Objectives (5 - 7 minutes)

1. Understand the concept of radicals and how it is applied in solving mathematical problems.
2. Learn to use the properties of radicals, such as the multiplication property and the power property, to simplify radical expressions.
3. Develop problem-solving skills by applying the knowledge acquired about radicals in practical situations.

Secondary Objectives:

1. Stimulate critical thinking and logical reasoning skills in students, encouraging them to seek different ways to solve problems.
2. Foster active participation of students, promoting discussions in the classroom and the completion of practical activities to deepen the understanding of the content.
3. Promote students' autonomy, encouraging them to seek additional knowledge about radicals outside the school environment through readings and complementary research.

Introduction (10 - 15 minutes)

1. Review of Previous Content: The teacher will start the lesson by reviewing the concepts of exponentiation and square root, which are fundamental for understanding radicals. Students will be asked to solve some expressions involving powers and roots to recall and consolidate this knowledge.

2. Problem-Solving Scenarios: The teacher will present two problem-solving scenarios to arouse students' interest and contextualize the lesson's theme:

   • Problem 1: "Suppose you need to make a square hole in a wooden board. How could you calculate the length of the square's side if you knew the area of the hole?"
   • Problem 2: "Imagine you are drawing a geometric figure on a poster. You need to calculate the length of one side of this figure, which is square. How would you do that if you knew the figure's area?"

3. Contextualization: The teacher will explain that radical is a fundamental mathematical tool in various areas of knowledge, such as physics, engineering, and architecture. Furthermore, it will be emphasized that the ability to simplify radical expressions is useful in everyday situations, such as solving problems involving areas and volumes.

4. Introduction to the Topic: The teacher will then introduce the concept of radicals, explaining that it is an operation inverse to exponentiation. To make the subject more interesting and appealing, the teacher may present some curiosities or practical applications of radicals, such as:

   • Curiosity 1: "Did you know that radicals were introduced in mathematics to solve problems involving geometric shapes?"
   • Curiosity 2: "Radicals are used to calculate the distance between two points on a Cartesian plane, in a concept known as 'Euclidean distance.'"

5. Engaging Students' Attention: To conclude the Introduction and capture students' attention, the teacher may propose two radical challenges:

   • Challenge 1: "What is the square root of 144?"
   • Challenge 2: "If the area of a square is 81 cm², what is the length of each side?"

The teacher should encourage students to try to solve the challenges, reinforcing the idea that radical is a powerful tool to simplify calculations and solve problems.

Development (20 - 25 minutes)

1. Activity 1: Roots in Nature (10 - 12 minutes)

   • The teacher will divide the class into groups of up to 5 students and provide each group with a sheet of paper, pencils, and a list of numbers.
   • The students' task will be to find the square root of each number on the list.
   • After completion, the groups should discuss and share with the class how they decided to calculate the roots, what strategies they used, and if they encountered any difficulties.
   • This activity aims to make students aware of the presence of square roots in their surroundings and develop their mental calculation skills.

2. Activity 2: Square Area Challenge (10 - 12 minutes)

   • The teacher will distribute a set of playing cards to each group of students. Each card will represent a square of different areas (for example, a square with the card 9 represents a square with an area of 9 cm²).
   • The challenge for students will be to organize the cards in ascending order according to the length of each square's side.
   • After completing the activity, the groups should present their solutions and explain the reasoning used. The teacher will correct and discuss the solutions with the class.
   • This activity aims to reinforce the concept of radicals and show students a practical application in a playful context.

3. Activity 3: Simplifying Radical Expressions (10 - 12 minutes)

   • The teacher will provide each group of students with a series of radical expressions to be simplified. The expressions may involve multiplication, division, and exponentiation operations.
   • Students must work together to simplify the expressions. They can use the radical properties learned in class to facilitate the process.
   • After completing the activity, the groups should share their solutions with the class and explain the reasoning used.
   • This activity aims to consolidate students' knowledge of radicals and their properties, as well as develop their problem-solving skills.

During the development of the activities, the teacher should circulate around the classroom, observing the groups' progress, offering help when necessary, and encouraging everyone's participation. Additionally, it is important for the teacher to promote the exchange of ideas and debate among students so that they can learn from each other and develop their communication and argumentation skills.

Return (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher will invite each group to share their solutions or conclusions from the activities carried out. Each group will have up to 3 minutes to present.
   • During the presentations, the teacher should encourage other students to ask questions and make comments to promote a rich and engaging discussion.
   • The teacher should highlight the different strategies used by the groups and reinforce the importance of thinking creatively and seeking different approaches to solving mathematical problems.

2. Connection to Theory (2 - 3 minutes)

   • After the presentations, the teacher will briefly review the properties of radicals and the technique of simplifying radical expressions, connecting them to the solutions presented by the groups.
   • The teacher should emphasize how understanding these concepts allowed students to simplify radical expressions and solve the problems proposed in the activities.
   • Furthermore, the teacher should reinforce that radical is a powerful mathematical tool useful in various everyday situations and other disciplines.

3. Individual Reflection (2 - 3 minutes)

   • To conclude the lesson, the teacher will propose that students make a brief reflection on what they have learned.
   • The teacher can ask questions such as: "What was the most important concept you learned today?", "What questions have not been answered yet?", and "How can you apply what you learned today in your daily life?".
   • Students will have a minute to think about the questions, and the teacher can ask some students to share their answers with the class.
   • This reflection stage is important for students to consolidate what they have learned, identify possible gaps in their understanding, and realize the relevance of the content learned for their lives.

During the Return, the teacher should encourage the participation of all students, ensuring that each one has the opportunity to share their ideas and doubts. Additionally, the teacher should be receptive to students' questions and comments, demonstrating that they value and respect each contribution. At the end of the lesson, the teacher should reinforce the most important concepts and clarify any remaining doubts.

Conclusion (5 - 7 minutes)

1. Summary and Recapitulation (2 - 3 minutes)

   • The teacher will start the Conclusion by recalling the main concepts covered in the lesson. This includes the definition of radicals, the properties of radicals (multiplication property and power property), and the technique of simplifying radical expressions.
   • Furthermore, the teacher should emphasize the importance of understanding and being able to apply these concepts, reinforcing that radical is an essential mathematical tool in various areas of knowledge and everyday life.

2. Connecting Theory with Practice (1 - 2 minutes)

   • The teacher should explain how the lesson connected the theory of radicals with practice, referring to the activities carried out and the problem-solving scenarios proposed.
   • The teacher should reinforce that the practical activities helped students visualize and better understand the theoretical concepts, while developing their problem-solving skills and logical reasoning.

3. Additional Materials (1 minute)

   • The teacher can suggest some additional study materials for students, such as books, videos, websites, and math apps that address the topic of radicals.
   • Additionally, the teacher can encourage students to practice more at home by solving radical exercises and seeking practical applications of this concept.

4. Importance of the Content (1 - 2 minutes)

   • To conclude the lesson, the teacher should emphasize the importance of the content learned for daily life and other disciplines.
   • The teacher can give examples of everyday situations where radical is useful, such as calculating areas and volumes, solving engineering and architecture problems, or understanding physics concepts.
   • The teacher can also mention how radical connects with other math topics, such as exponentiation, fractions, and algebra.

At the end of the Conclusion, the teacher should reinforce that the lesson was an Introduction to the topic of radicals and that students will have the opportunity to deepen their understanding of this concept in future lessons. The teacher should encourage students to continue studying and practicing, ensuring that they know the teacher is available to help with any questions or difficulties that may arise.