Lesson Plan | Traditional Methodology | Rotations in the Cartesian Plane

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to provide students with a clear understanding of the lesson objectives, preparing them for the content that will be covered. By detailing the objectives, students will know exactly what is expected of them and how it applies to practical problems, facilitating the assimilation of the content and its application in future exercises.

Main Objectives

1. Understand the concept of rotation of figures in the Cartesian plane.

2. Learn to identify figures after a 90-degree rotation around the origin.

3. Apply the acquired knowledge to solve practical problems involving rotations.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to introduce the topic clearly and engagingly, preparing students for the content that will be addressed. By providing initial context and sharing curiosities about the topic, students will be able to relate the content to everyday situations and perceive the relevance of studying rotations in the Cartesian plane. This will help capture students' attention and motivate them to actively participate in the lesson.

Context

To begin the lesson on rotations in the Cartesian plane, it is important to contextualize students about the concept of rotation. Explain that rotation is a circular movement around a fixed point. In the Cartesian plane, this fixed point is usually the origin (0,0). To illustrate, use visual examples such as the rotation of a gear, the movement of the hands of a clock, or even the spinning of a geometric figure in a dynamic geometry software. Use a graph on the board or in a presentation to show how a point or figure moves when rotated around the origin, emphasizing that the distance from the point to the origin remains constant during the rotation.

Curiosities

Did you know that rotations in the Cartesian plane are widely used in computer graphics and animations? When we watch an animated movie or play a video game, characters and objects often perform rotations and other geometric transformations to create realistic movements. Additionally, in engineering and design, rotations are essential for modeling and analyzing mechanical parts and structures.

Development

Duration: (50 - 60 minutes)

The purpose of this stage is to deepen students' understanding of the concept of rotation in the Cartesian plane. By addressing specific topics and providing practical examples, students will be able to visualize and apply the rotation formulas, consolidating the acquired knowledge. The guided problem-solving allows students to practice and validate their understanding, ensuring that they are capable of identifying and performing rotations of geometric figures in the Cartesian plane.

Covered Topics

1. Concept of Rotation in the Cartesian Plane: Explain that rotation is a geometric transformation that rotates a figure around a fixed point, usually the origin (0,0) in the Cartesian plane. Detail that a rotation does not change the size or shape of the figure, only its orientation. 2. Rotation Angles: Discuss common rotation angles, such as 90°, 180°, and 270°. Explain how to determine the direction of rotation (clockwise or counterclockwise) and its representation in the Cartesian plane. 3. Rotation Formulas: Introduce the formulas used to calculate the rotation of points in the Cartesian plane. For a 90° counterclockwise rotation, the formula is (x, y) -> (-y, x). Explain and demonstrate how these formulas are applied. 4. Practical Examples: Show examples of how to rotate points and specific geometric figures, such as triangles and squares, around the origin. Use graphs and drawings to illustrate each step of the rotation process. 5. Guided Problem Solving: Solve some example problems on the board, involving the rotation of geometric figures. Ask students to follow along and note the steps. Demonstrate how to verify if the rotated figure is correct by checking the coordinates of the vertices.

Classroom Questions

1. What is the new position of the point (3, 4) after a 90° counterclockwise rotation around the origin? 2. Rotate the triangle with vertices at (1,2), (3,4), and (5,2) by 180° around the origin. What are the new coordinates of the vertices? 3. Draw a square with vertices at (1,1), (1,3), (3,1), and (3,3). Rotate it 270° counterclockwise around the origin. What are the new coordinates of the vertices?

Questions Discussion

Duration: (20 - 25 minutes)

The purpose of this stage is to consolidate students' learning, allowing them to review and discuss the solutions to the presented questions. By engaging students in a detailed discussion, the teacher can clarify doubts, reinforce concepts, and ensure that everyone understands the steps and the logic behind rotations in the Cartesian plane. This moment of reflection and exchange of ideas also helps reinforce the practical application of the learned concepts.

Discussion

• Question 1: What is the new position of the point (3, 4) after a 90° counterclockwise rotation around the origin?

Answer: To rotate the point (3, 4) 90° counterclockwise, we use the formula (x, y) -> (-y, x). Thus, the new position will be (-4, 3).

• Question 2: Rotate the triangle with vertices at (1,2), (3,4), and (5,2) by 180° around the origin. What are the new coordinates of the vertices?

Answer: To rotate the triangle 180° around the origin, we use the formula (x, y) -> (-x, -y). Therefore, the new coordinates of the vertices are: (-1,-2), (-3,-4), and (-5,-2).

• Question 3: Draw a square with vertices at (1,1), (1,3), (3,1), and (3,3). Rotate it 270° counterclockwise around the origin. What are the new coordinates of the vertices?

Answer: To rotate the square 270° counterclockwise, we use the formula (x, y) -> (y, -x). Thus, the new coordinates of the vertices are: (3,-1), (1,-1), (3,-3), (1,-3).

Student Engagement

1.  Question 1: What was the biggest challenge you faced when applying the rotation formula? How can we overcome it? 2.  Question 2: Can you think of other everyday situations where we see rotations? What would they be and how do they relate to what we learned? 3.  Question 3: If we rotate a point 360°, where will it be? Why does this happen? 4.  Question 4: How do you verify if a figure has been rotated correctly? What is the importance of checking the coordinates of the vertices?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to provide a moment of review and reflection on the learned content. By summarizing the main points, connecting theory with practice, and highlighting the relevance of the topic, students reinforce their understanding and realize the importance of the acquired knowledge for future applications.

Summary

• Concept of rotation in the Cartesian plane as a geometric transformation that rotates a figure around a fixed point.
• Identification of common rotation angles: 90°, 180°, and 270°, and their direction (clockwise or counterclockwise).
• Rotation formulas: (x, y) -> (-y, x) for 90° counterclockwise, (x, y) -> (-x, -y) for 180°, and (x, y) -> (y, -x) for 270° counterclockwise.
• Practical examples of rotating points and geometric figures, such as triangles and squares.
• Guided problem solving to consolidate understanding of the presented concepts.

The lesson connected the theory of rotations in the Cartesian plane with practice through concrete examples and problem-solving. Students were able to visualize how the rotation formulas apply to points and geometric figures, making the learning more tangible and understandable.

The topic presented is of great importance for everyday life, as rotations are widely used in areas such as computer graphics, animations, engineering, and design. Understanding how to rotate figures in the Cartesian plane helps students develop spatial visualization skills and apply this knowledge in practical and relevant contexts.