the theory behind conic sections. They will reiterate the key points of the lesson, emphasizing the practical implications of understanding the equations of conic sections, and how these were reflected in the activities conducted in the class.

3. Reflection Questions: Next, the teacher will propose a moment of individual reflection. The students will be asked to ponder on:

   • What was the most significant concept they learned during the class?
   • What questions or doubts do they still have about conic sections?

4. Question and Answer Session: After the reflection, the teacher will open the floor for a question and answer session. The students will get an opportunity to clarify any doubts or questions they may have remaining. The teacher will answer these questions, ensuring that the students have a thorough understanding of the topic.

5. Feedback Collection: The teacher will then distribute a feedback form for the students to complete. The form will include questions about the students' understanding of the concepts, the effectiveness of the teaching methods used, and what the students felt about the flipped classroom methodology. This feedback will be valuable for the teacher to improve future lessons.

6. Homework Assignment: For the last part of the feedback session, the teacher will assign homework to the students. They will be asked to solve a set of problems involving the equations of conic sections, applying what they have learned in the class. The students will also be asked to prepare a small report on the real-world applications of their assigned conic section, enhancing their understanding of the topic's practicality.

7. Wrapping Up: Lastly, the teacher will conclude the class by summarizing the topics covered in the lesson. They will also give a brief introduction to the next topic in the curriculum, building anticipation and curiosity among the students.

By the end of this feedback session, the teacher will have a clear understanding of the students' comprehension of the concepts taught in the class. Furthermore, the students will have a firm grasp of the equations of conic sections, their applications, and their practical implications.

Conclusion (5 - 7 minutes)

1. Summary: The teacher will summarize the main concepts and ideas discussed during the lesson. The teacher will reinforce the understanding of conic sections – circles, ellipses, parabolas, and hyperbolas – and their respective equations and features. The teacher will also recap the primary application areas of these geometric shapes and the importance of understanding their mathematical representations.

2. Connecting Theory and Practice: The teacher will reflect on how the lesson's activities helped the students connect the theoretical information about conic sections and their equations to practical, real-world scenarios. The teacher will highlight how the group activities, especially the riddle-solving game, helped students apply the theoretical concepts in an engaging, practical, and fun manner.

3. Additional Resources: The teacher will suggest some additional resources for the students to expand their understanding of the topic. These resources could include recommended books, online tutorials, and educational websites that offer interactive exercises on conic sections. The teacher might also suggest some interesting documentaries related to the use of conic sections in different scientific fields.

4. Importance of the Topic: The teacher will conclude the lesson by stressing the importance of understanding conic sections and their equations in everyday life. The teacher will discuss how these geometric concepts are used in various fields such as architecture, astronomy, physics, and engineering. For instance, the teacher could mention how parabolic mirrors are used in telescopes and satellite dishes, how ellipses are used in the design of race tracks and whispering galleries, and how circles are fundamental in many designs and constructions.

5. Looking Ahead: Finally, the teacher will give a brief preview of the next lesson, linking the knowledge of conic sections to the upcoming topic. This will help keep the students curious and engaged in their future learning.

By the end of this conclusion, the students should have a clear understanding of the importance of the topic and how it was applied during the lesson. They will be equipped with additional resources to explore the topic further and be prepared for the next lesson.