# Standard 2: Understand similarity and congruence using models and transformations.

General Information
Number: MA.8.GR.2
Title: Understand similarity and congruence using models and transformations.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Grade: 8
Strand: Geometric Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.8.GR.2.AP.1
Given two figures on a coordinate plane, identify if the image is translated, rotated or reflected.
MA.8.GR.2.AP.2
Given a preimage and image describe the effect the dilation has on the two figures.
MA.8.GR.2.AP.3
Identify the coordinates of the vertices of a common polygon after a single translation, rotation or dilation on the coordinate plane.
MA.8.GR.2.AP.4
Use tools to solve mathematical problems using proportions between similar triangles.

## Related Resources

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

## Formative Assessments

Define a Translation:

Students are asked to develop a definition for translation.

Type: Formative Assessment

Define a Rotation:

Students are asked to develop a definition of rotation.

Type: Formative Assessment

Define a Reflection:

Students are asked to develop a definition of reflection.

Type: Formative Assessment

Angle Transformations:

Students are given the opportunity to experimentally verify the properties of angle transformations (translations, reflections, and rotations).

Type: Formative Assessment

Segment Transformations:

Students are given the opportunity to experimentally verify the properties of segment transformations (translations, reflections, and rotations).

Type: Formative Assessment

Parallel Line Transformations:

Students are given the opportunity to experimentally verify the properties of parallel line transformations (translations, reflections, and rotations).

Type: Formative Assessment

Translation Coordinates:

Students are asked to translate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Type: Formative Assessment

Rotation Coordinates:

Students are asked to rotate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Type: Formative Assessment

Reflection Coordinates:

Students are asked to reflect two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Type: Formative Assessment

Dilation Coordinates:

Students are asked to dilate two-dimensional figures in the coordinate plane and identify the coordinates of the vertices of the images.

Type: Formative Assessment

Transformations of Rectangles and Squares:

Students are asked to describe the rotations and reflections that carry a rectangle and a square onto itself.

Type: Formative Assessment

Transformations of Parallelograms and Rhombi:

Students are asked to describe the rotations and reflections that carry a parallelogram and rhombus onto itself.

Type: Formative Assessment

Transformations of Trapezoids:

Students are asked to describe the rotations and reflections that carry a trapezoid onto itself.

Type: Formative Assessment

Transformations of Regular Polygons:

Students are asked to describe the rotations and reflections that carry a regular polygon onto itself.

Type: Formative Assessment

## Lesson Plans

Transformations in the Coordinate Plane:

In this exploration activity of reflections, translations, and rotations, students are guided to discover general algebraic rules for special classes of transformations in the coordinate plane. This lesson is intended to be used after the development of formal definitions of rotations, translations, and reflections.

Type: Lesson Plan

"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.

Type: Lesson Plan

Reflecting on the Commute:

Students are given a set of coordinates that indicate a specific triangle on a coordinate plane. They will also be given a set of three reflections to move the triangle through. Students will then perform three other sequences of reflections to determine if the triangle ends up where it started.

Type: Lesson Plan

Transformation and Similarity:

Using non-rigid motions (dilations), students learn how to show that two polygons are similar. Students will write coordinate proofs confirming that two figures are similar.

Type: Lesson Plan

Sequence of Transformations:

This lesson will assist students in performing multi-step transformations. Students will follow a sequence of transformations on geometric figures using translations, reflections, and rotations.

Type: Lesson Plan

Regular Polygon Transformation Investigation:

This is an introductory lesson on regular polygon transformation for congruency using a hands-on approach.

Type: Lesson Plan

Construction of Inscribed Regular Hexagon:

A GeoGebra lesson for students to become familiar with computer based construction tools. Students work together to construct a regular hexagon inscribed in a circle using rotations. Directions for both a beginner and advanced approach are provided.

Type: Lesson Plan

Match That!:

Students will prove that two figures are congruent based on a rigid motion(s) and then identify the corresponding parts using paragraph proof and vice versa, prove that two figures are congruent based on corresponding parts and then identify which rigid motion(s) map the images.

