Lesson Plan | Active Learning | Direct Proportion Rule 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 - 7 minutes)

The Objectives stage is essential to direct students' focus and clearly outline what will be achieved during the class. This section aims to clarify the skills that students must develop to apply mathematical concepts to everyday situations, as well as to provide a solid foundation for solving practical problems involving direct proportionality. Thus, students have a clear view of what is expected of them and can better prepare for the upcoming practical activities.

Main Objectives:

1. Identify and understand the direct proportionality relationship between two quantities in various practical contexts.

2. Develop skills to solve problems involving the Rule of Three Direct, such as calculating the cost to fill a gasoline tank based on volume.

Side Objectives:

1. Encourage critical thinking and the application of mathematical knowledge in real situations.
2. Promote collaboration and discussion among students during practical activities.

Introduction

Duration: (20 - 25 minutes)

The introduction serves to engage students with the topic through problem situations they may have already encountered in their routines or in preparation for this class. Additionally, by contextualizing the relevance of the Rule of Three Direct with practical and historical examples, students can perceive the value of mathematical learning in their everyday and professional lives. This stage also establishes a direct connection between the theory studied and its practical application, laying the groundwork for the following activities.

Problem-Based Situations

1. As an initial example, propose to the students that they calculate how much it would cost to fill a 40-liter gasoline tank if the price per liter is R$ 6.50. This problem introduces the application of the Rule of Three Direct to determine costs based on volumes.

2. Next, present a scenario where a cook needs to adjust a recipe that serves 4 people to serve 10 people, maintaining the proportions of the ingredients. This example reinforces the understanding of proportionality in diverse contexts.

Contextualization

Start by contextualizing the importance of the Rule of Three Direct in everyday situations, such as planning family budgets or managing resources in small businesses. Share interesting facts, such as the historical use of this technique in ancient commerce to calculate costs and prices effectively, demonstrating its timeless and practical relevance.

Development

Duration: (65 - 75 minutes)

The Development stage is crucial to consolidate students' prior learning and apply it in practical and contextualized situations. This section is designed to be interactive and engaging, allowing students to work as a team to solve complex, everyday problems using the Rule of Three Direct. By the end of this stage, it is expected that students will not only have practiced mathematics but also developed logical reasoning, teamwork, and presentation skills.

Activity Suggestions

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

Activity 1 - Mathematical Journey: The Fuel Challenge

> Duration: (60 - 70 minutes)

- Objective: Develop calculation and critical analysis skills in solving practical problems of direct proportionality, stimulating collaboration and strategic reasoning.

- Description: Students, divided into groups of up to five people, will face the challenge of planning a car trip. They will need to calculate the amount of fuel required and the total cost, considering different gasoline prices in various cities along the way.

- Instructions:

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

• Distribute maps of a fictional route that includes five cities with different gasoline prices.

• Each group should calculate the amount of gasoline needed to cover the total kilometers, knowing that the car does 12 km per liter.

• Calculate the total cost of fuel, buying gasoline in the cheapest city possible along the route.

• Present the results in a comparative graph between the groups.

Activity 2 - Mathematical Master Cook: Scaling Recipes

> Duration: (60 - 70 minutes)

- Objective: Practice the Rule of Three Direct in a practical and fun context, improving mathematical skills and promoting teamwork.

- Description: In this exercise, students will be challenged to adjust proportions in culinary recipes for different numbers of people. Each group will receive a basic recipe and must calculate the necessary ingredients for 5, 10, and 20 people.

- Instructions:

• Form groups of up to five students.

• Give each group a recipe that serves exactly four people.

• Ask them to calculate the necessary ingredients to serve 5, 10, and 20 people, maintaining the proportions.

• Each group should present their adjusted recipes and explain the mathematical process used.

• Class discussion about the different strategies adopted by the groups.

Activity 3 - Entrepreneur Challenge: Price and Production

> Duration: (60 - 70 minutes)

- Objective: Apply direct proportionality concepts in the context of business management, stimulating critical thinking and the ability to make decisions based on mathematical calculations.

- Description: Students, organized in groups, will take on the role of entrepreneurs who must plan the production of a product. They will need to calculate the cost of production and the selling price based on material and labor cost data, applying the Rule of Three Direct to adjust production according to the budget.

- Instructions:

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

• Provide each group with fictional data on material and labor costs to produce 100 units of a product.

• Challenge them to calculate the cost to produce 250, 500, and 1000 units.

• Ask them to determine a fair selling price, considering a profit margin.

• Each group presents its production plan and justifies its economic choices.

Feedback

Duration: (10 - 15 minutes)

The purpose of this feedback stage is to consolidate students' learning, allowing them to reflect on the practical application of mathematics in everyday situations and share their experiences and learnings with peers. This discussion helps reinforce the understanding of mathematical concepts and develops essential communication and critical skills for collaborative learning and the effective application of mathematical knowledge in various situations.

Group Discussion

Start the group discussion by bringing all students together to share their experiences and conclusions from the activities carried out. Use the following script to guide the discussion: Begin by asking each group about the challenges they faced and the strategies they used to solve the proposed problems. Encourage students to explain how the Rule of Three Direct helped in solving the problems and ask them to share interesting insights or learnings that emerged during the process.

Key Questions

1. What were the main challenges your group faced when applying the Rule of Three Direct to the proposed problems?

2. How did understanding direct proportionality help to solve the problems more efficiently?

3. Was there any strategy or approach that proved particularly effective during the activities?

Conclusion

Duration: (5 - 10 minutes)

The purpose of this stage of the lesson plan is to ensure that students have a clear and in-depth understanding of the content covered, linking theoretical concepts to practice and highlighting the utility of mathematics in everyday life. This moment of recapitulation and reflection is crucial to reinforce learning and encourage students to apply the knowledge acquired in their lives.

Summary

In the conclusion of the class, the teacher should summarize the key concepts addressed about the Rule of Three Direct, recalling the practical examples used, such as the calculation of costs in trips and adjustments in recipes. This summary will help consolidate students' understanding of how directly proportional quantities are applied in different contexts.

Theory Connection

During the class, it was emphasized how the theory of direct proportionality connects with practice through activities that simulate real situations. The application of theoretical concepts to practical problems demonstrated to students the relevance of mathematical learning, facilitating understanding and retention of knowledge.

Closing

Finally, the teacher should highlight the importance of the Rule of Three Direct in daily life, reinforcing how this knowledge is essential for making informed and effective decisions in various situations, from financial planning to solving everyday problems.