Objectives (5 - 7 minutes)

1. Understanding Systems of Equations: The students should be able to understand what a system of equations is and how it is represented. This includes knowing that a system of equations is a set of two or more equations with the same variables, and that the solution to the system is the set of values that satisfy all the equations.

2. Solving Systems of Equations using Substitution: The students should be able to solve systems of equations using the substitution method. This involves substituting one equation into another in order to eliminate one variable, and then solving the resulting equation to find the value of the other variable.

3. Solving Systems of Equations using Elimination: The students should be able to solve systems of equations using the elimination method. This involves adding or subtracting the equations in the system in order to eliminate one variable, and then solving the resulting equation to find the value of the other variable.

Secondary Objectives:

• Recognizing Types of Solutions: The students should be able to recognize the different types of solutions that a system of equations can have: a unique solution, no solution, or infinitely many solutions.

• Applying Solving Methods: The students should be able to apply the substitution and elimination methods to solve various types of systems of equations. This includes systems with two variables, systems with three variables, and systems with more than three equations.

Introduction (10 - 15 minutes)

1. Recap of Previous Knowledge: The teacher starts the lesson by reminding students of the basic concepts of linear equations, such as variables, constants, and coefficients. The teacher also reviews the methods of solving linear equations, including the addition/subtraction and substitution methods. This serves as a foundation for understanding the new topic of systems of equations.

2. Problem Situations: The teacher presents two problem situations that can be solved using systems of equations. The first situation could be a business scenario, where the students have to determine the number of adult and children tickets sold at a movie theater based on total sales and the price of each type of ticket. The second situation could be a puzzle, where the students have to find the values of two unknown numbers based on their sum and product.

3. Real-world Applications: The teacher explains the importance of systems of equations in real-world applications. For instance, in physics, systems of equations are used to describe the motion of objects. In economics, they are used to model supply and demand. In computer science, they are used in cryptography. By understanding systems of equations, the students can see the practical value of what they are learning.

4. Topic Introduction: The teacher introduces the topic of systems of equations, explaining that it is a powerful tool for solving problems that involve more than one variable. The teacher also highlights that there are two main methods for solving systems of equations: substitution and elimination. The teacher assures the students that by the end of the lesson, they will be able to use these methods to solve a wide range of problems.

5. Curiosities and Fun Facts: As a way to pique the students' interest, the teacher shares a couple of curiosities or fun facts related to systems of equations. For instance, the teacher could mention that systems of equations have been used in ancient times to solve problems in astronomy and land surveying. The teacher could also share a puzzle that can be solved using systems of equations, challenging the students to solve it on their own. These elements help to make the lesson more engaging and interactive.

Development (20 - 30 minutes)

1. Theory of Systems of Equations (5 - 7 minutes):

   1. The teacher starts by introducing the concept of a system of equations. An example is projected on the board, and the teacher explains that it consists of two or more equations with the same variables.
   2. The teacher explains the terms "linear system" and "non-linear system" and their differences. The teacher highlights that in this lesson, the focus will be on linear systems, which are those where the highest power of the variable is 1.
   3. The teacher goes on to explain that a solution to a system of equations is a set of values that makes all the equations in the system true. The teacher uses the initial example to illustrate this concept.
   4. The teacher introduces the three types of solutions: a unique solution, no solution, or infinitely many solutions. The teacher exemplifies each type.

2. Solving Systems of Equations using Substitution (5 - 8 minutes):

   1. The teacher introduces the substitution method as a way to solve systems of equations. This method involves solving one equation for one variable and then substituting this expression into the other equation.
   2. The teacher demonstrates the method step-by-step with an example on the board. The example consists of a simple system of two linear equations.
   3. The teacher highlights that sometimes the method involves simplifying an equation first before performing the substitution. The teacher exemplifies this with another problem on the board.
   4. The teacher asks a volunteer student to solve a problem on the board using the substitution method. This allows the teacher to assess the understanding of the method by the students.

3. Solving Systems of Equations using Elimination (5 - 8 minutes):

   1. The teacher introduces the elimination method as another method to solve systems of equations. This method involves adding or subtracting the equations in the system to eliminate one variable.
   2. The teacher demonstrates the method step-by-step with an example on the board. The example consists of a system of two linear equations where the elimination method is more efficient than the substitution method.
   3. The teacher explains that sometimes the method involves multiplying one or both of the equations to get the coefficients of one of the variables to be the same. The teacher exemplifies this with another problem on the board.
   4. The teacher asks a volunteer student to solve a problem on the board using the elimination method. This allows the teacher to assess the understanding of the method by the students.

