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.2

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence 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.
• 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.
• Circle Sandwich
The purpose of this task is to apply knowledge about triangles, circles, and squares in order to calculate and compare two different areas.
• Congruent Rectangles
This task is designed to give students insight into the effects of translations, rotations, and reflections on geometric figures in the context of showing that two figures are congruent.
• Congruent Segments
When given two line segments with the same length this task asks students to describe a sequence of reflections that exhibits a congruence between them.
• Congruent Triangles
This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation.
• Cutting a rectangle into two congruent triangles
This task shows the congruence of two triangles in a particular geometric context arising by cutting a rectangle in half along the diagonal.
• Escaramuza: 2D Drawing
The real-life equestrian event known as Escaramuza is used to help student make 2D drawings to make triangles. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Escaramuza: Coordinates, Reflection, Rotation
A real-life equestrian event known as Escaramuza is used to demonstrate how to draw a two-dimensional diagram and then represent it on a coordinate plane. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• 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.
• Geometry in Tessellations
In this lesson students will learn about lines, angles, planes, and experiment with the area and perimeter of polygons.
• Grade 8 Math Module 2: The Concept of Congruence (EngageNY)
In this Grade 8 module, students learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence. Throughout Topic A, on the definitions and properties of the basic rigid motions, students verify experimentally their basic properties and, when feasible, deepen their understanding of these properties using reasoning. All the lessons of Topic B demonstrate to students the ability to sequence various combinations of rigid motions while maintaining the basic properties of individual rigid motions. Students learn that congruence is just a sequence of basic rigid motions in Topic C, and Topic D begins the learning of Pythagorean Theorem.
• 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.
• Identifying Unknown Transformations
This applet allows the student to drag a shape and then observe the changes to its behavior. They then determine whether the alteration is due to reflection, a rotation, or a translation/slide transformation.
• Introduction to Fractals: Infinity, Self-Similarity and Recursion
This lesson is designed to help students understand aspects of fractals, specifically self-similarity and recursion.
• 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.
• 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.
• Translation
This lesson opens with an animated short from the Utah Education Network to explain the concept of translation when a geometric figure changes location on a plane. Students are then asked to solve practice translation problems and explain their solution. 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.
• Triangle congruence with coordinates
This task gives students a chance to explore several issues relating to rigid motions of the plane and triangle congruence.

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.