Objectives (5 - 10 minutes)

1. Understand the Law of Cosines:

   • Identify the variables involved in the Law of Cosines: the sides of a triangle and the angle opposite one of these sides.
   • Understand the relationship between the variables: how the sum of the squares of the two sides adjacent to an angle is equal to the square of the side opposite that angle.
   • Apply the Law of Cosines to calculate the length of a side of a triangle when the lengths of the other two sides and the value of an internal angle are known.

2. Solve problems involving the Law of Cosines:

   • Interpret a problem involving the Law of Cosines, identifying the given information and what is sought.
   • Organize the information and apply the formula of the Law of Cosines to solve the problem.

3. Develop logical and mathematical reasoning skills:

   • Practice deduction and logical reasoning, identifying the necessary steps to solve a problem.
   • Reinforce the importance of studying geometry and trigonometry in solving practical problems.

Introduction (10 - 15 minutes)

1. Review of previous concepts:

   • The teacher should start the lesson by reviewing the concepts of triangle, internal and external angles, and the sum of the internal angles of a triangle, which were previously studied. This is essential for students to understand and apply the Law of Cosines.

2. Problem situations:

   • The teacher should propose two problem situations to arouse students' curiosity and demonstrate the importance of the Law of Cosines. For example:
     • "Suppose you need to calculate the distance between two points on a map, but cannot draw a straight line between them due to an obstacle. How could you use geometry to solve this problem?"
     • "Imagine you are in a boat in the middle of a lake and want to measure the distance between two islands. How could the Law of Cosines help you solve this problem?"

3. Contextualization:

   • The teacher should explain that the Law of Cosines is a fundamental tool in various areas, such as engineering, architecture, navigation, cartography, and even in physics and biology. It allows calculating the length of a side of a triangle when the lengths of the other two sides and the value of an internal angle are known.

4. Presentation of the topic:

   • The teacher should introduce the topic of the lesson, the Law of Cosines, explaining that it is a generalization of the Pythagorean Theorem, which only applies to right triangles. The Law of Cosines, on the other hand, is valid for any type of triangle, whether it is right-angled or not.

5. Curiosities:

   • The teacher can present some curiosities about the Law of Cosines to arouse students' interest. For example, he can talk about the fact that the Law of Cosines can be used to prove other important theorems, such as the Law of Sines and the Heron's Formula.
   • Another interesting curiosity is that the Law of Cosines is one of the few mathematical formulas named after a person who did not discover it. The Persian mathematician Nasir al-Din al-Tusi was the first to publish a version of the formula in the 13th century, but the Indians and the Chinese already knew it long before that.

Development (20 - 25 minutes)

1. Theory Presentation (10 - 15 minutes):

   • The teacher should start by explaining the definition of the Law of Cosines and highlighting its importance. The Law of Cosines states that, in a triangle, the square of one of the sides is equal to the sum of the squares of the other two sides, minus twice the product of these two sides by the cosine of the angle between them. This is a powerful statement, as it allows calculating the length of a side of a triangle when the lengths of the other two sides and the value of an internal angle are known.
   • Next, the teacher should present the formula of the Law of Cosines: c² = a² + b² - 2ab * cos(C), where c is the side opposite angle C, a and b are the other two sides, and C is the angle opposite side c.
   • The teacher should demonstrate the origin of the formula of the Law of Cosines, using the Pythagorean Theorem and the definition of cosine. This will help students understand the formula and memorize it.
   • The teacher should then present some examples of applying the Law of Cosines, solving them step by step and explaining each stage. The examples should be varied, involving different types of triangles and different configurations of information.
   • Finally, the teacher should emphasize the importance of verifying if the provided information is sufficient to apply the Law of Cosines before attempting to solve a problem, and of expressing the final answer with the appropriate precision and in the correct unit.

2. Guided Practice (5 - 10 minutes):

   • The teacher should now propose some guided practice exercises for students to apply the Law of Cosines. The exercises should be of progressive difficulty, starting with simple problems and advancing to more complex ones.
   • During the resolution of the exercises, the teacher should walk around the classroom, observing the students' work, answering questions, and providing immediate feedback. He should also encourage students to work together and discuss their strategies for problem-solving.

