Strand: GEOMETRY (8.G)

Understand congruence and similarity using physical models, transparencies, or geometry software (Standards 8.G.1-5). Understand and apply the Pythagorean Theorem and its converse (Standards 8.G.6-8). Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres (Standard 8.G.9).

Standard 8.G.4

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

• 3D Transmographer
This lesson contains an applet that allows students to explore translations, reflections, and rotations.
• A Scaled Curve
The goal of this task is to motivate and prepare students for the formal definition of dilations and similarity transformations. While these notions are typically applied to triangles and quadrilaterals, having students engage with the concepts in a context where they don't have as much training (these more "random" curves) lead students to focus more on the properties of the transformations than the properties of the figure.
• Are These Shapes Congruent?
This cool interactive will allow students to conceptualize whether two shapes are congruent my twisting and turning them. The student then applies an understanding of congruency by diagramming and building shapes on a graph in the accompanying classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Are They Similar?
This goal of this task is to provide experience applying transformations to show that two polygons are similar.
• Chapter 9 - Mathematical Foundation (UMSMP)
This is Chapter 9 of the Utah Middle School Math: Grade 8 textbook. It provides a Mathematical Foundation for Transformations, Congruence and Similarity.
• Chapter 9 - Student Workbook (UMSMP)
This is Chapter 9 of the Utah Middle School Math: Grade 8 student workbook. It focuses on these topics: Transformations, Congruence and Similarity.
• Creating Similar Triangles
The purpose of this task is to apply rigid motions and dilations to show that triangles are similar.
• Different Areas?
The goal of this task is to motivate a discussion of similarity and slope via a counterintuitive geometric construction where it appears as if area is not conserved by cutting and reassembling a simple shape.
• Escaramuza: Symmetry, Reflection, Rotation
The real-life equestrian event known as Escaramuza is used to teach students how to diagram 2D representations on an x-y graph and then reflect and rotate the figure. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Geometry (8.G) - 8th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 8 Geometry.
• Grade 8 Math Module 3: Similarity (EngageNY)
In 8th grade Module 3, students learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles. The module begins with the definition of dilation, properties of dilations, and compositions of dilations. One overarching goal of this module is to replace the common idea of same shape, different sizes with a definition of similarity that can be applied to geometric shapes that are not polygons, such as ellipses and circles.
• Grade 8 Unit 1: Transformations, Congruence, and Similarity (Georgia Standards)
In this unit students will develop the concept of transformations and the effects that each type of transformation has on an object; explore the relationship between the original figure and its image in regards to their corresponding parts being moved an equal distance which leads to concept of congruence of figures; learn to describe transformations with both words and numbers; relate rigid motions to the concept of symmetry and to use them to prove congruence or similarity of two figures; physically manipulate figures to discover properties of similar and congruent figures; and focus on the sum of the angles of a triangle and use it to find the measures of angles formed by transversals (especially with parallel lines), find the measures of exterior angles of triangles, and to informally prove congruence.
• Is this a rectangle?
The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle.
• Partitioning a Hexagon
The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.
• Reflecting a rectangle over a diagonal
The goal of this task is to give students experience applying and reasoning about reflections of geometric figures using their growing understanding of the properties of rigid motions.
• Rotation Symmetry
Exploring this interactive students are able to predict and find the angle of rotation for various figures by using rotation symmetry. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Same Size, Same Shape?
The purpose of the task is to help students transition from the informal notion of congruence as "same size, same shape" that they learn in elementary school and begin to develop a definition of congruence in terms of rigid transformations.
• Scaling
An interactive from Annenberg asks students to scale a picture by using the math strategies of multiplicative and additive relationships. Students then use those strategies to compare photocopies and rectangles in different scales. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Translations, Reflections, and Rotations
Students are introduced to the concepts of translation, reflection and rotation in this lesson plan.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.