The Concept of Transformations in a High School Geometry Course - PowerPoint Presentation

Slide1

The Concept of Transformations in a High School Geometry Course

A workshop prepared for the Rhode Island Department of Education

by

Monique

Rousselle

Maynard

momaynard86@gmail.comSlide2

Transformations

“

Means to an end

.”



Hung-Hsi

WuSlide3

Goals

Experience 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.Slide4

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.Slide5
Slide6

Rotation: in the Real WorldSlide7

Preliminary Notion of Rotation

https://

tube.geogebra.org/student/m50296

Slide8

How do I rotate a figure around a point?

http://

www.youtube.com/watch?v=U4Hv494HwrQ

Debrief.

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 pencilsSlide9

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?Slide10

Rotation

Sketch the image of

t

he figure L after it is rotated 60

 counterclockwise around point

O (optionally denoted RoO, 60

).ResourcesHandout with figure and point

O

on it

Compass

Protractor

PencilSlide11

Develop a Precise Definition of Rotation

In 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 Investigation

Definition Jigsaw (Poster Paper)Slide12

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 |mQOP| = 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 < |mQOP| < 180 lie in the fixed half-plane of that contains QThus 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 |mQOP| = |t| and Q is in the clockwise direction of P.Thus, we define Ro(P) = Q.

 Slide13

Reflection: in the Real WorldSlide14

Preliminary Notion of Reflection

http://www.harpercollege.edu/~

skoswatt/RigidMotions/reflection.htmlSlide15

Optional Additional Support: Performing a Reflection

http://

www.youtube.com/watch?v=nAt212f6UdsSlide16

Develop a Precise Definition of Reflection

In 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 Activity

Materials: Patty Paper, Straightedge, PencilSlide17

Precise Definition: Reflection

The

reflection

R

across a given line

l, where l is called the

line of reflection, assignsto 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

)

.Slide18

Translation: in the Real WorldSlide19

Preliminary Notion of Translation

http://

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

Slide20

Performing a Translation Along a Vector

Translation with Patty

Paper:

http

://

www.youtube.com/watch?v=aPo0X6u_W-I&list=PLl4sjkH9L9JBu1dG4UkAwdogsxMDB2VFeSlide21

Develop a Precise Definition of Translation Along a Vector

In developing a precise definition of Translation we need to distinguish between a vector with and without length.

Translation Along a Vector Activity

Slide22

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. Slide23

Rigid vs. Non-RigidTransformations:

What is the Difference?Slide24

Properties of Isometries

Rotations, Reflections, and Translations:

Map lines to lines, rays to rays, and segments to segments.

Are distance-preserving.

Are degree-preserving.Slide25

Connecting Today’s Workto the CCSS

During 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.Slide26

Teacher ResourcesSlide27

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/hsSlide28

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

http

://

www.illustrativemathematics.org/standards/hsSlide29

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.Slide30

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).Slide31

Activity. An opportunity

to demonstrate your understanding

of 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).Slide32

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.Slide33

Activity. An opportunity

to demonstrate one’s understanding

of the properties of reflection.

Graphical Activity.

Draw the image of the triangle, given as follows, under a reflection about the line

y = 4. y 4

x

Slide34

Properties of Translations

http://www.ixl.com/math/geometry

http://nrich.maths.org/5457

http://nrich.maths.org/public/leg.php?code=130Slide35

Activity. An opportunity

to demonstrate one’s understanding

of 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.

 Slide36

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

EquidistantEquilateral triangleExperimentFocusGeometricInscribed anglesInterior angle InterpretIsometryIsosceles triangleLineLine segmentLocusMedianMidpointParallelParallelogramPerimeterPerpendicularPerpendicular bisectorPlanePointPreserve anglePreserve distanceProofProportionPythagorean TheoremRadianRadiiRatioRectangleReflectionRegular hexagonRegular polygonRigid motionRotationScale factorSectorSequenceSkewSlopeSolutionSquare

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