# MA.912.GR.2.3

Identify a sequence of transformations that will map a given figure onto itself or onto another congruent or similar figure.

### Clarifications

Clarification 1: Transformations include translations, dilations, rotations and reflections described using words or using coordinates.

Clarification 2: Within the Geometry course, figures are limited to triangles and quadrilaterals and rotations are limited to 90°, 180° and 270° counterclockwise or clockwise about the center of rotation.

Clarification 3: Instruction includes the understanding that when a figure is mapped onto itself using a reflection, it occurs over a line of symmetry.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Coordinate Plane
• Dilation
• Origin
• Reflection
• Rigid Transformation
• Rotation
• Scale Factor
• Translation

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In the elementary grades, students learned about lines of symmetry. In grade 8, students learned about the effects of translations, rotations, reflections, and dilations on geometric figures. In Geometry, students use their knowledge of translations, dilations, rotations and reflections to identify a sequence or composition of transformations that map a triangle or a quadrilateral onto another congruent or similar figure or onto itself, and they connect reflections to lines of symmetry. In later courses, lines of symmetry are identified as key features in graphs of polynomials and trigonometric functions.
• To describe the sequence of transformations, students will need to know how to describe each one of the transformations in the composition using words or using coordinates. In each case, they will specify vertical and horizontal shifts, center and angle of the rotation, clockwise or counterclockwise, line of reflection, center of the dilation and scale factor, when needed. (MTR.3.1)
• Provide multiple opportunities for students to explore mapping a variety of triangles and quadrilaterals onto congruent or similar figures (given the preimage and the image) using both physical exploration (transparencies or patty paper) and virtual exploration when possible. This will allow students to experience multiple compositions of transformations and realize that more than one sequence can be used to map a figure onto another. (MTR.2.1)
• Instruction includes examples where preimages and images partially overlap each other.
• When a sequence includes a dilation, it may be helpful that students identify the dilation first, and then continue to identify any rigid motions that may be needed.
• Students can explore the sequence of a reflection over the $x$-axis followed by a reflection over the $y$-axis (or any sequence of two reflections over axes perpendicular to each other). To help students make the connection between different sequences of transformations, ask “Is there a single transformation that produces the same image as this sequence?” (MTR.5.1)
• To map a figure onto itself, explore the effect of each transformation. Discuss with students the possibilities of using translations or dilations.
• When a reflection maps a figure onto itself, the line of reflection is also a line of symmetry for the figure. Explore the lines of symmetries of isosceles and equilateral triangles, and rectangles, rhombi, squares, isosceles trapezoids and kites.
• When a rotation is used, explore the cases of regular polygons (equilateral triangles and squares) and how to determine the angles of rotation that will map them onto themselves. (MTR.5.1)
• Instruction includes discussing the case of a dilation with a scale factor of 1. Even if this case is considered trivial, it leads the conversation to the relationship between congruence and similarity. If a dilation is a similarity transformation, then it produces an image that is similar to the preimage. But if a dilation with a scale factor of 1 produces an image that is congruent to the preimage, then congruence is a case of similarity. In other words, when two figures are congruent, then they are necessarily similar to each other.
• An extension of this benchmark may be to explore the angle of rotation needed to map a regular polygon of 5 or more sides onto itself.

### Common Misconceptions or Errors

• Students may believe there is only one sequence that will lead to the image. Instead, students should explore the fact that multiple sequences will result in the same image.

• Part A. From the list provided, choose and order transformations that could be used to map ΔABC onto ΔA"'B"'C"'.

• Part B. Describe the transformation that maps ΔA"B"C" onto ΔA"'B"'C'".

### Instructional Items

Instructional Item 1

• Part A. What is the center of the rotation?
• Part B. If the rotation is counterclockwise, how many degrees is the rotation?
• Part C. Describe another transformation that maps quadrilateral ABCD onto quadrilateral A'B'C'D'.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.2.AP.3: Identify a given sequence of transformations, that includes translations or reflections, that will map a given figure onto itself or onto another congruent figure.

## Related Resources

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

## Formative Assessments

Justifying HL Congruence:

Students are asked to use rigid motion to explain why the HL pattern of congruence ensures right triangle congruence.

Type: Formative Assessment

Two Triangles:

Students are asked to describe the transformations that take one triangle onto another.

Type: Formative Assessment

Indicate the Transformations:

Students are asked to describe the transformations that take one triangle onto another.

Type: Formative Assessment

Similarity - 2:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Type: Formative Assessment

Similarity - 1:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Type: Formative Assessment

Similarity - 3:

Students are asked to describe a sequence of transformations that demonstrates two polygons are similar.

Type: Formative Assessment

Multistep Congruence:

Students are asked to describe a sequence of rigid motions to demonstrate the congruence of two polygons.

Type: Formative Assessment

Rigid Motion - 3:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Type: Formative Assessment

Rigid Motion - 2:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Type: Formative Assessment

Rigid Motion - 1:

Students are asked to describe a rigid motion to demonstrate that two polygons are congruent.

Type: Formative Assessment

## Lesson Plans

"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

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

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

Rotations of Regular Polygons:

This lesson guides students through the development of a formula to find the first angle of rotation of any regular polygon to map onto itself. Free rotation simulation tools such as GeoGebra, are used.

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

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

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

I Am Still Me Transformed.:

Students explore ways of applying, identifying, and describing reflection and rotation symmetry for both geometric and real-world objects, for them to develop a better understanding of symmetries in transformational geometry.

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

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.

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

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

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

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

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

## Original Student Tutorial

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.

Type: Original Student Tutorial

## 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.

Type: Perspectives Video: Professional/Enthusiast

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.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

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.

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.''

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. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

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.

## MFAS Formative Assessments

Indicate the Transformations:

Students are asked to describe the transformations that take one triangle onto another.

Justifying HL Congruence:

Students are asked to use rigid motion to explain why the HL pattern of congruence ensures right triangle congruence.

Multistep Congruence:

Students are asked to describe a sequence of rigid motions to demonstrate the congruence of two polygons.

Rigid Motion - 1:

Students are asked to describe a rigid motion to demonstrate that two polygons are congruent.

Rigid Motion - 2:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Rigid Motion - 3:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Similarity - 1:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Similarity - 2:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Similarity - 3:

Students are asked to describe a sequence of transformations that demonstrates two polygons are similar.

Two Triangles:

Students are asked to describe the transformations that take one triangle onto another.

## Original Student Tutorials Mathematics - Grades 9-12

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.

## Student Resources

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

## Original Student Tutorial

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.

Type: Original Student Tutorial

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.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

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.

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.''

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. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

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.

## Parent Resources

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

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.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

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.

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.''