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356 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines

Advance PreparationFor the optional Readiness activity in Part 3, set aside trapezoid pattern blocks.

Teacher’s Reference Manual, Grades 4–6 pp. 193, 196–199

Key Concepts and Skills• Plot, name, and label points in any of the

four quadrants of a coordinate grid. [Measurement and Reference Frames Goal 3]

• Identify congruent figures. [Geometry Goal 2]

• Practice and perform isometry transformations with geometric figures. [Geometry Goal 3]

• Classify a rotation by the number of degrees needed to produce a given image. [Geometry Goal 3]

Key ActivitiesStudents review and perform isometry transformations, including reflections, translations, and rotations.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 404). [Measurement and Reference Frames Goal 3]

Key Vocabularyisometry transformation � transformation �

translation (slide) � reflection (flip) � rotation (turn) � image � preimage � line of reflection

MaterialsMath Journal 1, pp. 178–181Student Reference Book, pp. 180 and 181Math Masters, p. 404Study Link 5�4transparent mirror � tracing paper (optional)

Making a Circle Graph with a ProtractorMath Journal 1, p. 182Geometry Template/protractor �

compass Students find fraction, decimal, and percent equivalencies and calculate degree measures of sectors to create a circle graph.

Math Boxes 5�5Math Journal 1, p. 183inch ruler Students practice and maintain skillsthrough Math Box problems.

Study Link 5�5Math Masters, p. 160 Students practice and maintain skillsthrough Study Link activities.

READINESS

Reviewing Degrees and Directions of RotationMath Masters, p. 161trapezoid pattern block (1 per student)Students use a full-circle protractor to practice rotating a figure about a point.

ENRICHMENTPerforming a Scaling TransformationMath Masters, p. 162rulerStudents perform size-change transformations.

ELL SUPPORT Building a Math Word BankDifferentiation Handbook, p. 131Students add the terms translation, reflection, and rotation to their Math Word Banks.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Isometry TransformationsObjective To review transformations that produce another figure while maintaining the same size and shape of the original figure.w

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

356_EMCS_T_TLG1_G6_U05_L05_576833.indd 356 2/9/11 1:27 PM

Isometry TransformationsLESSON

5�5

Date Time

Math Message

Translations (slides), reflections (flips), and rotations (turns) are basictransformations that can be used to move a figure from one place to another without changing its size or shape.

1. Study each transformation shown below.

Translation (Slide) Reflection (Flip) Rotation (Turn)

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A translation (slide) moveseach point of a figure a certaindistance in the same direction.

A reflection (flip) of a figuregives its mirror image over a line.

A rotation (turn) moves afigure around a point.

a. b. c. rotationtranslationreflection

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2. Identify whether the preimage (1) and image (2) are related by a translation, areflection, or a rotation. Record your answer on the line below each coordinate grid.

180 181

Math Journal 1, p. 178

Student Page

Adjusting the Activity

Lesson 5�5 357

Getting Started

Math MessageStudy Problem 1 and complete Problem 2 on journal page 178.

Study Link 5�4 Follow-UpReview answers.

Mental Math and Reflexes Students find the signed number that is halfway between each of the two given numbers. Suggestions:

between 5 and -5 0 between -7 and -3 -5

between -3 and 2 -0.5 between 1.5 and -4.5 -1.5

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 178; Student Reference Book, pp. 180 and 181)

Students worked with isometry transformations in Fifth Grade Everyday Mathematics, but some of the vocabulary in this lesson may be new. Provide students with multiple opportunities to read, write, and say the vocabulary words. Whenever possible, relate vocabulary to students’ experiences.

Briefly review each transformation (translation, reflection, and rotation) and go over the answers to Problems 2a–c. Draw students’ attention to the fact that each new figure—the image (2)—is the same size and shape as the original figure—the preimage (1). Point out that the distance between points also remains unchanged.

Language Arts Link The word isometry comes from the Greek words iso, meaning same, and metron, meaning measure.

Have students identify the following locations on a coordinate grid:• the origin (0,0)• the x-axis (or horizontal axis)• the y-axis (or vertical axis)Write several ordered number pairs on the board. Ask students to identify the x- and y-coordinates for each ordered number pair.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Mathematical PracticesSMP1, SMP2, SMP3, SMP5, SMP6, SMP7, SMP8Content Standards6.NS.6, 6.NS.6b, 6.NS.6c, 6.NS.8

357-361_EMCS_T_TLG1_G6_U05_L05_576833.indd 357 3/19/12 9:10 AM

TranslationsLESSON

5�5

Date Time

Example:Translate quadrangle ABCD 6 units to the right and 5 units up.

