The Concept of Transformations in a High School Geometry Course A workshop prepared for the Rhode Island Department of Education by Monique Rousselle Maynard.

The Concept of Transformations in a High School Geometry CourseA workshop prepared for the

Rhode Island Department of Education

by Monique Rousselle [email protected]

Transformations

“Means to an end.” Hung-Hsi Wu

GoalsExperience the transition from a hands-on and concrete experience with transformations to a more formalized and precise experience with transformations in a high school Geometry course.

Write precise definitions for Rotation, Reflection, and Translation. Distinguish the Properties of the Rigid Transformations.

Transformation Progression

Focus is on translation, reflection, rotation, and dilation.

Middle School High School Informal Formal Hands-on Definitions Descriptive Functions on the Plane*

Transformations of a function from plane to plane.

Rotation: in the Real World

Preliminary Notion of Rotation

https://tube.geogebra.org/student/m50296

How do I rotate a figure around a point?http://www.youtube.com/watch?v=U4Hv494HwrQDebrief.Q: What information does one require in order to perform a rotation?A: pre-image, center of rotation, degree (angle) of rotation, direction of rotation (in some cases)Q: What tools are required?A: paper, protractor, straightedge, pencil, colored pencils

A: the angle of rotation, direction of rotation, labels on points of the pre-image and image, and compass work

Q: If, on an assessment, you were asked to rotate a geometric shape, what evidence should you provide to support the location of the image?

RotationSketch the image of the figure L after it is rotated 60 counterclockwise around point O (optionally denoted RoO, 60).

Resources Handout with figure and point O on it Compass Protractor Pencil

Develop a Precise Definition of RotationIn developing a precise definition of rotation, we need to consider the effects of various centers of rotation, various degrees of rotation, and different directions of rotation. Thus, we will conduct a guided investigation to inform our definition.

Rotation Guided InvestigationDefinition Jigsaw (Poster Paper)

Precise Definition: Rotation

The rotation Ro of t degrees (180 t 180) around a given point O, called the center of the rotation, is a transformation of the plane.

Given a point P, the point Ro(P) is defined according to the following conditions.

The rotation is counterclockwise or clockwise depending on whether the degree is positive or negative, respectively.

For definiteness, we first deal with the case where 0 t 180.

If P = O, then by definition, Ro(O) = O.

If P is distinct from O, then by definition, Ro(P) is the point Q on the circle with center O and radius |OP| such that |mQOP| = t and such that Q is in the counterclockwise direction of the point P. We claim that this assignment is unambiguous (i.e., there cannot be more than one such Q).

If t = 180, then Q is the point on the circle so that is a diameter of the circle.

If t = 0, then Q = P; and Ro is the identity transformation I of the plane.

Hence, if 0 < t < 180, then all the Q's in the counterclockwise direction of the point P with the property 0 < |mQOP| < 180 lie in the fixed half-plane of that contains Q

Thus Ro is well-defined, in the sense that the rule of assignment is unambiguous.

Now, if t < 0, then by definition, we rotate the given point P clockwise on the circle that is centered at O with radius |OP|. Everything remains the same except that the point Q is now the point on the circle so that |mQOP| = |t| and Q is in the clockwise direction of P.

Thus, we define Ro(P) = Q.

Reflection: in the Real World

Preliminary Notion of Reflectionhttp://www.harpercollege.edu/~skoswatt/RigidMotions/reflection.html

Optional Additional Support: Performing a Reflection

http://www.youtube.com/watch?v=nAt212f6Uds

Develop a Precise Definition of ReflectionIn developing a precise definition of reflection, we need to distinguish between the reflections of points that do and do not lie on the line of reflection.

Use Reflection Definition-Writing ActivityMaterials: Patty Paper, Straightedge, Pencil

Precise Definition: Reflection

The reflection R across a given line l, where l is called the line of reflection, assigns to each point on line l, the point itself,

and to any point P not on line l, the point

R(P) that is symmetric to it with respect to line l, in the sense that line l is the perpendicular bisector of the segment joining P to R(P).

Translation: in the Real World

Preliminary Notion of Translation

http://tube.geogebra.org/material/show/id/18530

Performing a Translation Along a Vector

Translation with Patty Paper: http://www.youtube.com/watch?v=aPo0X6u_W-I&list=PLl4sjkH9L9JBu1dG4UkAwdogsxMDB2VFe

Develop a Precise Definition of Translation Along a VectorIn developing a precise definition of Translation we need to distinguish between a vector with and without length.

