Objectives (5-7 minutes)

1. Understanding of the concept of linear equations: Students should be able to understand what a linear equation is and how it is used to represent a comparison situation. They should grasp that a linear equation is an equality between two expressions, both linear, which can be true or false depending on the values assigned to the variables.

2. Application of the substitution method: Students need to be able to apply the substitution method in solving linear equations. They should understand how to replace a variable in an equation and how to simplify the resulting equation.

3. Solving comparison problems: Students should be able to solve comparison problems using linear equations. They should be able to translate a comparison situation into a linear equation, and then solve the equation to find the solution.

Secondary Objectives:

• Development of logical and analytical thinking: By working with linear equations, students will have the opportunity to develop their logical and analytical thinking skills. They will need to analyze the problem, identify the relevant variables, and use logic to solve the equation.

• Practice of basic math skills: Solving linear equations involves using several basic math skills, such as simplifying expressions, operating with negative numbers, and fractions. Students will have the opportunity to practice these skills as they work with linear equations.

Introduction (10-15 minutes)

1. Review of previous content: The teacher should start the lesson by briefly reviewing the previous content that is essential for understanding the current topic. This may include the definition of equations, variables, and linear expressions, as well as the methods of solving linear equations that have been studied before. This review should be done interactively, with questions and answers to engage students and ensure they have a solid understanding of these concepts. (3-5 minutes)

2. Presentation of problem situations: Next, the teacher should present two problem situations that involve comparing quantities. For example, a problem about comparing ages or a problem about comparing prices of products could be given. The teacher should explain that the aim is to use linear equations to solve these problems. (3-5 minutes)

3. Contextualization of the topic's importance: The teacher should then contextualize the importance of the topic by explaining that the ability to solve linear equations is fundamental in many areas of life and work. For instance, linear equations are used in economics to model supply and demand of products, in engineering to design structures, and in sciences to predict the behavior of physical systems. The teacher can give specific examples of how linear equations are used in different contexts. (2-3 minutes)

4. Introduction of the topic: Finally, the teacher should introduce the topic of the lesson - linear equations of comparison - explaining that this is a fundamental topic that students need to understand in order to continue studying algebra. The teacher should emphasize that while linear equations may seem complicated at first, they become easier with practice and that students will have many opportunities to practice during the lesson. (2-3 minutes)

Development (20-25 minutes)

1. Theory - Content Presentation (10-12 minutes)

   • Definition of linear equations of comparison: The teacher should start by explaining that a linear equation of comparison is a mathematical expression that represents a comparison situation between two or more quantities. He/she should give examples of linear equations of comparison, such as "2x + 3 = 7" (where x represents an unknown quantity) and "2y - 5 = 3y + 1" (where y and the unknown quantity).
   • Explanation of the substitution method: The teacher should then explain the substitution method, which is a technique used to solve linear equations. He/she should explain that the substitution method involves replacing an expression in one equation with another expression that is equal to it. The teacher should give examples of how to use the substitution method to solve linear equations, step by step.
   • Discussion about the importance of solving linear equations: The teacher should discuss why solving linear equations is important. He/she should explain that solving linear equations is a fundamental skill in mathematics and is used in many areas of life and work.

2. Practice - Content Application (10-13 minutes)

   • Linear equation solving activity: The teacher should then provide students with a number of linear equations of comparison to solve. The equations should be of increasing difficulty, starting with simple equations and gradually increasing the complexity. The teacher should circulate around the room, providing assistance and clarifying doubts as needed.
   • Group discussion: After the students have had sufficient time to solve the equations, the teacher should lead a group discussion about the solutions. He/she should ask students to explain how they arrived at their answers and what strategies they used to solve the equations. The teacher should encourage students to ask questions and to offer suggestions for improving the solutions.

3. Theory - Content Review (5-7 minutes)

   • Recap of the content: The teacher should then recap the main points of the lesson. He/she should highlight the definition of linear equations of comparison, the substitution method, and the importance of solving linear equations. The teacher can use additional examples to reinforce these concepts.
   • Questions and answers: The teacher should then open the floor for questions and answers. He/she should answer any questions students may have and clarify any misconceptions that may have arisen during the lesson.

Feedback (8-10 minutes)

1. Review and Reflection (3-4 minutes)

   • The teacher should begin this stage by asking students to revisit the solutions they found for the given problems. He/she should remind them that there is not just one correct way to solve a linear equation but rather several possible approaches. This encourages students to think critically about their own solutions and to consider other possibilities.
   • Next, the teacher should ask reflection questions, such as: "What was the most important concept you learned today?" and "What questions do you still have?" These questions encourage students to think about what they have learned and to identify any areas that may need further review.

2. Connection to the Real World (2-3 minutes)

   • The teacher should then explain how the topic of the lesson connects to the real world. For example, he/she could talk about how linear equations are used to model real-world situations, such as the relationship between time and distance in uniform motion, or the relationship between supply and demand in economics.
   • The teacher could also ask students to think of other examples of how linear equations can be used in everyday life or in different fields of study. This helps reinforce the relevance of the topic and stimulates students' critical thinking.

3. Feedback and Closure (3-4 minutes)

   • The teacher should wrap up the lesson by asking for students' feedback on the lesson. He/she could ask what they enjoyed about the lesson, what they found most challenging, and what they would like to see more of in future lessons. This allows the teacher to make adjustments to future lessons based on students' feedback.
   • Finally, the teacher should summarize the key points of the lesson and explain what will be covered in the next lesson. He/she should encourage students to review the lesson material at home and to ask any questions they may have. The teacher should also remind students of any homework or reading that may have been assigned.

Conclusion (5-7 minutes)

1. Content Recap (2-3 minutes)

   • The teacher should begin the Conclusion by recapping the main points of the lesson. This includes the definition of linear equations, the substitution method, and the application of solving linear equations in comparison situations.
   • He/she can do this through a quick interactive review, asking students to provide the definitions or explain the processes. This helps reinforce the knowledge gained and ensures that students have a solid understanding of the concepts.

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

   • Next, the teacher should highlight how the lesson connected theory, practice, and applications. He/she could mention how the theoretical presentation of the concept of linear equations of comparison was followed by practicing how to solve such equations.
   • Additionally, the teacher could point out how the skill of solving linear equations is applied in several areas of life and work, such as in economics, engineering, and sciences.

3. Extra Materials (1-2 minutes)

   • The teacher should then suggest extra materials for students who wish to enhance their knowledge of the topic. This could include recommended math books, math websites, explanatory videos, and linear equation solver apps.
   • For example, the teacher could suggest that students practice solving linear equations using a particular app or that they watch an online explanatory video that demonstrates the substitution method in a clear and detailed way.

4. Importance of the Topic in Everyday Life (1 minute)

   • Finally, the teacher should emphasize the importance of the lesson topic for everyday life. He/she could explain that while linear equations may seem abstract, they are used in many aspects of daily life.
   • For instance, linear equations are used to solve comparison problems like calculating discounts in stores, predicting the duration of a journey based on the average speed, or determining the amount of ingredients needed for a recipe. Therefore, the ability to solve linear equations is a practical and useful skill that students can apply in many real-life situations.