Objectives (5 - 7 minutes)

1. Understand and apply the compound rule of three in different contexts. This includes the ability to identify proportional quantities, establish the proportion, and calculate the unknown.

2. Develop the ability to solve practical problems using the compound rule of three. This implies applying the acquired knowledge to solve real-world issues, such as buying and selling situations, substance mixtures, among others.

3. Practice problem interpretation and formulation of strategies to solve them. Students should be able to understand the presented problem, identify relevant information, and decide the best way to use the compound rule of three to solve it.

Secondary objectives:

• Stimulate collaboration and communication among students through group work. This will be done through proposing activities that require discussion and sharing of ideas to solve problems.

• Foster critical thinking and argumentation skills. Students will be encouraged to justify their answers, explaining step by step how they arrived at them. This contributes to the development of valuable skills that go beyond the field of mathematics.

Introduction (10 - 15 minutes)

1. Review of previous content and contextualization:

   • The teacher should start the lesson by reviewing the concepts of simple rule of three, previously studied, and how they are applied in everyday situations.
   • Next, some problem situations involving the compound rule of three should be introduced, such as: "If 10 liters of water require 2 sachets of powdered juice, how many sachets will be needed for 20 liters of water?" or "If 3 workers take 4 days to build a wall, how many workers will be needed to build the same wall in 2 days?".
   • The teacher should emphasize the importance of understanding and knowing how to apply the compound rule of three, as it is frequently used in real situations, such as in cooking recipes, in substance dilution calculations, among others.

2. Problem situation to spark students' interest:

   • The teacher can propose the following situation: "Imagine you are in a cooking competition and need to adjust a recipe to serve 50 people, but the original recipe serves only 10 people. How would you calculate the amount of each ingredient that needs to be increased?". This situation involves the compound rule of three and the concept of proportionality, which will be addressed in the lesson.

3. Contextualization of the importance of the compound rule of three:

   • The teacher should explain that the compound rule of three is a very useful tool in various everyday situations and in various fields, such as engineering, architecture, medicine, economics, among others.
   • It should be highlighted that mastering this mathematical skill can facilitate the solution of practical problems and contribute to making more informed and efficient decisions.

4. Introduction of the topic with curiosities or practical applications:

   • The teacher can share the curiosity that the compound rule of three was developed by mathematicians from ancient Babylon, over 4,000 years ago, to solve practical problems in trade and engineering.
   • Some modern practical applications can also be mentioned, such as using the compound rule of three to calculate the correct dosage of medications, to determine the ideal proportion of ingredients in a mixture, as in the case of agricultural fertilizers, or to plan and control industrial production, adjusting the amount of inputs according to demand.

Development (20 - 25 minutes)

1. Activity 1 - "The great journey" (10 - 12 minutes):

   • The teacher should divide the class into groups of 4 to 5 students. Each group will receive a sheet with a map of a fictional city and a series of problems to be solved.
   • The problems will consist of hypothetical situations that require the application of the compound rule of three to be solved. For example: "If a car travels 10 km with 2 liters of gasoline, how many liters of gasoline will be needed to travel 30 km? And if the distance is 50 km?" or "If a bus takes 40 people from one city to another in 4 hours, how many hours will it take to take 80 people? And if it takes 120 people?".
   • Each group should discuss and solve the problems using the compound rule of three. The teacher should circulate around the room, assisting and guiding the students as necessary.
   • At the end of the activity, each group should present their solutions to the class, explaining step by step how they arrived at them. The teacher should provide feedback and clarify any remaining doubts.

2. Activity 2 - "The mystery of the secret formula" (10 - 12 minutes):

   • In this activity, students will continue working in their groups. Each group will receive a letter from a fictional scientist containing a formula for a mysterious substance and instructions to prepare a specific quantity of it.
   • The letter will also include a series of information about the substance and its preparation, but some of the information will be missing. Students will have to use the compound rule of three to fill in the gaps and complete the formula.
   • For example, the letter may say: "To prepare 1 liter of the substance, mix 2 parts of ingredient A with 3 parts of ingredient B. To prepare 5 liters, how many parts of each ingredient are needed?". Here, students will have to apply the compound rule of three to calculate the amount of each ingredient required.
   • The teacher should circulate around the room, helping the groups as needed and clarifying any doubts that may arise.
   • At the end of the activity, the groups should present their solutions to the class, explaining how they used the compound rule of three to solve the problem. The teacher should provide feedback and clarify any remaining doubts.