4. More Complex Systems and Additional Methods (5 - 10 minutes):

   1. The teacher explains that the same methods can be used to solve systems of equations with three or more variables. The teacher demonstrates this with a simple example on the board.
   2. The teacher also explains that there are other methods to solve systems of equations, like the graphical method and the matrix method, but these will not be covered in this lesson. The teacher briefly explains how these methods work.
   3. The teacher reinforces the importance of checking the solution of a system of equations by substituting it into the original equations. The teacher demonstrates this using one of the solved examples.
   4. The teacher encourages the students to practice the methods at home using their textbooks or online resources. The teacher reminds the students that solving systems of equations is a skill that requires practice to master.

Feedback (5 - 7 minutes)

1. Assessment of Learning (2 - 3 minutes):

   1. The teacher proposes a quick review of the main concepts learned in the lesson. The teacher asks the students to define what a system of equations is and what it means for a set of values to be the solution to a system of equations.
   2. The teacher reviews the two methods for solving systems of equations: substitution and elimination. The teacher asks the students to explain each method in their own words and when to use each one.
   3. The teacher asks the students to list the types of solutions that a system of equations can have and provide an example of each. The teacher emphasizes that it is important to always check the solution by substituting it into the original equations.
   4. The teacher asks the students to summarize the real-world applications of systems of equations that were discussed in the lesson. The teacher can also ask the students to come up with their own examples of situations where systems of equations could be used.

2. Reflection on the Lesson (2 - 3 minutes):

   1. The teacher encourages the students to take a moment to reflect on what they have learned in the lesson. The teacher proposes the following questions for the students to reflect on:
      1. What was the most important concept you learned today?
      2. Which method for solving systems of equations (substitution or elimination) do you feel most comfortable with? Why?
      3. Which parts of the lesson were the most challenging for you? Why?
   2. The teacher asks for volunteers to share their reflections. The teacher listens attentively and provides feedback on the students' understanding and progress. The teacher also assures the students that it is normal to find some parts of the lesson challenging, and that with practice, they will become more comfortable with solving systems of equations.

3. Addressing Unanswered Questions (1 - 2 minutes):

   1. The teacher asks the students if they have any remaining questions or doubts about the lesson. The teacher encourages the students to ask questions, assuring them that no question is too simple or too complex.
   2. The teacher writes down any unanswered questions and promises to address them in the next class or during office hours. The teacher also reminds the students that they can always ask questions by email or in the school's online learning platform.

4. Homework Assignment (1 minute):

   1. The teacher assigns homework for the students to practice the methods for solving systems of equations. The assignment consists of a set of problems from the textbook or an online resource. The teacher reminds the students to check their solutions by substituting them into the original equations.
   2. The teacher also suggests that the students look for real-world examples of systems of equations and think about how they could be solved. The teacher encourages the students to bring their examples and solutions to the next class for discussion.

Conclusion (3 - 5 minutes)

1. Summary and Recap (1 - 2 minutes):

   1. The teacher summarizes the main points of the lesson, recapping the definition of a system of equations and its solution, the methods for solving systems of equations (substitution and elimination), and the types of solutions that a system of equations can have.
   2. The teacher emphasizes the importance of checking the solution by substituting it into the original equations. The teacher also reminds the students that solving systems of equations is a skill that requires practice to master.

2. Connection of Theory, Practice, and Applications (1 minute):

   1. The teacher explains how the lesson connected theory, practice, and applications. The teacher emphasizes that the theory was presented in a clear and logical way, and was immediately applied to solve problems.
   2. The teacher highlights that the real-world applications of systems of equations were not only discussed, but also used as a context for the problems. The teacher encourages the students to continue to make these connections in their own learning.

3. Additional Materials (30 seconds):

   1. The teacher suggests additional resources for the students to deepen their understanding of systems of equations. This could include recommended sections of the textbook, online tutorials, or interactive learning tools.
   2. The teacher also recommends that the students practice more problems on their own to reinforce the concepts learned in the lesson.

4. Importance of the Topic (1 minute):

   1. The teacher concludes the lesson by emphasizing the importance of systems of equations in everyday life. The teacher explains that systems of equations are used in various fields, from physics and economics to computer science and cryptography.
   2. The teacher assures the students that by learning to solve systems of equations, they are gaining a powerful tool that can help them in many areas of their lives. The teacher encourages the students to continue to explore and apply what they have learned.