Objectives (5 - 7 minutes)

1. To understand the concept of systems of equations and the different methods for solving them, in this case, the method of substitution and elimination.
2. To explore the conditions under which systems of equations have one solution, no solution, or infinitely many solutions.
3. To apply the learned methods and concepts to solve real-world problems that can be represented by systems of equations.

Secondary Objectives:

1. To enhance critical thinking and problem-solving skills through the application of mathematical concepts.
2. To foster collaborative work and communication skills by working in pairs during the hands-on activities.
3. To promote a positive attitude towards mathematics through engaging and interactive learning activities.

Introduction (10 - 12 minutes)

1. Recalling Prior Knowledge: The teacher begins by reminding the students about the basic concepts of equations, such as variables, constants, and coefficients. The teacher also reviews the methods of solving single-variable equations, particularly the methods of substitution and elimination, which will be extended to systems of equations in the current lesson. This part should take around 2 - 3 minutes.

2. Problem Situations: The teacher then presents two problem situations that can be modeled using systems of equations. For instance, the teacher might present a scenario like: "A store sells apples and oranges. On a certain day, they sold a total of 20 fruits for $35. An apple costs $2 and an orange costs $3. How many of each fruit did they sell?" Another example could be: "A small and a large pipe can together fill a tank in 2 hours. The large pipe can fill the tank 3 times faster than the small one. How long would it take for each pipe to fill the tank separately?" This step should take about 3 - 4 minutes.

3. Real-World Applications: The teacher then explains the importance of understanding systems of equations by showing how they are used in real-world situations. For instance, the teacher might explain how these concepts are used in economics, physics, or engineering. The teacher can also show examples from everyday life, such as how a taxi fare can be calculated based on the distance traveled and the waiting time, which can be represented as a system of equations. This part should take about 2 - 3 minutes.

4. Topic Introduction: The teacher introduces the topic of systems of equations, emphasizing that they involve multiple variables and multiple equations. The teacher explains that the goal is to find values for the variables that satisfy all the equations in the system. The teacher also introduces the concept of the number of solutions, explaining that a system can have one unique solution, no solution, or an infinite number of solutions. The teacher emphasizes that understanding when and why each of these situations occurs is the main focus of the lesson. This part should take about 2 - 3 minutes.

5. Engaging Curiosities: To grab the students' attention and generate interest in the topic, the teacher could share some fun facts or curiosities related to the topic. For instance, the teacher might share that systems of equations were used by ancient Egyptians to divide inheritances or that they are essential in the development of computer graphics and video games. The teacher could also share a real-world puzzle that can be solved using systems of equations and challenge the students to solve it by the end of the lesson. This step should take about 1 - 2 minutes.

Development (20 - 25 minutes)

Activity 1: Substitution or Elimination Relay Race (10 - 12 minutes)

1. Preparation: The teacher divides the class into several teams of three students each. The teacher also prepares a set of systems of equations, each written on a separate card, with the same number of equations as the number of teams. Each equation in the system should be straightforward and solvable using the substitution or elimination method. This step should take about 2 - 3 minutes.

2. Activity Instructions: The teacher explains that this activity is a race to find the solutions to their designated system of equations using the substitution or elimination method. The teacher emphasizes that the goal is not only to solve the equations correctly but to do it as quickly as possible. This instruction should take about 1 - 2 minutes.

3. Implementation: The teacher distributes the system of equation cards to each team and starts the race. Each team member takes turns to solve one equation using the chosen method. The teacher checks the first solution, and if it is correct, the team member moves to the next equation. If it is incorrect, the team member must try again while the other teams continue solving their equations. This process continues until all the equations in the system are solved. This step should take about 5 - 7 minutes.

4. Reflection: The teacher concludes the activity by discussing the strategies used by the teams. The teacher also highlights common mistakes made during the race and provides necessary corrections. The teacher also emphasizes that in mathematics, speed is not the only goal, and accuracy is more important. This step should take about 2 - 3 minutes.

Activity 2: The Mystery of Missing Solutions (10 - 13 minutes)

1. Preparation: The teacher prepares a set of systems of equations, each written on a separate sheet of paper, with a mix of systems that have one solution, no solution, or infinitely many solutions. The teacher also prepares a set of clues that describe the type of solution the system has, but not the actual solution. This step should take about 3 - 4 minutes.

2. Activity Instructions: The teacher explains that this is a mystery-solving activity where the students are detectives trying to figure out the type of solution each system has. The clues will be their main tool for this. The teacher emphasizes that the goal is to understand the conditions under which a system has one solution, no solution, or infinitely many solutions. This instruction should take about 1 - 2 minutes.

3. Implementation: The teacher distributes the system of equation sheets and clue cards to each student. The students then have to solve the system and use the clues to deduce the type of solution. If they get a different answer from the clues, they have to go back and find their mistake. This step should take about 5 - 7 minutes.

