Lesson Plan | Active Learning | Spatial Geometry: Deformations in Projections

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 is crucial to guide students and the teacher about the main focus of the lesson. By clearly establishing the objectives, students can better direct their efforts and attention during investigation and discussion activities. This clarity helps maximize the effectiveness of class time, ensuring that everyone involved is aligned with the desired learning goals.

Main Objectives:

1. Understand the characteristics and implications of different geometric projections (cylindrical and conical) on the distortion of angles and areas in cartographic representations.

2. Develop skills to investigate and analyze how projections affect the accuracy of geographic representations in maps.

Side Objectives:

1. Encourage students' critical thinking and scientific curiosity through the analysis of distortions in different projections.

Introduction

Duration: (10 - 15 minutes)

The Introduction stage serves to engage students and make them think critically about the impact of projection choices in real maps. Through problem situations, students are challenged to apply their prior knowledge to solve practical issues, preparing them for deeper analysis activities in the classroom. The contextualization helps connect mathematical content with its applications in the real world, increasing interest and relevance of the topic for students.

Problem-Based Situations

1. Imagine you are a cartographer trying to represent the curved surface of the Earth on a flat map. How would you choose between a cylindrical or conical projection? What would be the implications of your choice on the accuracy of the angles and areas represented?

2. Consider that a geographer needs to create a detailed map of South America for a school atlas. What type of projection would you recommend to minimize distortion in the most extreme regions, such as Patagonia and the Amazon? Justify your choice based on the characteristics of the studied projections.

Contextualization

The choice of a cartographic projection can mean the difference between a useful map and one that leads to misinterpretations. For example, the Mercator projection, a cylindrical projection, is widely used for its ability to maintain angles but drastically distorts areas near the poles. This has led to the common misconception that Greenland is approximately the size of Africa, when in fact, Africa is about 14 times larger. These distortions have real implications in fields like geography, urban planning, and education.

Development

Duration: (70 - 80 minutes)

The Development stage is essential for students to practically and critically apply the knowledge they have acquired about geometric projections. With activities that challenge understanding and stimulate comparative analysis, students can concretely visualize the implications of projection choices, developing investigative and argumentative skills. Additionally, working in groups fosters the exchange of ideas and collaboration, key elements for effective and engaging learning.

Activity Suggestions

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

Activity 1 - Handmade World Map: A Journey of Projections

> Duration: (60 - 70 minutes)

- Objective: Visualize and understand how different geometric projections affect the representation of areas and angles on a map.

- Description: In this activity, students will manually create world maps using different types of projections (cylindrical and conical) to visually investigate how each projection affects the representation of areas and angles. The activity will involve the use of simple materials such as paper, colored pens, and rulers.

- Instructions:

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

• Distribute a large sheet of paper, colored pens, and a ruler to each group.

• Ask each group to draw two world maps, one using the cylindrical projection and another using the conical projection.

• Instruct them to label the continents, using a specific color for each.

• After completing the maps, each group must measure and record the observed distortions in angles and areas.

• Each group will present their findings to the class, discussing the differences between the projections.

Activity 2 - Regional Challenge: Projections and Perspectives

> Duration: (60 - 70 minutes)

- Objective: Analyze the practical implications of projection distortions in different geographical regions.

- Description: Students will be challenged to choose the most suitable projection to represent different geographical regions of the world. They will compare cylindrical and conical projections, considering specific distortions and their practical implications.

- Instructions:

• Form groups of up to 5 students.

• Assign each group a specific region of the world to analyze (for example, Northern Europe, Southeast Asia, etc.).

• Ask each group to create two representations of that region, one in each type of projection.

• Students should analyze the area and angle distortions in their projections and prepare a presentation.

• Each group will present their analysis and recommendation of which projection is more suitable for their region, with justifications based on the observed distortions.

Activity 3 - The Great Projection Tournament

> Duration: (60 - 70 minutes)

- Objective: Promote a critical understanding of the advantages and limitations of different cartographic projections through analysis and debate.

- Description: This playful activity will involve a competition among groups to identify which projection (cylindrical or conical) best preserves different geographical aspects. Students will use printed maps and measurement tools to argue their positions.

- Instructions:

• Organize the room into groups of up to 5 students.

• Provide each group with printed maps of both projections.

• Determine evaluation criteria such as angle accuracy, area representation, and overall readability.

• Ask the groups to analyze the maps based on the criteria and prepare a defense of their preferred projection.

• Hold a tournament-style debate, where groups defend their choices and vote on the best arguments.

Feedback

Duration: (15 - 20 minutes)

This stage is vital to consolidate students' learning, allowing them to articulate and reflect on the knowledge acquired during the practical activities. By sharing their experiences and conclusions, students develop a deeper understanding of the impact of cartographic projections and improve their communication and critical argumentation skills.

Group Discussion

To start the group discussion, the teacher should gather all students and ask each group to share their findings and learnings. Encourage students to reflect on how the practical activities helped them understand the implications of cartographic projections. The teacher can encourage interaction between groups by asking how the experiences of one might inform or complement the observations of others.

Key Questions

1. What were the main differences you observed between the cylindrical and conical projections in terms of angle and area distortion?

2. How could the choice of projection impact the utility of a map in real situations, such as in urban planning or environmental studies?

3. What did you find most challenging about representing geographical regions in the projections and why?

Conclusion

Duration: (5 - 10 minutes)

The Conclusion stage is essential to ensure that students have a clear and integrated understanding of what was learned during the lesson. It allows the teacher to reinforce key concepts and developed skills, as well as highlight the applicability and importance of this knowledge in the real context. This moment of reflection and synthesis helps students to connect the dots between theory and practice, consolidating learning and preparing them for future academic and professional applications.

Summary

In the conclusion of the lesson, the teacher should summarize the main points discussed about cylindrical and conical projections, emphasizing how they affect the representation of angles and areas in maps. It is important to highlight the observations made by students during practical activities, consolidating the understanding of geometric deformations and their implications in cartographic representations.

Theory Connection

This stage also serves to reinforce how the theory studied was applied in practice through the drawing and analysis of maps activities, linking mathematical and geographic knowledge with real-world situations such as urban planning and environmental studies.

Closing

Finally, the teacher should emphasize the importance of studying projections in understanding maps and correctly interpreting geographical information, highlighting the relevance of these skills in everyday life and in various professional areas.