 # Standard 1 : Understand congruence and similarity using physical models, transparencies, or geometry software. (Major Cluster)

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Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

### General Information

Number: MAFS.8.G.1
Title: Understand congruence and similarity using physical models, transparencies, or geometry software. (Major Cluster)
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Geometry

#### Related Standards

This cluster includes the following benchmarks
 Code Description MAFS.8.G.1.1: Verify experimentally the properties of rotations, reflections, and translations: Lines are taken to lines, and line segments to line segments of the same length.Angles are taken to angles of the same measure.Parallel lines are taken to parallel lines. MAFS.8.G.1.2: 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. MAFS.8.G.1.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. MAFS.8.G.1.4: 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. MAFS.8.G.1.5: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

#### Related Access Points

This cluster includes the following access points.

#### Access Points

 Access Point Number Access Point Title MAFS.8.G.1.AP.1a: Perform rotations, reflections, and translations using pattern blocks. MAFS.8.G.1.AP.1b: Draw rotations, reflections, and translations of polygons. MAFS.8.G.1.AP.2a: Demonstrate that two-dimensional polygons that are rotated, reflected, or translated are still congruent using area, perimeter, and length of sides on a coordinate plane. MAFS.8.G.1.AP.3a: Dilate common polygons using graph paper and identifying the coordinates of the vertices. MAFS.8.G.1.AP.4a: Recognize congruent and similar figures. MAFS.8.G.1.AP.4b: Identify two-dimensional figures as similar or congruent given coordinate plane representations. MAFS.8.G.1.AP.4c: Compare area and volume of similar figures. MAFS.8.G.1.AP.5a: Use angle relationships to find the value of a missing angle. MAFS.8.G.1.AP.3b: Given two figures on a coordinate plane, identify if the image is dilated, translated, rotated, or reflected.

#### Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

#### Original Student Tutorial

 Name Description Home Transformations: Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

#### Assessments

 Name Description Sample 4 - Eighth Grade Math State Interim Assessment: This is a State Interim Assessment for eighth grade. Sample 3 - Eighth Grade Math State Interim Assessment: This is a State Interim Assessment for eighth grade. Sample 2 - Eighth Grade Math State Interim Assessment: This is a State Interim Assessment for eighth grade. Sample 1 - Eighth Grade Math State Interim Assessment: This is a State Interim Assessment for eighth grade.

#### Educational Game

 Name Description Transformation Complete: Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.

#### Educational Software / Tools

 Name Description Transformations Using Technology: This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image. Glossary: This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

#### Image/Photograph

 Name Description Angles (Clipart ETC): This large collection of clipart contains images of angles that can be freely used in lesson plans, worksheets, and presentations.

