# MA.912.GR.2.1

Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

### Examples

Example: Given a triangle whose vertices have the coordinates (-3,4), (2,1.7) and (-0.4,-3). If this triangle is reflected across the y-axis the transformation can be described using coordinates as (x,y)→(-x,y) resulting in the image whose vertices have the coordinates (3,4), (-2,1.7) and (0.4,-3).

### Clarifications

Clarification 1: Instruction includes the connection of transformations to functions that take points in the plane as inputs and give other points in the plane as outputs.

Clarification 2: Transformations include translations, dilations, rotations and reflections described using words or using coordinates.

Clarification 3: Within the Geometry course, rotations are limited to 90°, 180° and 270° counterclockwise or clockwise about the center of rotation, and the centers of rotations and dilations are limited to the origin or a point on the figure.

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 grade 8, students developed an understanding of single transformations. In Algebra 1, students extended this knowledge of transformations to transforming functions using tables, graphs and equations. In Geometry, students will understand transformations as functions and will transform figures, using two or more transformations, using words and coordinates. In later courses, types of transformations will be expanded to include stretches that can transform functions and conic sections and conversions between rectangular and polar coordinates.
• Instruction includes describing transformations using words and coordinates. Common transformations are provided below.
• Rotations can be described algebraically as the following:
90° counterclockwise about the origin ($x$, $y$) → (−$y$, $x$)
90° clockwise about the origin ($x$, $y$) → ($y$, −$x$)
180° counterclockwise about the origin ($x$, $y$) → (−$x$, −$y$)
180° clockwise about the origin ($x$, $y$) → (−$x$, −$y$)
270° counterclockwise about the origin ($x$, $y$) → ($y$, −$x$)
270° clockwise about the origin ($x$, $y$) → (−$y$, $x$)
• Reflections can be described algebraically as the following:
Over the $x$-axis ($x$, $y$) → ($x$, −$y$)
Over the $y$-axis ($x$, $y$) → (−$x$, $y$)
Over the line $y$$x$ ($x$, $y$) → ($y$, $x$)
Over the line $y$$x$ ($x$, $y$) → (−$y$, −$x$)
• Dilations
Dilation by a factor of a, where a is a real number ($x$, $y$) → ($a$$x$, $a$$y$)
• Translations
• Horizontal translation by $h$ units, where $h$ is a real number ($x$, $y$) → ($x$ + $y$, $y$) Vertical translation by $k$ units, where $k$ is a real number ($x$, $y$) → ($x$, $y$ + $k$) Horizontal translation by $h$ units, where $h$ is a real number, and vertical translation by $k$ units, where $k$ is a real number ($x$, $y$
• Instruction includes examining the effect of transforming coordinates by adding, subtracting, or multiplying the $x$- and $y$-coordinates with real-number values to make the connection between functions and transformations.
• For example, ΔPQR, with vertices P(−1, 4), Q(3, 4) and R(1, 7) can be transformed using the coordinates representation ($x$, $y$) → ($x$, $y$ − 1).
• For example, if the vertices of ΔABC are (4, −2), (4, 5) and (3, 3), respectively, and the vertices of ΔABC′ are (8, −4), (8, 10) and (6, 6), respectively, the coordinate representation can be determined.
• Instruction includes the use of hands-on manipulatives and geometric software for students to explore transformations.
• Instruction includes using a variety of ways to describe a transformation using coordinates. (MTR.2.1)
• For example, the same translation can be described using words as 2 units to the right and 4 units down, using coordinates as ($x$, $y$) → ($x$ + 2, $y$ − 4) or as $T$$x$,$y$ = ($x$ + 2, $y$ − 4), or .
• For example, a reflection over the $x$-axis can be represented as ($x$, $y$) → ($x$, −$y$) or as $r$$x$-axis ($x$, $y$) = ($x$, −$y$).
• For example, a 90°rotation counterclockwise about the origin can be represented as ($x$, $y$) → (−$y$, $x$) or as $R$0,90($x$, $y$) = (−$y$, $x$) where $O$ is the origin.
• The discussion of translations can be extended to include vectors when describing translations.
• For example, if a point is translated 3 units to the left and 4 units up the translation vector is . The vector summarizes the horizontal and vertical shifts.($x$ + $h$, $y$ + $k$)

### Common Misconceptions or Errors

• Students may believe the orientation of a figure would be conserved in a rotation in the same way that the orientation of a car, or gondola, is preserved when rotating on a Ferris wheel).
• Student may not be able to visualize some transformations like rotations. To address this, instruction includes using folding paper (e.g., patty paper) or interactive geometric software to allow students hands-on experiences and flexibility in exploration.

• Use the graph to the below to answer the following questions.

• Part A. Describe the transformation that maps ABCD to A'B'C'D'
• Part B. Represent the transformation described in Part A algebraically.
• Part C. Algebraically represent the transformation needed to map A"B"C"D" onto ABCD
• Part D. Describe the transformation that maps A"B"C"D" onto A"'B"'C""D""
• Part E. How is the transformation described in Part D related to the transformation needed to map A"'B"'C""D"" onto A"B"C"D"

• Part A. Ask students to plot A, B and C and A', B' and C' on the coordinate plane. What do you notice?
• Part B. How can you describe the transformation using words? Explore the patterns among the coordinates of the points of the preimages and the images.
• Part C. How can you describe the transformation using coordinate notation??

### Instructional Items

Instructional Item 1
• A triangle whose vertices are located at ($\frac{\text{2}}{\text{7}}$, −1), (−4,− $\frac{\text{14}}{\text{5}}$ ) and (3,1) is shifted to the right 2 units.
• Part A. What are the coordinates of the triangle after the translation?
• Part B. Describe the transformation that would map the preimage to the image algebraically.

*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.1a: Given a preimage and image, identify the transformation.
MA.912.GR.2.AP.1b: Select the algebraic coordinates that represent the transformation.

## Related Resources

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

## Formative Assessments

Transformations And Functions:

Students are given examples of three transformations and are asked if each is a function.

Type: Formative Assessment

Demonstrating Translations:

Students are asked to translate a quadrilateral according to a given vector.

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

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

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

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

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

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

## MFAS Formative Assessments

Demonstrating Translations:

Students are asked to translate a quadrilateral according to a given vector.

Transformations And Functions:

Students are given examples of three transformations and are asked if each is a function.

## Student Resources

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

## Parent Resources

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