Contextualization

Introduction to Dilations

Dilations are a type of geometric transformations that change the size of a figure without changing its shape. They are similar to the scaling you might do on a computer screen or when you use a photocopier to enlarge or reduce the size of a document.

In dilations, a fixed point called the "center of dilation" and a scale factor are used. Each point on the figure is moved along a line that passes through the center of dilation and the point itself. The distance of the point on the original figure to the center of dilation is multiplied by the scale factor to determine the distance of the corresponding point on the image from the center of dilation.

Real-World Application of Dilations

Dilations have numerous real-world applications. They are used in architecture when designing buildings, in cartography for map scaling, in the medical field when taking X-rays, in graphic design for resizing images, and even in our everyday life when we use a zoom function on our phones or cameras.

Importance of Dilations

Understanding dilations is not only important for the field of geometry but it also plays a crucial role in other areas of mathematics, such as trigonometry, calculus, and even in physics and engineering.

In this project, we'll delve deeper into the concept of dilations, explore their properties, and apply them to real-world situations. Through this, we'll not only develop a better understanding of the mathematical concept but also enhance our problem-solving, critical thinking, and collaboration skills.

Resources for Further Study

1. Dilations in Geometry: Definition & Scale Factor - This study.com article provides a clear and in-depth explanation of dilations.
2. Geometric Transformations: Dilations - Mathisfun.com is an excellent resource for understanding mathematical concepts. This link provides an easy-to-understand explanation of dilations.
3. Dilations: Khan Academy - Khan Academy offers an extensive library of video lessons. This link directs you to their section on dilations.
4. Math Antics - Dilations - Math Antics provides engaging video lessons on various math topics. This video on dilations is a great visual aid.
5. Geometry - Dilations - Math.com offers comprehensive lessons on dilations. This link takes you to their section on the topic.

Practical Activity

Activity Title: "Scaling the World: A Dilation Exploration"

Objective of the Project:

To understand the concept of dilations, the scale factor, and its effect on the size of a figure, and to apply these concepts in a real-world context.

Detailed Description of the Project:

In this project, each group will choose a real-world object or image to dilate. The students will identify the center of dilation and use a specific scale factor to create an enlarged or reduced version of the original figure. The final step will be to create a 3D model of the dilated figure using simple materials.

Necessary Materials:

1. Rulers
2. Graph paper
3. Colored pencils/markers
4. Scissors
5. Glue
6. Cardboard or foam board

Detailed Step-by-Step for Carrying Out the Activity:

1. Planning Stage: Each group should select a real-world figure or object they wish to dilate. It can be anything from a logo to a building. The group should draw the original figure on a piece of graph paper, using a scale they choose.

2. Dilation Stage: Using the ruler, each group should identify the center of dilation on the figure and determine the scale factor. Then, they should dilate the figure by multiplying the distance of each point from the center of dilation by the scale factor. Draw the dilated figure on another piece of graph paper.

3. Modeling Stage: Cut out the original and dilated figures. Glue the figures onto the cardboard or foam board to create a 3D model. The cardboard can be folded or cut to create the depth of the object.

4. Presentation Stage: Each group will present their final dilated model to the class. During the presentation, they will explain the process they went through, the center of dilation, the scale factor, and the real-world application of their chosen figure.

Project Deliverables:

1. Written Document: This document should include the following sections:

   • Introduction: The student should contextualize the theme (dilations), its importance, and real-world application as well as explaining the objective of the project.
   • Development: This section should detail the theory behind dilations, explain the activity in detail, indicate the methodology used, and finally present and discuss the obtained results.
   • Conclusion: The student should revisit the main points of the project, explicitly stating what they have learned from the theory and the practical activity. They should also discuss the conclusions drawn about their chosen real-world figure after dilating it.
   • Bibliography: The resources used for the project should be cited in this section. These resources could include the books, web pages, videos, etc. that helped the students understand dilations and complete the project.

2. Dilated Model: Each group will present their 3D model of the dilated figure during their presentation. They should be prepared to discuss the process they went through to create the model, the center of dilation, the scale factor, and the real-world application of their chosen figure.

3. Presentation: Each group will give a brief presentation to the class. This presentation should provide an overview of the project, including the original and dilated figures, the chosen scale factor, the process of creating the 3D model, and the real-world application of their chosen figure.

Project Duration:

This project should take approximately one week to complete, with a recommended 2-3 hours of work per student. The first two days should be dedicated to the planning and dilation stages, the next three days to the modeling stage, and the final day for the presentation and finalization of written document.