GEOMETRY HOUSTON ISD PLANNING GUIDE 4

GEOMETRY HOUSTON ISD PLANNING GUIDE

4TH SIX-WEEKS

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned to Upcoming State Readiness Standard

- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)

© Houston ISD Curriculum 2012 – 2013 Page 1 of 4

Planning Guide User Information Unit 10: Transformations

Time Allocations

Unit 6 lessons (90-minutes each)

or 12 lessons (45-minutes each)

Unit Overview

Transformations – Students apply transformations to various geometric shapes and make conjectures about coordinate notation after a transformation. TEKS/SEs (district clarifications/elaborations in italics)

Ⓡ GEOM.2B Make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. Ⓢ GEOM.5C Apply properties of transformations: reflections, translations, rotations, and glide reflections to make connections between mathematics and the real world, such as tessellations. Ⓢ GEOM.7A Use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures. Ⓢ GEOM.10A Use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane. Ⓢ GEOM.11B Apply ratios to solve problems involving similar figures. English Language Proficiency Standards

ELPS C.1b Monitor oral and written language production and employ self-corrective techniques or other resources.

ELPS C.1e Internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment.

ELPS C.2d Monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed.

ELPS C.3e Share information in cooperative learning interactions.

ELPS C.3h Narrate, describe, and explain with increasing specificity and detail as more English is acquired.

ELPS C.5g Narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.

College and Career Readiness Standards

CCRS 3.A1 Identify and represent the features of plane and space figures.

CCRS 3.A2 Make, test, and use conjectures about one-, two-, and three-dimensional figures and their properties.

CCRS 3.B1. Identify and apply transformations to figures.

CCRS 3.B3 Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.

CCRS 10.B1 Use multiple representations to demonstrate links between mathematical and real world situations.

Key Concepts

composite transformation matrix

symmetry transformation

vector

Academic Vocabulary

apply rule

Content-Specific Vocabulary

image pre-image

GEOMETRY HOUSTON ISD PLANNING GUIDE

4TH SIX-WEEKS

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned to Upcoming State Readiness Standard

- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)

© Houston ISD Curriculum 2012 – 2013 Page 2 of 4

Essential Understandings / Guiding Questions

Conjectures regarding transformations may be expressed in coordinate notation for reflections, translations, rotations, and dilations.

1. How are various transformations similar to or different from each other? 2. Why is coordinate notation important for expressing transformations?

Composite transformations can be used to design tessellations for art and architecture. 1. What polygons can be used in tessellations and how are those polygons related? 2. How are tessellations used in art and architecture?

Assessment Connections

Performance Expectation o Students will use concrete objects and pictorial and verbal representations to model properties and attributes

with respect to transformations of geometric figures on a coordinate plane. o Students will express and compare the pre-image and image in coordinate notation, and communicate their

mathematical ideas using graphical, numerical, physical, algebraic, or verbal representation. Media Script -– students create a short story about transformations. The final product should include the story and a

graph that details the story. Formative Assessment – Transformation Dance – students participate in a modification of the “Function Dance.”

Student groups will use their arms to demonstrate geometric figures and properties. SpringBoard® Geometry – Embedded Assessment #1: “In Mutatio Nos Fides” STAAR Sample Item – Item #1 (GEOM.2B) and Item #5 (GEOM.5C)

Texas English Language Proficiency Assessment System (TELPAS): End-of-year assessment in listening, speaking, reading, and writing for all students coded as LEP (ELL) and for students who are LEP but have parental denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide five writing samples – one narrative about a past event, two academic (from science, social studies, or mathematics), and two others.

Instructional Considerations

Information in this section is provided to assist the teacher with the background knowledge needed to plan instruction that facilitates students to internalize the Key Concepts and Essential Understandings for this unit. It is recommended that teachers thoroughly read this section before implementing the strategies and activities in the Instructional Strategies section. Prerequisites and/or Background Knowledge for Students Transformations are consistent throughout the middle school geometry curriculum and Laying the Foundations (LTF). Students graphed and concretely explored these transformations with patty paper. (Ⓡ MATH.7.7B, Ⓡ MATH.8.6A) Background Knowledge for Teacher Critical Content Perform translations, reflections, and rotations on a coordinate plane; Perform composite transformations; Alternate methods for reflections and rotations.

Introduction

The introductory translation, reflection, and rotation activities in this unit all begin with a brief class discussion about transformations so that the teacher can assess students’ background knowledge (GEOM.5C).

Students have an increasingly stronger use of vocabulary and properties to show a more sophisticated use of words for communications as well a good knowledge of geometric tools. Use clarifying activities to develop visual understanding of transformations – see Resources. ( MATH.8.15A Communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models.)

GEOMETRY HOUSTON ISD PLANNING GUIDE

4TH SIX-WEEKS

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned to Upcoming State Readiness Standard

- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)

© Houston ISD Curriculum 2012 – 2013 Page 3 of 4

Instructional Considerations

Note word variations of “Engage, Explore, Explain, Elaborate, and Evaluate” that imply the 5E Lesson Model.

Instructional Accommodations for Diverse Learners The textbook shows an algebraic method using matrices for determining the image matrix that represents the translated or reflected polygon. Use concrete methods to enhance student learning. Use a graphing calculator to perform matrix multiplication – see Matrix Multiplication in Resources. (GEOM.2B, GEOM.7A) Extensions for Pre-AP Note that transformations from Laying the Foundation Geometry involve transforming piecewise functions instead of transforming geometric figures as in the McDougal-Littell textbook. Instructional Strategies / Activities

The strategies and activities in this section are designed to assist the teacher to provide learning experiences to ensure that all learners achieve mastery of the TEKS SEs for this unit. It is recommended that the strategies and activities in this section be taught in the order in which they appear.

