Objectives (5 - 7 minutes)

• Understand the Compound Rule of Three: The main objective of this lesson is for students to understand the concept and application of the Compound Rule of Three. They should be able to identify when it is necessary to use this rule and how it differs from the Simple Rule of Three.

• Practice Applying the Compound Rule of Three: In addition to understanding the theory, students should be able to apply the Compound Rule of Three in practical situations. They should be able to solve problems involving inversely proportional and mixed quantities, using the compound rule of three efficiently and accurately.

• Develop Logical Reasoning Skills: The use of compound rules of three involves the application of logical reasoning. Therefore, a secondary objective is for students to develop their logical thinking skills when solving compound rule of three problems.

Secondary objectives:

• Promote Group Collaboration: Through group activities, students should be encouraged to work as a team, discussing and solving problems together.

• Encourage Active Participation: The lesson should be structured to encourage active participation from students, whether through group discussions, problem-solving on the board, or other interactive activities.

• Foster Autonomous Learning: Students should be encouraged to seek problem-solving autonomously, using the tools and resources available to them. The teacher should act as a facilitator of the learning process, assisting students when necessary.

Introduction (10 - 15 minutes)

• Review of Previous Content: The teacher should start the lesson by reviewing the concepts of the Simple Rule of Three, which were previously covered. This can be done through a quick oral quiz, where the teacher asks students to recall what the Simple Rule of Three is and how it is applied. This step is crucial, as the Compound Rule of Three is an extension of the Simple Rule of Three.

• Initial Problem Situations: Next, the teacher should present two problem situations involving inversely proportional and mixed quantities. For example:

  1. "If 5 workers work 8 hours a day and produce 1,500 pieces in 10 days, how many pieces will 12 workers produce in 6 days, if they work 6 hours a day?"
  2. "If 3 workers build a wall in 6 days, working 8 hours a day, how many workers will be needed to build the same wall in 4 days, if they work 12 hours a day?"

• Subject Contextualization: The teacher should then contextualize the importance of the Compound Rule of Three, explaining that it is frequently used in various areas such as engineering, economics, physics, and chemistry. For example, in engineering, the compound rule of three is used to calculate the speed of a fluid in a pipe, based on the pressure and diameter of the pipe.

• Topic Introduction: To introduce the topic and spark students' interest, the teacher can present some curiosities or interesting applications of the Compound Rule of Three. For example:

  1. "Did you know that the Compound Rule of Three was used to calculate the speed of light for the first time? Scientists of the time used the speed of sound, the length of a day, and the distance from Earth to the Sun to arrive at an approximate value of the speed of light!"
  2. "Another interesting application of the Compound Rule of Three is in gastronomy! Have you ever thought about how recipes are adapted when we increase or decrease the amount of ingredients? This also involves the Compound Rule of Three!"

Development (20 - 25 minutes)

• Activity 1 - "The Workers' Challenge" (10 - 12 minutes): In this activity, students will be divided into groups of 4 to 5 people. Each group will receive a problem situation involving inversely proportional and mixed quantities, similar to those presented in the Introduction. The challenge is for them to solve the problem, applying the Compound Rule of Three, and present the solution to the class. The teacher should circulate around the room, assisting groups that have difficulties and encouraging discussion and logical reasoning. At the end of the activity, each group should present their solution, explaining step by step how they arrived at it.

  • Activity step by step:

    1. The teacher divides the class into groups and gives each group a problem situation.
    2. The students, in their groups, discuss the problem situation and plan the best strategy to solve it.
    3. Each group solves the problem situation, applying the Compound Rule of Three. They should show all calculations and justifications.
    4. The groups present their solutions to the class. The teacher and other students can ask questions and make comments.
    5. The teacher leads a brief discussion on the different strategies used by the groups and the possible difficulties encountered. He also reinforces the concepts learned and clarifies any remaining doubts.

• Activity 2 - "Building a Wall" (10 - 12 minutes): In this activity, students will continue working in groups. The teacher will present a new problem situation involving inversely proportional and mixed quantities, this time related to building a wall. The challenge is for students to create their own problem situation, based on the presented situation, and solve it applying the Compound Rule of Three. They should think of a situation that could occur in real life and that involves the same logic as the original situation. For example, "If 5 painters can paint a wall in 8 hours, how many hours would 3 painters take to paint the same wall, if they worked in a lower temperature that decreases painting efficiency?" At the end of the activity, each group should present their problem situation and the solution to the class.