Type: Lesson Plan

Isometries with GeoGebra:

In this lesson, students explore transformations with GeoGebra and then apply concepts using a straightedge on paper. Students apply rules for each isometry. There is a teacher-driven opening followed by individual student activity.

Type: Lesson Plan

How Did It Get There? A Series of Transformation Events:

Students will perform a series of transformations in order to determine how the pre-image will map onto the final image of a given figure. Students will use patty paper to manipulate their pre-image onto the image. Students will also work in collaborative groups to discuss their findings and will have the opportunity to share their series of transformations with the class. The class discussion will be used to demonstrate that there are several ways for the students to map their pre-image onto the final image.

Type: Lesson Plan

Dancing For Joy:

We have danced our way through reflections, rotations, and translations; now we are ready to take it up a notch by performing a sequence of transformations. Students will also discover the results when reflecting over parallel lines versus intersecting lines.

Type: Lesson Plan

Turning to Congruence:

This lesson uses rigid motions to prove the ASA and HL triangle congruence theorems.

Type: Lesson Plan

Product of Two Transformations:

Students will identify a sequence of steps that will translate a pre-image to its image. Students will also demonstrate that the sequence of two transformations is not always commutative.

Type: Lesson Plan

Slip, Slide, Tip, and Turn: Corresponding Angles and Corresponding Sides:

Using the definition of congruence in terms of rigid motion, students will show that two triangles are congruent.

Type: Lesson Plan

How to Land Your Spaceship:

Teach your students to maneuver a "spaceship" through a sequence of transformations that will successfully carry it onto its landing pad. GeoGebra directons are provided.

Type: Lesson Plan

Where Will I Land?:

This is a beginning level lesson on predicting the effect of a series of reflections or a quick review of reflections for high school students.

Type: Lesson Plan

Exploring Congruence Using Transformations:

This is an exploratory lesson that elicits the relationship between the corresponding sides and corresponding angles of two congruent triangles.

Type: Lesson Plan

How do your Air Jordans move?:

In this inquiry lesson, students are moving their individually designed Air Jordans around the room to explore rigid transformations on their shoes. They will Predict-Observe-Explain the transformations and then have to explain their successes/failures to other students.

Type: Lesson Plan

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.

Type: Lesson Plan

Discovering Dilations:

This resource is designed to allow students to discover the effects of dilations on geometric objects using the free online tools in GeoGebra.

Type: Lesson Plan

Transformations... Geometry in Motion:

Students will practice and compare transformations, and then 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 their reasoning,

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Type: Lesson Plan

Reflections Hands On:

Students will use a protractor/ruler to construct reflections and a composite of reflections. They will create transformations using paper cut-outs and a coordinate plane. For independent practice, students will predict and verify sequences of transformations. The teacher will need an LCD Projector and document camera.

Type: Lesson Plan

A Transformation's Adventure with Patty Paper: Exploring Translations, Reflections and Rotations.:

Students are introduced to isometric transformations using patty paper. Translations, reflections, and rotations will be explained and practiced, emphasizing the properties preserved during those transformations and, without sacrificing precision, allowing students to differentiate between these isometries. The lesson can also be taught using GeoGebra free software.

Type: Lesson Plan

Flipping Fours:

Students will translate, rotate and reflect quadrilaterals (Parallelogram, Rectangle, Square, Kite, Trapezoid, and Rhombus) using a coordinate grid created on the classroom floor and on graph paper. This activity should be used following guided lessons on transformations.

Type: Lesson Plan

Let's Reflect On This...:

Students will use parallel and intersecting lines on the coordinate plane to transform reflections into translations and rotations.

Type: Lesson Plan

Dilation Transformation:

Students identify dilations, verify that polygons are similar, and use the dilation rule to map dilations. Task cards are provided for independent practice. The PowerPoint also includes detailed illustrations for constructing a dilation using a compass and a straight edge.

Type: Lesson Plan

Fundamental Property of Reflections:

This lesson helps students discover that in a reflection, the line of reflection is the perpendicular bisector of any segment connecting any pre-image point with its reflected image.