Cooperative Learning To facilitate a large-group discussion while students are seated in small groups, establish the following guidelines: Each group is responsible for at least one response. This helps to establish a classroom culture where participation

in small-group and whole-class discussion is expected. One member of each group will serve as reporter. This encourages everyone to learn from his or her group

members. One person speaks at a time, whether it is a student or the teacher, to prevent inappropriate side conversations. Establish a culture of respect for everyone’s opinions. C.1b, C.2d, C.3e

For additional strategies for cooperative learning, see Marzano’s Handbook for Classroom Instruction That Works. Summarizing and Note-Taking

KWL and Think-Pair-Share (Turn The Light On) The clarifying activities Translations, Translations on the Coordinate Plane, Triangle Reflections, and Rotations

engage students with a brief class discussion about transformations so the teacher can assess existing knowledge. Have groups or pairs “popcorn” what information they remember from each transformation. From the discussion, write any questions that students may want answered or clarified.

For instance, students may receive help with reading translation notation and drawing translations in Translations on the Coordinate Plane. This may also be the first opportunity that students encounter vectors. For students having difficulty visually completing the transformation, have patty paper available to trace the pre-image and physically complete the indicated transformation. Use the activity notes to assist with delivery and scaffolding questions. Keep chart papers listing properties and ideas for transformations for future reference. (SpringBoard® Mathematics with Meaning: Geometry, Activity 5.1 “Transformations on the Coordinate Plane) (GEOM.5C, GEOM.7A, GEOM.10A)

MATH.8.14A Identify and apply mathematics to everyday experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics.

Instructional Accommodations for Diverse Learners Students write a paragraph on how to use coordinate notation and other tools to perform transformations.

(SpringBoard® Mathematics with Meaning: Geometry, Activity 5.7 “Transformations with Matrices”) C.1e, C.3h, C.5g

For additional activities on basic transformations, use Critical Thinking Activities in Patterns, Imagery, and Logic – see Resources.

For additional strategies to assist diverse learners, access Recommendations for Accommodating Special Needs Students: Geometry, Cycle 4, Unit 10.

GEOMETRY HOUSTON ISD PLANNING GUIDE

4TH SIX-WEEKS

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned to Upcoming State Readiness Standard

- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)

© Houston ISD Curriculum 2012 – 2013 Page 4 of 4

Instructional Considerations

Formative Assessment

For a kinesthetic activity, use Formative Assessment Transformation Dance to review students on functional transformations studied in Algebra I.

Elaborate and have student teams of three or four to demonstrate ways to show any figure and any transformation on demand. On the classroom floor, superimpose a grid with tape for the coordinate plane.

Once students have mastered a single transformation, elaborate by having the students complete composite transformations. In a composite transformation, a student will complete two types of transformations such as a vertical shift and a reflection.

Extensions for Pre-AP Students explore transformations using piecewise functions in the LTF activity “Transformations of Piecewise Functions,” and apply vectors in “Vectors in Geometry.” Resources

Adopted Instructional Materials McDougal-Littell, Geometry: “Translate Figures and Use

Vectors,” pp. 572 – 575 “Use Properties of Matrices,” pp.580

– 587 “Investigating Geometry: Reflections

in a Plane,” p. 588 “Perform Reflections,” pp. 589 – 592 “Perform Rotations,” pp. 598 – 601 “Using Alternative Methods,” p. 606

Laying the Foundation, Connecting Geometry: “Transformation of Piecewise

Functions,” p. 92 “Vectors in Geometry,” pp 220 –

222 (1–2b); p. 224 (1–2)

SpringBoard® Mathematics with

Meaning: Geometry 5.1 “Transformations on the

Coordinate Plane” 5.7 “Transformations with Matrices”

Supporting Resources

Translations Translations on the Coordinate

Plane Triangle Reflections Rotations Recommendations for

Accommodating Special Needs Students: Geometry, Cycle 4, Unit 10

Contributing Lesson:

Transformation Lesson Plan and PowerPoint

Matrix Multiplication

Dale Seymour, Critical Thinking Activities in Patterns, Imagery, and Logic

“Reflections – Reflect on This,” p. 76

“Line of Symmetry Mirror Images,” p. 65

“Rotation – It’s Your Turn,” p. 95

“Scale Drawings,” p. 69

Professional Texts

Pearson, Handbook for Classroom Instruction That Works

Formative Assessment: Transformation Dance for Geometry

In groups, students can demonstrate geometric figures using their arms and bodies. Students may serve as vertices and their arms sides of the figure. If your classroom has tiled floors, use the tile floors as a giant coordinate plane. Superimpose a grid on the floor using masking tape. Either make index cards with specific figures and coordinates of the figures’ vertices or have student teams challenge other student teams to demonstrate that figures. Write the transformations that will fit the size of your classroom or open-area. As a written portion of the assessment, ask students to sketch the original geometric figure than the transformation on graph paper. Name the new coordinates of the figure. At the end, students write a summary statement of generic rules or ideas that they discovered in the transformation of the geometric figures. Students work may be evaluated with the table provided or with the Region IV Student Rubrics that are attached to this document.

Transformation Dance for Geometry Observations and Summary

Figure Coordinates Shifts Reflection Dilation New

CoordinatesTriangle A(0,0), B(3,4),

C(6,0)

Square A(0,0), B(0,3), C(3,3), D(3,0)

Rectangle E(0,0), F(0, 5), G(3, 5), H(3,0)

Parallelogram E(0,0), F(2,3) G(7,3), H(5,0)

Student Summary:

These released questions represent selected TEKS student expectations for each reporting category. These questions are samples only and do not represent all the student expectations eligible for assessment.

Copyright © 2011, Texas Education Agency. All rights reserved. Reproduction of all or portions of this work is prohibited without express written permission from the Texas Education Agency.

2011 Released Test Questions

Geometry

State of Texas Assessments of

Academic Readiness

STAARTM

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 2

1 The figure below was formed by joining 2 segments of equal length at common endpoint Y .

X

Y

Z

If points X , Y , and Z are non-collinear , which of the following statements regarding XZ must always be true?

A XZ І XY

B XZ І 2( XY)

C XZ > 2( XY)

D XZ < 2( XY )

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 3

2 A geometry student concluded:

If two sides and a non-included angle of one triangle are congruent to two sides and a non-included angle of another triangle, then the two triangles are congruent.

Which diagram can be used as a counterexample to the student’s conclusion?