Translation Along a Vector Activity

Precise Definition: Translation

The translation T along a given vector assigns the point D to a given point C.Let the starting point and endpoint of be A and B, respectively. Assume C does not lie on . Draw the line l parallel to passing through C.* The line passing through B and parallel to then intersects line l at a point D; we call the line .**

By definition, T assigns the point D to C; that is, T(C) = D.

If C lies on , then the image D is by definition the point on to such that the direction from C to D is the same direction as from A to B such that |CD| = |AB|.If the vector is , the zero vector (i.e., the vector with zero length), then the translation along is the identify transformation I.

Rigid vs. Non-RigidTransformations:

What is the Difference?

Properties of Isometries

Rotations, Reflections, and Translations: Map lines to lines, rays to rays, and

segments to segments. Are distance-preserving. Are degree-preserving.

Connecting Today’s Workto the CCSSDuring the course of today’s session, our activities have connected with several CCSS for Geometry and Mathematical Practices.

Can you identify the standards and briefly explain the connection.

Teacher Resources

Illustrative Math Activity:Defining Rotations (G-CO.A.4)

*Alternative: Provide students with these definitions and ask them to critique their accuracy. MP3

http://www.illustrativemathematics.org/standards/hs

Illustrative Math Activity:Defining Reflections (G-CO.A.4)

http://www.illustrativemathematics.org/standards/hs

Properties of Rotations, Reflections, and Translations

Activities can be similarly developed that will lead students to visualize or develop the properties of the individual rigid transformations.

Properties of Rotations

The distance of a point on the pre-image from the center of rotation is equal to the distance of its corresponding point on the image from the center.

** Although demonstrated to be the most difficult transformation for students, it has been observed that spatial imagery cognitive style can significantly improve performance in rotation tasks (Xenia & Demetra, 2009).

Activity. An opportunityto demonstrate your understandingof the properties of rotation.

Analytic Activity. Find the coordinates of the image of the triangle after a 90 clockwise rotation about the point (3, 5).

Properties of Reflections

A reflection is a transformation of a plane having the following properties: The line joining the pre-image and corresponding

image is perpendicular to the line of reflection (which is a perpendicular bisector of the line joining any two corresponding points).

Any point on the reflected pre-image is the same distance as its corresponding image point from the line of reflection.

All points on the line of reflection are unchanged or are not affected by the reflection.

The pre-image and the image are oppositely congruent to each other.

Activity. An opportunityto demonstrate one’s understandingof the properties of reflection.

Graphical Activity. Draw the image of the triangle, given as follows, under a reflection about the liney = 4. y

4

x

Properties of Translations

http://www.ixl.com/math/geometry http://nrich.maths.org/5457 http://nrich.maths.org/public/leg.php?co

de=130

Activity. An opportunityto demonstrate one’s understandingof the properties of translation.

Algebraic Activity.The vertices of a triangle are A(4, 1), B(2, 1), and C(4, 5). If ABC is translated by vector , find the coordinates of the vertices of its image.

**In a study carried out by Xenia & Demetra (2009) it emerged that students perform better in translation tasks than the other types.

Representative CCSS Vocabulary for HS Geometry Algebraic Alternate interior

angles Arc Area Base angles Central angle Chords Circle Circumference Circumscribed angle Collinear Compass Complete the square Cone Congruent Constructions Coordinate geometry Coordinate plane Coplanar Corresponding sides Corresponding angles Cross-section Cylinder Derive Diagonal

Dilation Directrix Distance formula Distinct Endpoint Equidistant Equilateral triangle Experiment Focus Geometric Inscribed angles Interior angle Interpret Isometry Isosceles triangle Line Line segment Locus Median Midpoint Parallel Parallelogram Perimeter Perpendicular Perpendicular

bisector

Plane Point Preserve angle Preserve distance Proof Proportion Pythagorean

Theorem Radian Radii Ratio Rectangle Reflection Regular hexagon Regular polygon Rigid motion Rotation Scale factor Sector Sequence Skew Slope Solution Square Sphere Straightedge

Symmetry Tangent Theorem Three dimensional Transformation Translation Transversal Trapezoid Triangle Triangle congruence Trig ratios Two dimensional Undefined Vector Vertical angles Volume xy-Coordinate axis