  • Activity step by step:

    1. The teacher presents the original problem situation.
    2. The students, in their groups, discuss the problem situation and think of a similar situation that could occur in real life.
    3. Each group creates their own problem situation and writes it on paper, along with the solution.
    4. The groups exchange their problem situations with each other. They should solve the other group's problem situation, applying the Compound Rule of Three.
    5. The groups present their problem situations and solutions to the class. The teacher and other students can ask questions and make comments.
    6. The teacher leads a brief discussion on the different problem situations created by the groups and the possible difficulties encountered. He also reinforces the concepts learned and clarifies any remaining doubts.

• Discussion and Synthesis (5 - 7 minutes): After the activities, the teacher should lead a classroom discussion to synthesize what was learned. He should reinforce the main concepts of the Compound Rule of Three, problem-solving strategies, and common difficulties. The teacher should also clarify any remaining doubts and prepare students for the next stage of the lesson.

Return (8 - 10 minutes)

• Group Discussion (3 - 4 minutes): The teacher should promote a group discussion, where each team shares their solutions and conclusions about the activities carried out. At this stage, the teacher should encourage students to explain how they arrived at their answers, what strategies they used, and what difficulties they encountered. The teacher should encourage the participation of everyone, asking questions to the groups and promoting interaction among them.

  • Activity step by step:

    1. The teacher asks each group to share their solutions and conclusions with the class.
    2. The students in each group have a limited time to present their ideas.
    3. The teacher and other students can ask questions and make comments to each group.
    4. The teacher highlights the most relevant ideas and clarifies any remaining doubts.

• Connection with Theory (2 - 3 minutes): After the group discussion, the teacher should make a connection between the practical activities and the theory presented at the beginning of the lesson. He should explain how the strategies used by students to solve the problems relate to the Compound Rule of Three and how they applied logical reasoning to reach the solutions. The teacher should also reinforce the main concepts of the Compound Rule of Three, clarifying any doubts that may have arisen during the activities.

  • Activity step by step:

    1. The teacher makes a connection between the practical activities and the theory.
    2. The teacher reinforces the main concepts of the Compound Rule of Three and clarifies any remaining doubts.

• Final Reflection (2 - 3 minutes): To conclude the lesson, 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?"

  The teacher should encourage students to share their reflections with the class, if they feel comfortable. This reflection activity is important to consolidate learning and identify any comprehension gaps that may need to be addressed in future lessons.

  • Activity step by step:

    1. The teacher asks students to reflect for a minute on the proposed questions.
    2. Students who wish to can share their reflections with the class.
    3. The teacher makes a brief comment on the students' reflections and concludes the lesson.

Conclusion (5 - 7 minutes)

• Summary of Covered Content (2 - 3 minutes): The teacher should summarize the main points discussed during the lesson. This includes the definition of the Compound Rule of Three, the difference between directly and inversely proportional quantities, and the application of the Compound Rule of Three in practical situations. The teacher can use the blackboard or a presentation slide to illustrate the main points and help students visualize the ideas.

• Connection between Theory, Practice, and Applications (1 - 2 minutes): The teacher should emphasize how the lesson connected the theory, practice, and applications of the Compound Rule of Three. He can mention the problem situations presented and how they illustrate the application of the rule. Additionally, the teacher can reinforce how group activities helped students better understand the theory and apply it in a practical manner.

• Extra Materials (1 - 2 minutes): The teacher should suggest extra materials for students who wish to deepen their knowledge of the Compound Rule of Three. This may include websites, videos, books, and online exercises. For example, the teacher can suggest an explanatory video about the Compound Rule of Three, a website with various examples and solved exercises, and a mathematics book with a section dedicated to this topic.

• Importance of the Subject in Daily Life (1 minute): Finally, the teacher should highlight the importance of the Compound Rule of Three in daily life. He can mention some everyday situations where the rule can be applied, such as in adapting recipes, calculating economies of scale, or solving engineering problems. The goal is to show students that mathematics has practical and relevant applications in their lives.