A

B

C

D

STAAR Geometry 2011 ReleaseReleased Test Questions

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3 Which set of statements represents a valid deductive argument?

A All quadrilaterals have 4 angles. All parallelograms have 4 angles. All quadrilaterals are parallelograms.

B All parallelograms have diagonals that bisect each other. All parallelograms have opposite sides that are parallel. All polygons whose diagonals bisect each other have opposite sides that are parallel.

C All rectangles have 4 right angles. All squares have 4 right angles. All rectangles are squares.

D All parallelograms have 4 sides. All polygons with 4 sides are quadrilaterals. All parallelograms are quadrilaterals.

4 In each of the circles below, four angles are formed by the intersection of 2 secant lines. The measures of two intercepted arcs and one angle are shown for the first three circles.

AC

B

86° 77° 68°

32°

39°

25°

91°100° 82°

(5x + 2)°

(4x + 4)°

Which expression can be used to represent m Є�� ABC in degrees?

1A [(5 2x x+ −) (4 + 4)]2

1 B [(5 2x x+ +) (4 + 4)]2

C 2 5[( x x+ −2) (4 + 4)]

D 2 5[( x x+ +2) (4 + 4)]

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 5

5 Jake took pictures of Ana’s flag while she was practicing her routine for the football game, as shown below.

1 32 4

Which of the following best describes the movement of the flag from picture to picture?

A Reflection, rotation, translation

B Rotation, translation, translation

C Rotation, translation, dilation

D Reflection, translation, translation

6 When viewed from above, the base of a water fountain has the shape of a hexagon composed of a square and 2 congruent isosceles right triangles, as represented in the diagram below.

40 ft

10 ft

Base ofFountain

Top View

Which of the following measurements best represents the perimeter of the water fountain’s base in feet?

A ( )20 + 20 2 ft C ( )40 + 20 2 ft

B ( )20 + 40 2 ft D ( )40 + 40 2 ft

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 6

7 A side view of the intersection of a plane and a square pyramid is modeled below.

Plane

Base of pyramid

Which of the following best represents the shape formed by this intersection?

A

B

C

D

STAAR Geometry 2011 ReleaseReleased Test Questions

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8 The three-dimensional figure shown is composed of 11 identical cubes.

Front

Which of the following could not represent a top, front, or side view of the figure?

A

B

C

D

STAAR Geometry 2011 ReleaseReleased Test Questions

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9 RG is graphed on the coordinate grid below.

−5

−4

−6

−7

−8

−9

−10

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

−1 1−2−3−4−5−6−7−8−9−10 2 3 4 5 6 7 8 9 10

y

x

R

G

Which of the following equations best represents the perpendicular bisector of RG?

1 A y x= − 23

C y x= −3 10

1 B y x= −3 8+ D y x= − + 1 3

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 9

10 Half of an international basketball court is shown below. The shaded region is composed of an isosceles trapezoid and a semicircle. The diameter of the semicircle is 3.6 meters.

6.0 m

5.8 m

3.6 m

If 1 meter is approximately equal to 3.28 feet, which of the following is closest to the area of the shaded region in square feet?

A 32.9 ftầ C 354 ftầ

B 409 ftầ D 108 ftầ

11 In quadrilateral ABCD , AB � CD, ,∠ A ≅ ∠B and AB CD. Which of the following statements is a reasonable conclusion?

A m A∠ ≅ m∠C

B Quadrilateral ABCD is a rectangle.

C Quadrilateral ABCD is an isosceles trapezoid.

D AD � BC

STAAR Geometry 2011 ReleaseReleased Test Questions

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12 Triangles RST and VSU are shown below.

R

S

T

V

U

R V≅ , and RT ≅∠ ∠ VU. Which additional condition is sufficient to prove that RS ≅ SV?

A TS ≅ SU

B VS ⊥ RU

C RS ≅ SU

D ∠VUS ≅ ∠RST

STAAR Geometry 2011 ReleaseReleased Test Questions

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13 Triangle RST was dilated to create triangle R ′ S ′ T ′, as shown on the coordinate grid below.

−5

−4

−6

−7

−8

−9

−10

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

−1 1−2−3−4−5−6−7−8−9−10 2 3 4 5 6 7 8 9 10

y

x

R

R'

S S'

T'

T

Which statement appears to be true?

A The center of dilation used to create Ј R ′ S ′ T′ w as (−10, 8).

B Ј RST and Ј R ′ S ′ T′ are congruent.

C The scale factor used to create Ј R′ S ′ T ′ is 2.5.

D Ј RST was reduced in size to create Ј R ′ S ′ T ′.

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 12

14 A tree’s shadow is 4.8 m long on level ground, as shown in the diagram.

4.8 m

50°

The angle of elevation from the tip of the shadow to the sun is 50°. Based on this information, which of the following is closest to the height of the tree?

A 3.6 m

B 5.7 m

C 3.1 m

D 7.5 m

STAAR Geometry 2011 ReleaseReleased Test Questions

Page 13

15 A company packages their product in two sizes of cylinders. Each dimension of the larger cylinder is twice the size of the corresponding dimension of the smaller cylinder.

d

h

2d

2h

Based on this information, which of the following statements is true?

A The volume of the larger cylinder is 2 times the volume of the smaller cylinder.

B The volume of the larger cylinder is 4 times the volume of the smaller cylinder.

C The volume of the larger cylinder is 8 times the volume of the smaller cylinder.

D The volume of the larger cylinder is 6 times the volume of the smaller cylinder.

STAAR Geometry 2011 ReleaseAnswer Key

Item Reporting Readiness or Content Student Correct Number Category Supporting Expectation Answer

1 1 Readiness G.2(B) D

2 1 Readiness G.3(C) D

3 1 Supporting G.3(E) D

4 2 Readiness G.5(A) B

5 2 Supporting G.5(C) A

6 2 Readiness G.5(D) D

7 3 Supporting G.6(A) D

8 3 Supporting G.6(C) C

9 3 Readiness G.7(B) A

10 4 Supporting G.8(F) C

11 4 Supporting G.9(B) C

12 4 Readiness G.10(B) B

13 5 Supporting G.11(A) C

14 5 Readiness G.11(C) B

15 5 Readiness G.11(D) C

For more information about the new STAAR assessments, go to www.tea.state.tx.us/student.assessment/staar/.

