Lesson Plan | Traditional Methodology | Polygon Transformations

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage of the lesson plan is to establish a clear and understandable base for students regarding what will be learned during the class. By defining specific objectives, students can focus their attention on the main points and understand the importance of transformations in polygons in the Cartesian plane. This will also help guide the teacher in structuring and conducting the lesson, ensuring that all crucial points are covered.

Main Objectives

1. Understand and apply basic geometric transformations (dilation and contraction) on polygons in the Cartesian plane.

2. Calculate the area, perimeter, and side lengths of the resulting polygons after the transformations.

3. Develop the skill to multiply the coordinates of the vertices of polygons by a specific value to perform transformations.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to capture students' attention and motivate them for the lesson's topic. By presenting a rich and relevant context, students can see the practical application of polygon transformations in real-world situations. This not only makes learning more interesting but also helps students understand the importance and utility of geometric transformations in everyday life and various professions.

Context

Start the lesson by introducing students to the concept of polygon transformations in the Cartesian plane. Explain that geometry is not just a set of mathematical rules, but also a powerful tool for understanding and modeling the world around us. Show how geometric transformations, such as dilation and contraction, are used in various fields, from art and design to engineering and computer science. To illustrate, present images of mosaics, modern architecture, and computer graphics, highlighting how shapes are manipulated to create different visual and functional effects.

Curiosities

Did you know that geometric transformations are widely used in digital animation? For instance, in animated films, characters and scenes are often transformed through dilations and contractions to create realistic movements and special effects. Additionally, architects use transformations to design buildings that can expand or contract as needed, such as mobile or adaptable structures for different weather conditions.

Development

Duration: (50 - 60 minutes)

The purpose of this stage is to deepen students' understanding of geometric transformations of polygons in the Cartesian plane. By detailing the concepts and performing practical examples, students will be able to visualize and apply the transformations, as well as calculate the areas and perimeters of the resulting polygons. The proposed questions will allow students to practice and consolidate the knowledge acquired.

Covered Topics

1. Definition of Geometric Transformations: Explain that geometric transformations include operations like translation, rotation, reflection, and dilation/contraction. Emphasize that this lesson will focus on dilation and contraction, which involve multiplying the coordinates of the vertices of polygons by a specific value. 2. Dilation and Contraction: Detail how these transformations affect the coordinates of the vertices of polygons. For example, for a dilation with scale factor k, each coordinate (x, y) of the vertex is multiplied by k, resulting in (kx, ky). For contraction, the scale factor is less than 1. 3. Application in the Cartesian Plane: Demonstrate how to apply these transformations to polygons in the Cartesian plane. Use practical examples to illustrate how the coordinates of the vertices change. Draw an initial polygon and show step by step how it is transformed. 4. Calculation of Area and Perimeter: Explain how to calculate the area and perimeter of the resulting polygons after the transformations. Use specific examples to show how the area and perimeter formulas are applied to the new polygons. 5. Practical Examples: Present various practical examples of transformations. For instance, show a triangle in the Cartesian plane, perform a dilation and a contraction, and calculate the new areas and perimeters. Allow students to follow these examples in their notebooks. 6. Guided Problem Solving: Solve problems together with the class, guiding students step by step. Provide examples of different types of polygons and show how to apply the transformations and calculate the resulting measures.

Classroom Questions

1. Given a triangle with vertices at coordinates (1, 2), (3, 4), and (5, 6), apply a dilation with scale factor 2. What are the new coordinates of the vertices? 2. A square has vertices at coordinates (0, 0), (0, 3), (3, 0), and (3, 3). After a contraction with scale factor 0.5, what are the new coordinates of the vertices and what is the new area of the square? 3. Given a pentagon with vertices at coordinates (1, 1), (2, 3), (4, 3), (5, 1), and (3, -1), apply a dilation with scale factor 1.5. Calculate the perimeter of the resulting pentagon.

Questions Discussion

Duration: (15 - 20 minutes)

The purpose of this stage is to consolidate the knowledge acquired by the students during the lesson, ensuring that they understand geometric transformations and know how to apply this knowledge in solving practical problems. The discussion of the questions and engagement through reflective questions help reinforce the concepts and stimulate critical thinking.

Discussion

• Discussion of the Presented Questions:

• 1. Question: Given a triangle with vertices at coordinates (1, 2), (3, 4), and (5, 6), apply a dilation with scale factor 2. What are the new coordinates of the vertices?
• • Answer: To apply the dilation with scale factor 2, we multiply each coordinate by 2. The new coordinates of the vertices will be: (2, 4), (6, 8), and (10, 12).
• 2. Question: A square has vertices at coordinates (0, 0), (0, 3), (3, 0), and (3, 3). After a contraction with scale factor 0.5, what are the new coordinates of the vertices and what is the new area of the square?
• • Answer: To apply the contraction, we multiply each coordinate by 0.5. The new coordinates of the vertices will be: (0, 0), (0, 1.5), (1.5, 0), and (1.5, 1.5). The new area of the square is calculated as (side length)^2, resulting in (1.5)^2 = 2.25 square units.
• 3. Question: Given a pentagon with vertices at coordinates (1, 1), (2, 3), (4, 3), (5, 1), and (3, -1), apply a dilation with scale factor 1.5. Calculate the perimeter of the resulting pentagon.
• • Answer: After applying the dilation with scale factor 1.5, the new coordinates of the vertices will be: (1.5, 1.5), (3, 4.5), (6, 4.5), (7.5, 1.5), and (4.5, -1.5). To calculate the perimeter, we sum the distances between the adjacent vertices.

Student Engagement

1. Questions and Reflections to Engage Students: 2. 1. How can you verify if you correctly applied dilation or contraction to a polygon? 3. 2. Why is it important to understand geometric transformations in the Cartesian plane? 4. 3. Think of a practical application in your daily life or in a profession you know where geometric transformations are used. Can you identify that application? 5. 4. If you apply a dilation to a polygon and then a contraction with the same scale factors, does the polygon return to its original coordinates? Why? 6. 5. How did you calculate the new area of the square after the contraction? What steps did you follow and why?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to review and consolidate the content presented during the lesson, ensuring that students have a clear and complete understanding of geometric transformations of polygons. By summarizing the main points, connecting theory and practice, and highlighting the relevance of the topic, students are encouraged to reflect on what they have learned and how they can apply that knowledge.

Summary

• Introduction to the concept of geometric transformations, including dilation and contraction in the Cartesian plane.
• Detailed explanation of how to multiply the coordinates of the vertices of a polygon to perform dilations and contractions.
• Practical demonstration of transformations in polygons in the Cartesian plane.
• Calculation of the area and perimeter of the resulting polygons after the transformations.
• Guided problem-solving involving geometric transformations of polygons.

During the lesson, it was shown how the theory of geometric transformations applies to practical situations, such as digital animation and architecture. Concrete examples were used to illustrate how the coordinates of the vertices of polygons are multiplied by scale factors to perform dilations and contractions, connecting abstract mathematics with real-world applications.

Understanding geometric transformations is crucial for various daily activities and professions. For example, architects use these transformations when designing buildings that can adapt to different conditions. In the field of digital animation, transformations are essential for creating realistic movements. Knowing how to apply these transformations allows students to develop skills that are useful in various areas, making learning meaningful and relevant.