Plot and label the vertices of the image that would result from the translation.

Plot and label the vertices of the image that would result from each translation.

1. Translate triangle DEF 4 units 2. Translate pentagon LMNOPto the right and 3 units down. 0 units to the right and 7 units down.

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3. Square WXYZ has the following vertices:W (�3,�2), X (�1,�2), Y (�1,�4), Z (�3,�4)

Without graphing the preimage, list the vertices of image W�X�Y�Z� resulting fromtranslating each vertex 3 units to the right and 2 units up.

W� ( , ); X� ( , ); Y� ( , ); Z� ( , )�2

180–181

Math Journal 1, p. 179

Student Page

ReflectionsLESSON

5�5

Date Time

Reflect each figure over the indicated axis or line of reflection. Then plot and label thevertices of the image that results from the reflection. Use a transparent mirror to checkyour placement of each image.

1. Reflect triangle TAM over the x-axis. 2. Reflect rectangle STUV over the y-axis.

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3. Reflect triangle PQR over the y-axis. 4. Reflect square DEFG over line m.

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180

Math Journal 1, p. 180

Student Page

358 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines

▶ Translating Geometric Figures

WHOLE-CLASS ACTIVITY

(Math Journal 1, p. 179)

When a figure is translated, each point on the preimage slides the same distance in the same direction to create the image. If the translation is done on a coordinate plane, the image of a point can be found by adding translation numbers to the coordinates of the point being translated. In the example on journal page 179, each point of Figure ABCD is translated 6 units to the right (+6) and 5 units up (+5). Each point of the preimage has a corresponding point in the image. Point out that to distinguish between the image and the preimage, students should label an image point with the same letter as the preimage point and the additional symbol ('). For example, the image of A is A', read “A prime.”

Ongoing Assessment: Exit Slip �Recognizing Student Achievement

Use the example on journal page 179 to assess students’ ability to name ordered number pairs in the third quadrant. Students are making adequate progress if they can correctly name the vertices of preimage ABCD. Ask students to record the number pairs on an Exit Slip (Math Masters, p. 404). [Measurement and Reference Frames Goal 3]

List the coordinates of the vertices of ABCD and A'B'C'D'. Show students how they can produce the image (A'B'C'D') from the preimage (ABCD) by adding +6 units to each x-coordinate of a vertex and +5 units to each y-coordinate.

A = (-3,-1) A' = (3,4)B = (-2,-2) B' = (4,3)C = (-3,-4) C' = (3,1)D = (-4,-2) D' = (2,3)

Ask students to pay attention to the coordinates of the vertices of preimages and images as they complete journal page 179.

▶ Reflecting Geometric Figures

WHOLE-CLASS ACTIVITY

(Math Journal 1, p. 180)

The reflection of a figure appears to be a reversal, or mirror image of the preimage. Each point on the preimage is the same distance from the line of reflection as the corresponding point on the image. When reflecting geometric figures over an axis on the coordinate grid, the axis is the line of reflection. The preimage and image are on opposite sides of the axis. If the figure is reflected across the x-axis, the x-coordinates stay the same and the y-coordinates differ only by sign. If the figure is reflected across the y-axis, the y-coordinates stay the same and the x-coordinates differ only by sign.

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EM3cuG6TLG1_357-361_U05L05.indd 358 12/18/10 1:47 PM

RotationsLESSON

5�5

Date Time

Rotate each figure around the point in the direction given. Then plot and label thevertices of the image that results from that rotation.

1. Rotate triangle XYZ 180� clockwise 2. Rotate quadrangle BCDE 90�

about Point X. counterclockwise about the origin.

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3. Rotate trapezoid MNOP 90� 4. Rotate triangle SEV 270� clockwisecounterclockwise about point M. about (0,2).

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180

Math Journal 1, p. 181

Student Page

LESSON

5�5

Date Time

1. One way to convert a percent to the degree measure of a sector is to multiply 360� by the decimal equivalent of the percent.

Example: What is the degree measure of a sector that is 55% of a circle?55% of 360� � 0.55 � 360� � 198�

Complete the table below.

2. The table below shows the elective courses taken by a class of seventh graders. Complete the table. Then, in the space to the right, use a protractor to make a circle graph to display the information. Do not use the Percent Circle. (Reminder: Use a fraction or adecimal to find the degree measure of each sector.)Write a title for the graph.