Page 14

The 5 E Learning Cycle Model

Engage Objects, events, or questions are used to engage students. Connections are

made between what students know and can do.

Explore Objects and phenomena are explored through hands-on activities, with

guidance.

Explain

Students explain their understanding of concepts and processes. New

concepts and skills are introduced as conceptual clarity and cohesion are

sought.

Elaboration Activities allow students to apply concepts in contexts, and build on or

extend understanding and skill.

Evaluation Students assess their knowledge, skills, and abilities. Activities permit

evaluation of student development and lesson effectiveness.

Engage:

Learner Teacher

calls up prior knowledge poses problems

has an interest asks questions

experiences doubt or disequilibrium reveals discrepancies

has a question(s) causes disequilibrium or doubt

identifies problems to solve, decisions to be

made, conflicts to be resolved

assess prior knowledge

writes questions, problems, etc.

develops a need to know

self reflects and evaluates

Explore:

Learner Teacher

hypothesizes and predicts questions and probes

explores resources and materials models when needed

designs and plans makes open suggestions

collects data provides resources

builds models provides feedback

seeks possibilities assesses understandings and processes

self reflects and evaluates

Explain:

Learner Teacher

clarifies understandings provides feedback

shares understandings for feedback asks questions, poses new problems and

issues

forms generalizations models or suggests possible modes

reflects on plausibility offers alternative explanations

seeks new explanations enhances or clarifies explanations

employs various modes for explanation

(writing, art, etc)

evaluates explanations

Elaborate:

Learner Teacher

applies new knowledge asks questions

solves problems provides feedback

makes decisions provides resources

performs new related tasks makes open suggestions

resolves conflicts models when necessary

plans and carries out new project

asks new questions

seeks further clarification

Evaluate:

Learner Teacher

self-assess their own learning and

understanding of new concepts

evaluates effectiveness of the instruction

provide feedback to the teacher on lesson

effectiveness

assesses student learning and understanding

reflect with adults and their peers uses information about student learning to

guide subsequent instruction

communicate, in a variety of ways (e.g.

journals, reporting drawing, graphing,

charting) their level of understanding of

concepts that t hey have developed to date

asks open-ended questions to examine

students’ thinking

create and use quality indicators to assess

their own work

employs a rubric on which to give students

feedback on their learning

report and celebrate their strengths and

identify what they'd like to improve upon

Source: http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm

Triangle Reflections Items 1 – 7: Reflect each triangle across its reflection line. Items 8 – 9: Construct the reflection lines, showing your process clearly.

65

4

3

21

98

7

When you have successfully completed all nine items write a new definition for reflection, including the relationship between the reflection line and the segments connecting pre-image and image points.

Notes to the Teacher Appendices Materials One copy of Blackline Master for each student Blackline Master consists of two pages which can be printed back-to-back. Patty paper should be made available for students who need to trace and fold. One 4'' by 6'' mirror for checking can be held by the teacher and used by the student as needed. Students should not be allowed to keep the mirror to use for the whole activity. The lesson consists of nine items, building in difficulty. Vocabulary: Define, draw or give and example of: Reflection Horizontal Vertical Congruence Mapping

Triangle Reflections Introduce the lesson with a brief discussion about what students know about reflections. Reinforce the behavior expectations for whole class discussion: • Each group is responsible for at least one response. • One member from each group will be asked to provide the

group’s response. • One person speaks at a time, whether it is a student or the

teacher. Side conversations are inappropriate during group discussion.

• What is a reflection? • Can you give me an example? • Can you demonstrate? Have students give a “visual” level definition of reflection. Encourage students to use mathematical vocabulary. You might expect to hear: • What you see in the mirror. • The image is the same as (or congruent to) the pre-image, but

back-to-front, or switched around. • The image is facing the pre-image. Hand out Blackline Master to each student. Throughout the activity emphasize the fact that the images are congruence mappings of the pre-images. This will be fundamental to later proofs by transformation. Items 1-4 are reflections across horizontal and vertical lines. Most students should be able to perform these reflections by counting spaces on either side of the reflection lines. As you move around the room monitoring, ask students in each group how they accomplished the task. If you see several students struggling, call for a class discussion. Encourage students to share different strategies, to establish a culture of trust and sharing. Give students who are struggling with these problems patty paper. Trace the pre-image and the reflection line. Fold the paper across the reflection line. Trace the image. Reproduce the result on the activity sheet. Encourage them to look for ways that they could accomplish the task without the patty paper. Let them hold the mirror on the reflection line so that the image can be seen and corrected if necessary. They should not be allowed to work the whole set with the mirror. Scaffolding questions where individual students need assistance: • Which problem of the first four was done correctly? Let’s verify it

with the mirror. Is it correct? Students will look at the reflection of the pre-image, and then across the mirror at the drawn image.

• Draw in the line segments that connect pre-image points to their corresponding image points. Place the mirror on the reflection line. What do you notice about the connecting line segments? They are perpendicular to the mirror and are bisected by the mirror.

• Now try to do the same thing with the new problem. Can you see where to draw in the connecting segment? (Pause) How

long should you make this segment? (Pause) The length to the mirror from the pre-image is extended the same distance to the image.

• When you draw all the connecting segments, what do you notice about the structure? The connecting segments are all parallel to one another, and perpendicularly bisected by the mirror line.

Items 5 – 6: Some students will be able to perform these reflections by counting spaces across the diagonals. Where you see errors, assist individual students with scaffolding questions. Item 7: The grid is now a hindrance. Most students will struggle with this item. Have them draw in the pre-image to image segments for items 5 and 6. When they see this structure they will be able to complete item 7 successfully. Items 8 – 9: All students must draw in the connecting segments. This builds their understanding of the properties of reflections which will be used to justify and prove important ideas later. Using a mirror they can check for correct placement of the reflection line.

