Objectives (5 - 7 minutes)

1. Understand the concept of inequalities: The teacher will introduce the concept of inequalities, highlighting that they are mathematical expressions that compare two quantities, indicating whether one is greater than, less than, or equal to the other. This will be done by revisiting previous lessons on equations and comparisons.

2. Learn how to solve one-step inequalities: The teacher will refresh the students' memory on solving one-step inequalities. This will involve reminding them of the inverse operations (addition/subtraction and multiplication/division) used in solving these types of inequalities.

3. Learn how to solve two-step inequalities: The teacher will introduce the main topic of the lesson - two-step inequalities. The students will be informed that these inequalities require two inverse operations to be solved and that they are commonly represented on a number line.

Secondary Objectives:

• Enhance critical thinking: The students will be encouraged to think critically about the inequalities and the outcomes of the different steps in the problem-solving process.

• Improve problem-solving skills: By practicing solving two-step inequalities, the students will enhance their problem-solving abilities in a mathematical context.

Introduction (10 - 12 minutes)

1. Recap of Previous Lessons: The teacher will begin by reminding the students of the basic concepts necessary for understanding two-step inequalities. This includes a brief review of one-step inequalities, the concept of inverse operations, and how to represent inequalities on a number line. The teacher will use a few example problems to ensure that the students are familiar with these concepts.

2. Problem Situations: The teacher will then present two problem situations to the class.

   • Problem 1: "Imagine you have a snack bar and you want to sell at least 50 snacks. Each snack costs $3.50. How many snacks do you need to sell to make at least $50?"
   • Problem 2: "You are going on a road trip and you want to drive at least 200 miles in 4 hours. If you drive at a constant speed, what is the minimum speed you need to maintain?" The teacher will relate these problems to the real world, explaining that inequalities can be used to make decisions about real-life situations.

3. Real-world Applications: The teacher will discuss how inequalities are used in various fields such as economics, engineering, and even in video game design. For instance, in a game, if a character has to score at least 100 points to win, the inequalities can be used to determine the different possible scores.

4. Introduction of the Topic: The teacher will then introduce the topic of two-step inequalities. The teacher will explain that two-step inequalities are used when more than one operation is needed to solve an inequality. The teacher will also mention that solving two-step inequalities involves a step-by-step process and the solutions are often depicted on a number line.

5. Attention-Grabbing Facts: To pique the students' interest, the teacher will share a couple of fun facts related to inequalities.

   • Fact 1: "Did you know that inequalities can be used to describe the growth of populations in biology? For example, if the population of rabbits in a forest increases by at least 10 every year, we can use inequalities to model this."
   • Fact 2: "Inequalities are also used in music! When musicians create a crescendo or decrescendo, they are essentially creating an inequality in volume."

The teacher will conclude the introduction by stating the importance of understanding and applying inequalities in solving problems both in and out of the classroom.

Development (20 - 25 minutes)

1. Introducing Two-Step Inequalities (5 - 7 minutes)

   • The teacher will begin this stage of the lesson by outlining what two-step inequalities are. They will explain that these inequalities are a combination of inequalities with two steps, meaning they require two inverse operations to be solved.
   • Then, the teacher will use a few examples to illustrate the structure of two-step inequalities: "2x + 5 < 15" or "3y - 7 ≥ 8". The teacher will underline the roles of the inequality symbols, operations, and variables in these examples.
   • To further elucidate the concept, the teacher can break down a two-step inequality into its two parts, each representing a single step. For instance, for the inequality "2x + 5 < 15", the teacher will explain that the first step is "2x + 5" and the second step is the comparison of this to the number 15.

2. Solving Two-Step Inequalities (10 - 12 minutes)

   • The teacher will then explain the step-by-step process for solving two-step inequalities, highlighting that the goal is to isolate the variable on one side of the inequality.
   • The teacher will demonstrate the process by solving an example on the board. For instance, the teacher can use the inequality "3x + 4 > 10". The teacher will begin by subtracting 4 from both sides, then dividing by 3, and finally getting "x > 2".
   • To ensure the students grasp the process, the teacher will solve another example, involving multiplication or division. For instance, the teacher can use "2y/3 - 1 ≥ 4". The teacher will start by adding 1 to both sides, then multiplying by 3/2, and finally getting "y ≥ 9".
   • The teacher will remind students that the inequality symbol remains the same if they multiply or divide both sides by a negative number. This point is crucial to avoid any confusion or errors during the solving process.
   • The teacher will emphasize the importance of checking the solution: plugging the value back into the inequality to ensure it satisfies the inequality.

3. Practice with Two-Step Inequalities (5 - 6 minutes)

   • The teacher will distribute a worksheet that contains a variety of two-step inequalities for the students to solve independently.
   • The teacher will walk around the room, checking on students' progress, and offering assistance where needed.
   • The teacher will remind the students to write down each step and explain each step as they solve the inequalities to reinforce the concept in their minds.