4. Reflection: The teacher concludes the activity by discussing the solutions and the clues. The teacher also asks each student to explain how they determined the type of solution from the clues, promoting class discussion and peer learning. This step should take about 2 - 3 minutes.

Activity 3: Real-World Puzzle (5 - 7 minutes)

1. Preparation: The teacher prepares a set of real-world problems that can be modeled using systems of equations. This could include problems from various areas, such as business, physics, or engineering. The teacher also prepares a set of hints for each problem, which can guide the students in setting up and solving the systems. This step should take about 1 - 2 minutes.

2. Activity Instructions: The teacher explains that this is a puzzle-solving activity where the students have to solve real-world problems using systems of equations. The hints will help them with the first steps, but they have to figure out the rest on their own. The teacher emphasizes that the goal is to apply the learned concepts to practical situations. This instruction should take about 1 - 2 minutes.

3. Implementation: The teacher distributes the problem cards and hint cards to each student. The students then work independently or in pairs to solve the problems. The students can refer to their notes and textbooks or use calculators if necessary. This step should take about 2 - 3 minutes.

4. Reflection: The teacher concludes the activity by discussing the solutions and the strategies used by the students. The teacher also asks each student to share how they translated the real-world problem into a system of equations, promoting class discussion and peer learning. This step should take about 1 - 2 minutes.

Feedback (8 - 10 minutes)

1. Group Discussion: The teacher initiates a group discussion about the activities carried out during the lesson. Each group is given a chance to share their solutions and strategies for each activity. The teacher encourages students to express their thoughts, doubts, and any difficulties they encountered. This step should take about 3 - 4 minutes.

2. Linking Theory to Practice: The teacher then connects the solutions and strategies presented by the students to the theoretical concepts taught at the beginning of the lesson. The teacher explains how the methods of substitution and elimination were applied in the activities, and how the conditions for one solution, no solution, or infinite solutions were identified from the clues and problem contexts. This step should take about 2 - 3 minutes.

3. Student Reflection: The teacher then asks the students to take a moment to reflect on their learning. The students are asked to think about the most important concept they learned in this lesson and any questions or doubts they still have. The teacher can provide guiding questions to help the students in their reflection, such as: "What was the most challenging part of today's lesson?" or "How can you apply what you learned today in real life?" This step should take about 2 - 3 minutes.

4. Closing Remarks: The teacher concludes the lesson by summarizing the key points and concepts learned. The teacher also addresses any common questions or doubts that arose during the reflection. The teacher emphasizes that understanding the concept of systems of equations and the number of solutions is crucial in solving real-world problems. The teacher also encourages the students to continue practicing these concepts at home using the resources provided. This step should take about 1 - 2 minutes.

5. Homework Assignment: The teacher assigns homework that involves practicing the methods of solving systems of equations and identifying the number of solutions for different systems. The homework could include a set of problems similar to the ones used in the class activities. This step should take about 1 minute.

Conclusion (5 - 7 minutes)

1. Summary and Recap: The teacher begins the conclusion by summarizing the main points covered in the lesson. The teacher recaps the definition of systems of equations and the methods of solving them - substitution and elimination. The teacher then revisits the concept of the number of solutions in a system, explaining again the conditions for one solution, no solution, or infinitely many solutions. This part should take about 2 minutes.

2. Connecting Theory, Practice, and Applications: The teacher then explains how the lesson connected theory, practice, and applications. The teacher emphasizes that the hands-on activities, such as the relay race and the mystery-solving activity, allowed students to apply the theoretical concepts in a practical context. The real-world puzzle further demonstrated the usefulness of systems of equations in solving everyday problems. This part should take about 1 - 2 minutes.

3. Additional Materials: The teacher suggests additional materials for students to deepen their understanding of the topic. This could include online resources, such as interactive websites or video tutorials, books or chapters in textbooks that provide more examples and practice problems, and mobile apps that allow students to practice solving systems of equations on their phones or tablets. The teacher could also recommend a few interesting problems or puzzles that students can solve for fun. This step should take about 1 - 2 minutes.

4. Real-World Relevance: The teacher concludes the lesson by emphasizing the importance of understanding systems of equations for everyday life. The teacher explains that these concepts are not only relevant in academic settings but also in various professional fields, such as economics, engineering, and computer science. The teacher also points out that being able to solve systems of equations can help in making decisions and understanding the world around us. This final part should take about 1 minute.

5. Curiosity: To end on a high note, the teacher shares a curiosity related to the topic. For instance, the teacher might share that the concept of systems of equations is not only used in mathematics but also in computer science, where it forms the basis of many algorithms. The teacher could also share a fun fact about the history of systems of equations, such as how they were first used by ancient Babylonians over 4,000 years ago. This final touch should leave the students with a positive and curious attitude towards the topic. This final step should take about 1 - 2 minutes.