Objectives (5 - 7 minutes)

1. Understand the definition and usefulness of the Transformation of Product into Sum in trigonometry.
   • Learn about the applications of the Transformation of Product into Sum in real situations and in other areas of mathematics.
2. Master the application of the Transformation of Product into Sum in problem-solving.
   • Practice solving exercises involving the Transformation of Product into Sum to develop the ability to apply this concept in different contexts.
3. Develop critical thinking and logical-mathematical reasoning skills.
   • Through problem-solving, students will be encouraged to think logically and enhance their problem-solving skills.

Secondary objectives:

• Promote interaction among students, encouraging discussion and exchange of ideas during problem-solving.
• Develop self-learning skills, encouraging students to seek solutions to problems autonomously.
• Encourage studying at home by providing support materials for students to prepare for the lesson.

Introduction (10 - 15 minutes)

1. Review of previous content:

   • The teacher should start the lesson by reviewing fundamental concepts of trigonometry, such as addition and subtraction formulas for angles, and duplication formulas for angles. These concepts are essential for understanding the Transformation of Product into Sum.
   • The teacher may propose some quick review exercises to verify if students are ready to move on to the new content.

2. Problem situations:

   • The teacher can present two situations involving the transformation of product into sum, but without indicating that this is the technique that will be used to solve them. For example:
     • Situation 1: 'Suppose you need to calculate the value of sin(3x) * sin(4x). How would you do that?'
     • Situation 2: 'Imagine you have the expression cos(3x) * cos(4x). How could you simplify it?'

3. Contextualization:

   • The teacher should explain that the Transformation of Product into Sum is an important tool in trigonometry and is frequently used in areas such as physics and engineering to simplify calculations and solve complex problems.
   • Some examples of real-world applications can be mentioned, such as the use of the transformation of product into sum in the analysis of sound waves and in solving engineering problems.

4. Introduction to the topic:

   • The teacher should introduce the topic of the lesson, explaining that the Transformation of Product into Sum is a technique that allows expressing the product of two trigonometric functions as the sum of two trigonometric functions.
   • The teacher can show the general formula of the Transformation of Product into Sum and explain that it can be used to simplify complex expressions and facilitate the resolution of trigonometric equations.
   • To spark students' interest, the teacher can share some curiosities or interesting applications of the Transformation of Product into Sum. For example, it can be mentioned that this technique was developed by Isaac Newton and is widely used in physics to analyze wave motion.

Development (20 - 25 minutes)

1. Playful activity: 'Trigonometric Treasure Hunt' (10 - 12 minutes)

   • The teacher should divide the class into groups of 3 to 4 students and give each group a set of trigonometric puzzles involving the Transformation of Product into Sum. Each puzzle should consist of a series of trigonometric equations that students must simplify using the Transformation of Product into Sum to find the correct answer.
   • Each puzzle should be presented on a separate card, and the next card should only be given to the group when they have correctly solved the equation on the previous card.
   • The last card of each set of puzzles should lead the students to a 'treasure,' which can be a candy, a sticker, or any other small prize.
   • The teacher should circulate around the room, assisting groups that are struggling and ensuring that all students are engaged in the activity.
   • This playful activity aims to make learning the Transformation of Product into Sum more fun and engaging, as well as to provide students with the opportunity to practice applying this concept in a practical and meaningful way.

2. Group discussion: 'Applications of the Transformation of Product into Sum' (5 - 7 minutes)

   • After the conclusion of the playful activity, the teacher should promote a group discussion on the applications of the Transformation of Product into Sum in real life and in other areas of mathematics, such as physics and engineering.
   • The teacher can start the discussion by asking open-ended questions, such as 'Can you think of a daily life situation where the Transformation of Product into Sum could be useful?' or 'How can the Transformation of Product into Sum be used to simplify calculations in other areas of mathematics?'.
   • The goal of this discussion is for students to realize the relevance and usefulness of what they are learning, as well as to encourage them to make connections between theory and practice.