By the end of this development stage, the students should be able to identify and solve two-step inequalities. They should also understand the importance of careful and methodical problem-solving, especially when dealing with multi-step problems.

Feedback (8 - 10 minutes)

1. Discussion and Reflection (4 - 5 minutes)

   • The teacher will initiate a class discussion, asking students to share their answers and solutions to the problems they worked on during the practice phase. The teacher will encourage students to explain their thought processes and the steps they took to solve the inequalities. This will help the students to understand the variety of methods that can be used to solve two-step inequalities.
   • The teacher will then ask the students to reflect on the skills and knowledge they gained from the lesson. The teacher can guide this reflection by asking questions such as:
     1. "How did your understanding of inequalities evolve during this lesson?"
     2. "What strategies did you use to solve the two-step inequalities?"
     3. "Why is it important to check the solution in an inequality?"
   • The teacher will acknowledge the students' efforts and progress, highlighting the areas where the students demonstrated a good understanding of the lesson's objectives. The teacher will also address any common misconceptions or errors that arose during the practice phase.

2. Connection to Real Life (2 - 3 minutes)

   • The teacher will then facilitate a discussion on how the students can apply their knowledge of two-step inequalities in real-life situations. They can use examples such as:
     1. "If you are planning to save money for a new phone, you need to consider your income and other expenses. You can use inequalities to determine how much you need to save each month."
     2. "In sports, inequalities can be used to determine the minimum number of games a team needs to win to make it to the playoffs."
     3. "In cooking, if a recipe serves 4 people and you need to feed at least 8, you can use inequalities to adjust the quantities of the ingredients."
   • The teacher will encourage the students to come up with their own examples of how inequalities can be used in real life, fostering a deeper understanding and appreciation of the subject matter.

3. Feedback and Conclusion (2 minutes)

   • The teacher will conclude the lesson by summarizing the key points and skills learned, and how they relate to the broader context of mathematics and problem-solving.
   • The teacher will then ask the students to provide feedback on the lesson, either orally or in writing. The teacher can use guiding questions such as:
     1. "What was the most important concept you learned today?"
     2. "What questions do you still have about two-step inequalities?"
     3. "What parts of the lesson did you find most helpful? Why?"
   • The teacher will collect the feedback and use it to improve future lessons and address any lingering questions or misconceptions in the next class.

By the end of this feedback stage, the students should have a solid understanding of two-step inequalities, their relevance in real-world applications, and their own progress and areas of improvement in the subject.

Conclusion (5 - 7 minutes)

1. Recap (2 - 3 minutes)

   • The teacher will begin by summarizing the main points of the lesson. This includes the definition of inequalities, the process of solving two-step inequalities, and the importance of checking the solutions. The teacher will remind the students that inequalities are mathematical expressions used to compare quantities and that they can be represented on a number line.
   • The teacher will reiterate the step-by-step process of solving two-step inequalities, emphasizing the role of inverse operations and the need to isolate the variable. The teacher will also remind the students about the importance of checking the solution by substituting it back into the original inequality.
   • The teacher will remind the students of the real-world applications of inequalities, such as in business, sports, and cooking. The teacher will also mention how inequalities are used in various academic disciplines like biology and music.

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

   • The teacher will then explain how the lesson connected theory, practice, and applications. The theoretical part of the lesson involved understanding the concept of inequalities and the step-by-step process of solving two-step inequalities.
   • The practice phase allowed the students to apply these theoretical concepts in a hands-on way. By solving inequalities on their own, the students got a chance to practice the skills they learned and to solidify their understanding of the topic.
   • The discussion of real-world applications helped the students to see the relevance of the lesson. It also provided them with examples of how inequalities can be used in different contexts, which can help them in future problem-solving situations.

3. Additional Materials (1 minute)

   • The teacher will suggest some additional materials for the students to further their understanding of two-step inequalities. These could include online math games, worksheets, and videos that explain the concept in a different way. The teacher could also recommend some books that contain more complex examples and problems for the students to practice.
   • The teacher will remind the students that these materials are not mandatory but are there to support their learning and to provide them with extra practice if they need it.

4. Everyday Life Relevance (1 - 2 minutes)

   • Finally, the teacher will conclude the lesson by highlighting the importance of understanding inequalities in everyday life. The teacher will remind the students that inequalities are not just a theoretical concept but a practical tool that can be used to make decisions in various situations.
   • The teacher will mention that whether it's planning a budget, setting goals, or making decisions in a game, understanding inequalities can help them to think critically and to solve problems more effectively.
   • The teacher will also encourage the students to be aware of the inequalities they encounter in their daily life and to think about how they can use their mathematical skills to make sense of these inequalities.

By the end of the conclusion, the students should feel confident in their understanding of two-step inequalities, and they should be able to see its relevance in their everyday life. They should also be aware of the resources available to them for further learning and practice.