Context
Mathematics is an incredibly useful discipline, providing us with ways to understand and solve problems in our everyday lives. The study of geometric transformations in the Cartesian plane, our central theme, is no exception. To start, let's understand what these transformations are.
Geometric transformations involve changing the position, orientation, shape, or size of a geometric figure. In 7th grade, you'll learn about two main types of transformations: translations and reflections. In a translation, a figure is moved in a specific direction without changing its size or shape. In a reflection, the figure is flipped or mirrored in a specific way. Performing these transformations involves multiplying the coordinates of the polygon's vertices by an integer.
For example, to translate a polygon 3 units to the right and 2 units up on the Cartesian plane, you would add 3 to all x-coordinates of its vertices and add 2 to all y-coordinates. To reflect a polygon over the x-axis, you would multiply all y-coordinates by -1.
But why is this important? Transformations in the Cartesian plane have many practical applications in the real world. For instance, they are frequently used in graphic design and animation to move, resize, or flip images. In the field of physics, transformations are used to understand how objects change position, orientation, and shape over time.
Moreover, learning how to perform and visualize geometric transformations can help develop valuable skills such as spatial thinking, problem-solving, and understanding abstract mathematical concepts. This project will challenge you to practice these skills as you work together to explore and understand transformations in the Cartesian plane.
To aid in the project's development and facilitate the comprehension of such concepts, we recommend the following research sources:
- Khan Academy: Transformations in the Cartesian plane
- Khan Academy: Exercises on transformations in the Cartesian plane
- Book: "Fundamentos da Matemática Elementar" - Volume 4: Geometria Plana - Gelson Iezzi (in Portuguese)
Practical Activity
Activity Title: "Transforming Polygons"
Project Objective:
The objective of this project is to enable students to apply theoretical concepts of geometric transformations using polygons in the Cartesian plane and recognize the importance of these transformations in their daily lives.
Detailed Project Description:
Students will be divided into groups of three to five members and will have to choose or create a polygon on a Cartesian plane. The groups will perform three types of transformations: translation, reflection about the x-axis, and reflection about the y-axis. For each transformation, students will record the new coordinates of the polygon's vertices.
After performing the transformations, groups should prepare a report including graphs of the polygon before and after each transformation, as well as a detailed description of the procedures used. The report's last section should include a reflection on practical applications of geometric transformations in their daily lives.
Required Materials:
- Graph paper or a printed Cartesian plane.
- Pencil and eraser.
- Ruler.
- Textbooks or online resources for research.
- Graphic editing software (optional). Ex: Desmos, Geogebra.
Detailed Step-by-Step:
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Meet with your group and choose or create a polygon on the Cartesian plane. Draw this polygon on graph paper or in the geometric drawing software, if available. Write down the coordinates of each vertex.
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Perform a translation of the polygon by moving it three units to the right and two units up. Draw the resulting polygon and record the new coordinates of each vertex.
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Perform a reflection of the polygon about the x-axis. Draw the resulting polygon and record the new coordinates of each vertex.
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Perform a reflection of the polygon about the y-axis. Draw the resulting polygon and record the new coordinates of each vertex.
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Compare the original and transformed polygons. Discuss with your group how the coordinates of each vertex have changed, and how these changes relate to the transformation performed.
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Research applications of geometric transformations in different fields, such as art, design, engineering, physics, etc. Choose one application that is relevant to you and describe it in detail.
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Prepare the project report.
Project Report:
Introduction: Contextualize the concept of geometric transformations. Explain their relevance and real-world applications and the project's objective.
Development: Thoroughly explain the theory of geometric transformations. Describe the original polygon chosen and the transformations performed, including the new coordinates of each vertex for each transformation. Include graphs of the polygon before and after each transformation. Describe the methodology that your group used to carry out this project, from choosing the polygon to preparing the report.
Practical Application: Describe the practical application of geometric transformations that you researched. Discuss how this application relates to what you learned through this project.
Conclusion: Highlight the main takeaways and conclusions that your group obtained from this project. How can these transformations be used in your lives? What did you learn about teamwork and time management?
References: List all sources that you and your group used for the project, including books, websites, videos, etc.
The project should be completed within 30 days, and each group member should dedicate five to ten hours to its completion.