3. Group Discussion (5 - 10 minutes):

   • After the guided practice, the teacher should propose a group discussion so that students can share their problem-solving strategies and difficulties.
   • The teacher should guide the discussion by asking questions that stimulate students to think deeply about the material and make connections with what they already know. For example, he can ask: "How do you decide which formula to use in each problem? What are the steps to apply the Law of Cosines? How do you know if the answer is correct?".
   • The teacher should also take advantage of the discussion to clarify any misunderstandings and reinforce the most important concepts.

4. Practical Application (5 - 10 minutes):

   • To conclude the Development stage, the teacher should propose a practical application activity, in which students will have to solve a real-world problem involving the Law of Cosines.
   • The teacher should choose a problem that is relevant to students' lives and that requires the application of the learned concepts. For example, he can propose that students use the Law of Cosines to calculate the height of a tree, the distance between two points on a map, or the distance between two islands in a lake, as in the problem situations from the Introduction.
   • During the problem-solving process, the teacher should walk around the classroom, observing the students' work, answering questions, and providing immediate feedback. He should also encourage students to discuss their problem-solving strategies and difficulties.

Return (10 - 15 minutes)

1. Group Discussion (5 - 7 minutes):

   • The teacher should promote a group discussion, encouraging students to share the solutions they found for the problems proposed in the Practical Application stage.
   • During the discussion, the teacher should question students about the processes they used to arrive at their answers, any difficulties they encountered, and how they overcame them.
   • The teacher should also take advantage of the discussion to clarify any misunderstandings and reinforce the most important concepts.

2. Connection with Theory (3 - 5 minutes):

   • The teacher should make the connection between the practice carried out and the theory presented in the Theory Presentation stage.
   • He should highlight how the Law of Cosines was applied to solve the proposed problems, reinforcing the importance of understanding the formula and knowing how to apply it correctly.
   • The teacher should also take this opportunity to reinforce the most important concepts and clarify any remaining doubts.

3. Individual Reflection (2 - 3 minutes):

   • The teacher should propose that students reflect individually for a minute on the answers to the following questions:
     1. What was the most important concept you learned today?
     2. What questions have not been answered yet?
   • After a minute of reflection, the teacher should ask some students to share their answers with the class. This will allow the teacher to assess what was learned and identify any gaps in students' understanding that need to be addressed in future classes.

4. Feedback and Evaluation (2 - 3 minutes):

   • Finally, the teacher should ask for feedback from students about the lesson. He can ask questions like: "What did you think of today's lesson? What worked well? What could be improved?".
   • The teacher should also evaluate the students' performance during the lesson and their understanding of the material. This can be done through observation, oral questions, or a small written task. The teacher's feedback and evaluation will help guide the planning of future lessons and ensure that students' needs are met.

Conclusion (5 - 10 minutes)

1. Recapitulation (2 - 3 minutes):

   • The teacher should start the Conclusion of the lesson by recapping the main points covered. He should review the definition of the Law of Cosines, the associated formula, and how it is applied to solve geometry problems involving triangles.
   • The teacher should connect the theory presented with the practice carried out, reinforcing the importance of understanding theoretical concepts for the efficient resolution of practical problems.

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

   • Next, the teacher should explain how today's lesson managed to connect theory, practice, and applications. He should show how the theory of the Law of Cosines was applied in practice, in the guided practice exercises, in the group discussion, and in the practical application activity.
   • The teacher should also highlight the real-world applications of the Law of Cosines, showing how it can be used in everyday situations, such as solving the problems of the islands in the lake or the distance between two points on a map.

3. Supplementary Materials (1 - 2 minutes):

   • The teacher should suggest some additional study materials for students who wish to deepen their knowledge of the Law of Cosines. These materials may include math books, explanatory videos online, interactive math websites, or math learning apps.
   • The teacher should encourage students to explore these materials at their own pace and to bring any questions or difficulties to the next lesson.

4. Importance of the Subject (1 - 2 minutes):

   • To conclude the lesson, the teacher should emphasize the importance of the subject presented. He should explain that the Law of Cosines is an essential tool in various areas of science and engineering, and that the ability to apply it correctly can be very valuable.
   • The teacher should also emphasize that, in addition to being useful in practical applications, the study of the Law of Cosines and other geometry and trigonometry concepts can help develop valuable skills, such as logical reasoning, deduction, and problem-solving.
   • Finally, the teacher should motivate students to continue to strive and believe in their own potential, reminding them that mathematics can be challenging, but with practice and study, they will be able to master these concepts and apply them successfully.