Objectives (5 - 7 minutes)

1. Understanding the concept of direct and inverse proportion:

   • Students should be able to understand what direct and inverse proportion is, recognizing their characteristics and differences.
   • It should be emphasized that in a direct proportion, when one quantity increases, the other also increases in the same proportion, and vice versa. In an inverse proportion, when one quantity increases, the other decreases, and vice versa.

2. Solving problems of direct and inverse proportion:

   • Students should be able to apply the corresponding formulas to solve problems of direct and inverse proportion.
   • They should also be able to identify the type of proportionality relationship in a given problem and apply the correct strategy to solve it.

3. Performing calculations involving directly and inversely proportional quantities:

   • Students should be able to perform simple calculations involving directly and inversely proportional quantities.
   • It should be emphasized that to perform these calculations, it is necessary to understand and correctly apply the corresponding formulas.

Introduction (10 - 15 minutes)

1. Review of previous concepts:

   • The teacher should start the lesson by reviewing the concepts of ratio and proportion, which are fundamental for understanding the topic 'Proportionality Relations'.
   • Examples of real-life situations where these concepts are applied can be brought, such as the proportion of ingredients in a recipe, the relationship between speed and time in uniform motion, among others.
   • It is important for students to understand these concepts well, as they will be the basis for understanding direct and inverse proportionality relationships.

2. Contextualization of the topic:

   • The teacher should then contextualize the importance of direct and inverse proportionality relationships, showing how they are present in various everyday situations and in different areas of knowledge.
   • Examples of real situations can be brought, such as the relationship between the amount of fuel and the distance traveled in a vehicle, the relationship between the number of workers and the time needed to perform a task, among others.
   • It is important for students to realize that understanding these concepts is fundamental for solving practical problems and for understanding real-world phenomena.

3. Presentation of the problem situation:

   • The teacher should present two problem situations to the students involving direct and inverse proportionality relationships.
   • For example, questions like: 'If a motorcycle travels a distance at a constant speed, what can we say about the relationship between the speed of the motorcycle and the time it takes to travel that distance?' or 'If the number of workers on a construction site is doubled, the time needed to complete the task is halved. What kind of proportionality relationship can we establish between the number of workers and the time needed to complete the task?'
   • These problem situations will serve to arouse the students' interest and introduce them to the topic of the lesson.

4. Introduction to the topic:

   • To introduce the topic in an engaging way, the teacher can tell the story of how proportionality relationships were discovered and how they have been used throughout history.
   • For example, it can be mentioned how the ancient Egyptians used proportionality relationships to build their pyramids, or how the ancient Greeks deeply studied proportions and used them in their works of art and architecture.
   • These historical curiosities will help capture the students' attention and show the importance and relevance of the topic.

Development (20 - 25 minutes)

1. Theory - Direct Proportionality

   • The teacher should start by explaining the concept of direct proportionality.
   • It should be emphasized that in a direct proportionality relationship, when one quantity increases, the other also increases in the same proportion, and vice versa.
   • The example of speed and travel time can be used: if a vehicle's speed increases, the time required to travel a certain distance decreases, and if the speed decreases, the time increases.
   • The teacher should show the formula for direct proportionality: if A is directly proportional to B, then A = k * B, where k is the proportionality constant.

2. Practical Examples - Direct Proportionality

   • The teacher should then present some practical examples of situations involving direct proportionality.
   • Examples of real situations can be brought, such as the relationship between the amount of fuel and the distance traveled in a vehicle, the relationship between the number of workers and the time needed to perform a task, among others.
   • For each example, the teacher should explain how to identify that it is a direct proportionality relationship and how to solve the problem by applying the corresponding formula.

3. Theory - Inverse Proportionality

   • The teacher should then move on to the theory of inverse proportionality.
   • It should be emphasized that in an inverse proportionality relationship, when one quantity increases, the other decreases, and vice versa.
   • The example of the number of workers and the time to perform a task can be used: the more workers, the less time is needed to perform the task, and vice versa.
   • The teacher should show the formula for inverse proportionality: if A is inversely proportional to B, then A = k / B, where k is the proportionality constant.