7

65

4

3

21

When you have successfully completed all nine items write a new definition for reflection, including the relationship between the reflection line and the segments connecting pre-image and image points. • The reflection line perpendicularly bisects the segments

connecting pre-image and image points. • The image is the same (perpendicular) distance behind as the

pre-image is in front of the reflection line. Student Practice exercises 7 - 9 utilize the congruence mapping property along with algebraic equation practice. Problem 24 has students look at the reflection structure from a slightly different angle.

98

7

a. Rotations Rotate ∆CDE 180o about point D. Rotate ∆FGH 180o about point J. Rotate ∆KLM 180o about point O. Rotate ∆PQR 180o about point S.

S

O

R

Q

PM

K

L

J

G

HD

EC F

Draw in the construction segments for each pre-image—rotation point—image point set. Each set should be a different color; three colors per triangle. Write down your observations. Be prepared to share these with the class.

b. Rotations Rotate ∆CDE 90o counterclockwise about point D. Rotate ∆FGH 90o counterclockwise about point J. Rotate ∆KLM 90o counterclockwise about point O. Rotate ∆PQR 90o counterclockwise about point S.

S

O

R

Q

PM

K

L

J

G

HD

EC F

Draw in the construction segments in three different colors for each triangle. Measure the angles formed within each colored set. Write down your observations. Be prepared to share these with the class. Write a definition for rotation. Be sure to include the special relationships between the pre-image—rotation point—image segments.

c. Rotations – Coordinate Considerations The following coordinate work will be done on the figures on sheets a and b. Select at least two rotation sets on each sheet. For each set, draw in coordinate axes, and use the rotation point as the origin. The gridlines are one unit apart. Find the coordinates for each pre-image point and its corresponding image point.

Sheet a (180o Rotations) Sheet b (90 o Rotations) Coordinates Coordinates

D (0,0) D (0,0) D (0,0) D (0,0) C (0,3) C' C (0,3) C' E E' E E' J (0,0) J (0,0) J (0,0) J (0,0) F F’ F F' G G' G G' H H' H H' O (0,0) O (0,0) O (0,0) O (0,0) K K' K K' L L' L L' M M' M M' S (0,0) S (0,0) S (0,0) S (0,0) P P' P P' Q Q' Q Q' R R' R R'

1. Find a general relationship that always applies for 180o rotations about the origin:

(x, y) → 2. Find a general relationship that always applies for 90o counter-clockwise rotations

about the origin:

(x, y) →

Notes to the Teacher Appendices Materials One copy of Blackline Masters a

and b for each student. These can be copied back-to-back.

One transparency copy of a One blank transparency One copy of Blackline Master c for

each student Colored pencils Rulers Patty paper, if needed Protractors Vocabulary: Define, draw or give and example

of: Rotation Rotation point / Point of rotation Clockwise Counterclockwise Straight angle The level of difficulty increases,

starting with at least one triangle side lying along the grid lines, to all three at non-horizontal or non-vertical orientations. The structure behind the rotation is brought out by drawing in colored line segments for each pre-image—rotation point—image set.

Scaffolding steps are included in the appendix to assist those who might be running into difficulty.

Student Practice: For students

who are able to accomplish this task without assistance, including the construction segments, provide them with another copy of the activity, but change the rotation angle to 45o. Construction angles can be measured with a protractor or with the right angled corner of a piece of paper folded in half.

Rotations As in the previous two lessons begin with a short discussion of what students know about rotations. • What is a rotation? • Can you demonstrate? • Can you show an example? Ask a volunteer to stand facing the class and then turn on the spot to face the board. Ask a second volunteer to stand at arm’s length next to the first volunteer, and slowly walk around the first volunteer keeping arm’s length distance away. Explain that both represent rotations; the first rotating about a point attached to the person (the point of rotation), the second rotating about a point not attached to the person (the point of rotation). Worksheet a Remind students that in line segments connecting pre-image and image points passed through the reflection line. In this lesson the construction lines from pre-image to image must first be connected to the rotation point. In Worksheet “a” the congruent construction segments form straight angles at the rotation point. Each set should be done in a different color to show the rotation property clearly. For example, in ∆FGH, FJF’ in red; HJH’ in blue; GJG’ in green. The 180o rotations can be done counterclockwise or clockwise. Most students will struggle for a few minutes with this activity. This is a necessary step for their spatial visualization growth. As you move around the room monitoring the work you may need to stop and help individual groups. This way you can keep an eye on each student as they complete the activity. Another option is to call a student up to the overhead projector. You will need a transparency copy of Worksheet “a” and a blank transparency to mimic patty paper. Use the scaffolding directions for the volunteer and the class to follow. Scaffolding directions to assist students having difficulty: • Trace the triangle in a sheet of patty paper. Label the vertices. • On the patty paper, along one of the grid lines draw an arrow

pointing up (or right). • Place your pencil point on the rotation point. Rotate the patty

paper counterclockwise until the arrow points down (left). • Trace the triangle back onto the activity sheet.

R'

Q'

P'

M'

G'

H'

F'

E'

C'

S

O

R

Q

PM

K

L

J

G

HD

EC F

Worksheet b Point out to students that the activity calls for counter-clockwise rotations. The 90o rotations must all be counter-clockwise. This reinforces positive rotation vs. negative (clockwise) rotation, which will be encountered in Pre-calculus. Explain this convention to students. This also reinforces the counter-clockwise direction we use in numbering coordinate plane quadrants. • What does counter-clockwise mean? Turning to the left. Ask a

volunteer to demonstrate a 180o counter-clockwise turn. Ask another to demonstrate a 180o clockwise turn.

• Why was the direction not specified in activity B8a? The image would be exactly the same. Now have them do counter-clockwise and clockwise 90o turns.

• Why is it important to specify direction for a 90o turn? The images would face the different directions for clockwise and counter-clockwise 90o turns. Scaffolding directions follow: • Trace the triangle on a sheet of patty paper. Label the vertices. • On the patty paper, along one of the grid lines draw an arrow

pointing up (or right). • Place your pencil point on the rotation point. Rotate the patty

paper counter-clockwise until the arrow points left (up). • Trace the triangle back onto the activity sheet.

TEKS: 8.7.D Locate and name points on a coordinate plane using ordered pairs of rational numbers.

