Objectives (5 - 7 minutes)

1. Understand the concept of proportion and its application in everyday life: Students should be able to define proportion and understand how it applies to various everyday situations. This includes identifying when something is directly or inversely proportional.

2. Solve proportion problems: Students should be able to apply the formulas and methods learned to solve practical problems involving proportions. They should be able to identify the quantities involved and establish the proportional relationship between them.

3. Understand the difference between ratio and proportion: One important skill to be developed is the ability to distinguish between a ratio and a proportion. Students should understand that a proportion is an equality between two or more ratios.

Secondary Objectives

• Develop critical thinking and problem-solving skills: Through the practice of solving proportion problems, students should enhance their logical and critical thinking skills.

• Promote active and collaborative learning: The lesson plan should encourage active participation from students, through classroom discussions and group activities. Collaboration among students should be valued for knowledge building.

Introduction (10 - 15 minutes)

1. Review of previous lessons: The teacher should begin the class by reviewing concepts that are prerequisites for the understanding of the current topic. This would include a review of basic math concepts, such as the definition of ratio and the division operation. The teacher could, for example, propose a problem involving the division of a quantity into equal parts, leading the students to think about the idea of proportion.

2. Problem situations: The teacher could present two problem situations that involve the idea of proportion. For instance, he or she could ask students how they could divide a cake equally among three people, or how they could calculate the distance that a person would travel if they walked at a constant speed for a certain amount of time. These situations should serve to arouse the interest of students and to introduce the idea of proportion in a practical and contextualized way.

3. Contextualization: The teacher should then contextualize the importance of the topic, showing how proportion is used in various areas of life, such as cooking, engineering, economics, among others. He or she could, for example, mention how proportion is used to adjust recipes, to calculate the amount of material needed for a construction, or to determine the relationship between supply and demand of a product.

4. Introduction to the topic: To gain the attention of the students, the teacher could start the introduction of the topic of proportion in an intriguing way. For example, he or she could mention that the idea of proportion is as old as human civilization, and that it was used by ancient Egyptians to calculate the area of their lands, and by ancient Greek civilizations to build their temples. Another interesting fact that can be mentioned is that proportion is one of the most universal mathematical ideas, present in all cultures and time periods of history.

Development (20 - 25 minutes)

1. Presentation of theory (10 - 12 minutes): The teacher should present the theory about proportion, beginning with the definition of the term. The teacher could use practical examples and visual illustrations to help explain the concept.

   1.1. Definition of Proportion: The teacher should explain that proportion is an equality between two ratios. For example, if we have two ratios, a/b and c/d, they are proportional if a/b = c/d.

   1.2. Properties of Proportion: The teacher should explain that in a proportion, if we multiply or divide the terms of one ratio, we must do the same with the terms of the other ratio. In addition, if the terms of a proportion are switched in position, the proportion remains valid.

   1.3. Ratio and Proportion: The teacher should reinforce the difference between ratio and proportion. While ratio is a comparison between two quantities, proportion is an equality between two or more ratios.

   1.4. Types of Proportion: The teacher should introduce the idea of direct and inverse proportion. He or she should explain that, in a direct proportion, the quantities increase or decrease in the same proportion. In an inverse proportion, however, one quantity increases while the other decreases, or vice versa.

2. Solving Examples (5 - 7 minutes): The teacher should then solve some examples step by step to demonstrate how to apply the theory. The examples should be varied, including problems of direct and inverse proportion. The teacher should explain each step of the solving process, highlighting the most important steps.

3. Group Discussion (5 - 6 minutes): After solving the examples, the teacher should propose a group discussion. The students should be encouraged to share their solving strategies, identify possible mistakes, and correct them. The teacher should act as a mediator, guiding the discussion and clarifying doubts.

4. Practical Activity (5 - 7 minutes): To consolidate the learning, the teacher should propose a practical activity. The students should work in groups to solve a set of proportion problems. The teacher should circulate through the room, offering support and guidance as needed. At the end of the activity, the teacher should review the answers with the class, clarifying any remaining doubts.

Review (8 - 10 minutes)

1. Discussion and Reflection (3 - 4 minutes): The teacher should start the Review by promoting a classroom discussion about the solutions found by the groups for the proportion problems. He or she could ask the students to share their solving strategies, difficulties encountered, and the connection between the theory presented and the practice. The teacher should encourage students to ask questions and explain their answers, fostering an environment of collaborative learning.

2. Connection to Theory (2 - 3 minutes): The teacher should then make the connection between the practical activities and the theory presented. He or she could, for instance, highlight how the idea of direct and inverse proportion was applied to solve the problems. The teacher could also reinforce the importance of understanding the difference between ratio and proportion, and how it influences the problem-solving process.

3. Concept Review (2 - 3 minutes): The teacher should ask the students to summarize in their own words the main concepts learned in the lesson. He or she could, for example, ask: “What is a proportion?” and “How can you tell if two quantities are directly or inversely proportional?”. The teacher should correct any misunderstandings and reinforce the key concepts.

4. Final Reflection (1 - 2 minutes): The teacher should end the lesson by asking the students to reflect for a minute about what they learned. He or she could ask questions such as: “What was the most important concept you learned today?” and “What questions still need to be answered?”. Students should have the opportunity to share their reflections, either verbally or in writing.

5. Feedback (1 minute): The teacher should finally collect feedback from the students about the lesson. He or she could ask what the students liked most and least, and whether they felt that their questions and doubts were adequately answered. The feedback from the students can be used to improve future lessons and to ensure that the students’ needs and expectations are met.

Conclusion (5 - 7 minutes)

1. Lesson Summary (2 - 3 minutes): The teacher should begin the Conclusion of the lesson by summarizing the main points covered. This includes the definition of proportion, the difference between ratio and proportion, and the types of proportion (direct and inverse). The teacher could, for example, remind the students that in a direct proportion, the quantities increase or decrease in the same proportion, while in an inverse proportion, one quantity increases while the other decreases, or vice versa.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should then emphasize how the lesson connected theory, practice, and applications. He or she could, for instance, mention how the theory of proportion was applied to solve the practical problems proposed. In addition, the teacher should reinforce the importance of proportion in various areas of life, showing how it is used in everyday situations, such as cooking, engineering, economics, among others.

3. Extra Materials (1 - 2 minutes): The teacher should then suggest extra materials for students who wish to deepen their understanding of the topic. These materials could include math textbooks, educational websites, explanatory videos, among others. The teacher could, for example, recommend the use of an interactive math website, where students can practice solving proportion problems in a playful and engaging way.

4. Relevance of the Topic (1 minute): Finally, the teacher should reinforce the relevance of the topic covered to the students' everyday lives. He or she could, for example, mention how the ability to solve proportion problems can be useful in various practical situations, such as adjusting cooking recipes, calculating distances, areas, volumes, rates, among others. The teacher should emphasize that mathematics, although it often seems abstract, has real and concrete applications and is a powerful tool to understand and interact with the world around us.