#### Lesson Plans

 Name Description Coding Geometry Challenge #10 & 11: This set of geometry challenges focuses on scaled drawings and area as students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor. Coding Geometry Challenge #23 & 24: This set of geometry challenges focuses on using transformations to show similarity and congruence of polygons and circles. Students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor. "Triangle Congruence Show" Starring Rigid Transformations: Students will be introduced to the definition of congruence in terms of rigid motion and use it to determine if two triangles are congruent. Triangles on a Lattice: In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible. Rotations and Reflections of an Equilateral Triangle: Students will apply simple transformations (rotation and reflection) to an equilateral triangle, then determine the result of the action of two successive transformations, eventually determining whether the action satisfies the commutative and associate properties. Transformations... Geometry in Motion: Transformations... Geometry in Motion is designed for students to practice their knowledge of transformations. Students will represent transformations in the plane, compare transformations, and determine which have isometry. Students should have a basic understanding of the rules for each transformation as they will apply these rules in this activity. There is a teacher-led portion in this lesson followed by partner-activity. Students will be asked to explain and justify reasoning, as well. A Transformation's Adventure with Patty Paper: Exploring Translations, Reflections and Rotations.: This lesson is an introduction to isometric transformations explained using patty paper. Meant for students with limited access to technology, it uses patty paper as a principal resource. Translations, reflections and rotations will be explained and practiced with this inexpensive element, emphasizing the properties preserved during those transformations and, without sacrificing precision, allowing students to differentiate between these isometries in a simple way. Slide to the Left... Slide to the Right!: Students will identify, review, and analyze transformations. They will demonstrate their understanding of transformations in the coordinate plane by creating original graphs of polygons and the images that result from specific transformations. Identifying Similar Triangles: This 105-minute lesson series helps teachers assess how students reason about geometry, including how they use facts about the angle sum and exterior angles of triangles to calculate missing angles, apply angle theorems to parallel lines cut by a transversal, and interpret geometrical diagrams using mathematical properties to identify similarity of triangles. In order to complete this lesson, students will need whiteboards, pens, wipes, copies of the assessment tasks, pencils, markers, scissors, glue sticks, and poster paper. Scientific calculations from a distant planet: Students will act as mathematicians and scientists as they use models, observations and space science concepts to perform calculations and draw inferences regarding a fictional solar system with three planets in circular orbits around a sun. Among the calculations are estimates of the size of the home planet (using a method more than 2000 years old) and the relative distances of the planets from their sun. Shape It Up: Students will draw diagonals for different polygons, separating the polygons into triangles. Using the fact that the sum of the measures of the interior angles of a triangle is 180 degrees, and the fact the angles of the triangles are used to form the angles of the polygons, students will derive the formula for finding the sum of the measures of the angles of a polygon with n sides. Students will also learn to use this formula, along with the fact that all angles of a regular polygon are congruent, to find the measures of the angles of a regular polygon. Dilly Dallying with Dilations: Students will understand the concept of dilation by constructing similar polygons on a coordinate grid using coordinate notation of dilation . Students use similar figures to determine the scale factor. Students use proportions to determine side lengths of similar figures. The Ins and Outs of Polygons: In this lesson, students will explore how to find the sum of the measures of the angles of a triangle and then be able to find the sum of the measures of the angles of other polygons. They will also be able to find the sum of the exterior angles of triangles and other polygons. Using both of these concepts, they will be able to find missing measurements. Triangles: Finding Interior Angle Measures: In this lesson plan, students will start with a hands-on activity and then experiment with a GeoGebra-based computer model to investigate and discover the Triangle Angle Sum Theorem. Then they will use the Triangle Angle Sum Theorem to write and solve equations and find missing angle measures in a variety of examples. Polygon Transformers: This guided discovery lesson introduces students to the concept that congruent polygons can be formed using a series of transformations (translations, rotations, reflections). As a culminating activity, students will create a robot out of transformed figures. Special Angle Pairs Discovery Activity: This lesson uses a discovery approach to identify the special angles formed when a set of parallel lines is cut by a transversal. During this lesson students identify the angle pair and the relationship between the angles. Students use this relationship and special angle pairs to make conjectures about which angle pairs are considered special angles. How Many Degrees?: This lesson facilitates the discovery of a formula for the sum of the interior angles of a regular polygon. Students will draw all the diagonals from one vertex of various polygons to find how many triangles are formed. They will use this and their prior knowledge of triangles to figure out the sum of the interior angles. This will lead to the formulation of a formula for finding the sum of interior angles and the measure of one interior angle. Help me Find my Relationship!: In this lesson, students will investigate the relationship between angles when parallel lines are cut by a transversal. Students will identify angles, find angle measures, and they will use the free application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of angles relationships. Exploring Rotations with GeoGebra: This lesson will help students understand the concept of a geometric rotation. The teacher/students will use a GeoGebra applet to derive the rules for rotating a point on the coordinate plane about the origin for a 90 degree, 180 degree, and a 270 degree counterclockwise rotation. An Investigation of Angle Relationships Formed by Parallel Lines Cut by a Transversal Using GeoGebra: In this lesson, students will discover angle relationships formed (corresponding, alternate interior, alternate exterior, same-side interior, same-side exterior) when two parallel lines are cut by a transversal. They will establish definitions and identify whether these angle pairs are supplementary or congruent. Rotations and Reflections of an Equilateral Triangle: Students will apply simple transformations (rotation and reflection) to an equilateral triangle, then determine the result of the action of two successive transformations, eventually determining whether the action satisfies the commutative and associate properties. Triangles on a Lattice: In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible.