Making a Circle Graph with a Protractor

55%

Music

Elective Courses Taken by

Seventh Graders

Art

Compu

ters

Photography

Percent Decimal Degree Measureof Circle Equivalent of Sector

40% 0.4 0.4 � 360° �

90% 0.9 0.9 � 360� � 324�

65% 0.65 0.65 � 360� � 234�

5% 0.05 0.05 � 360� � 18�

1% 0.01 0.01 � 360� � 3.6�

DegreeCourse

Number Fraction Decimal PercentMeasureof Students of Students Equivalent of Students of Sector

Music 6 0.2 20% 72�

Art 9 0.3 30% 108�

Computers 10 0.33– 33 �13�% 120�

Photography 5 0.16– 16 �23�% 60�

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59 60147

Math Journal 1, p. 182

Student Page

Adjusting the Activity

Adjusting the Activity

Lesson 5�5 359

Have students complete journal page 180. Bring the class together to discuss answers. Ask a volunteer to read the preimage coordinates and the image coordinates for Problem 1. Record them on the board, listing corresponding coordinates next to each other. Ask: What do you notice about the coordinates of the preimage and the reflected image? The x-coordinates stayed the same and the y-coordinates changed signs. Repeat this procedure for Problem 2. The y-coordinates stayed the same and the x-coordinates changed signs. Ask: Look at Problem 3. Describe a pattern you notice about the coordinates of figures that are reflected across an axis. The preimage and image of a figure reflected across the x-axis have the same x-coordinates and their y-coordinates differ only by sign. The preimage and image of a figure reflected across the y-axis have the same y-coordinates and their x-coordinates differ only by sign.

Ask students to look closely at Problem 2 and decide whether the same image could be obtained by a horizontal translation of the preimage. No. The rectangles have the same orientation in relation to the x-axis, but the order of the vertices changes from the preimage to the image. When a translation is performed, the order of the vertices stays the same.

▶ Rotating Geometric Figures

WHOLE-CLASS ACTIVITY

(Math Journal 1, pp. 178 and 181)

Draw attention again to Problem 2c of the Math Message on journal page 178. Point out that to rotate a figure, the following are needed:

� a specific point about which the figure is to rotate.

� a number of degrees the figure is to rotate about that specific point.

� a specific direction of rotation (clockwise or counterclockwise).

To support English language learners, discuss the meaning of clockwise and counterclockwise.

In Problem 2c, the preimage (1) is rotated 90° counterclockwise about the point (0,-3) to produce the image (2). A clockwise rotation of 270° of the preimage (1) will also produce the same image.

Have students work in partnerships on journal page 181.

Make copies of journal page 181 from which students can cut out each preimage and perform each rotation manually. Students can also trace the preimage onto another sheet of paper and rotate the paper to help determine the orientation of the image.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

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ELL

Have students set a transparent mirror on the line of reflection to more clearly see the reversal of the points on a reflected image.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

EM3cuG6TLG1_357-361_U05L05.indd 359 12/18/10 1:47 PM

Math Boxes LESSON

5�5

Date Time

4. Multiply or divide.

a. 45 � �15� �

b. 60 � 4 �

c. � 56 � �18�

d. � 108 � 1297

159

5. Find the value that makes each numbersentence true.

a. 32 � n � 52 n �

b. y � 15 � 20 y �

c. (2 � m) � 5 � 17 m � 63520

1. Rectangle MNOP has sides parallel to theaxes. What are the coordinates of point O?

Coordinates of point O: ( , ) �23

0

y

x

M (�3,2)

P (�3,�2) O

N (3,2)

3. a. Draw a line segment that is 3�156� inches long.

b. By how many inches would you need to extend the line segment you drew to make it 5 inches long?

85 86

242 243

234

88

2. Solve mentally.

a. 20% of 50 �

b. �38� of 48 �

c. � 75% of 64

d. � �59� of 90 50

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49 5087

Math Journal 1, p. 183

Student Page

STUDY LINK

5�5 Transforming Patterns

180 181

Name Date Time

A pattern can be translated, reflected, or rotated to create many different designs. Consider the pattern at the right.

The following examples show how the pattern can be transformed to create different designs:

1. Translate the pattern at the right across 2 grid squares. Then translate the resulting pattern (the given pattern and its translation) down 2 grid squares.

2. Rotate the given pattern clockwise 90�

around point X. Repeat 2 more times.

3. Reflect the given pattern over line JK.Reflect the resulting pattern (the given pattern and its reflection) over line LM.

