Objectives (5 - 7 minutes)

1. Understand the concept of equations with two variables: The teacher should introduce the basic concept of equations with two variables, explaining how they work and why they are useful in solving mathematical problems.

2. Learn how to solve equations with two variables: The teacher should teach students the steps involved in solving equations with two variables, including the need to isolate a variable before solving the equation.

3. Apply the knowledge to solving real-world problems: The teacher should emphasize the importance of applying the knowledge gained to solving real-world problems. Students should be able to identify and solve problems that can be represented by equations with two variables.

Secondary objectives:

• Develop critical thinking and problem-solving skills: Through solving equations with two variables, students should be encouraged to develop critical thinking and problem-solving skills, which are valuable skills in various areas of life.

• Promote active student participation: The teacher should encourage students to actively participate in the lesson by asking questions, solving problems, and sharing their solutions and thoughts.

• Stimulate interest in mathematics: The teacher should use interesting and relatable examples and applications to maintain students' interest in the subject.

Introduction (10 - 12 minutes)

1. Review of previous content: The teacher should start the lesson by briefly recalling the concepts of equations, variables, and systems of equations, including simple examples of each. This will serve as the starting point for introducing the new topic. (2 - 3 minutes)

2. Presentation of problem situations: The teacher can present two problem situations that involve equations with two variables. For example, "Imagine you are buying apples and oranges at a fair. An apple costs $2 and an orange costs $3. You spend $20 in total and buy 8 pieces of fruit altogether. How many apples and how many oranges did you buy?" and "Suppose you are riding a bike at a constant speed of 10 km/h. At the same time, your friend starts walking in a straight line at 3 km/h. After one hour, how far will you be from your friend?" These situations will help contextualize the topic and pique students' interest. (3 - 4 minutes)

3. Contextualization of the importance of the subject: The teacher should explain that equations with two variables are widely used in various fields, such as physics, engineering, economics, and even in everyday situations. For example, they can be used to model mixing situations, such as mixing different components in a solution. Moreover, the ability to solve equations with two variables is fundamental to solving systems of equations, which is an important topic in mathematics. (2 - 3 minutes)

4. Introduction of the topic with curiosities or applications: The teacher can share two curiosities or applications of the topic to capture students' attention. For example, "Did you know that equations with two variables can be used to determine the location of a point on a plane? This is very useful in cartography and navigation." and "Have you ever wondered how meteorologists predict the temperature? They use equations with two variables that relate atmospheric pressure to temperature." (2 - 3 minutes)

Development (20 - 25 minutes)

1. Presentation of theory (10 - 12 minutes):

   1.1. Definition of an equation with two variables: The teacher should start the theoretical part by defining what an equation with two variables is. They should explain that an equation with two variables is a mathematical expression that has two unknowns and that the solution to the equation is an ordered pair of values that satisfy the expression. (2 - 3 minutes)

   1.2. Examples of equations with two variables: Next, the teacher should present examples of equations with two variables. For example, "2x + 3y = 10" and "5x - 2y = -7". They should emphasize that in an equation with two variables, the goal is to find the values of x and y that make the equality true. (2 - 3 minutes)

   1.3. Graphical representation of an equation with two variables: The teacher should explain that an equation with two variables can be represented graphically as a straight line on a Cartesian plane. They should show how to plot the points that satisfy the equation and how to draw the line that passes through those points. (2 - 3 minutes)

   1.4. Intersection of two lines: The teacher should explain that the solution to a system of equations with two variables is the point of intersection of the two lines that represent the equations. They should show how to find this point of intersection and how to verify if it is the correct solution. (2 - 3 minutes)

2. Solving problems (10 - 13 minutes):

   2.1. Explanation of the steps to solve an equation with two variables: The teacher should explain the steps involved in solving an equation with two variables. They should emphasize the importance of isolating one variable before solving the equation, and they should demonstrate how to do this in several examples. (3 - 4 minutes)

   2.2. Solving examples of equations with two variables: The teacher should solve a few examples of equations with two variables, step by step, on the board. They should explain each step of the process and should allow time for students to follow along and take notes. (4 - 5 minutes)

   2.3. Solving the problem situations from the Introduction: The teacher should return to the problem situations presented at the beginning of the lesson and should solve each of them, step by step, on the board. They should explain how the situation can be modeled by an equation with two variables and should demonstrate how to find the solution. (3 - 4 minutes)