#### Perspectives Video: Professional/Enthusiast

 Name Description All Circles Are Similar- Especially Circular Pizza!: What better way to demonstrate that all circles are similar then to use pizzas! Gaines Street Pies explains how all pizza pies are similar through transformations. Download the CPALMS Perspectives video student note taking guide.

#### Student Center Activity

 Name Description Edcite: Mathematics Grade 8: Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

#### Tutorials

 Name Description Rotating polygons 180 degrees about their center: Students will investigate symmetry by rotating polygons 180 degrees about their center. Proving congruent angles: In this tutorial, students are asked to prove two angles congruent when given limited information. Students need to have a foundation of parallel lines, transversals and triangles before viewing this video. Introduction to Transformations: This video introduces the concept of rigid transformation and congruent figures. Scaling Down a Triangle by Half: This video demonstrates the effect of a dilation on the coordinates of a triangle. Testing Similarity Through Transformations: This video shows testing for similarity through transformations. Sum of measures of triangles proof: This video gives the proof of sum of measures of angles in a triangle. This video is beneficial for both Algebra and Geometry students.

#### Virtual Manipulatives

 Name Description Congruent Triangles: This manipulative is a virtual realization of the kind of physical experience that might be available to students given three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles (SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle, students can experience congruence by actually moving one on the top of the other. Transformations - Translation: The user can demonstrate or explore translation of shapes created with pattern blocks, using or not using a coordinate axes and lattice points background, by changing the translation vector.(source: NLVM grade 6-8 "Transformations - Translation") Transformations - Reflections: The user clicks and drags a shape they have constructed to view its reflection across a line. A background grid and axes may or may not be used. The reflection may by examined analytically using coordinates. Symmetry may be displayed. Transformations - Dilation: Students use a slider to explore dilation and scale factor. Students can create and dilate their own figures. (source: NLVM grade 6-8 "Transformations - Dilation") Transformations - Rotation: Rotate shapes and their images with or without a background grid and axes. Rotation of a Point: This virtual manipulative is an interactive visual presentation of the rotation of a point around the origin of the coordinate system. The original point can be dragged to different positions and the angle of rotation can be changed with a 90° increment.

#### Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

#### Original Student Tutorial

 Title Description Home Transformations: Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

#### Educational Game

 Title Description Transformation Complete: Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.

#### Educational Software / Tools

 Title Description Transformations Using Technology: This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image. Glossary: This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

#### Student Center Activity

 Title Description Edcite: Mathematics Grade 8: Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

#### Tutorials

 Title Description Rotating polygons 180 degrees about their center: Students will investigate symmetry by rotating polygons 180 degrees about their center. Proving congruent angles: In this tutorial, students are asked to prove two angles congruent when given limited information. Students need to have a foundation of parallel lines, transversals and triangles before viewing this video. Introduction to Transformations: This video introduces the concept of rigid transformation and congruent figures. Scaling Down a Triangle by Half: This video demonstrates the effect of a dilation on the coordinates of a triangle. Testing Similarity Through Transformations: This video shows testing for similarity through transformations. Sum of measures of triangles proof: This video gives the proof of sum of measures of angles in a triangle. This video is beneficial for both Algebra and Geometry students.

#### Virtual Manipulatives

 Title Description Congruent Triangles: This manipulative is a virtual realization of the kind of physical experience that might be available to students given three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles (SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle, students can experience congruence by actually moving one on the top of the other. Transformations - Translation: The user can demonstrate or explore translation of shapes created with pattern blocks, using or not using a coordinate axes and lattice points background, by changing the translation vector.(source: NLVM grade 6-8 "Transformations - Translation") Transformations - Reflections: The user clicks and drags a shape they have constructed to view its reflection across a line. A background grid and axes may or may not be used. The reflection may by examined analytically using coordinates. Symmetry may be displayed. Transformations - Dilation: Students use a slider to explore dilation and scale factor. Students can create and dilate their own figures. (source: NLVM grade 6-8 "Transformations - Dilation") Transformations - Rotation: Rotate shapes and their images with or without a background grid and axes. Rotation of a Point: This virtual manipulative is an interactive visual presentation of the rotation of a point around the origin of the coordinate system. The original point can be dragged to different positions and the angle of rotation can be changed with a 90° increment.

#### Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

#### Educational Software / Tool

 Title Description Glossary: This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.