Lesson Plan | Technical Methodology | Translations: Advanced

Objectives

Duration: 10 - 15 minutes

This stage of the lesson plan aims to introduce students to advanced concepts of translations, isometric and homothetic transformations, with a focus on developing practical skills. By understanding and applying these concepts, students will be better prepared to face challenges in the job market, especially in areas that require mathematical precision and spatial visualization skills, such as engineering, architecture, and design.

Main Objectives

1. Master the execution of translations of objects in a plane or in space.

2. Calculate accurately the distance between initial and final points of a translation.

3. Apply isometric and homothetic transformations to construct figures and analyze elements of nature and human productions.

Side Objectives

1. Develop practical skills in maker activities.
2. Connect mathematical concepts with applications in the job market.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to introduce students to the concept of translations in an engaging and relevant manner, connecting mathematical theory with practical applications in the job market. By sparking students' interest with real-world examples and an intriguing initial activity, a solid foundation is created for the development of practical skills that will be explored throughout the lesson.

Contextualization

Translations are geometric transformations that move figures or points in a plane or in space without changing their shapes or sizes. Imagine moving a chess piece from one square to another on the board – the shape of the piece does not change, only its position. Translations are fundamental in many areas, such as in creating computer animations where objects need to be moved precisely to create smooth movements.

Curiosities and Market Connection

Did you know that translations are used in civil engineering to design complex structures? Engineers use these transformations to calculate displacements of forces and materials. Additionally, in computer graphics, translations are essential for manipulating images and creating amazing visual effects. Architects also apply these concepts to draw building plans, ensuring that each element is in the correct place.

Initial Activity

To start, show students a short 3-minute video demonstrating how translations are used in the animation of a video game character. After the video, ask the provocative question: 'How do you think developers manage to move characters so precisely on the screen?'

Development

Duration: (50 - 55 minutes)

The purpose of this stage is to deepen students' understanding of translations and other isometric transformations through practical activities and reflections. By engaging in practical challenges and fixation exercises, students will be able to apply the concepts learned concretely, enhancing their spatial visualization and mathematical precision skills, which are essential for various professions in the job market.

Covered Topics

1. Concept of translations in the plane and in space
2. Calculation of distances between initial and final points of a translation
3. Isometric transformations: translation, reflection, rotation, and compositions of these
4. Homothetic transformations
5. Practical applications in engineering, architecture, computer graphics, and other fields

Reflections on the Theme

Guide students to reflect on how translations and other isometric transformations are used in different professional fields. Ask them how these techniques might be important in designing a bridge or animating a character in a movie. Encourage them to think about the need for precision and the usefulness of these skills in solving real-world market problems.

Mini Challenge

Practical Challenge: Building a Mosaic with Translations

Students will be divided into groups and challenged to create a mosaic using translations. They should use graph paper and a ruler to draw geometric figures and then translate them to create a repetitive pattern that forms a mosaic.

Instructions

1. Divide students into groups of 3 to 4 members.
2. Distribute graph paper, rulers, and pencils to each group.
3. Explain that each group must draw a basic geometric figure (triangle, square, hexagon, etc.) on the graph paper.
4. Guide the groups to translate the geometric figure in a specific direction while maintaining the same shape and size to create a repetitive pattern.
5. Instruct the groups to calculate the distance between the initial and final points of each translation and record these values.
6. Ask the groups to present their mosaics and explain the creation process, including the calculations made.

Objective: Develop the ability to apply translations to create visual patterns, calculate distances between translated points, and understand the practical application of geometric transformations in design and engineering.

Duration: (30 - 35 minutes)

Evaluation Exercises

1. Given figure A on a Cartesian plane, translate figure 5 units to the right and 3 units up. Draw the new position of figure A and calculate the total distance traveled.
2. Consider points P(2,3) and Q(5,7). After a translation of 4 units to the left and 2 units down, what are the new coordinates of P and Q? Calculate the distance between the initial and final points of P and Q.
3. In a three-dimensional space, figure B is translated 2 units along the x-axis, -3 units along the y-axis, and 1 unit along the z-axis. Draw the new position of figure B and calculate the total distance traveled.
4. Use isometric transformations to create a mosaic pattern that includes translations, reflections, and rotations. Describe the process and calculations involved.

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage of the lesson plan is to consolidate students' learning, providing a moment of reflection and discussion about the concepts covered. By recapping the content and connecting theory with practice, students will be able to internalize the knowledge acquired and understand its relevance and application in the job market and daily life.

Discussion

Promote a final discussion with students about the subject covered. Ask students how they felt performing the practical activities and facing the proposed challenges. Encourage them to reflect on the importance of translations and other isometric transformations in their future careers, and how these skills can be applied in different areas such as engineering, architecture, graphic design, among others. Also question them about the fixation exercises: which were the most challenging and why? What was the experience of calculating distances and translating figures?

Summary

Recap the main concepts presented during the lesson, such as translations in the plane and in space, calculation of distances between points, isometric transformations (translation, reflection, rotation, and compositions of these), and homothetic transformations. Emphasize how these concepts were applied in practical activities, such as creating the mosaic and solving the fixation exercises.

Closing

Explain to students how the lesson connected mathematical theory with practical applications in the job market. Highlight the importance of a deep understanding of translations and other isometric transformations to solve real problems in various professions. Conclude the lesson by stressing the relevance of this knowledge for everyday life and how it can be applied to create innovative and efficient solutions in different contexts.