3. Activity 3 - "Proportion challenge" (5 - 7 minutes):

   • To conclude the Development stage, the teacher can propose a quick challenge to the class. The challenge will consist of a series of compound rule of three problems that students must solve individually.
   • The teacher should distribute an activity sheet with the problems and give students about 5 minutes to solve them. The problems should be of progressive difficulty, starting with simpler problems and gradually increasing in complexity.
   • After the time is up, the teacher should collect the sheets and correct them. The correct solutions will be discussed in the review stage. The challenge will serve to consolidate students' learning and to identify possible difficulties that need to be clarified.

Return (8 - 10 minutes)

1. Sharing solutions in groups (3 - 4 minutes):

   • Each group will have up to 3 minutes to share their solutions or conclusions from the activities carried out. The teacher should guide the students to explain step by step how they applied the compound rule of three to solve each problem.
   • During the presentations, the teacher should encourage the participation of all students, asking questions and requesting explanations to ensure that everyone understood the solutions presented.
   • The teacher should also take this opportunity to correct any conceptual errors that may have arisen and to reinforce the key points of the compound rule of three.

2. Connection between practice and theory (2 - 3 minutes):

   • The teacher should then promote a discussion on how the practical activities carried out connect with the theory of the compound rule of three. Students should be encouraged to identify and explain the theoretical concepts that were applied in solving the problems.
   • The teacher should reinforce the importance of understanding the theory behind the compound rule of three in order to apply it correctly in different situations.
   • For example, the teacher may ask: "How did you apply the idea of proportions in the problems you solved?" or "How does the simple rule of three differ from the compound rule of three?".

3. Final reflection (2 - 3 minutes):

   • The teacher should propose that students reflect for a minute on the following questions:
     1. "What was the most important concept learned today?"
     2. "What questions have not been answered yet?"
   • After this minute of reflection, some students may be invited to share their answers with the class.
   • The teacher should take into account the students' responses to plan the next lessons and to adjust the teaching approach, if necessary.
   • Additionally, the teacher should encourage students to bring their doubts to the next class or to the office hours, if further clarification is needed.

4. End of the lesson:

   • The teacher should thank everyone for their participation and effort, encouraging them to continue practicing and studying the topic at home.
   • The teacher should also briefly recap the main points covered in the lesson and the learning objectives achieved.
   • Finally, the topic of the next lesson should be announced along with any necessary preparations or readings.

Conclusion (5 - 7 minutes)

1. Summary of key topics (2 - 3 minutes):

   • The teacher should start the Conclusion by recalling the main topics covered in the lesson. This includes the definition of compound rule of three, identification of proportional quantities, establishment of proportions, and calculation of the unknown.
   • The importance of understanding the theory behind the compound rule of three to apply it correctly in different contexts and situations should be emphasized.
   • The teacher can give a quick summary of the problems solved in groups and the solutions presented, reinforcing the theoretical concepts that were applied in practice.

2. Connection between theory, practice, and applications (1 - 2 minutes):

   • The teacher should explain how the lesson managed to connect the theory of the compound rule of three with practice, through the activities carried out in groups.
   • The practical application of the compound rule of three in everyday situations and in various fields, such as engineering, architecture, medicine, economics, among others, should be highlighted.
   • The teacher may, for example, mention again the problem situations presented in the Introduction of the lesson and show how the compound rule of three can be used to solve them.

3. Extra study materials (1 minute):

   • The teacher should suggest extra materials for students who wish to deepen their knowledge of the compound rule of three. This may include math books, educational websites, explanatory videos, among others.
   • For example, the teacher may suggest reading a specific chapter of a math book or watching a video that explains the compound rule of three in a different way.

4. Importance of the subject for daily life (1 - 2 minutes):

   • Finally, the teacher should reinforce the importance of the compound rule of three for students' daily lives. It should be emphasized that the ability to solve problems using the compound rule of three can facilitate decision-making in various life situations.
   • The teacher may, for example, give examples of everyday situations where the compound rule of three can be useful, such as in recipe preparation, calculating discounts on purchases, planning trips, among others.
   • Students should be encouraged to observe and identify new situations where the compound rule of three can be applied, as a way to reinforce learning and the relevance of the subject.