Objectives (5 - 7 minutes)

1. Understanding of basic operations with real numbers: Students should be able to understand and perform basic operations (addition, subtraction, multiplication, and division) with real numbers, correctly applying the rules of sign and order of operations.

2. Identification of properties of operations with real numbers: Students should be able to identify and apply properties of operations with real numbers, such as commutativity, associativity, and distributivity.

3. Solving numerical expressions: Students should be able to solve numerical expressions involving operations with real numbers, using the acquired skills to simplify the expressions and obtain the final result.

   Secondary objectives:

   • Development of critical thinking and problem-solving: Through the practice of operations with real numbers and the resolution of numerical expressions, students will be encouraged to develop critical thinking and problem-solving skills, essential for the study of mathematics and for everyday life.

   • Strengthening of logical-mathematical reasoning: Solving operations with real numbers requires the use of logical-mathematical reasoning, which will contribute to strengthening these skills in students.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher should start the lesson by reviewing previous concepts, such as the system of real numbers, properties of real numbers, and the basic operations of addition, subtraction, multiplication, and division. This review can be done through a quick oral quiz or a written review exercise that students must solve individually.

2. Problem Situations: To introduce the topic and arouse students' interest, the teacher can propose two problem situations:

   • Situation 1: "If we add -7 and 3, what will be the result? And if we subtract 7 from 3? How can we represent these operations mathematically?"
   • Situation 2: "Imagine you have a debt of -50 reais. If someone gives you 20 reais, will your debt increase or decrease? And if you win the lottery and receive another 50 reais, will your debt increase or decrease? How can we represent these situations mathematically?"

3. Contextualization: The teacher should explain that operations with real numbers are used in various everyday situations, such as in finance, physics, chemistry, among others. For example, in finance, we can use the addition and subtraction of real numbers to calculate profits and losses. In physics, we can use the multiplication and division of real numbers to calculate speeds and accelerations. In chemistry, we can use operations with real numbers to calculate quantities of reactants and products in a chemical reaction.

4. Curiosities: To spark students' interest, the teacher can share some curiosities about real numbers and their operations:

   • Curiosity 1: "Did you know that the addition of real numbers is commutative? This means that no matter the order in which we add the numbers, the result will always be the same."
   • Curiosity 2: "And what about distributivity in multiplication? When multiplying a real number by a sum or difference of other real numbers, we can distribute the multiplication to each number in the sum or difference. It's as if the multiplication 'passes through' the plus or minus sign."

Development (20 - 25 minutes)

1. Theory Presentation (10 - 12 minutes):

   • Definition of Operations with Real Numbers: The teacher should start the theoretical part by explaining that operations with real numbers are the same as operations with natural, integer, and rational numbers: addition, subtraction, multiplication, and division. However, it is important to emphasize that when performing these operations, we must take into account the rules of signs. For example, when adding two real numbers, we must add their absolute values and keep the sign of the number with the greater absolute value.

   • Rules of Sign: The teacher should review the rules of sign, explaining that the sum of two real numbers with opposite signs results in a negative number, and that the sum of two real numbers with the same sign results in a positive number. Additionally, it should be emphasized that the sign of a real number can be changed by multiplying or dividing it by a negative number.

   • Commutativity, Associativity, and Distributivity: The teacher should explain that operations with real numbers are commutative (the order of numbers does not alter the result), associative (the way of grouping numbers does not alter the result), and distributive (multiplication distributes to each number in the sum or difference). To illustrate these properties, the teacher can use practical examples and/or drawings.

2. Problem Solving (10 - 13 minutes):

   • Example 1: The teacher should present an example of a numerical expression involving the four operations with real numbers, such as: (-5 + 3) x (2 - 4) / 2. The teacher should explain step by step how to solve this expression, using the rules of sign, the order of operations (parentheses, exponentiation and root extraction, multiplication and division, and finally, addition and subtraction) and the properties of operations with real numbers. The teacher should encourage students to participate in the resolution, asking questions and requesting them to explain their reasoning.

   • Example 2: The teacher should present a second example, but this time, the teacher should pause after each step of the resolution, so that students can try to solve the expression on their own. The teacher should walk around the classroom, observing the students' work and offering help when needed.

3. Practical Activities (5 - 7 minutes):

   • Practice Exercises: The teacher should hand out a list of exercises involving operations with real numbers to the students. The exercises should vary in difficulty and format, including numerical expressions, real-world problems, and theoretical questions. Students should work individually or in small groups to solve the exercises. The teacher should walk around the classroom, observing the students' work and offering help when needed.

   • Correction of Exercises: At the end of the activities, the teacher should correct the exercises together with the class, explaining the most common errors and reinforcing key concepts. The teacher should take this opportunity to clarify any doubts that may have arisen during the activity.

Return (8 - 10 minutes)

1. Review of Key Concepts (3 - 4 minutes): The teacher should start this stage by reviewing the key concepts covered during the lesson, such as the basic operations with real numbers, the rules of sign, the properties of operations (commutativity, associativity, and distributivity), and the resolution of numerical expressions. The teacher can do this through a quick oral quiz, asking students to explain each concept in their own words.

2. Connection to Practice (2 - 3 minutes): The teacher should then connect theory to practice by asking students how they could apply what they have learned in everyday situations or in other disciplines. For example, the teacher can ask: "How could you use operations with real numbers to calculate change in a store?" or "In which subject do you believe operations with real numbers are most useful and why?" The teacher should encourage students to think critically and express their opinions.

3. Individual Reflection (2 - 3 minutes): The teacher should then propose that students reflect individually on what they have learned during the lesson. For 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 for you yet?"

   Students should write down their answers on a piece of paper or in their notebook. The teacher should remind students that there are no right or wrong answers and that the goal of this activity is to help them consolidate their learning and identify areas that may need review or additional study.

4. Sharing and Conclusion (1 minute): Finally, the teacher should invite some students to share their reflections with the class. The teacher should praise the students' efforts and reinforce the importance of continuous study and critical thinking. The teacher should end the lesson by highlighting the main points covered and suggesting additional materials for students who wish to deepen their studies.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes): The teacher should start the Conclusion of the lesson by providing a brief summary of the main points covered. This includes the definition of operations with real numbers, the rules of sign, the properties of operations (commutativity, associativity, and distributivity), the order of operations, and the resolution of numerical expressions. The teacher should emphasize the importance of understanding and correctly applying these concepts for solving mathematical problems and for everyday life.

2. Connection of Practice and Theory (1 - 2 minutes): Next, the teacher should reinforce the connection between the presented theory and practice. This can be done through examples of how operations with real numbers are used in everyday situations and in other disciplines. The teacher should emphasize that mathematics is not just a theoretical discipline, but a powerful tool for solving problems and understanding the world around us.

3. Additional Materials (1 - 2 minutes): The teacher should then suggest some additional materials for students who wish to deepen their studies. This may include math books, educational websites, explanatory videos, math games, among others. For example, the teacher may recommend the use of educational game websites that offer interactive activities with operations with real numbers, such as Khan Academy, Math Playground, and Math Games.

4. Relevance of the Subject (1 minute): Finally, the teacher should summarize the relevance of the subject presented for the students' daily lives. The teacher can reinforce the fact that operations with real numbers are used in various everyday situations, such as calculating change, solving financial problems, interpreting scientific data, and in many other situations. Additionally, the teacher should emphasize that studying operations with real numbers helps develop important skills, such as critical thinking, problem-solving, and logical-mathematical reasoning.