Lesson Plan | Active Learning | Compound Rule of Three Problems

Assumptions: This Active Lesson Plan assumes: a 100-minute class, prior student study with both the Book and the start of Project development, and that only one activity (among the three suggested) will be chosen to be conducted during the class, as each activity is designed to take up a significant portion of the available time.

Objectives

Duration: (5 - 10 minutes)

The Objectives stage in the lesson plan is essential to establish a clear direction for learning. By outlining specific and measurable objectives, students are guided on what they need to achieve and how it will be evaluated. This ensures that the class is focused and efficient, maximizing the use of time in the classroom for practical application of previously acquired knowledge.

Main Objectives:

1. Empower students to solve problems involving the composed rule of three, especially when there is an inversion of proportional quantities.

2. Develop analytical skills in students to identify and apply the concept of inversion of quantities in different mathematical contexts.

Side Objectives:

1. Encourage critical thinking and students' curiosity through practical and real problems that require the application of the composed rule of three.

Introduction

Duration: (20 - 25 minutes)

The introduction of the class is designed to engage students and connect the theoretical content they studied previously with practical and real situations. By presenting problems based on everyday situations, students are encouraged to apply knowledge practically and understand the relevance of the topic in their lives. The contextualization helps solidify learning, showing how mathematics is integrated into various activities and highlighting the historical and current importance of the composed rule of three.

Problem-Based Situations

1. Imagine that an ice cream factory needs to calculate how long it will take to fill a 5000-liter tank if they use three hoses, each with a different flow rate, and it is known that two of them fill the tank in 3 hours, while the third, more powerful one, takes only 2 hours.

2. Consider a scenario where a construction team needs to determine how many days it will take to finish a project if, working together, they can complete the work in 8 days, but a new material would allow it to be finished in just 4 days.

Contextualization

The composed rule of three is a powerful mathematical tool that applies to everyday situations and practical challenges in various fields such as engineering, economics, and logistics. For example, understanding how different production speeds can affect the total time of a process, or how variations in resource usage impact the final outcome, are common applications of this rule. Furthermore, the history of the rule of three dates back to ancient times when it was used to solve trade and proportion problems, showing its relevance and usefulness over time.

Development

Duration: (75 - 80 minutes)

The Development stage is designed to put students in practical and challenging situations that require the direct application of the composed rule of three. By working in groups, students have the opportunity to discuss and collaborate in solving complex problems, which not only reinforces learning of theoretical content but also develops communication and critical thinking skills. By choosing one of the proposed activities, students will apply prior knowledge in problematic and realistic contexts, thereby increasing retention and understanding of the content.

Activity Suggestions

It is recommended to carry out only one of the suggested activities

Activity 1 - The Challenge of the Mathematics Alchemists

> Duration: (60 - 70 minutes)

- Objective: Apply the concept of the composed rule of three to solve a real problem and understand the importance of proportions in chemistry and everyday life.

- Description: Students will be divided into groups of up to 5 people and will receive the mission to help an ancient guild of alchemists prepare a magical potion. The potion requires a mixture of three mystical ingredients, each with different proportions of potency. Students will need to use the composed rule of three to determine the exact quantities of each ingredient needed to produce a potion that cures a rare disease.

- Instructions:

• Divide the class into groups of up to 5 students.

• Present the description of the challenge and the mystical ingredients, each with its potency and proportion.

• Each group must calculate the necessary quantities of each ingredient for the ideal potion mixture, using the composed rule of three.

• Students should present their calculations and justify the quantities chosen.

• Hold a class discussion on the different approaches and solutions found by the groups.

Activity 2 - The Engineers' Race

> Duration: (60 - 70 minutes)

- Objective: Use the composed rule of three to solve a resource optimization problem in a practical and dynamic context.

- Description: In this scenario, students are engineers of a racing team that needs to optimize fuel usage in different parts of a long-distance race. They must decide how much fuel to take for each section, given that efficiency varies between different types of terrain and speeds.

- Instructions:

• Form groups of up to 5 students and designate each group as a racing engineering team.

• Present maps with different terrain segments and their respective fuel efficiencies.

• Students must use the composed rule of three to determine the ideal amount of fuel to be used for each section, considering the distance and expected speed.

• Each group will present their fuel strategy and how the composed rule of three was applied.

• Debate in class the different strategies and results.

Activity 3 - The Mystery of the Lost Travelers

> Duration: (60 - 70 minutes)

- Objective: Develop logic skills and the application of mathematics in a real and urgent problem-solving scenario.

- Description: Students, in groups, will assume the role of investigators trying to unravel the disappearance of a group of travelers. They will have access to information about the last known locations of the travelers and the average speed of each group in different terrains. The challenge is to use the composed rule of three to estimate where the travelers might be and plan an efficient rescue route.

- Instructions:

• Divide the class into groups of up to 5 students, each representing a rescue team.

• Provide the data on the last sightings and the average speeds of the travelers in different terrains.

• Groups must use the composed rule of three to calculate the possible routes and current locations of the travelers.

• Present a detailed rescue plan, including the possible locations of the travelers and the resources needed for the rescue.

• Discuss in class the different scenarios proposed by the groups and the calculation strategies used.

Feedback

Duration: (15 - 20 minutes)

The purpose of this feedback stage is to consolidate learning, allowing students to articulate what they have learned and reflect on the problem-solving process. Group discussion helps identify gaps in understanding and clarify confusing concepts, as well as promote communication and collaboration skills. This moment also serves for the teacher to assess students' understanding and can guide future revisions or explorations of the content.

Group Discussion

Start the group discussion by inviting each group to share their discoveries and thought processes throughout the activities. Encourage them to discuss not only the answers found but also the challenges faced and how they overcame them. Use questions like 'What was the most difficult part when applying the composed rule of three in this scenario?' and 'How did you decide the order of calculations to arrive at the solution?' to guide the conversation and deepen the students' understanding.

Key Questions

1. What were the main difficulties you encountered when applying the composed rule of three to the presented problems?

2. How did the inversion of proportional quantities impact the way you approached the calculations?

3. Was there any specific strategy or technique that helped your group reach the solutions more easily?

Conclusion

Duration: (5 - 10 minutes)

The purpose of the Conclusion stage is to ensure that students have a clear and consolidated view of what they learned during the lesson. By summarizing the main points, we reinforce the connection between theory and practice, as well as highlight the importance of learning for future applications. This recap helps reinforce knowledge and ensures that students can transfer what they learned to real situations and new challenges.

Summary

To conclude, we will summarize the essential concepts of the composed rule of three, highlighting how it is applied in everyday and practical situations. We revisit the idea of inversion of proportional quantities and how it affects problem-solving. Students had the opportunity to apply these concepts in simulated scenarios ranging from production calculations to rescue strategies, demonstrating the versatility and importance of the composed rule of three.

Theory Connection

During the lesson, we connected theory with practice through interactive activities that simulated real and challenging situations. These activities allowed students to directly apply prior theoretical knowledge in practical contexts, reinforcing understanding and the utility of the composed rule of three. This approach not only facilitates comprehension but also prepares students to use mathematical concepts in their professional and personal lives.

Closing

Understanding the composed rule of three is essential, as it is not only a crucial mathematical tool in various professional areas but also helps develop analytical and critical skills. The ability to apply this knowledge in varied contexts gives students a significant advantage, both in academia and in the job market.