Objectives (5 - 7 minutes)

1. Understand the concept of radicals, highlighting the relationship between radicand, radical, and index.
   • Identify the radicand as the number or expression under the radical sign.
   • Recognize the index as the number indicating which root is being extracted.
   • Understand that the radical is the sign indicating the root extraction operation.
2. Learn to simplify radical expressions, reducing the index and radicand to their simplest form.
   • Apply the power property to simplify the radicand.
   • Use factorization to simplify the radicand.
3. Practice solving problems involving radicals, applying the rules of operations with radicals.
   • Perform addition, subtraction, multiplication, and division operations with radicals.
   • Solve equations and inequalities involving radicals.

Secondary Objectives:

1. Develop logical and critical thinking skills when solving problems involving radicals.
2. Reinforce understanding of previous mathematical concepts, such as exponentiation and operations with polynomials.
3. Promote students' self-confidence and autonomy in solving complex mathematical problems.

Introduction (10 - 12 minutes)

1. Review of Previous Concepts:

   • The teacher begins the lesson with a brief review of exponentiation concepts, emphasizing the meaning of a root and the difference between radicand and index. This review is essential for students to understand the new concept that will be presented: radicalization. (3 - 4 minutes)

2. Problem Situations:

   • The teacher proposes two problem situations involving the concept of radicalization to stimulate students' reasoning:
     • Situation 1: 'Imagine you have a square area of 16m². How long is the side of this square?' (2 - 3 minutes)
     • Situation 2: 'If the volume of a cube is 27m³, how long is the side of this cube?' (2 - 3 minutes)

3. Contextualization:

   • The teacher contextualizes the importance of studying radicalization, explaining that it is essential for solving many practical problems, such as those presented in the problem situations. Additionally, it emphasizes that radicalization is widely used in various areas of science and engineering, such as physics and computing. (1 - 2 minutes)

4. Topic Introduction:

   • The teacher introduces the topic of the lesson, radicalization, explaining that it is an operation inverse to exponentiation, meaning that while exponentiation multiplies a number by itself several times, radicalization extracts the root of a number. (1 - 2 minutes)

5. Curiosities and Applications:

   • The teacher shares with students some curiosities and applications of radicalization to spark their interest:
     • Curiosity 1: 'Did you know that the radical symbol (√) was introduced by the German mathematician Christoph Rudolff in the 16th century? Before him, the letter R was used for the root.'
     • Application 1: 'Radicalization is widely used in physics. For example, the speed of light in a vacuum, which is approximately 299,792,458 meters per second, can be represented as √(8,987 x 10^16).' (1 - 2 minutes)

Development (20 - 25 minutes)

1. Radicalization Theory (5 - 7 minutes)

   • The teacher begins the theoretical explanation of radicalization by reinforcing the idea that radicalization is the inverse operation of exponentiation.
   • Next, it explains that radicalization is composed of three elements: the radical (√), the index, and the radicand.
   • The teacher clarifies that the radical (√) indicates that we are performing the radical operation, the index indicates which root we are taking, and the radicand is the number or expression under the radical sign.
   • Using practical examples, the teacher demonstrates how to identify each of these elements in a radical expression.

2. Simplification of Radical Expressions (5 - 7 minutes)

   • The teacher explains that one of the main objectives of radicalization is to simplify radical expressions, that is, to reduce the index and radicand to their simplest form.
   • The teacher demonstrates how to do this, using the power property and factorization.
   • First, the teacher explains that if the index is an even power, we can simplify the radicand by dividing the index by the power.
   • Next, the teacher shows how factorization can be used to simplify the radicand.
   • The teacher uses step-by-step examples to illustrate each of these processes.

3. Operations with Radicals (5 - 7 minutes)

   • After explaining the simplification of radical expressions, the teacher moves on to operations with radicals.
   • The teacher explains that we can only add or subtract radicals if they have the same index and the same radicand.
   • The teacher demonstrates how to do this, using step-by-step examples.
   • Next, the teacher explains that we can multiply and divide radicals normally, as if they were numbers.
   • The teacher uses examples to illustrate each of these processes.

4. Problem Solving with Radicals (5 - 7 minutes)

   • Finally, the teacher moves on to solving problems with radicals.
   • The teacher explains that to solve a problem with radicals, we must follow the order of operations, simplify the radical expressions, and, if necessary, apply the rules of operations with radicals.
   • The teacher demonstrates this process, using step-by-step examples.
   • To reinforce learning, the teacher proposes some problems for students to solve in pairs.

