Objectives (5 - 7 minutes)

1. Understanding the Concept of Equations and Inequalities: The teacher should aim to ensure that students comprehend the basic definitions of equations and inequalities in one variable. They should be able to differentiate between the two concepts and understand the role of the variable in these mathematical statements.

2. Solving Simple Equations and Inequalities: The teacher should guide the students to solve simple linear equations and inequalities in one variable. This includes understanding the process of isolating the variable and finding the solution set.

3. Real-World Application of Equations and Inequalities: The teacher should help students understand the practical importance of equations and inequalities in various real-world situations. This can be achieved by providing relatable examples and problem-solving tasks during the lesson.

Secondary Objectives:

• Promoting Active Participation: The teacher should encourage students to actively engage in the lesson by asking questions, solving problems on the board, and participating in class discussions. This will help create a collaborative learning environment and enhance the students' understanding of the topic.
• Developing Problem-Solving Skills: The teacher should ensure that students not only learn the procedural aspects of solving equations and inequalities but also develop their problem-solving skills. The teacher can achieve this by providing challenging problems and guiding the students through the process of understanding and solving them.
• Cultivating Mathematical Thinking: The teacher should aim to cultivate the students' mathematical thinking and reasoning abilities. This can be done by guiding the students to understand the concepts behind the procedures and encouraging them to think critically about the solutions. The teacher can also provide opportunities for the students to explain their thinking and solutions to the class.

Introduction (8 - 10 minutes)

• The teacher begins by reminding the students of the fundamental concepts they have learned in previous lessons that are necessary for understanding equations and inequalities in one variable. This includes the concept of variables, constants, and operations such as addition, subtraction, multiplication, and division. The teacher can use a quick refresher activity or a brief review of these concepts on the board. (2 - 3 minutes)

• The teacher then presents the students with two problem situations to serve as a starting point for the lesson:

  1. "You run a lemonade stand and want to know how many cups of lemonade you need to sell in order to break even. Each cup costs $0.50 to make, and you sell them for $1. Your fixed costs for the day are $10. How many cups do you need to sell to cover your costs?"
  2. "You are saving for a new video game that costs $60. You currently have $30, and you can save $5 a week. How many weeks will it take you to save enough to buy the game?" (2 - 3 minutes)

• The teacher contextualizes the importance of equations and inequalities in real life. They explain that these mathematical concepts are used in a wide range of fields, from business and finance to engineering and computer science. For example, businesses use equations and inequalities to calculate profits and losses, while engineers use them to design structures that meet certain criteria. (1 - 2 minutes)

• The teacher grabs the students' attention by sharing two interesting facts or stories related to the topic:

  1. "Did you know that the concept of equations and inequalities can be traced back to ancient civilizations? The Babylonians, who lived over 4,000 years ago, used a form of linear equations to solve problems related to trade and commerce."
  2. "Have you ever wondered how video game designers create the challenges in your favorite games? They often use inequalities in their designs to ensure that the game is not too easy or too difficult for the players." (2 - 3 minutes)

• Finally, the teacher formally introduces the topic of the lesson: "Today, we will be learning about equations and inequalities in one variable. We will understand what they are, how to solve them, and how they can be used in real life." (1 minute)

Development (20 - 25 minutes)

• Defining Equations and Inequalities (5 - 7 minutes)

  1. The teacher presents the basic definition of an equation: a mathematical statement that shows two expressions are equal. They write on the board: 2x + 3 = 7. This is an equation because both sides of the equation are equal to 7.
  2. The teacher presents the basic definition of an inequality: a mathematical statement that shows a relationship between two expressions that may not be equal. They write on the board: 2x + 3 < 7. This is an inequality because the left side of the inequality is less than the right side.
  3. The teacher highlights how the presence of a variable in these statements distinguishes them as equations and inequalities.

• Solving Equations (7 - 10 minutes)

  1. The teacher introduces the concept of solving equations. They explain that the goal is to find the value of the variable that makes the equation true.
  2. The teacher demonstrates the process of solving a simple linear equation step-by-step on the board. They choose an example similar to the one used in the introduction: 2x + 3 = 7.
     • Step 1: The teacher explains that the first step is to isolate the variable term. In this case, they subtract 3 from both sides of the equation: 2x = 4.
     • Step 2: The teacher explains that the next step is to solve for the variable. In this case, they divide both sides of the equation by 2: x = 2.
     • Step 3: The teacher explains that the last step is to check the solution by substituting it back into the original equation. They show that when x = 2, the equation becomes 2(2) + 3 = 7, which is true. Therefore, the solution is x = 2.
  3. The teacher provides another example for the students to solve together. They choose an equation that has a negative number or a fraction, such as -3x - 2 = 7 or 3/4x + 2 = 5.