Type: Lesson Plan

Transformers 3:

Students will learn the vocabulary of three rigid transformations, reflection, translation, and rotation, and how they relate to congruence. Students will practice transforming figures by applying each isometry and identifying which transformation was used on a figure. The teacher will assign students to take pictures of the three transformations found in the real world.

Type: Lesson Plan

Rotation of Polygons about a Point:

Students will rotate polygons of various shapes about a point. Degrees of rotation vary but generally increase in increments of 90 degrees. Points of rotation include points on the figure, the origin, and points on the coordinate plane. The concept of isometry is addressed.

Type: Lesson Plan

Transform through the Maze:

In this fun activity, students will use rigid transformations to move a triangle through a maze. The activity provides applications for both honors and standard levels. It requires students to perform rotations, translations, and reflections.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

Mirror, Mirror on the ... Ground?:

This activity allows students to go outdoors to measure the height of objects indirectly. Similar right triangles are formed when mirrors are placed on the ground between the object that needs to be measured and the student observing the object in the mirror. Students work in teams to measure distances and solve proportions.

This activity can be used as a review or summative assessment after teaching similar triangles.

Type: Lesson Plan

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.

Type: Lesson Plan

Exploring Rotations with GeoGebra:

This lesson will help students understand the concept of geometric rotations. 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.

Type: Lesson Plan

How does scale factor affect the areas and perimeters of similar figures?:

In this lesson plan, students will observe and record the linear dimensions of similar figures, and then discover how the values of area and perimeter are related to the ratio of the linear dimensions of the figures.

Type: Lesson Plan

## Perspectives Video: Professional/Enthusiast

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.

Type: Perspectives Video: Professional/Enthusiast

## Perspectives Video: Teaching Idea

Measuring Height with Triangles and Mirrors:

<p>Reflect for a moment on how to measure tall objects with mirrors and mathematics.</p>

Type: Perspectives Video: Teaching Idea

## Problem-Solving Tasks

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Type: Problem-Solving Task

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 in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

Reflecting Reflections:

In this resource, students experiment with the reflection of a triangle in a coordinate plane.

Type: Problem-Solving Task

Point Reflection:

The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of two-dimensional figures are found by reflecting individual points.

Type: Problem-Solving Task

Reflecting a Rectangle Over a Diagonal:

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections.

Type: Problem-Solving Task

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Type: Problem-Solving Task

## Student Resources

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

## Problem-Solving Tasks

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Type: Problem-Solving Task

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 in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

Reflecting Reflections:

In this resource, students experiment with the reflection of a triangle in a coordinate plane.

Type: Problem-Solving Task

Point Reflection:

The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of two-dimensional figures are found by reflecting individual points.

Type: Problem-Solving Task

Reflecting a Rectangle Over a Diagonal:

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections.

Type: Problem-Solving Task

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Type: Problem-Solving Task

## Parent Resources

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

## Problem-Solving Tasks

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Type: Problem-Solving Task

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 in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

Reflecting Reflections:

In this resource, students experiment with the reflection of a triangle in a coordinate plane.

Type: Problem-Solving Task

Point Reflection:

The purpose of this task is for students to apply a reflection to a single point. The standard asks students to apply the effect of a single transformation on two-dimensional figures. Although this problem only applies a reflection to a single point, it has high cognitive demand if the students are prompted to supply a picture. This is because the coordinates of the point (1000,2012) are very large. If students try to plot this point and the line of reflection on the usual x-y coordinate grid, then either the graph will be too big or else the point will lie so close to the line of reflection that it is not clear whether or not it lies on this line. A good picture requires a careful choice of the appropriate region in the plane and the corresponding labels. Moreover, reflections of two-dimensional figures are found by reflecting individual points.

Type: Problem-Solving Task

Reflecting a Rectangle Over a Diagonal:

The task is intended for instructional purposes and assumes that students know the properties of rigid transformations. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. This means that students need to approximate and this provides an extra challenge. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. However, this choice of grid also makes it easier to reason about the reflections.

Type: Problem-Solving Task

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

Type: Problem-Solving Task