Illustrative Math Alignment: Grade 8 Unit 2

Overview

Overview

Overview

Overview

Overview

In this program we look at applications of transformations. We do this in the context of three real-world applications. In the first, we look at translations and rotations in the context of roller coaster rides. In the second example we look at translations in three-dimensional space in the context of cargo ships. In the third example, we look at the design of observatories to look at rotations, reflections, and symmetry.

Cargo ships transport tons of merchandise from one country to another and accounts for most of the global economy. Loading and unloading these ships requires a great deal of organization and provides an ideal example of three-dimensional translations.

The Gemini telescope in Hawaii is an example of architecture that moves. All observatories rotate in order to follow objects in the sky. This also provides an opportunity to explore rotations, reflections, and symmetry.

Roller coasters provide an ideal opportunity to explore translations and rotations. Displacement vectors are also introduced.

The Eiffel Tower includes quite a number of exposed triangular trusses. The properties of triangles are used to explore and explain the frequent use of triangular trusses in many building. In particular, isosceles and equilateral triangular trusses are explored. In addition triangle postulates and similarity are explored and analyzed.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is part of a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

The following section includes background information on the Cartesian coordinate plane.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is part of a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

The following section includes background information on the Cartesian coordinate plane.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

This is a collection of definitions related to coordinate systems and related topics. This includes general definitions for coordinate systems, as well as examples of specific coordinate systems and properties of those systems.

Topic

Geometry Basics

Definition

Congruence refers to figures or shapes that have the same size and shape.

Description

Congruence is a fundamental concept in geometry, indicating that two figures are identical in form and dimensions. This concept is essential for understanding geometric transformations and proving the properties of shapes. For instance, two triangles are congruent if their corresponding sides and angles are equal. Congruence is used in various real-world applications, such as engineering and architecture, to ensure precision and accuracy in design and construction.

Topic

Geometry Basics

Definition

Similarity refers to figures that have the same shape but not necessarily the same size.

Description

Similarity is a key concept in geometry, used to compare shapes and solve problems involving proportions. For example, two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. Understanding similarity helps in solving problems related to scaling, resizing, and geometric transformations. This concept is widely used in fields such as art, design, and architecture to create proportional and aesthetically pleasing structures.

Topic

Geometry Concepts

Description

This image depicts a square undergoing a dilation transformation. Dilation changes the size of a figure while maintaining its shape, resulting in a similar figure. The scale factor of the dilation determines whether the square is enlarged or reduced.

In this transformation, all sides of the square are scaled by the same factor, and all angles remain congruent to the original. This preserves the square's shape while changing its size, demonstrating the concept of similarity in geometry.

Topic

Geometry Concepts

Description

This image shows a square undergoing a combination of dilation and translation. The dilation changes the size of the square while maintaining its shape, and the translation moves the dilated square to a new position without changing its size or orientation.

This composite transformation demonstrates how multiple transformations can be applied sequentially. The resulting figure is similar to the original square but different in size and position.

Topic

Geometry Concepts

Description

This image illustrates a rectangle undergoing a combination of dilation, translation, and rotation. The dilation changes the size of the rectangle, the translation moves it to a new position, and the rotation changes its orientation.

This complex transformation demonstrates how multiple transformations can be combined to create a figure that is similar to the original but different in size, position, and orientation. The resulting figure maintains the proportions of the original rectangle.

Topic

Geometry Concepts

Description

This image shows a rectangle undergoing a combination of dilation and rotation. The dilation changes the size of the rectangle while maintaining its proportions, and the rotation changes its orientation.

This transformation demonstrates how a figure can be both resized and reoriented while maintaining its shape. The resulting figure is similar to the original rectangle but different in size and orientation.

Topic

Geometry Concepts

Description

This image depicts a triangle undergoing a rotation transformation. The rotation changes the orientation of the triangle without altering its size or shape.

This transformation demonstrates how a figure can be moved around a fixed point (the center of rotation) while maintaining its congruence to the original shape. The resulting triangle is identical to the original in all aspects except its orientation.

Topic

Geometry Concepts

Description

This image shows a triangle undergoing a translation transformation. The translation moves the triangle to a new position without changing its size, shape, or orientation.

This transformation demonstrates how a figure can be moved in a straight line without any other changes. The resulting triangle is congruent to the original and maintains all its properties, only differing in its position.

Topic

Geometry Concepts

Description

This image illustrates a triangle undergoing a combination of translation and rotation. The translation moves the triangle to a new position, and the rotation changes its orientation.

This composite transformation demonstrates how multiple transformations can be applied sequentially. The resulting triangle is congruent to the original but different in both position and orientation.

Topic

Geometry Concepts

Description

This image shows a triangle undergoing a combination of dilation and translation performed twice. The dilation changes the size of the triangle while maintaining its shape, and the translations move it to new positions.

This complex transformation demonstrates how multiple transformations can be applied repeatedly. The resulting triangles are similar to the original but differ in size and position.