• Solving Inequalities (7 - 10 minutes)

  1. The teacher moves on to solving inequalities. They explain that, similar to equations, the goal is to find the value of the variable that makes the inequality true. However, the solution is a set of values rather than a single value.
  2. The teacher demonstrates the process of solving a simple linear inequality step-by-step on the board. They choose an example similar to the one used in the introduction: 2x + 3 < 7.
     • Step 1: The teacher explains that, like with equations, the first step is to isolate the variable term. In this case, they subtract 3 from both sides of the inequality: 2x < 4.
     • Step 2: The teacher explains that, unlike with equations, when they divide both sides of the inequality by a negative number, the direction of the inequality changes. They divide both sides by 2: x < 2.
  3. The teacher provides another example for the students to solve together. They choose an inequality that has a negative number or a fraction, such as -3x - 2 > 7 or 3/4x + 2 <= 5.

• Practical Applications and Additional Examples (1 - 3 minutes)

  1. After discussing the theory and process of solving equations and inequalities, the teacher revisits the problem situations presented at the beginning of the lesson. They explain how the process of solving equations and inequalities can help in finding the solutions to these problems.
  2. The teacher also provides additional real-life examples where equations and inequalities are used. For example, in the manufacturing industry, equations are used to determine the optimal amount of a product to produce based on cost and demand, while inequalities can be used to determine the maximum or minimum level of a resource to use or produce.
  3. The teacher concludes this section by highlighting the importance of the topic, as it is a foundational skill in algebra, and it has numerous applications in various fields of study and careers.

Feedback (10 - 12 minutes)

• Assessment of Learning (3 - 4 minutes)

  1. The teacher assesses the students' understanding of the lesson by asking a series of quick questions. These questions can involve simple equations and inequalities that the students can solve mentally or on the board.
  2. The teacher can also ask the students to solve the problem situations presented at the beginning of the lesson. This will provide an opportunity for the students to apply their newly acquired knowledge and skills.
  3. The teacher observes the students' problem-solving processes and provides constructive feedback. They correct any misconceptions and guide the students to the correct solutions if needed.

• Reflection (3 - 4 minutes)

  1. The teacher encourages the students to take a moment to reflect on what they have learned. The teacher can pose reflective questions such as:
     1. "What was the most important concept you learned today?"
     2. "What questions do you still have about equations and inequalities in one variable?"
  2. The teacher asks for a few volunteers to share their reflections with the class. This promotes a collaborative learning environment and allows the students to learn from each other's perspectives.

• Connecting Theory, Practice, and Applications (2 - 3 minutes)

  1. The teacher discusses how the lesson connected theory (the definitions and concepts of equations and inequalities), practice (the process of solving them), and applications (real-life examples and problem situations).
  2. The teacher emphasizes that understanding the theory is crucial for correctly applying the process of solving equations and inequalities. They also explain that the real-life examples and problem situations help to contextualize the theoretical concepts and make them more relevant and understandable.
  3. The teacher encourages the students to continue practicing the skills they learned in the lesson and to look for more examples of how equations and inequalities are used in their daily lives and in different fields of study and careers.

• Closing the Lesson (2 minutes)

  1. The teacher wraps up the lesson by summarizing the main points and key concepts discussed. They also remind the students of the importance of equations and inequalities in one variable and their applications in various real-world situations.
  2. The teacher thanks the students for their active participation and encourages them to keep exploring and learning about mathematics.

Conclusion (5 - 7 minutes)

• Summary and Recap (2 - 3 minutes)

  1. The teacher summarizes the main points of the lesson. They reiterate the definitions of equations and inequalities in one variable, the process of solving them, and the difference between the two.
  2. The teacher recaps the steps involved in solving equations and inequalities, emphasizing the importance of isolating the variable and checking the solution.
  3. The teacher reminds the students of the problem situations and real-life examples discussed during the lesson, reinforcing the practical application of the concepts learned.

• Connecting Theory, Practice, and Applications (1 - 2 minutes)

  1. The teacher explains how the lesson connected theoretical concepts, practical skills, and real-world applications. They reiterate that understanding the theory of equations and inequalities is essential for solving them correctly.
  2. The teacher emphasizes that the practical skill of solving equations and inequalities is not only useful in mathematics but also in various fields and everyday life situations. They mention that the ability to translate real-world problems into mathematical equations and inequalities is a valuable skill.

• Additional Materials (1 minute)

  1. The teacher suggests additional resources for students who want to further their understanding of the lesson's content. This could include online tutorials, interactive games, and problem-solving websites.
  2. The teacher recommends that students practice solving equations and inequalities on their own or with a study group. They can start with simple problems and gradually move on to more complex ones.
  3. The teacher reminds the students that they can always ask questions or seek help if they encounter difficulties in their self-study.

• Relevance to Everyday Life (1 - 2 minutes)

  1. The teacher concludes the lesson by emphasizing the importance of equations and inequalities in everyday life. They explain that these mathematical concepts are not just theoretical ideas but tools that can help in making decisions and solving problems.
  2. The teacher gives examples of how equations and inequalities are used in various professions and situations, such as in calculating expenses and profits in business, in designing structures in engineering, and in setting and solving problems in video games.
  3. The teacher encourages the students to be aware of the applications of what they learn in school and to see the relevance of mathematics in their daily lives.