Objectives (5 - 7 minutes)

1. To understand the concept of multi-step inequalities and how they are different from linear equations.
2. To be able to solve multi-step inequalities using the addition, subtraction, multiplication, and division properties of inequality.
3. To apply knowledge and skills acquired to real-world problem-solving.

Secondary Objectives:

1. To foster critical thinking and problem-solving skills in the context of multi-step inequalities.
2. To encourage peer collaboration and cooperative learning during activity and discussion sessions.
3. To improve students' confidence in their mathematical abilities and reduce anxiety related to problem-solving.

Introduction (10 - 15 minutes)

1. The teacher begins the lesson by reminding students of the basic concepts of inequalities and their symbols. This will include a quick review of the symbols for 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to'. The teacher will also remind students of the properties of inequality, especially the ones involving addition, subtraction, multiplication, and division.

2. The teacher presents two problem situations to the students:

   • "You have $40 to spend on a new pair of shoes, and you want to buy a pair that costs $25. How much could you spend at most on a second pair of shoes?"
   • "You have to run at least 5 miles a day to prepare for a race. You've already run 2 miles. How many more miles do you need to run?"

   These problems serve as a transition to the topic of multi-step inequalities. The teacher points out that these problems involve multiple steps and can be represented using inequalities.

3. The teacher contextualizes the importance of the topic by explaining its real-world applications. For instance, multi-step inequalities can be used to determine how many hours one needs to work to afford an item, or to find the possible range of values in a scientific experiment.

4. To introduce the topic in an engaging way, the teacher uses the following strategies:

   • The teacher presents a "Guess the Rule" game, where the students are shown a sequence of numbers, and they have to determine the rule that governs the sequence. The teacher then reveals that the rule can be represented by a multi-step inequality.
   • The teacher presents a short video clip or story about the use of multi-step inequalities in real life, such as in the stock market or in predicting the weather. This will help students to see the relevance and applicability of the topic.

5. The teacher then formally introduces the topic of 'Multi-Step Inequalities' and explains that it involves inequalities with more than one step or operation. The teacher emphasizes that the techniques used to solve multi-step inequalities are similar to those used in solving linear equations, but with specific adjustments due to the nature of inequalities.

Development (20 - 25 minutes)

Activity 1: 'Walk the Inequality Line' (10 - 12 minutes)

1. The teacher divides the students into small groups of 3 or 4 and distributes a 'Walk the Inequality Line' worksheet to each group. This worksheet contains a series of 'multi-step inequality stations' represented by number lines.

2. The teacher explains that the goal of the activity is for the students to 'walk' along the number lines, starting from the first station and following the inequality rules, until they reach the final station.

3. At each station, the students will find a multi-step inequality problem. They must solve this problem to determine the direction and number of 'steps' they need to take on the number line.

4. The teacher emphasizes that the students must adhere to the rules of inequalities throughout the activity. For example, if they encounter a subtraction step that involves flipping the inequality sign, they must remember to flip the sign on the number line as well.

5. The teacher expects the students to work collaboratively within their groups, using their combined knowledge and skills to navigate the number lines. This encourages peer learning and engagement.

6. Once all the groups have completed the activity, the teacher facilitates a class discussion, asking each group to explain how they solved one of the problems and 'walked' along the number line. This is an opportunity for the students to articulate their understanding of multi-step inequalities and for the teacher to assess their comprehension.

Activity 2: 'Inequality Problem Solvers' (10 - 13 minutes)

1. The teacher instructs the students to remain in their groups and distributes a set of 'Inequality Problem Solvers' task cards. Each card presents a real-world problem that can be solved using multi-step inequalities.

2. The teacher explains that the students' task is to solve these problems, using the guidelines and techniques they have learned. The problems range in difficulty to challenge the students and accommodate different learning styles.

3. The teacher encourages the students to discuss and reason through the problems, reminding them that the process is just as important as the final answer. They should pay attention to how they apply the multi-step inequality rules, especially when there are multiple operations involved.

4. As the students work, the teacher circulates the room, providing support and guidance as needed. The teacher asks probing questions to guide the students' thinking and to ensure they are grasping the concept.