Note: Additional work should be

done with fractional coordinates.

R'

Q'

P'

L'

M'

K'

G'F'

H'

E'

C'

S

O

R

Q

PM

K

L

J

G

HD

EC F

. The construction segments from pre-image point to rotation point to corresponding image point are congruent and form 90o angles. These should also be done in different colors. For example FJF’ in red; HJH’ in blue; GJG’ in green. Have students measure the angles formed within each construction set with a protractor or with the corner of an index card or piece of paper. For Worksheet “b” all of the angles should be 90o. Write a definition for rotation. Be sure to include the special relationships between the pre-image—rotation point—image segments. • The pre-image—rotation point segment is congruent to the rotation

point—image segment. These two segments form an angle equal to the rotation angle.

• If the pre-image—image points are connected directly, not through the rotation point, an isosceles triangle forms with the two segments connecting the rotation point, except when the angle of rotation is 180o.

Worksheet c As soon as Worksheet “a” and Worksheet “b” have been completed successfully, hand students Worksheet “c”. The goal is to generalize what happens when a figure is rotated about the origin. This can be completed for homework after directions have been given. Instruct them to draw in coordinate axes with the origin at each rotation point. Emphasize that each set of rotations stands in its own coordinate plane, even though they are all drawn on the same grid sheet. Demonstrate by drawing the axes on the transparency. You probably need to show only ∆CDE and ∆FGH. Each student should select at least two sets from each activity sheet. ∆KLM is the most difficult because the origin, O, is not located on a grid intersection

3. Find a general relationship that always applies for 180o rotations about the origin: (x, y) →(-x, -y)

4. Find a general relationship that always applies for 90o counter-clockwise rotations about the origin: (x, y) →(-y, x)

Sheet a (180o Rotations) Sheet b (90o Rotations) Coordinates Coordinates

D (0, 0) D (0, 0) D (0, 0) D (0, 0) C (0, 3) C’ (0, -3) C (0, 3) C' (-3, 0) E (2, 2) E’ (-2, -2) E (2, 2) E' (-2, 2) J (0, 0) J (0, 0) J (0, 0) J (0, 0) F (1, 5) F' (-1, -5) F (1, 5) F' (-5, 1) G (1, 1) G' (-1, -1) G (1, 1) G' (-1, 1) H (3, 2) H' (-3, -2) H (3, 2) H' (-2, 3) O (0, 0) O (0, 0) O (0, 0) O (0, 0) K (0, 2.5) K' (0, -2.5) K (0, 2.5) K' (-2.5, 0) L (0, -2.5) L' (0, 2.5) L (0, -2.5) L' (2.5, 0) M (2, .5) M' (-2, -.5) M (2, .5) M' (-2, .5) S (0, 0) S (0, 0) S (0, 0) S (0, 0) P (-3, 0) P' (3, 0) P (-3, 0) P' (0, -3) Q (-1, -3) Q' (1, 3) Q (-1, -3) Q' (3, -1) R (2, -2) R' (-2, 2) R (2, -2) R' (2, 2)

Translations on the Coordinate Plane 1. Translate ΔECD with vector FG

uuuur. Show the translation vectors for each point to

image point. Another way to describe this translation is by the rule (x, y) → (x+3, y+1), which is also called coordinate notation.

2. Translate ΔHIJ according to the rule (x, y) → (x – 1, y – 4). 3. List the coordinates of all points on the given table. Describe the relationships

between the pre-image coordinates and the image coordinates.

J

I

H

G

F

D

C

E

Pre-image Image C D E H I J

Notes to the Teacher Appendices Materials One copy of Blackline Master for each student Straightedge Student Practice exercises review this exercise. Problems 9 -11 ask students to describe the translations using words or coordinate notation, which is a necessary extension. TEKS 8.6.B Graph dilations, reflections, and translations on a coordinate plane.

Translations on the Coordinate Plane When individual students have completed B5 successfully, hand them B6. Explain to the class that they will perform translations on these triangles using coordinates as a way to describe the transformation. Make sure that they understand that the vector is to be used only on ΔECD.

FGuuur

J'

H'

I'

D'

E'

C'

J

I

H

G

F

D

C

E

3.

Pre-Image Image C (-3, 2) (0, 3) D (1, 1) (4, 2) E (-2, 5) (1, 6) H (-1, -3) (-2, -7) I (4, 1) (3, -3) J (5, -6) (4, -10)

Describe the relationships between the pre-image coordinates and the image coordinates. For ΔCDE, the x coordinates change to x + 3, the y-coordinates change to y + 1. For ΔHIJ, the x-coordinates change to x – 1, the y-coordinates change to y – 4.

Translations

H'

G'

F'

F

G

H

C'

B'

A'

A

B

C

L

M

NQ

R

“Translation.” Be prepared to shar

. with its image point. For

o share

. A vector is represented by an arrow drawn between two points to show its

1. Triangles ABC and FGH have both been translated. Write a definition for e your definition with the class.

2 With a straightedge connect each point of ΔABCexample, A connects to A’, B connects to B’. Do the same for ΔFGH. Measure each segment. Write down your observations. Be prepared tthese with the class. Add the observations to your definition.

3

size and its direction. Transform ΔLMN using vector QRuuuur

. Each point on triangle LMN will be translated in direction QR, the same distance as Q toBe sure to show the connecting segments from L, M and N to the image points L’, M’, and N’ to demonstrate the accuracy of your translation.

R.

Notes to the Teacher Appendices Materials One copy of Blackline master Ruler for each student Students’ entry knowledge of

transformations is usually at a visual level. These properties will be used extensively as a tool for developing and justifying properties of other planar figures during this semester.

Translations are the result of a

distance and direction motion. Usually the translation is defined by a given vector. At the visual level the vector is given as a ray, showing magnitude (size) and direction.

Vocabulary Define, draw or give and example

of: Pre-image Image The prime symbol (example A' ) Vector

Translations Transformations will be used to develop the properties of a variety

of planar figures. In this unit the basic transformation properties are developed using triangles. In later lessons the properties of other figures will also be developed using transformations.

