# MA.912.GR.2.5

Given a geometric figure and a sequence of transformations, draw the transformed figure on a coordinate plane.

### Clarifications

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

Clarification 2: Instruction includes two or more transformations.

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 learned about the effects of translations, rotations, reflections, and dilations on geometric figures. In Algebra 1, students extended this knowledge of transformations to transforming functions using tables, graphs and equations. In Geometry, students are given the preimage and a sequence of two or more transformations (a composition) and draw the image on a coordinate plane. 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.
• The purpose of this benchmark is that students produce an image on the coordinate plane as the result of combining transformations (translations, dilations, rotations and reflections).
• Instruction includes transformations described using words and using coordinates.
• Instruction includes understanding that a single transformation can be done as a sequence of transformation. Likewise, some sequences can be described as a single transformation.
• For example, if the transformation sequence is to translate 1 unit to the left, then rotate 90-degrees clockwise about the origin, then translate 1 unit to the right; the result is the same as rotating the figure 90-degrees clockwise about the point (1,0). This can be described in coordinates as ($x$, $y$) → ($x$ − 1, $y$), then ($x$ − 1, $y$) → ($y$, −($x$ − 1)), then ($y$, −($x$ − 1)) → ($y$ + 1, −($x$ − 1)), and the result of rotating the figure 90-degrees clockwise about the point (1,0) can be described in coordinates as ($x$, $y$) → ($y$ + 1, −$x$ + 1).
• Instruction includes discussing the resulting images using the same preimage and a set of transformations in different orders and understanding that transformations may or not be commutative based on the sequence and type of transformations.
• For example, if the transformation is the result of a vertical and a horizontal translation, the sequence is commutative.
• For example, if the transformation is the result of a 180° rotation and a reflection over the $x$-axis, the sequence is commutative.
• For example, if the transformation is the result of a 90° rotation and a reflection over the $x$-axis, the sequence is not commutative.
• Instruction includes the understanding that if a dilation is part of a sequence, it can be applied at any time within the sequence of transformations without changing the final result.

### Common Misconceptions or Errors

• Students may think order of transformations doesn’t matter.

• Part A. On the coordinate plane, draw the resulting figure after transforming quadrilateral ABCD through the following sequence below.
• Reflect quadrilateral ABCD over the line $y$ = $x$ .
• Translate horizontally and vertically the resulting figure using ($x$, $y$) → ($x$ + 3, $y$ − 2).
• Part B. Would the resulting figure be the same if the transformations were reversed? How did you come to your conclusion?

### Instructional Items

Instructional Item 1
• Perform the following sequence of transformations on the polygon ABCDEF on the coordinate plane.
• Rotate 180° counterclockwise about the origin.
• Then, translate horizontally 2 units to the left and vertically 3 units down.

Instructional Item 2
• Draw the resulting figure after quadrilateral ABCD is transformed using ($x$, $y$) → (−$x$, $y$ − 3).

*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.5: Given a geometric figure and a sequence of transformations, select the transformed figure on a coordinate plane.

## Related Resources

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

## Formative Assessments

Dilation of a Line: Factor of Two:

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Type: Formative Assessment

Dilation of a Line: Factor of One Half:

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Type: Formative Assessment

Dilation of a Line Segment:

Students are asked to dilate a line segment and describe the relationship between the original segment and its image.

Type: Formative Assessment

Reflect a Semicircle:

Students are asked to reflect a semicircle across a given line.

Type: Formative Assessment

Dilation of a Line: Center on the Line:

Students are asked to graph the image of two points on a line after a dilation using a center on the line and to generalize about dilations of lines when the line contains the center.

Type: Formative Assessment

Type: Formative Assessment

Repeated Reflections and Rotations:

Students are asked to describe what happens to a triangle after repeated reflections and rotations.

Type: Formative Assessment

Demonstrating Rotations:

Type: Formative Assessment

Demonstrating Reflections:

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

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

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

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

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

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

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

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

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

## Perspectives Video: Professional/Enthusiast

3D Modeling with 3D Shapes:

<p>Complex 3D shapes are often created using simple 3D primitives! Tune in and shape up as you learn about this application of geometry!</p>

Type: Perspectives Video: Professional/Enthusiast

## MFAS Formative Assessments

Demonstrating Reflections:

Demonstrating Rotations:

Dilation of a Line Segment:

Students are asked to dilate a line segment and describe the relationship between the original segment and its image.

Dilation of a Line: Center on the Line:

Students are asked to graph the image of two points on a line after a dilation using a center on the line and to generalize about dilations of lines when the line contains the center.

Dilation of a Line: Factor of One Half:

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Dilation of a Line: Factor of Two:

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Reflect a Semicircle:

Students are asked to reflect a semicircle across a given line.

Repeated Reflections and Rotations:

Students are asked to describe what happens to a triangle after repeated reflections and rotations.