5. When a group solves a task, the teacher asks them to share their solution with the class. This promotes a sense of achievement and encourages the students to learn from each other.

6. At the end of the activity, the teacher holds a brief discussion, asking the students to reflect on the process, any challenges they faced, and how they overcame them. This reflection helps consolidate the learning and prepares the students for the next steps – application of the concept and independent practice.

These hands-on activities provide students with the opportunity to apply the concept of multi-step inequalities in a dynamic and engaging way. They also foster collaborative problem-solving, critical thinking, and communication skills - all essential for their mathematical development.

Feedback (5 - 7 minutes)

1. The teacher initiates a group discussion, asking each group to share their solutions or conclusions from the activities. Each group is given up to 2 minutes to present their findings. This allows for the sharing of different approaches and solutions, promoting a deeper understanding of multi-step inequalities.

2. The teacher then facilitates a class-wide discussion, connecting the solutions presented by the groups to the theory of multi-step inequalities. The teacher points out the similarities and differences between the groups' solutions, reinforcing the key concepts and techniques.

3. The teacher asks the students to reflect on the activities and the lesson as a whole. The teacher can pose guiding questions such as:

   • "What was the most important concept you learned today?"
   • "Which parts of the lesson were the most challenging for you, and why?"
   • "How did working in groups help you understand multi-step inequalities better?"

4. The teacher encourages the students to share their reflections, promoting a culture of open communication and reflection. This not only helps the students consolidate their learning but also provides valuable feedback for the teacher to improve future lessons.

5. The teacher concludes the feedback session by summarizing the key points of the lesson and highlighting the progress made by the students. The teacher reassures the students that it's normal to find some concepts challenging and that with continued practice and effort, they will become more proficient in solving multi-step inequalities.

6. Finally, the teacher assigns homework that involves solving additional multi-step inequality problems. The teacher emphasizes that the purpose of the homework is to provide further practice and reinforcement of the concepts learned in class. The teacher also encourages the students to seek help if they have any difficulties with the homework.

This feedback stage is crucial for assessing the students' understanding of multi-step inequalities, identifying any misconceptions, and planning for future lessons. It also provides an opportunity for the students to reflect on their learning, fostering a growth mindset and self-directed learning.

Conclusion (3 - 5 minutes)

1. The teacher begins the conclusion by summarizing the main points of the lesson. They remind the students that multi-step inequalities involve inequalities with more than one step or operation, and that the techniques used to solve them are similar to those used in solving linear equations, although with some adjustments due to the nature of inequalities. The teacher also highlights the importance of adhering to the rules of inequalities, such as flipping the inequality sign when multiplying or dividing by a negative number.

2. The teacher then explains how the lesson connected theory, practice, and applications. They point out that the initial theory discussion and problem-solving activities provided the students with a solid understanding of multi-step inequalities. The hands-on activities, 'Walk the Inequality Line' and 'Inequality Problem Solvers', allowed the students to put this theory into practice in an engaging and interactive way. The real-world problem situations presented in the activities and the discussion of their solutions during the feedback session demonstrated the practical applications of multi-step inequalities.

3. The teacher suggests additional materials for the students to further enhance their understanding of the topic. This could include relevant sections from the textbook, online video tutorials, interactive math games, and extra practice worksheets. The teacher reminds the students to make use of these resources and to seek help if they encounter any difficulties or have any questions.

4. Finally, the teacher briefly discusses the importance of the topic for everyday life. They explain that multi-step inequalities are not just abstract mathematical concepts, but are actually used in many real-life situations. They provide a few examples, such as in financial planning (e.g., determining how much one can spend based on their income and expenses), in physics (e.g., calculating the range of a projectile), and in sports (e.g., determining the number of goals needed to win a game). They stress that understanding and being able to solve multi-step inequalities can therefore be very useful in many practical contexts.

5. The teacher ends the lesson by encouraging the students to continue practicing and applying the concept of multi-step inequalities, and by reminding them that the key to mastering this, like any other skill, is through consistent effort and practice.