Objectives (5 - 7 minutes)

1. Understand and define what the critical angle is in optics, and how it relates to the refraction of light.

2. Solve practical problems involving the critical angle and the refraction of light, using the laws of refraction and the criterion for total internal reflection.

3. Apply the acquired knowledge about the critical angle to analyze and explain everyday optical phenomena, such as the formation of rainbows and optical illusions in prisms.

Secondary Objectives:

• Stimulate critical thinking and problem-solving skills of students through practical and theoretical questions.

• Promote interaction and collaboration among students, encouraging discussion and exchange of ideas during the resolution of proposed problems.

• Develop research skills and autonomous learning, using digital and bibliographic resources to deepen the understanding of the topic.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by reviewing the concepts of light refraction, refractive index, and the laws of refraction. This can be done through directed questions to activate prior knowledge. The teacher should also review the definition of angle of incidence and angle of refraction. (3 - 5 minutes)

2. Problematic situations: The teacher should then present two problematic situations to arouse students' interest:

   • Situation 1: 'Imagine you are inside a pool looking up towards the water surface. Why do you see the world outside the pool appearing to be in a different position?'

   • Situation 2: 'Why, when sunlight passes through a prism, does it separate into a variety of colors?' (3 - 5 minutes)

3. Contextualization: The teacher should explain that geometric optics is a discipline that studies the propagation of light in transparent media, and that the critical angle is a key concept to understand these phenomena. It should be emphasized that knowledge about the critical angle has practical applications, such as in the construction of lenses and prisms used in glasses, cameras, and telescopes. (2 - 3 minutes)

4. Capturing students' attention: To capture students' attention, the teacher can share some curiosities and applications of the critical angle:

   • Curiosity 1: 'Did you know that the formation of a rainbow is an example of light refraction and reflection? And that the critical angle plays an important role in the formation of this optical phenomenon?'

   • Curiosity 2: 'Did you know that optical fibers, widely used in telecommunications, work due to the total reflection of light inside the fiber, and that this is also related to the critical angle?' (3 - 4 minutes)

Development (20 - 25 minutes)

1. Theory of the critical angle (10 - 12 minutes): The teacher should introduce the theory of the critical angle, explaining that it is the minimum angle of incidence for total reflection of light to occur inside a medium and that, beyond it, the light is refracted. To do this, the teacher should:

   • Formally define the critical angle, emphasizing that it depends on the refractive indices of the involved media.

   • Explain that if the angle of incidence is less than the critical angle, refraction and reflection will occur. If the angle of incidence is equal to or greater than the critical angle, total reflection will occur.

   • Show the mathematical formula that relates the critical angle to the refractive indices of the media, usually presented as n₁sinθ₁ = n₂sinθ₂, where n₁ and n₂ are the refractive indices of the media and θ₁ and θ₂ are the angles of incidence and refraction, respectively.

2. Analysis of problematic situations (5 - 7 minutes): The teacher should then return to the problematic situations presented in the Introduction and guide the students in applying the theory of the critical angle to explain these phenomena. For this, the teacher should:

   • For the first problematic situation, explain that what the students observe is an example of light refraction. Light travels more slowly in water than in air, so when passing from one medium to another, it undergoes a change in direction. If the angle of incidence is greater than the critical angle, total reflection will occur, and this is what makes the students see the world outside the pool in a different position.

   • For the second problematic situation, explain that sunlight is composed of various colors, each with a different wavelength. When passing through the prism, the different colors of light are refracted at different angles due to the difference in their refractive indices. If the angle of incidence is greater than the critical angle, total reflection will occur, and this is what makes the students see the separated colors.

3. Exercise resolution (5 - 6 minutes): The teacher should then propose some exercises for the students to solve, in order to deepen their understanding of the critical angle concept and light refraction. The exercises may involve determining the critical angle between two media with given refractive indices, determining the refractive index of a medium from the critical angle, among others. The teacher should circulate around the classroom, assisting students who have difficulties and clarifying any possible doubts.

