Objectives (5 - 7 minutes)

1. Understanding the concept of Transformation of Product into Sum: The teacher must ensure that students fully understand what the Transformation of Product into Sum in trigonometry is. This involves understanding the relationship between the products of trigonometric functions and their corresponding sums.

2. Identification of Practical Applications: The teacher should guide students to identify and understand real-life situations where the Transformation of Product into Sum can be applied. This may include problems in physics, engineering, architecture, among others.

3. Resolution of Practical Exercises: Students should be able to apply the concept of Transformation of Product into Sum to solve practical exercises. This involves identifying the type of problem and applying the appropriate formula to obtain the correct solution.

   • Secondary Objective: In addition to understanding and solving practical exercises, students should also be able to explain step by step how the Transformation of Product into Sum was applied.

The teacher should clarify that these Objectives are fundamental for a comprehensive understanding of trigonometry and for the ability to solve complex problems involving this area of mathematics.

Introduction (10 - 15 minutes)

1. Review of Previous Content:

   • The teacher should start the lesson by reviewing the concepts of trigonometric functions, focusing specifically on the sine and cosine functions, and their properties.
   • It is important to reinforce the idea that the sine and cosine functions are periodic, with a period of 2π, and vary between -1 and 1.

2. Initial Problem Situation:

   • The teacher can present two problem situations that will serve as a starting point for the explanation of the theory:
     1. "Imagine you are building a bridge and need to calculate the force that the tide exerts on it. How could you use trigonometry to solve this problem?"
     2. "Suppose you are studying ocean waves and need to determine the maximum height a wave can reach. How could trigonometry help you in this situation?"

3. Contextualization of the Subject's Importance:

   • The teacher should explain that trigonometry is an essential tool in various areas, such as engineering, physics, architecture, among others, to solve problems involving spatial and periodic relationships, as presented in the problem situations.
   • It is important to emphasize that the ability to transform products into sums is particularly useful for simplifying complex calculations involving trigonometric functions.

4. Introduction to the Topic:

   • The teacher can introduce the topic of the lesson by explaining that the Transformation of Product into Sum is a technique that allows transforming a product of trigonometric functions into a sum of trigonometric functions.
   • To spark students' interest, the teacher can share some curiosities or practical applications of the Transformation of Product into Sum, such as the fact that this technique was developed by Euler, one of the greatest mathematicians in history, and is widely used in fields like quantum physics and wave theory.

Development (20 - 25 minutes)

1. Activity "Building a Bridge" (10 - 12 minutes):

   • Students will be divided into groups of up to 5 people.
   • Each group will receive a set of materials (toothpicks, balloons, glue, etc.) and a set of instructions to build a small bridge.
   • The instructions will contain a problem related to the bridge's resistance that can only be solved using the Transformation of Product into Sum.
   • Students will have to apply the concept of Transformation of Product into Sum to solve the problem and, thus, complete the construction of the bridge.
   • The teacher will circulate around the room, assisting groups that encounter difficulties and ensuring that everyone understands the concept and applies the technique correctly.
   • At the end of the activity, each group will present their bridge, explaining the problem they had to solve and how they used the Transformation of Product into Sum to find the solution.

2. Activity "Height of Waves" (10 - 12 minutes):

   • After the previous activity, students will already have a practical understanding of the Transformation of Product into Sum. Now, they will be challenged to apply this concept in a new context.
   • The teacher will present a problem related to the height of ocean waves that can only be solved using the Transformation of Product into Sum.
   • Students, again in groups, will have to work together to solve the problem, applying the learned concept.
   • The teacher will circulate around the room, assisting groups and clarifying any doubts that may arise.
   • At the end of the activity, each group will present their solution, explaining the problem they had to solve and how they used the Transformation of Product into Sum to find the solution.

3. Discussion and Synthesis (5 - 6 minutes):

   • After the group presentations, the teacher will lead a classroom discussion to reinforce the concept of Transformation of Product into Sum and clarify any remaining doubts.
   • The teacher must ensure that all students understand the relevance of the topic for trigonometry and for the practical applications presented.
   • To conclude the Development of the lesson, the teacher will provide a brief summary of what was learned, highlighting the most important points and emphasizing the importance of continuous practice for improving trigonometry skills.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher should gather all students and promote a group discussion, where each team will have up to 2 minutes to share their solutions or conclusions from the activities "Building a Bridge" and "Height of Waves".
   • The focus should be on the practical application of the concept of Transformation of Product into Sum and how the problems were approached and solved.
   • The teacher should encourage students to ask questions and make comments, thus promoting the exchange of ideas and deepening everyone's understanding.

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

   • After the discussion, the teacher should bridge the gap between the activities carried out and the theory presented at the beginning of the lesson.
   • The goal is for students to understand how the Transformation of Product into Sum, which was initially presented as a mathematical rule, can be applied in a practical and useful way in different contexts.
   • The teacher can reinforce the idea that mathematics is not just a theoretical discipline, but a powerful tool for solving real-world problems.

3. Individual Reflection (2 - 3 minutes):

   • To conclude the lesson, the teacher should propose that students reflect individually on what was learned.
   • The teacher can ask questions like: "What was the most important concept learned today?" and "What questions have not been answered yet?".
   • The goal is for students to internalize the acquired knowledge and identify any areas that may need review or further exploration.
   • The teacher can collect students' responses in writing or through a brief classroom discussion, depending on the available time and the level of student participation.

4. Teacher's Feedback (1 minute):

   • To conclude, the teacher should provide overall feedback on the lesson, highlighting strengths and areas for improvement.
   • The teacher can praise students' efforts, active participation, and understanding demonstrated during the activities.
   • Additionally, the teacher can remind students about the importance of reviewing the material at home and practicing regularly to consolidate learning.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes):

   • The teacher should start the Conclusion of the lesson by giving a brief summary of the main points covered. This includes recalling the definition of Transformation of Product into Sum, its importance in trigonometry, and its practical applications.
   • It is important for the teacher to make connections between theory, practical activities, and examples, reinforcing how the Transformation of Product into Sum can be used to solve real and complex problems.
   • The teacher should also highlight the skills and competencies developed by students during the lesson, such as the ability to work in groups, apply mathematical knowledge in practical contexts, and explain step by step the reasoning used to solve the proposed problems.

2. Additional Materials (1 - 2 minutes):

   • The teacher should suggest some complementary study materials for students who wish to deepen their knowledge of the Transformation of Product into Sum. This may include textbooks, explanatory videos, math websites, and additional exercises.
   • It is important for the teacher to guide students to use these materials autonomously, as part of continuous and self-directed study.

3. Connection with Everyday Life (1 - 2 minutes):

   • To conclude, the teacher should reinforce the importance of the Transformation of Product into Sum in everyday life, making connections with real or common practical situations.
   • For example, the teacher can mention how this technique is useful for understanding and predicting natural phenomena, such as tides or ocean waves, or for solving common problems in various areas, such as engineering, physics, or architecture.
   • The goal is for students to realize that mathematics is not just an abstract discipline distant from reality, but a powerful and relevant tool for understanding and interacting with the world around us.

4. Closure (1 minute):

   • To end the lesson, the teacher should thank the students for their participation and encourage them to continue studying and practicing. For example, the teacher can say: "Congratulations to everyone for today's work. Remember that practice is fundamental for learning mathematics. Keep studying and practicing, and do not hesitate to seek help if you have any doubts. See you in the next lesson!".