The introductory translation, reflection and rotation activities in this unit all begin with a brief class discussion about transformations for students to share what they already know so the teacher can assess existing knowledge.

To facilitate large-group discussion while students are seated in small groups, establish the following guidelines:

• Each group is responsible for at least one response. This helps establish a classroom culture where participation in small group and whole class discussion is expected.

• One member of each group will be asked to provide the group’s response. This way everyone has a responsibility to learn from other group members.

• One person speaks at a time, whether it is a student or the teacher, which means that side conversations are not appropriate.

• Put-down comments are also not appropriate. Ask the following questions encouraging students to recall what

they know, whether it is correct or not. If an incorrect comment is made and other students wish to provide correction, then allow for the correction. In general, you should not provide feedback at this stage, even in the form of body language. • What is a transformation? Possible answers: A transformation

is when you change a figure’s position; when you change its size; when you change its direction.

• Can you give me an example? Allow students to demonstrate. If necessary call on a student to stand in front of the class. Direct her/him to move to another position in the room. Pretend that he is able to grow to twice his size.

Hand out Blackline Master to each student. Allow time to work in small groups. For item 1 ask what is meant by the notation A' (or B', or C'). Make sure that they all understand that A, B and C are points on the pre-image and that A', B' and C' are the corresponding image points that result from the transformation (named “A prime”, “B prime”, “C prime”). If you then performed another transformation on the image, the second image’s points would be labeled A'', B'', C'' (named A double-prime, B double-prime, C double-prime).

1. Write a definition for “Translation.” You are expecting a “visual” (what you see) definition. Ask each group to share their definition with the class. You might write these on the board or accept them orally. If an incorrect definition is provided, ask the class if they agree. Hopefully someone will provide a correction. If not then wait for the other definitions and make sure that the incorrect definition is not kept. After all groups have shared, have the class choose the best definition, or allow them to keep their own if it is correct.

ote that the copying process might change the relative size of the images in successive copies, so the given measurements may vary from those given. However, all wchanged in the same propand all will be equal for a given copy.

You might expect to heais exactly the same as the pre-image but in a different

place.

ments AA’, BB’ and CC’ measure approximately 4.5 cm. AA’, d

.

iptive” level.

e end . Instruct

er/him to translate according to the given floor vector (the arrow

on. Repeat the exercise facing a different direction. he end position will still be at the arrow end of the vector, but

N

ill be ortion,

r: • The image

• The image is congruent to the pre-image and points in the same direction.

2. Seg

BB’ and CC’ are all parallel to each other. Segments FF’, GG’ anHH’ measure approximately 1.5 cm. and are parallel to each other The purpose for drawing in the pre-image to image segments is to raise students’ understanding to the “descr• Each point moves the same distance and the same direction. • The transformation preserves congruence (the image is

congruent to the pre-image). 3. Make sure students understand the definition of vector. To

reinforce the definition you might have a student stand at onof a meter stick or length of masking tape on the floorhis at the other end). (S)he should move to the other end, but keepher/his orientatiTfacing the new direction. Another student can stand in a different place on the floor, away from the vector, and perform her/his translation in the same way, but this time disconnected from the actual vector.

C'

B'

A'

B

C H'

G'

F'

F

A G

H

L'

N'

M'

L

M

NQ

R

Geometry HOUSTON ISD PLANNING GUIDE

4th SIX-WEEKS

Recommendations for Instructional Enhancements for

Students with Special Needs

Unit 10: Transformations

Content-specific Accommodations for this Unit

When teaching students how to represent a reflection on a graph, provide them with patty paper and teach them to fold their paper over the line of reflection to check the accuracy of the reflected coordinates. Patty paper can also be used to represent rotations and translations.

When students are working with graphs, have them cut a piece of graph paper into four pieces and glue the graph paper side by side with their notes, work, etc.

Assist students to construct a visual aide of the coordinate plane as shown below. Allow students to use the aide when they transform figures.

(-x, y) (x, y)

II I

III IV (-x, -y) (x, -y)

General Accommodations for this Unit

Remind students to erase marks clearly so they are not misled by previous errors. Encourage students to use map colors when graphing to denote contrast between

various lines and figures on the same graph. Teach students to mark places in their notes where they did not understand the

explanations, directions, instructions, etc. and to refer to these marked places when they formulate their questions.

Permit students to correct errors for partial credit recovery on assignments. Require students to clean out their binders after each six-week period. Place the

corresponding papers in a large manila envelope with the correct cycle written on the outside. Store the envelopes until review time at the end of each semester. However, ensure that students keep their notes in their binders throughout the year.

5E Lesson Plan Transformations MATH.8.6B

Time Period

Stage of the Lesson (Engage, Explore, Explain, Elaborate,

Evaluate)

Teacher Activities Student Activities TAKE Notes Evidence of Success

7 min

Warm up - Plot 3 points; connect to form a right triangle. Determine the length for each side. Pythagorean Theorem will be used.

Give directions for warm up, and circulate while students are working on problem. Take attendance

Work on warm up, Plot points, & connect the points. Use calculator. a2 +b2 = c2

Recommendation: allow a student to perform drawing at the board as teacher facilitates. Note: Relation to areas of squares in the P.T. diagram worked well with students. Note: Time ran a little long.

5 min

Engage: Transformation Puzzle Match reflection, rotation, dilation, & translation with a visual rep and a student friendly work

Instruct the students to match the puzzle pieces

Matching puzzle pieces (12) and collaborating with partner

Recommendation: Don’t discuss contents of bag with students, lets them form their own conclusions. Recommendation: Create a transparent set and let the first pair done complete the puzzle on the overhead. Recommendation: Inform the students that the bag contains 4 groups of three to preserve time. Ask students if they can identify the concept.

10 min Explore: Guided Instruction

Use questions and cuesto elicit prior knowledge of translation. Discuss Image and Preimage. Shapes stay congruent.Instruct S to describe a given translation.

In pairs-describe to a partner the translation. ThinkPairShare. One S does A while the other S does B

Note: During teacher explanation, ask a student to track the teachers’ movements on a coordinate system to model translation. Note: For English language learners, perform a translation example using 2 locations in the room, move around the room using steps to the right, left, up, and down. Have students tell you how you would get to another location.

