Objectives (5 - 7 minutes)

1. Understand the concept of Inequalities: Students should be able to understand what inequalities are and how they differ from equations. They should realize that an inequality represents an inequality and not an equality.

2. Learn to represent Inequalities on the number line: Students should learn to represent inequalities on a number line. They should understand that the number line is a useful tool for visualizing and solving inequalities.

3. Practice solving Inequalities: Students should practice solving inequalities, using the number line as a guide to help them visualize the solution. They should be able to identify and mark the solution correctly.

Secondary Objectives:

• Develop critical thinking skills: Through solving inequalities, students should also develop critical thinking skills, such as the ability to analyze information, make connections, and solve problems.

• Promote teamwork skills: During the lesson, students should be encouraged to work in pairs or small groups to solve problems. This will promote collaboration and communication among students.

Introduction (10 - 15 minutes)

1. Review of previous content: The teacher should start the lesson by quickly reviewing the concepts of equations, inequalities, and the number line. This will help prepare students for the new content, reinforcing the importance of these concepts in mathematics. (3 - 5 minutes)

2. Problem situation: The teacher can propose two problem situations to spark students' interest. For example:

   • "Suppose you are organizing a party and need to buy cans of soda. Each can costs R$ 2.50 and you only have R$ 10.00. How many cans of soda can you buy at most?"

   • "Imagine you have homework to do, but also want to play video games. You can only play if you finish your homework before 8 p.m. If it is 5 p.m. now and the homework takes 2 hours to complete, will you be able to play video games today?" (5 - 7 minutes)

3. Contextualization: The teacher should explain that inequalities are used to represent situations of inequality or restriction in everyday problems and in many areas of science and engineering. For example, in economics, inequalities are used to represent inequality relationships between prices and quantities of goods. In physics, inequalities are used to represent constraints in motion problems. (2 - 3 minutes)

4. Curiosities: To capture students' attention, the teacher can share some curiosities about inequalities. For example:

   • "Did you know that inequalities are also used in gambling games, such as poker? The cards you have in hand and those on the table can be seen as an inequality representing the probability of winning the game."

   • "Another curiosity is that, although inequalities are represented by symbols like < (less than) and > (greater than), they can be read in a different way. For example, x > 3 can be read as 'x is greater than 3', but it can also be read as '3 is less than x'." (2 - 3 minutes)

Development (20 - 25 minutes)

1. Modeling Activity 1 - Inequalities in Real Life (10 - 12 minutes)

   • The teacher should divide the class into groups of up to 5 students.
   • Each group will receive a set of cards with different everyday situations involving inequalities. For example: "You have R$ 100.00 to spend at a mall. Which stores can you enter and what can you buy?" or "How many friends can you serve a cake to if each can eat at most 2 pieces?".
   • Students should discuss and decide how to represent these situations as inequalities. They should write the inequalities on paper and then mark them on a number line.
   • After the activity, each group should present their situations and how they solved them to the class. The teacher should correct any errors and reinforce the correct concepts.

2. Modeling Activity 2 - Inequalities Game (10 - 12 minutes)

   • Still in groups, students will receive a game board drawn with a number line.
   • The teacher will provide each group with a set of cards with written inequalities, such as "x > 3", "2x ≤ 8", "5 - x < 7", etc.
   • The goal of the game is for students to place their pieces on the board in a way that satisfies the inequality written on the card. For example, if the card says "x > 3", the group's piece should be placed on any number greater than 3 on the number line.
   • The teacher should monitor the game, clarifying doubts and ensuring that the inequalities are being correctly represented on the board.
   • After a few rounds, the teacher can make the game more challenging by introducing cards with compound inequalities, such as "2 < 3x + 1 ≤ 5".
   • At the end of the game, the teacher should lead a discussion about the strategies used by the groups and the difficulties encountered in solving the inequalities.

3. Problem Solving Activity (5 - 7 minutes)

   • Each group will receive a set of problems involving inequalities to solve. These problems can be taken from textbooks or math websites.
   • Students should work together to solve the problems, using the strategies they learned during the previous activities.
   • The teacher should circulate around the room, helping groups that are struggling and challenging groups that are progressing quickly.
   • After completing the activity, the teacher should correct the problems with the class, discussing the solutions and clarifying any remaining doubts.

Feedback (8 - 10 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should group all students for a group discussion. Each group should share their solutions or conclusions from the activities carried out.
   • During the discussion, the teacher should encourage students to explain their solving strategies and how they arrived at their answers. This will allow students to learn from each other and see different approaches to solving inequalities.
   • The teacher should ask targeted questions to ensure that students have understood the concepts and procedures correctly. For example, "How did you decide which side of the number line to mark for the solution of an inequality?" or "Why do we sometimes need to invert the sign of an inequality when multiplying or dividing both sides by a negative number?".

2. Connection with Theory (2 - 3 minutes)

   • The teacher should review the theoretical concepts presented in the lesson, highlighting how they were applied in the practical activities. For example, the teacher can recall how inequalities were represented on the number line and how students used this representation to solve problems.
   • The teacher can also reinforce the importance of critical thinking and teamwork skills, which were developed during the practical activities.
   • The teacher should clarify any misunderstandings or confusion that arose during the group discussion, ensuring that all students have understood the concepts and procedures correctly.

3. Individual Reflection (2 - 3 minutes)

   • The teacher should suggest that students reflect individually on what they learned in the lesson. They should think about the following questions:
     1. "What was the most important concept I learned today?"
     2. "What questions have not been answered yet?"
   • Students should write down their answers on a piece of paper. This reflection activity will help students consolidate their learning and identify any areas where they still have doubts or difficulties.
   • Students can share their answers with the class if they wish. The teacher should listen carefully to students' answers and take note of any questions that have not been answered, to be addressed in future lessons.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes)

   • The teacher should start the Conclusion by recalling the main points of the lesson. This includes the definition of inequalities, the representation of inequalities on the number line, and the solving of inequalities.
   • The teacher should reinforce that inequalities are used to represent inequalities, unlike equations that represent equalities.
   • The teacher should remind students how the number line can be used as a visual tool to solve inequalities, facilitating the visualization of the solution.

2. Connection between Theory and Practice (1 - 2 minutes)

   • The teacher should highlight how the lesson connected theory with practice. For example, the group activities allowed students to apply the theoretical concepts of inequalities and the number line in everyday situations, making learning more meaningful.
   • The teacher should emphasize that theoretical understanding is essential for the practical resolution of inequalities, and vice versa. Both aspects are necessary for a complete understanding of the topic.

3. Extra Materials (1 - 2 minutes)

   • The teacher should suggest some extra materials for students who want to deepen their knowledge of inequalities. This may include math books, educational websites, explanatory videos, and online exercises.
   • The teacher can also indicate some practical applications of inequalities in different areas, such as economics, physics, engineering, among others. This will help students see the relevance of the learned content beyond the school environment.

4. Importance of the Subject (1 minute)

   • In conclusion, the teacher should emphasize the importance of the subject presented for students' daily lives. It should be remembered that the ability to solve inequalities is useful in various situations, from solving math problems to making decisions in everyday life.
   • The teacher can give some examples of how inequalities are used in everyday situations, such as managing personal finances, scheduling, solving logistics problems, among others.
   • In the end, the teacher should encourage students to continue practicing and exploring the topic, reminding them that mathematics is a discipline that is built over time and practice.