5. Review and Clarification of Doubts (3 - 5 minutes)

   • The teacher concludes the lesson's Development by reviewing the main points covered and clarifying any doubts students may have.

This Development of the lesson allows students to understand the concept of radicalization, learn to simplify radical expressions, perform operations with radicals, and solve problems involving radicals. Additionally, the use of practical examples and the opportunity to solve problems in pairs help make learning more meaningful and reinforce the concepts learned.

Return (8 - 10 minutes)

1. Concept Review (3 - 4 minutes):

   • The teacher begins this stage by asking students to share their perceptions of the lesson, highlighting the points they liked the most or found challenging. This allows the teacher to assess the impact of the lesson and make adjustments for future classes.
   • Next, the teacher briefly reviews the key concepts covered in the lesson, reinforcing the idea that radicalization is the inverse operation of exponentiation, and that it is composed of three elements: the radical (√), the index, and the radicand. The teacher also reviews the main rules for simplification and operations with radicals.
   • The teacher encourages students to ask questions and clarify any doubts they may have.

2. Connection to Practice (2 - 3 minutes):

   • The teacher suggests that students reflect on how they can apply what they learned in the lesson to everyday situations or other subjects. For example:
     • Situation 1: 'Can you think of a situation in everyday life where you would need to simplify a radical expression? How would you solve this situation now, after learning about radicalization?'
     • Situation 2: 'Can you think of a situation where radicalization could be useful in other subjects, such as physics, chemistry, or biology?'
   • The teacher encourages students to share their reflections, promoting a classroom discussion.

3. Individual Reflection (2 - 3 minutes):

   • The teacher suggests that students engage in individual reflection on what they learned in the lesson. To do this, the teacher can ask the following questions:
     1. 'What was the most important concept you learned today?'
     2. 'What questions have not been answered yet?'
   • The teacher gives students a minute to think about these questions. Then, asks some students to share their answers with the class.

4. Feedback and Closure (1 minute):

   • The teacher thanks the students for their participation and asks them to provide feedback on the lesson, highlighting what they liked the most and what they think can be improved.
   • The teacher concludes the lesson by reinforcing the importance of studying radicalization and encouraging students to continue practicing and studying the subject at home.

This final Return allows students to consolidate what they learned in the lesson, reflect on the application of concepts in real situations, and clarify any doubts they may have. Additionally, students' feedback helps the teacher improve their lessons and adapt their teaching methods to the class's needs.

Conclusion (5 - 7 minutes)

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

   • The teacher begins the Conclusion of the lesson by summarizing the covered contents. Recaps the concept of radicalization, highlighting the relationship between radicand, radical, and index. Reminds the importance of simplifying radical expressions and the rules for operations with radicals.
   • The teacher reinforces that radicalization is a powerful tool for solving complex mathematical problems, and that mastering this skill is essential for success in many areas of science and engineering.

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

   • The teacher emphasizes how the lesson connected theory, practice, and applications.
   • Explains that by learning the theory of radicalization, students were able to understand the logic behind operations with radicals.
   • By practicing the simplification of radical expressions and solving problems with radicals, students were able to apply this theory in a practical way.
   • By discussing the applications of radicalization, students were able to understand how these concepts are relevant to everyday life and other subjects.

3. Extra Materials (1 minute):

   • The teacher suggests some extra materials for students who want to deepen their knowledge of radicalization.
   • It could be a math book, an online education site, a math channel on YouTube, among other resources.
   • The teacher can also provide a list of exercises for students to practice what they have learned.

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

   • Finally, the teacher emphasizes the importance of the subject.
   • Explains that radicalization is not just another mathematical rule, but a powerful tool for solving real-world problems.
   • Can give examples of how radicalization is used in various areas, such as physics, engineering, computing, and economics.
   • The teacher encourages students to continue studying and practicing, and reinforces that, with effort and dedication, they can master radicalization and become math masters.

This Conclusion allows students to consolidate what they learned in the lesson, understand the relevance of the presented concepts, and be motivated to continue studying and practicing. Additionally, by suggesting extra materials and reinforcing the importance of the subject, the teacher demonstrates commitment to students' learning and encourages them to seek knowledge autonomously.