# MA.912.GR.2.2

Identify transformations that do or do not preserve distance.

### Clarifications

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

Clarification 2: Instruction includes recognizing that these transformations preserve angle measure.

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 were introduced to transformations and whether they preserve congruence or similarity. In Geometry, students determine which transformations preserve distance (rigid transformations, or rigid motions) and which do not (non-rigid transformations, or nonrigid motions), while understanding that all the transformations in this benchmark preserve angle measures. This discussion leads to the definition of congruence and the definition of similarity in terms of rigid and non-rigid motions. (MTR.3.1, MTR.5.1) In later courses, other non-rigid motions are studied, including stretches in one coordinate direction.
• Instruction includes the comparison of a variety of geometric figures before and after a single transformation (including reflections, translations, rotations and dilations) to solidify that each of these transformations preserves angle measure. (MTR.4.1
• Instruction includes the student understanding that in order for a transformation based on stretching and shrinking to preserve angle measure, it must have a stretch or shrink of the same scale factor in both the $x$-direction and the $y$-direction.
• For example, the transformation ($x$, $y$) → (2.5$x$, 2.5$y$) would preserve angle measure, but the transformation ($x$, $y$) → (2.5$x$, 3.5$y$) would not.
• While the intent of the benchmark is to focus on dilations, instruction includes the introduction of other non-rigid transformations to showcase that not all non-rigid transformations preserve angle measure.
• For example, “stretch” or “shrink” in the direction of one of the axes, such as ($x$, $y$) → (5$x$, $y$) or ($x$, $y$) → ($x$, $\frac{\text{y}}{\text{6}}$), does not preserve angle measure. Students can visualize this using geometric software to see how each affects angle measures of a triangle.
• Transformations should be presented using words and using coordinates.
• Instruction includes the use of folding paper (e.g., patty paper) for hands-on experiences, as needed, and interactive geometry software to allow more flexibility in exploration, when possible.

### Common Misconceptions or Errors

• Students may incorrectly assume that dilations change angle measures.
• Students may assume that dilations only occur in either the $x$-direction or the $y$-direction based on knowledge of function transformations.
• Students may assume that every non-rigid motion preserves angle measures since the only non-rigid motions in the benchmark are dilations, which do preserve angle measures. But many non-rigid motions preserve neither length nor angle measure.

• Penelope made the following statement in Geometry class, “Figure A is a rotation of Figure B about the origin.” Chalita disagreed because distance and angle measures are not preserved between the two figures.
• Figure A has the coordinate points (1, −1), (3, −1) and (1, −2).
• Figure B has the coordinate points (0.8, 1), (1, 3) and (2, 0.8).
• Part A. What does “distance and angle measures are not preserved” mean in relation to the two figures?
• Part B. Determine whether angle measures were preserved from Figure A to Figure B.
• Part C. Determine whether distance measures were preserved from Figure A to Figure B.
• Part D. Based on your answers from Part B and Part C, determine which student is correct.

• Sort the following transformations into preserves distance and does not preserve distance.

### Instructional Items

Instructional Item 1
• Circle the transformations that can be used when it is important to preserve angle measure.
Horizontal Translations Reflections Clockwise Rotations
Dilations Vertical Translations Counterclockwise Rotations

Instructional Item 2
• Circle the transformations that can be used when it is important to preserve distance.
Horizontal Translations Reflections Clockwise Rotations
Dilations Vertical Translations Counterclockwise Rotations

Instructional Item 3
• Write a transformation, or sequence of transformations, that preserves angle measure but does not preserve distance.

*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.2: Select a transformation that preserves distance.

## Related Resources

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

## Formative Assessment

Comparing Transformations:

Students are asked to determine whether or not dilations and reflections preserve distance and angle measure.

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

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

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

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

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

## MFAS Formative Assessments

Comparing Transformations:

Students are asked to determine whether or not dilations and reflections preserve distance and angle measure.

## 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.