5E Lesson Plan Transformations MATH.8.6B

Time Period

5E Lesson Stage (Engage, Explore, Explain, Elaborate,

Evaluate)

Teacher Activities Student Activities TAKE Notes Evidence of Success

10 min

Explain: Introduce transformation notation (x + ,y - )

Guide students to connect words with transformation notation and graph observation

Working on G.I. connection notation with words + right, - left, and + up, -down

Note: During observations, both the explain and explore sections last about 20 minutes total (when combined). Suggest merging the two sections into one if possible.

5 min

Elaborate: Write problem A & B from explore in new notation A(x-2, y+4) B (x+5, y)

Teacher will instruct S to write problems A & B from Explore in new notation. No words, justan ordered pair.

Perform specified translation. S will write 2 problems in new notation.

Students may go to the board to show what they came up with for their transformation notation.

10 min Evaluate: Mystery Message Translation Activity

Give instructions, circulate, pass out a transparency if needed to help S perform the move

Will work in pairs to decode mystery phrase using new notation

Note: One teacher used the conclusion as an opportunity to make up the “Think Pair Share”. Have a few new notation problems available and have the students share with each other what the notation represents, for example, (x+1,y+4), (x, y-3). Have the grid on the screen when explaining how to read the location of points and letters. If you run out of time, assign for homework. As an extension, relate the concept back to TAKS problems. 8.6B

RotationReflection

Dilation

Tran

slatio

n

Turn MirrorImage

EnlargeOr

ReduceSlide

2010

2010

RotationReflection

DilationTr

ansla

tion

Turn MirrorImage

EnlargeOr

ReduceSlide

2010

2010

Lets Get it Started!! #__Plot the following points, C (-3,-1), A(-3,6), T(2,6). Connect the points. Determine the length of CA, AT, & CT.

CA =

AT =

CT =

Geo 1/8

Obj 7 8.7CObj 6 8.7D

Rotation

ReflectionDilation

Tran

slatio

n

TurnMirrorImage

EnlargeOr

ReduceSlide

2010

2010

Transformations

Explore

Translation - ____________ or _________

Describe the translation that occurred from Figure 1 to produce Figure 2.

Fig. 1

Fig. 2

A

Slide Move

G’

H’

F’

Fig.1Fig.2

B

Explain

What type of translation can occur for the x and y coordinates? ( , )

X coordinate -Directions:

________ = _____ ( )

or

_______= ________ ( )

Y coordinate - Directions:

______ = ____ ( )

or

______ = ________ ( )

X Y

Right Add +

Left Subtract –UP Add +

Down Subtract –

Perform the specific Translation

A. Graph: M (2,4), N (-3,0), P (1, - 4)

(x + 2, y – 3)

M’ ( , )

N’ ( , )

P’ ( , )

Explain

B. Graph: D (0,-2), E (3,-4), F (2, 6)

(x – 3 , y – 4)

D’ ( , )

E’ ( , )

F’ ( , )

Explain

Elaborate

Write the translation for A and B in new notation

A. ( ) B. ( )

G’

H’

F’

Fig.1Fig.2

B

Fig. 1

Fig. 2

A

Bumble Bee Has left the Building ! !

ExploreTranslation - ________________ or ___________

Describe the translation that occurred from Figure 1 to produce Figure 2.

Fig. 1

Fig. 2 G’

H’

F’

Fig.1Fig.2

A B

Write the translation for A and B in new notation

A. ( ) B. ( )

Elaborate

What type of translation can occur for the x and y coordinates?

( , )

X coordinate Directions:

__________ = ____ ( )

or

________= ________ ( )

Perform the specific Translation

A. Graph: M (2,4), N (-3,0), P (1, - 4)

(x + 2, y – 3)

M’ ( , )

N’ ( , )

P’ ( , )

B. Graph: D (0,-2), E (3,-4), F (2, 6)

(x – 3 , y – 4)

D’ ( , )

E’ ( , )

F’ ( , )

Explain

Y coordinate Directions:

________ = ____ ( )

or

________= ________ ( )

Obj. 8.6B –Practice

What are the ordered pairs that best representspoint K’ and point Q’?

1.

2.

3.

What are the ordered pairs that best representspoint K’ and point Q’?

1.

2.

3.

Obj. 8.6B –Practice

Evaluation Activity Instructions

Each point corresponds to the letter at the top right of the given point. For example, point (2, 2) corresponds to letter I. 1. Use the transformation notation to write the new coordinates. 2. Write the letter of the new coordinate to create a famous quote by

Muhammad Ali.

(X,Y) (X-3, Y+5)

(0, 1) (5, 0) (1, -5) (0, -1)

(__, __) (__,__) (__,__) (__,__)

____ ____ ____ ____

(X,Y) (X-3, Y-5)

(-2,3) (0,1) (3, 1) (3,4) (2, 4)

(__,__)(__,__)(__,__) (__,__)(__,__)

____ ____ ____ ____ ____

(X,Y) (X+4, Y)

(-5, -1), (-5, 5), (0, 2)

(__, __) (__,__) (__,__)

____ ____ ____

(X,Y) (X-2, Y+3)

(-2, 1), (-2, -1), (1, -2), (5, 2)

(__, __) (__,__) (__,__) (__,__)

____ ____ ____ ____ ,

(X,Y) (X-2, Y+4)

(0, 3) (2, 3) (5, -2) (-1, -3)

(__, __) (__,__) (__,__) (__,__)

____ ____ ____ ____

(X,Y) (X-2, Y+2)

(1, -3), (1, -6) (-1, -3),

(__, __) (__,__) (__,__)

____ ____ ____

(X,Y) (X, Y+3)

(-1, 1) (2, 1) (4, -6) (-3, -1)

(__, __) (__,__) (__,__) (__,__)

____ ____ ____ _____

(X,Y) (X-1, Y-5)

(-5, 2) (1, -1) (1, 1) (6, 5) (-1, 8)

(__,__)(__,__)(__,__)( __,__)(__,__)

____ ____ ____ ____ ____