A

B

C D

Translations Rotations Reflections

4. 26 � 5. 35 � 6. 70 � 7. 43 � 64124364Practice

X

J

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Math Masters, p. 160

Study Link Master

Links to the Future

360 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines

Offer an example of a transformation that is not an isometry transformation. In Unit 8 of Sixth Grade Everyday Mathematics, students will study similar figures, which are produced by scaling transformations. A scaling transformation produces a figure that is the same shape as the original figure but not necessarily the same size. The following shows a scaling (but not an isometry) transformation:

preimage image

2 Ongoing Learning & Practice

▶ Making a Circle Graph

INDEPENDENT ACTIVITY

with a Protractor(Math Journal 1, p. 182)

Students find fraction, decimal, and percent equivalencies; calculate degree measures of sectors; and use a compass and a protractor to create a circle graph to display elective course information.

▶ Math Boxes 5�5

INDEPENDENT ACTIVITY

(Math Journal 1, p. 183)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 5-7. The skills in Problems 4 and 5 preview Unit 6 content.

Writing and Reasoning Have students write a response to the following: Explain how to rewrite Problems 4b and 4d as multiplication problems. Sample answer: The division

problem 60 ÷ 4 can be expressed as the fraction 60 _ 4 . Because 60 _ 4 = 60 ∗ 1 _ 4 , 60 ÷ 4 and 60 ∗ 1 _ 4 are equivalent expressions. Similarly, 108 ÷ 12 can be expressed as 108 _ 12 or 108 ∗ 1 _ 12 .

▶ Study Link 5�5

INDEPENDENT ACTIVITY

(Math Masters, p. 160)

Home Connection Students practice performing transformations on a grid. If necessary, review the examples at the top of page 160 with the class.

EM3cuG6TLG1_357-361_U05L05.indd 360 12/13/10 8:48 AM

LESSON

5 �5

Name Date Time

Degrees and Directions of Rotation

When a figure is rotated, it is turned a certain number of degrees around a particular point. A figure can be rotated clockwise or counterclockwise.

Position a trapezoid pattern blockon the center point of the anglemeasurer as shown at the right. Then rotate the pattern block as indicated and trace it in its new position.

Example: Rotate 90° clockwise.

For each problem below, rotate and then trace the pattern block in its new position.

1. Rotate 90° counterclockwise. 2. Rotate 270° clockwise.

12

6

11

5

10

4

1

7

2

839

degrees

12

6

11

5

10

4

1

7

2

839

degrees

12

6

11

5

10

4

1

7

2

839

degrees

Math Masters, p. 161

Teaching Master

LESSON

5�5

Name Date Time

Scaling Transformations

Some scaling transformations produce a figure that is the same shape as the originalfigure but not necessarily the same size. Enlargements and reductions are types of scaling transformations.

Enlargement: Follow the steps to draw a triangle D�E�F� with angles that are congruent totriangle DEF and sides that are twice as long as triangle DEF.

Step 1 Draw rays from P through each vertex. The first ray P���D has been drawnfor you.

Step 2 Measure the distance from point P to vertex D. Then locate the point onP���D that is 2 times that distance. Label it D�.

Step 3 Use the same method from Step 2 to locate point F� on P���F andpoint E � on P���E.

Step 4 Connect points D�, E�, and F�.

Reduction: Change Steps 2 and 3 to draw a triangle DE F with angles that are congruent to triangle DEF and sides that are half as long as triangle DEF.

P

D

F

E

Math Masters, p. 162

Teaching Master

Lesson 5�5 361

3 Differentiation Options

READINESS

INDEPENDENT ACTIVITY

▶ Reviewing Degrees and 5–15 Min

Directions of Rotation(Math Masters, p. 161)

To provide experience using an angle measurer and rotating a figure, have students complete Math Masters, page 161. Students use an angle measurer with an embedded clock face to practice rotating a trapezoid pattern block a given number of degrees in a clockwise or counterclockwise direction. If necessary, provide additional opportunities to rotate the trapezoid.

ENRICHMENT

INDEPENDENT ACTIVITY

▶ Performing a Scaling 5–15 Min

Transformation(Math Masters, p. 162)

To deepen students’ understanding of transformations and to introduce a size-change transformation, have students complete Math Masters, page 162. Students follow steps to perform scaling transformations to produce images that are twice and half the size of preimages. Have students describe the relationships between the original figure and the enlarged or reduced figure. Encourage them to use the vocabulary they have developed in this unit.

ELL SUPPORT

INDEPENDENT ACTIVITY

▶ Building a Math Word Bank 5–15 Min

(Differentiation Handbook, p. 131)

To provide language support for transformational geometry terms, have students use the Word Bank template found on Differentiation Handbook, page 131. Ask students to write the terms translation, reflection, and rotation, draw pictures depicting each term, and write other related words. See the Differentiation Handbook for more information.

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