# Cluster 1: Experiment with transformations in the plane. (Geometry - Supporting Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.G-CO.1
Title: Experiment with transformations in the plane. (Geometry - Supporting Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Geometry: Congruence

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.G-CO.1.AP.3a
Describe the rotations and reflections of a rectangle, parallelogram, trapezoid, or regular polygon that maps each figure onto itself.
MAFS.912.G-CO.1.AP.5a
Transform a geometric figure given a rotation, reflection, or translation using graph paper, tracing paper, or geometric software.
MAFS.912.G-CO.1.AP.1a
Identify precise definitions of angle, circle, perpendicular line, parallel line and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
MAFS.912.G-CO.1.AP.2a
Represent transformations in the plane using, e.g., transparencies and geometry software.
MAFS.912.G-CO.1.AP.2b
Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
MAFS.912.G-CO.1.AP.4a
Using previous comparisons and descriptions of transformations, develop and understand the meaning of rotations, reflections, and translations based on angles, circles, perpendicular lines, parallel lines, and line segments.
MAFS.912.G-CO.1.AP.5b
Create sequences of transformations that map a geometric figure on to itself and another geometric figure.

## Related Resources

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

## Educational Software / Tools

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

Free Graph Paper:

A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Type: Educational Software / Tool

## Formative Assessments

Reflect a Semicircle:

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

Type: Formative Assessment

Two Triangles:

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

Type: Formative Assessment

Rotation of a Quadrilateral:

Students are asked to rotate a quadrilateral around a given point.

Type: Formative Assessment

Indicate the Transformations:

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

Type: Formative Assessment

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

Demonstrating Rotations:

Students are asked to rotate a quadrilateral 90 degrees clockwise.

Type: Formative Assessment

Define a Reflection:

Students are asked to develop a definition of reflection.

Type: Formative Assessment

Definition of Line Segment:

Students are asked to draw, label, and give a precise definition of a line segment.

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

Definition of a Circle:

Students are asked to draw, label and give a precise definition of the term circle.

Type: Formative Assessment

Definition of Parallel Lines:

Students are asked to draw, label, and give a precise definition of parallel lines.

Type: Formative Assessment

Definition of Perpendicular Lines:

Students are asked to draw, label, and give a precise definition of perpendicular lines.

Type: Formative Assessment

Definition of an Angle:

Students are asked to draw, label, and give a precise definition of an angle.

Type: Formative Assessment

Transformations And Functions:

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

Type: Formative Assessment

Comparing Transformations:

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

Type: Formative Assessment

Demonstrating Translations:

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

Type: Formative Assessment

Demonstrating Reflections:

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

Type: Formative Assessment

## Lesson Plans

Coding Geometry Challenge #23 & 24:

This set of geometry challenges focuses on using transformations to show similarity and congruence of polygons and circles. Students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor.

Type: Lesson Plan

Coding Geometry Challenge # 12 & 13:

This set of geometry challenges focuses on creating circles and calculating area/circumference as students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor

Type: Lesson Plan

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

Basic Definitions in Geometry:

A set of basic definitions in geometry (line segment, ray, angle, perpendicular lines, and parallel lines) is addressed. The notation used in naming each defined term is also emphasized.

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 see whether the triangle ends up at the same location or not.

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

Musical Chairs with Words and a Ball:

This lesson introduces students to concepts and skills that they will use throughout the year. Students will learn that the terms point and line are considered "undefined." Students will play musical chairs while learning to develop precise definitions of circle, angle, parallel line, and perpendicular line, using counterexamples at different classroom stations. Students will identify models, use notation, and make sketches of these terms.

Type: Lesson Plan

Isometries with GeoGebra:

In this lesson, students explore transformations with GeoGebra 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 in order 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 that will find the first angle of rotation of any regular polygon that would map that polygon onto itself by using premade examples from the program Geogebra.

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.

Type: Lesson Plan

Product of Two Transformations:

Students will identify a sequence of two-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

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.

Type: Lesson Plan

Pick Your Words Precisely!:

Students will develop precise definitions of the terms angle, circle, line segment, parallel line and perpendicular line by completing a graphic organizer under the Frayer model through small and whole group discussion.

Type: Lesson Plan

I Am Still Me Transformed.:

This lesson is intended for students to explore several ways of applying, identifying, and describing reflection and rotation symmetry for both geometric and real-world objects, in order 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

Transformations... Geometry in Motion:

Transformations... Geometry in Motion is designed for students to practice their knowledge of transformations. Students will represent transformations in the plane, compare transformations, and 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 reasoning, as well.

Type: Lesson Plan

Reflections Hands On:

Students will use a protractor/ruler to construct reflections and 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

Sage and Scribe - Points, Lines, and Planes:

Students will practice using precise definitions and words while they describe and draw images of Points, Lines, and Planes. Students will work in pairs taking turns describing an image while their partner attempts to accurately draw the image.

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

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

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 work with transformations to move a triangle through a maze. The activity has an honors level and standard level application. It requires students to perform rotations, translations, and reflections.

