Transformations5-Day Unit Plan

10TH Grade

byCarrie J. Magee

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Table of Contents

Unit Objectives………………………………………………………………..3

NCTM Standards……………………………………………………………..3

NYS Standards………………………………………………………………..3

Resources……………………………………………………………………..3

Materials………………………………………………………………………3

Day 1: Rigid Motion in a Plane………………………………………………4

Day 2: Reflections……………………………………………………………10

Day 3: Rotations………………………………………………………………14

Day 4: Translations and Vectors………………………………………………18

Day 5: Glide Reflections and Compositions…………………………………..24

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Unit Objectives- Students will be able to:• identify types of rigid transformations.• use properties of reflections.• relate reflections and line symmetry.• relate rotations and rotational symmetry.• use properties of translations and glide reflections.

NCTM Standards for Grades 9-12

• Geometry Standard• Communication Standard• Connections Standard• Representation Standard

NYS Standards for Math A

• Key Idea 3-Operations• Key Idea 4-Modeling/Multiple Representation

Resources

• Textbook, Geometry by McDougall-Littell, 2001

Materials

• Compass, protractor & straight edge• Mira• Geometer’s Sketchpad

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Day 1: Rigid Motion in a Plane

Objective: Students will be able to:• identify the three basic rigid transformations.• identify isometries.• use transformations in real-life.

MaterialsMiras, overhead projector, transparencies

Opening Activity- Have students copy definitions of image, preimage, transformation andisometry.

Activity #1- Discuss the above definitions. Discuss the three basic rigid transformations-reflections, rotations and transformations.

Activity #2- Had out worksheet #1. Using the mira, have the students reflect ABC overthe y-axis and then answer the questions. Discuss the answers to the questions.

Activity #3- Discuss rigid transformations. Have the students decide which ones of thefollowing are isometries.A.)

B.)

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C.)

A and C are isometries while B is not. Explain why this is true. This is because the anglemeasure and lengths are preserved.

Activity #4- Identifying and using transformations in real-life. Discuss examples 4 and 5in the book on page 398. Have the students think of at least one other transformation.

Closing Activity- Investigate the second example on worksheet #1.

Homework- pp. 399-400; 12-17, 21-25, 26-30 even, 34, 35

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Rigid Motion in a Plane

Image- the new figure that results from the transformationof a figure in a plane.

Preimage- the original figure in the transformation of afigure in a plane.

Transformation- the operation that maps, or moves, apreimage onto an image.

Isometry- a transformation that preserves lengths.

Three Basic Transformations

1. Reflections- a type of transformation that uses a line thatacts like a mirror, called a line of reflection, with animage reflected in the line.

2. Rotations- a figure is turned about a fixed point, called acenter of rotation.

3. Translations- maps every two points P and Q in theplane to P` and Q` so that the following two propertiesare true:

• PP` = QQ`.• PP` || QQ`, or line segments PP` and QQ` are

collinear.

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Worksheet #1

1. Name and describe the transformation.

2. Name the coordinates of the vertices of the image.

3. Is DABC congruent to its image? Why or whynot?

C'E'

D'D

C E

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m

nF

G

E

H

C

B

D

A

4. Describe the motion that moves ABCD onto EFGH.

5. Reflect EFGH over line m and name thecorresponding vertices of the new figure JKLM. IsEFGH congruent to JKLM?

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m

n

M

L

K

J

F

G

E

H

C

B

D

A

6. Describe the motion that maps ABCD onto JKLM.Is ABCD congruent to JKLM?

7. Can one “flip” be used to move ABCD ontoJKLM? Explain why or why not?

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Day 2: Reflections

Objective- Students will be able to:• identify and use reflections in a plane.• identify relationships between reflections and line symmetry.

Materials- Miras, graph paper, overhead projector, transparencies.

Opening Activity- Have the students copy definitions of reflection and line of reflection.Discuss the properties of a reflection.

Activity #1- Have the students graph H(2,2) and G(5,4). They should then reflect H in thex-axis and G in the line y=4. Then discuss the properties of reflections in the coordinateaxes.

Activity #2- Explain Theorem 7.1 and then prove it. The proof is in the book on page 405example 2.

Closing Activity- Define line of symmetry. Have the students use the miras to find thelines of symmetry in the figures on worksheet #2.

Homework- pp. 407-409; 15-17, 22-29, 31-33, 48, 49

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Reflections

Reflection- a type of transformation that uses a line whichacts like a mirror, with an image reflected in the line.

Line of Reflection- the mirror line.

A reflection maps every point P in the plane to a point P`,so that the following properties are true:• If P is not on m, then m is the perpendicular bisector of

PP`.• If P is on m, then P = P`.

P`P

m

P`

mP

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Reflections in the coordinate axes have the followingproperties:• If (x,y) is reflected in the x-axis, its image is the point (x,-

y).• If (x,y) is reflected in the y-axis, its image is the point (-x,y).

Theorem 7.1 Reflection TheoremA reflection is an isometry.

Line of Symmetry- a line that a figure in the plane has ifthe figure can be mapped onto itself by a reflection in theline.

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G` y=4

H`

H

G

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Worksheet #2

Using the Mira, find all line of symmetry of the following figures.

1.) 2.)

3.)

4.) 5.)

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Day 3: Rotations

Objectives- Students will be able to:• identify rotations in a plane.• identify rotational symmetry.• use Geometer’s Sketchpad to reflect and rotate images.• use a compass and protractor to rotate a figure.

Materials- compasses, protractors, computers equipped with Geometer’s Sketchpad,overhead projector, transparency.