4. Practical Examples - Inverse Proportionality

   • The teacher should then present practical examples of situations involving inverse proportionality.
   • Examples of real situations can be brought, such as the relationship between the number of workers and the time needed to perform a task, the relationship between the number of students and the time needed to correct exams, among others.
   • For each example, the teacher should explain how to identify that it is an inverse proportionality relationship and how to solve the problem by applying the corresponding formula.

5. Summary and Recap

   • At the end of the theoretical explanation and practical examples, the teacher should summarize the main points covered, reinforcing the concepts of direct and inverse proportionality, and the corresponding formulas.
   • Students should also be given time to ask questions and clarify any doubts.
   • The teacher should then propose a brief review activity, where students will have to identify if the situations presented are of direct or inverse proportionality, and solve the corresponding problems.

Return (10 - 12 minutes)

1. Group Discussion (3 - 4 minutes):

   • The teacher should start this stage by promoting a group discussion on the solutions to the proposed problems.
   • Students should be encouraged to share their solutions and explain how they arrived at them, thus promoting the exchange of ideas and the collective construction of knowledge.
   • The teacher should be attentive to correct possible mistakes and clarify doubts that may arise.

2. Connection with Theory (3 - 4 minutes):

   • After discussing the problems, the teacher should make the connection between the solutions found by the students and the theory presented.
   • It should be highlighted how the concepts of direct and inverse proportionality were applied in solving the problems, thus reinforcing the importance of understanding these concepts.
   • For example, the teacher can ask: 'How did you identify that this situation was one of direct proportionality?' or 'Why did you solve this problem using the formula for inverse proportionality?'.

3. Individual Reflection (2 - 3 minutes):

   • The teacher should then propose that students make a brief individual reflection on what they learned in the lesson.
   • Students can be asked to write down on a paper or in their notebooks the answers to the following questions: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • The goal of this activity is for students to realize what they have learned and what their doubts are, allowing the teacher to adjust the planning of the next lessons according to the students' needs.

4. Teacher's Feedback (2 - 3 minutes):

   • Finally, the teacher should give feedback to the students about the lesson.
   • Positive points, such as active participation of students, understanding of concepts, problem-solving, should be praised, and areas that need improvement, such as attention to details, clarity in presenting concepts, among others, should be pointed out.
   • The teacher should also address the questions that were raised during the lesson, or take note of them to be clarified in the next lessons.

Conclusion (5 - 8 minutes)

1. Summary of Key Points (2 - 3 minutes):

   • The teacher should start the Conclusion by summarizing the key points covered in the lesson.
   • The concept of direct and inverse proportionality, the corresponding formulas, and how to identify and solve problems involving these relationships should be reinforced.
   • This can be done interactively, asking students to contribute to the summary, thus reinforcing what was learned.

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

   • The teacher should then explain how the lesson connected theory, practice, and applications.
   • It should be emphasized that understanding theoretical concepts is fundamental for the practical resolution of problems and for the application of these concepts in real situations.
   • Examples of how direct and inverse proportionality relationships are applied in different areas of knowledge and in everyday situations can be brought, reinforcing the importance and relevance of these concepts.

3. Extra Materials (1 - 2 minutes):

   • The teacher should then suggest some extra materials for students who wish to deepen their knowledge on the subject.
   • Books, websites, videos, and online exercises that address the topic of proportionality relationships in more detail can be recommended.
   • The teacher can also propose that students research and bring examples of real situations involving proportionality relationships to the next lesson, thus stimulating autonomous study and practical application of the concepts learned.

4. Relevance of the Topic (1 minute):

   • Finally, the teacher should emphasize the importance of the topic for daily life and for learning in mathematics.
   • It should be highlighted that the ability to recognize and solve proportionality problems is fundamental for problem-solving in various areas of knowledge, for making data-based decisions, and for understanding real-world phenomena.
   • The teacher should encourage students to apply the knowledge acquired in the lesson in their daily lives and in their future careers, thus reinforcing the relevance and usefulness of mathematics.