3. Practical activities (5 - 7 minutes):

   3.1. Activities for solving equations with two variables: The teacher should distribute some activities for solving equations with two variables for the students to solve individually. The activities should be varied and should include examples that reinforce the concepts introduced in the lesson. The teacher should circulate around the room to help students who are struggling and to check on everyone's progress. (3 - 4 minutes)

   3.2. Discussion and correction of the activities: After allowing time for students to solve the activities, the teacher should lead a classroom discussion in which students share their solutions and thoughts. They should correct the activities on the board, explaining each step of the process and clarifying any doubts that may have arisen. (2 - 3 minutes)

Feedback (8 - 10 minutes)

1. Review of theory (3 - 4 minutes):

   1.1. Connection with the real world: The teacher should review the main points of the theory, emphasizing how solving equations with two variables is relevant and applicable in various real-world situations. They should recall the problem situations presented in the Introduction, re-explaining how equations with two variables were used to solve each one. (1 - 2 minutes)

   1.2. Connection with other concepts: The teacher should explain how the ability to solve equations with two variables connects with other mathematical concepts. For example, they can mention that solving equations with two variables is a fundamental skill for solving systems of equations, which is an important topic in mathematics. They can also discuss how the graphical representation of an equation with two variables relates to the geometry of the Cartesian plane. (1 - 2 minutes)

2. Reflection on learning (2 - 3 minutes):

   2.1. Class discussion: The teacher should ask students to reflect on what they learned in the lesson and to share their reflections with the class. They can ask questions such as, "What was the most important concept you learned today?" and "What questions are still unanswered?" This will allow the teacher to assess the students' level of understanding and identify any areas that may need review or reinforcement in future lessons. (1 - 2 minutes)

   2.2. Self-assessment: The teacher can ask students to evaluate their own learning by giving themselves a grade from 1 to 5 for how well they think they understood the content of the lesson. This can help students become more aware of their own progress and identify any areas of difficulty that they may need to focus on. (1 minute)

3. Preparation for the next lesson (1 - 2 minutes):

   3.1. Anticipation of the next topic: The teacher should preview the topic of the next lesson, explaining how it relates to the content of the current lesson. They can also suggest any additional reading or activities that students can do to prepare for the next lesson. (1 minute)

   3.2. Review of content for homework: The teacher can suggest that students review the concepts and techniques introduced in the lesson by doing practice exercises at home. They can also provide a list of exercises for students to work on, if they feel it is appropriate. (1 minute)

Conclusion (5 - 7 minutes)

1. Summary of main points (2 - 3 minutes):

   1.1. Review of theory: The teacher should recap the main concepts covered in the lesson, recalling the definition of equations with two variables, the graphical representation of these equations, and the solving of problems using this type of equation.

   1.2. Steps for solving an equation with two variables: He/she should reinforce the steps involved in solving an equation with two variables, emphasizing the importance of isolating a variable before solving the equation.

   1.3. Practical applications: The teacher should reiterate the practical applications of equations with two variables, reminding students that they are used in various areas, such as physics, engineering, economics, and everyday situations.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   2.1. Demonstration on the board: The teacher should explain how the lesson connected theory, practice, and applications. He/she should emphasize how solving practical problems, such as the problem situations presented, helped illustrate the application of theory in practice.

   2.2. Practical exercises: The teacher should highlight how the exercises for solving equations with two variables allowed students to practice applying the theoretical concepts.

3. Extra materials (1 - 2 minutes):

   3.1. Suggestion of readings: The teacher can suggest extra materials for students who wish to deepen their understanding of the topic. This may include textbook readings, online articles, or explanatory videos.

   3.2. Practice sites: The teacher can also suggest websites or apps that offer interactive practice exercises, so that students can continue practicing solving equations with two variables outside the classroom.

4. Importance of the topic for daily life (1 minute):

   4.1. Relevance of the topic: Finally, the teacher should highlight the importance of the topic for everyday life. He/she can give examples of how equations with two variables can be used in everyday situations, such as solving mixture problems, planning trips, or calculating trajectories. This will help reinforce the relevance of the topic and motivate students to continue studying and practicing it.