3. Problem-solving: 'Transformation of Product into Sum Challenge' (5 - 6 minutes)

   • To consolidate learning, the teacher should propose a problem-solving challenge involving the application of the Transformation of Product into Sum.
   • The challenge can be presented in the form of a real or hypothetical problem that students must solve using the learned technique. For example, 'Suppose you are designing an amusement park and need to calculate the maximum height that a Ferris wheel can reach without passengers feeling uncomfortable due to centrifugal force. How could you use the Transformation of Product into Sum to solve this problem?'.
   • The teacher should encourage students to work together to solve the challenge, promoting collaboration and exchange of ideas among group members. The teacher should circulate around the room, assisting groups that are struggling and ensuring that all students are actively participating in solving the problem.

Return (8 - 10 minutes)

1. Group discussion: 'Sharing Solutions' (3 - 4 minutes)

   • The teacher should gather all groups and promote a classroom discussion about the solutions found by each group for the proposed challenge.
   • Each group should have the opportunity to share their problem-solving strategies, the difficulties encountered, and how they overcame them.
   • The teacher should encourage the participation of all students, asking questions and requesting clarifications to ensure that everyone understood the solutions presented.
   • This activity promotes the exchange of experiences among students, stimulates reflection on the problem-solving process, and allows the teacher to identify possible gaps in students' understanding that need to be addressed.

2. Connection with theory: 'What did we learn?' (2 - 3 minutes)

   • After discussing the solutions, the teacher should summarize what was learned, reinforcing the main concepts and the importance of the Transformation of Product into Sum in trigonometry and in other areas of mathematics.
   • The teacher can revisit the situations presented in the Introduction of the lesson and explain how the Transformation of Product into Sum can be used to simplify trigonometric expressions, solving the proposed problems.
   • The teacher should also reinforce the importance of critical thinking and logical-mathematical reasoning in problem-solving, highlighting how these skills were developed during the lesson.

3. Individual reflection: 'Thinking about the Lesson' (2 - 3 minutes)

   • To conclude the lesson, the teacher should propose a moment of individual reflection, where students will have the opportunity to think about what they learned and what questions they still have.
   • The teacher should ask questions that encourage reflection, such as 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • Students should be encouraged to write down their reflections, which can be useful to guide their autonomous study and provide feedback to the teacher on the progress of the class's learning.
   • This reflection activity allows students to consolidate what they have learned, identify possible doubts, and reflect on the learning process. Additionally, it provides valuable feedback to the teacher on the effectiveness of the lesson and the learning needs of the students.

Conclusion (5 - 7 minutes)

1. Recap of Contents (2 - 3 minutes):

   • The teacher should summarize the main points covered during the lesson, recalling the definition of the Transformation of Product into Sum and its application in simplifying trigonometric expressions and solving problems.
   • It should also reinforce the importance of the concepts reviewed at the beginning of the lesson, such as addition and subtraction formulas for angles, and duplication formulas for angles, for understanding the Transformation of Product into Sum.
   • A board or slide can be used to present a visual summary of the concepts, highlighting the formulas and the steps of the Transformation of Product into Sum.

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

   • The teacher should emphasize how the lesson connected the theory, practice, and applications of the Transformation of Product into Sum.
   • It can be highlighted how the playful activity allowed students to apply the theory in a practical and meaningful way, and how the problem-solving challenge provided the opportunity to explore the applications of the Transformation of Product into Sum.
   • Additionally, it can be mentioned how the group discussion allowed students to share their approaches and learn from each other's strategies.

3. Extra Study Materials (1 - 2 minutes):

   • The teacher should suggest additional study materials for students who wish to deepen their knowledge of the Transformation of Product into Sum.
   • These materials may include textbooks, educational videos online, interactive math websites, and practice exercises.
   • The teacher should emphasize that continuous practice is essential for understanding and effectively applying the Transformation of Product into Sum.

4. Importance of the Subject in Daily Life (1 minute):

   • To conclude the lesson, the teacher should emphasize the relevance of the Transformation of Product into Sum in daily life.
   • It can be mentioned how the ability to simplify trigonometric expressions and solve trigonometric equations is useful in various areas, such as physics, engineering, architecture, and computer science.
   • Additionally, it can be highlighted that trigonometry, and the Transformation of Product into Sum in particular, is an essential component of the mathematics curriculum and is often tested in standardized exams and college entrance exams.