Type: Lesson Plan

Building Blocks of Geometry:

Students will be introduced to the undefinable concepts of points, lines, and planes that are the building blocks of geometry and recognize that these three terms become the basis for many other geometric definitions. Students will participate in a Building Block Scavenger Hunt, using cameras to photograph examples of specified terms that they find outside of the math classroom. The students will compose a power point to display their photographs of the required terms.
This lesson is adapted from a presentation that included an activity by Dianne Olix, 1995.

Type: Lesson Plan

Virtually Possible:

This is a ray drawing activity to aid students in their understanding of how virtual images are formed by plane mirrors, and how the image size and distance from the mirror compare to those of the object.

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

## Perspectives Video: Experts

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Knot Theory Entangled in Cellular Biology:

This FSU professor describes how knot theory and cellular biology are intertwined. Researchers are still trying to determine how enzyme bridges are able to un-knot long strands of DNA to mitigate potential cell destruction.

Type: Perspectives Video: Expert

The Geometry of DNA Replication:

A discussion of the applications of Knot Theory, replication of DNA, enzymes, and fluid dynamics.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Reflections, Rotations, and Translations with Additive Printing:

Transform your understanding of 3D modeling when you learn about how shapes are manipulated to arrive at a final 3D printed form!

Type: Perspectives Video: Professional/Enthusiast

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Seven Circles II:

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

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

## Tutorials

Developing a Definition of a Reflection:

Click "View Site" to open a full-screen version. In this tutorial, the development of a formal definition of a reflection is presented.

Type: Tutorial

Developing a Definition of a Rotation:

Click "View Site" to open a full-screen version. In this tutorial, the development of a formal definition of a rotation is presented.

Type: Tutorial

Developing a Definition of a Translation:

Click "View Site" to open a full-screen version. In this tutorial, the development of a formal definition of a translation is presented.

Type: Tutorial

Rotational Symmetries of Regular Polygons:

Click "View Site" to open a full-screen version. The world contains many examples of figures with rotational symmetry. Some of these occur in art, others occur in nature. Knowledge of rotational symmetry can be helpful in understanding both the human and the natural world. In this tutorial, rotations that carry a regular polygon onto itself are explored and suggestions for teaching this topic are offered.

Type: Tutorial

Rotating polygons 180 degrees about their center:

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Line of reflection:

Students are shown, with an interactive tool, how to reflect a line segment. Students should have an understanding of slope and midpoint before viewing this video.

Type: Tutorial

Line of reflection:

This tutorial uses the midpoint of two lines to find the line of reflection.

Type: Tutorial

Points after rotation:

Students will see what happens when a figure is rotated about the origin -270 degrees. Having a foundation about right triangles is recommended before viewing this video.

Type: Tutorial

Specifying planes in three dimensions:

In this tutorial, students are introduced to the concept that three non-collinear points are necessary to define a unique plane.

Type: Tutorial

The language of geometry:

Before learning any new concept it's important students learn and use common language and label concepts consistently. This tutorial introduces students to th point, line and plane.

Type: Tutorial

Identifying parallel and perpendicular lines:

This tutorial is great practice for help in identifying parallel and perpendicular lines.

Type: Tutorial

Basic Geometry Language and Labels:

In this tutorial we will learn the basics of geometry, such as identifying a line, ray, point, and segment.

Type: Tutorial

## Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

## Virtual Manipulative

Combining Transformations:

In this manipulative activity, you can first get an idea of what each of the rigid transformations look like, and then get to experiment with combinations of transformations in order to map a pre-image to its image.

Type: Virtual Manipulative

## Student Resources

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

## Educational Software / Tool

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

## Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Expert

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Seven Circles II:

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

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

## Tutorials

Rotating polygons 180 degrees about their center:

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Line of reflection:

Students are shown, with an interactive tool, how to reflect a line segment. Students should have an understanding of slope and midpoint before viewing this video.

Type: Tutorial

Line of reflection:

This tutorial uses the midpoint of two lines to find the line of reflection.

Type: Tutorial

Points after rotation:

Students will see what happens when a figure is rotated about the origin -270 degrees. Having a foundation about right triangles is recommended before viewing this video.

Type: Tutorial

Specifying planes in three dimensions:

In this tutorial, students are introduced to the concept that three non-collinear points are necessary to define a unique plane.

Type: Tutorial

The language of geometry:

Before learning any new concept it's important students learn and use common language and label concepts consistently. This tutorial introduces students to th point, line and plane.

Type: Tutorial

Identifying parallel and perpendicular lines:

This tutorial is great practice for help in identifying parallel and perpendicular lines.

Type: Tutorial

Basic Geometry Language and Labels:

In this tutorial we will learn the basics of geometry, such as identifying a line, ray, point, and segment.

Type: Tutorial

## Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

## Virtual Manipulative

Combining Transformations:

In this manipulative activity, you can first get an idea of what each of the rigid transformations look like, and then get to experiment with combinations of transformations in order to map a pre-image to its image.

Type: Virtual Manipulative

## Parent Resources

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

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.