4. Discussion and Conclusion (2 - 3 minutes): At the end of the exercise resolution, the teacher should promote a discussion in the classroom, asking students about their solutions and encouraging them to share their resolution strategies. The teacher should then conclude the lesson, reinforcing the main points addressed and clarifying any remaining doubts. The teacher should also suggest to the students to research more about the critical angle and its associated optical phenomena, as a complementary activity for the next lesson.

Return (8 - 10 minutes)

1. Review of concepts (3 - 4 minutes): The teacher should start the Return stage by reviewing the key concepts covered in the lesson. This can be done through directed questions to students to verify their understanding of the critical angle and light refraction. For example, the teacher can ask students to define the critical angle, explain how it relates to light refraction, and give examples of practical situations involving the critical angle.

2. Connection to practice (2 - 3 minutes): The teacher should then ask students to reflect on the connection between the learned theory and practice. For this, the teacher can ask questions like:

   • 'How can the concept of the critical angle be applied to explain the formation of a rainbow?'
   • 'How can the understanding of the critical angle help us understand why we see the world outside a pool in a different position?'
   • 'How can knowledge about the critical angle be useful in the construction of lenses and prisms used in cameras and telescopes?'

   The teacher should encourage students to answer the questions, promoting a classroom discussion. Additionally, the teacher should reinforce that understanding the critical angle is fundamental to comprehend various everyday optical phenomena.

3. Individual reflection (2 - 3 minutes): The teacher should then propose to students to individually reflect on what they learned in the lesson. For this, the teacher can ask questions like:

   • 'What was the most important concept you learned today about the critical angle?'
   • 'What questions have not been answered for you about the critical angle?'

   The teacher should give a minute for students to think about the questions and then can ask some students to share their answers with the class. This not only allows the teacher to assess students' understanding of the topic but also helps students consolidate what they have learned.

4. Feedback and Closure (1 minute): Finally, the teacher should thank the students for their participation and dedication. The teacher should also inform the students about the content of the next lesson and remind them of any homework or additional reading that may be necessary. Additionally, the teacher should be open to receiving feedback from students about the lesson, encouraging them to share any doubts or suggestions they may have. This helps build a collaborative learning environment and strengthens the teacher-student relationship.

Conclusion (5 - 7 minutes)

1. Summary of contents (2 - 3 minutes): The teacher should recap the main points covered during the lesson, reinforcing the concept of the critical angle and its relation to light refraction. The teacher should also recall the discussed everyday optical phenomena, such as light refraction in water and the separation of colors when passing through a prism. This can be done through a brief theoretical review or through an interactive conversation with the students.

2. Connection between theory, practice, and applications (1 - 2 minutes): The teacher should highlight how the lesson connected theory, practice, and applications. It should be emphasized that the theory of the critical angle was explained clearly and concisely, and that students had the opportunity to apply this knowledge in problem-solving and analysis of practical situations. Additionally, the teacher should reinforce the applications of the critical angle, explaining how it is used in the construction of everyday optical devices, such as lenses and prisms.

3. Extra materials (1 - 2 minutes): The teacher should suggest some extra materials for students who wish to deepen their understanding of the critical angle. These materials may include physics textbooks, scientific articles, educational videos on the internet, and online physics simulators. The teacher should encourage students to use these resources autonomously, as a way to complement learning in the classroom.

4. Importance of the topic (1 minute): Finally, the teacher should emphasize the importance of the critical angle for everyday life. The teacher should explain that knowledge about the critical angle is not only useful for understanding complex optical phenomena, but also can help understand simple everyday situations, such as the reason why we see distorted images when looking at water at an angle. Additionally, the teacher should highlight that understanding the critical angle is fundamental for various practical applications, from the construction of lenses and prisms used in optical devices, to understanding how light behaves in optical fibers, widely used in telecommunications.