Opening Activity- Using Geometer’s Sketchpad, have the students construct a scalenetriangle, then label the vertices A, B, and C. Next draw two lines that intersect but do notintersect the triangle. Label the lines k and m. Label the point of intersection P. ReflectDABC in line k to obtain DA`B`C`. Reflect DA`B`C` in line m to obtain DA``B``C``. Howis DABC related to DA``B``C``?

Activity #1- Have students copy definitions of rotation, center of rotation, and angle ofrotation. Discuss the above definitions.

Activity #2- Rotate a figure using a compass and protractor. First draw a triangle ABCand a point P. Then draw a segment connecting vertex A and the center of rotation point P.Next use a protractor to measure a 60 degree angle counterclockwise and draw a ray. Thenplace the point of the compass at P and draw an arc from A to locate A`. Repeat these stepsfor each vertex. Connect the vertices to form the image. Label the image A`B`C`.

Activity #3- Rotations in a coordinate plane. Using Geometer’s Sketchpad, draw a figureusing the points A(2,-2), B(4,1), C(5,1), and D(5,-1). Rotate the figure 90 degrees counterclockwise about the origin. What are the coordinates of the new vertices? A`(2,2), B`(-1,4),C`(-1,5) and D`(1,5).

Closing Activity- Discuss rotational symmetry. Have students identify rotationalsymmetry.

Homework- pp.416-417; 1-5, 13-19, 20, 22-27.

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Rotations

Rotation- a transformation in which a figure is turnedabout a fixed point.

Center of Rotation- the fixed point.

Angle of Rotation- the angle formed when rays are drawnfrom the center of rotation to a point and its image.

A rotation about a point P through x degrees is atransformation that maps every point Q in to plane to apoint Q`, so that the following properties are true:• If Q is not point P, then QP = Q`P and m–QPQ`=x°.• If Q is point P, then Q = Q`.

This transformation can be described as (x,y)Æ(-y,x).

m

p

kB''

C''

A''

C'A'

B'B

AC

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Rotational Symmetry- the figure can be mapped ontoitself by a rotation of 180 or less.

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D`

D

C

A'

C

B'

A

B

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Worksheet #3

Which of the following have rotational symmetry and why?

A.) B.) C.)

A can be mapped onto itself by a clockwise or counterclockwise rotation of45°, 90°, 135°, or 180° about its center so therefore it has rotationalsymmetry.

B can be mapped onto itself by a clockwise or counterclockwise rotation of180° about its center so therefore it also has rotational symmetry.

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Day 4: Translations and Vectors

Objectives- Students will be able to:• identify and use translations in the plane.• identify vector components.• find vectors.

Materials- overhead projector, transparencies, Geometer’s Sketchpad.

Opening Activity- Define translation. Discuss Theorem 7.5.

Activity #1- Using Geometer’s Sketchpad, sketch a triangle with vertices A(-1,-3), B(1,-1), and C(-1,0). Then sketch the image of the triangle after the translation(x,y)Æ(x-3,y+4). Shift each point 3 units to the left and 4 units up. The translated verticesshould be A`(-4,1), B`(-2,3), and C`(-4,4).

Activity #2- Define vector, initial point, terminal point and component form. Havestudents identify vector components.

Closing Activity- Perform a translation using vectors.

Homework- pp.425-426; 15-30.

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Translations and Vectors

Translation- a transformation that maps every two pointsP and Q in a plane to points P` and Q`, so that thefollowing properties are true:• PP` = QQ`• PP` || QQ` or PP` and QQ` are collinear.

Theorem 7.5- If lines k and m are parallel, then a reflectionin line k followed by a reflection in line m is a translation. IfP`` is the image of P, then the following are true:• PP`` is perpendicular to k and m.• PP`` =2d, where d is the distance between k and m.

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Activity #1

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A

B`

C`

A

B

C

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Vector- quantity that has both direction and magnitude.

Initial point- starting point.

Terminal point- ending point.

Component form- a vector combines the horizontal andvertical components.

Identify the vector components.

A.)6

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K

J

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B.)

C.)

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x

N

M

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S

T

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The component form of GH is <4,2>. Use GH to translatethe triangle whose vertices are A(3,-1), B(1,1) and C(3,5).

First graph ABC. GH is <4,2> so the vertices should beshifted to the right 4 units and up 2 units. Label thevertices A`, B`, and C`.

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C`

B`

A`

C

B

A

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Day 5: Glide Reflections and Compositions

Objectives- Students will be able to:• identify glide reflections in a plane.• represent transformations as compositions of simpler transformations.

Materials- Geometer’s Sketchpad, overhead projector.

Opening Activity- Define glide reflection. Use the following information to find theimage of a glide reflection.

A(-1,-3), B(-4,-1), and C(-6,-4)Translation: (x,y)Æ(x+10,y)Reflection: in the y-axis

Activity #1- Define compostion. Find the image of a composition using the following:P(2,-2), Q(3,-4)Rotation: 90 degrees counterclockwise about the origin.Reflection: in the y-axis.

Closing Activity- Repeat Activity #1 but reverse the order. Perform the reflection firstand then the rotation. What do you notice?

Homework- pp. 433-434; 9-21.

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Glide Reflections and Compositions

Glide Reflection- a transformation in which every point Pis mapped onto a P`` by the following steps:• a translation maps P onto P`.• a reflection in a line k parallel to the direction of the

translation maps P` onto P``.

Composition- when two or more transformations arecombined to produce a single transformation.

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C''

A''

B''

A'

C'

B'

A

B

C

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Activity #16

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Q``

P``

Q`

P`

Q

P

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Closing Activity

The order which the transformations are performed affectsthe final image.

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Q``

P``

Q`

P`

Q

P