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

Advance PreparationFor Part 1, draw and cut out two congruent polygons similar to those on page 178 of the Student Reference Book. Size them so both polygons fit on an overhead screen at the same time.

Teacher’s Reference Manual, Grades 4–6 pp. 42–44, 192–199

Key Concepts and Skills• Identify congruent figures. 

[Geometry Goal 2]

• Use a straightedge and compass to construct a figure that is congruent to the original. [Geometry Goal 2]

• Apply the knowledge that a preimage and its image are congruent. [Geometry Goal 3]

Key ActivitiesStudents explore the properties of congruent line segments, angles, and other figures. They also use drawing tools to construct congruent segments, angles, and 2-dimensional figures.

Key Vocabularycongruent � corresponding (sides, angles) �

rough sketch � accurate drawing

MaterialsMath Journal 1, pp. 184–186Student Reference Book, p. 178Study Link 5�5transparency of Study Link 5�5 (optional) �

compass � ruler � Geometry Template/protractor � board compass and protractor (for demonstration purposes)

Using a Grid MapMath Journal 1, pp. 187A and 187Bstraightedge Students draw a grid map of a carnival and use the map to find distances.

Math Boxes 5�6Math Journal 1, p. 187 Students practice and maintain skillsthrough Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1. [Measurement and Reference Frames Goal 3; Geometry Goal 3]

Study Link 5�6Math Masters, p. 163tracing paper (optional) Students practice and maintain skillsthrough Study Link activities.

READINESS

Using Congruence to Sort ShapesMath Masters, pp. 164 and 165scissors � Geometry Template/protractorStudents sort quadrangles according to angles and congruent sides.

ENRICHMENTExploring Isometry Transformations and CongruenceMath Masters, p. 4185 square tiles � large sheets of paper �

scissors � tape/glueStudents explore isometry transformations using pentominoes.

EXTRA PRACTICE 5-Minute Math 5-Minute Math™, pp. 57, 61, 64, 220, and 229Students practice using vocabulary terms to describe polygons and other shapes.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Congruent FiguresObjectives To explore the meaning of congruence; and to usedrawing tools to construct congruent figures.d

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

362_EMCS_T_TLG1_G6_U05_L06_576833.indd 362 2/16/11 1:50 PM

Congruent FiguresLESSON

5�6

Date Time

Math Message

Carefully examine the figures in examples a–f.

The following pairs of figures The following pairs of figures areare congruent to each other. not congruent to each other.

a. d.

b. e.

c. f.

1. Write a definition of congruent polygons. Then compare your definition to thedefinition on page 178 in the Student Reference Book.

Line segments are congruent if they Angles are congruent if they have the have the same length. same degree measure.

2. Draw wavy lines to connect each pair of 3. Draw wavy lines to connect each paircongruent line segments below. Use a of congruent angles below. Use a protractorruler to measure line segments if needed. to measure angles if needed.

Answers vary.

178

Math Journal 1, p. 184

Student Page

Lesson 5�6 363

Getting Started

10% of 100 10 5% of 100 5 1% of 100 1 6% of 100 6

10% of 360 36 5% of 360 18 1% of 360 3.6 6% of 360 21.6

10% of 90 9 5% of 90 4.5 1% of 90 0.9 6% of 90 5.4

Math MessageComplete the Math Message on journal page 184.

Study Link 5�5 Follow-UpStudents compare the designs resulting from the transformations. Consider making a transparency of Study Link 5-5 and have volunteers share their designs.

Mental Math and Reflexes Students mentally calculate 10%, 5%, 1%, and 6% of a number. Suggestions:

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 184; Student Reference Book, p. 178)

Begin by discussing the need for definitions in mathematics. For example, definitions help avoid confusion when people make mathematical statements. You may also want to identify some characteristics of a good definition:

� It should name the term being defined.

� It should include only words that are easily understood or words that have been defined previously.

� It should be accurate and concise.

Ask students to share their definitions of congruent polygons. Then discuss the Student Reference Book definition. Congruent figures are figures that have the same size and the same shape. The word shape is difficult to define. Students should know intuitively that all squares have the same shape, all circles have the same shape, and so on. Discuss why all squares have the same shape but not all rectangles do. Some rectangles have sides that differ greatly in length; others have sides that are almost the same length.

Have students use the definition of congruent to explain why the pairs of polygons in Examples d, e, and f are not congruent. Explanations should include the following:

� The sides of the polygons in Example d are the same length, but the polygons are not the same shape—they don’t have the same number of sides.

� The equilateral triangles in Example e are the same shape, but they are not the same size—their sides are not the same length.

� The polygons in Example f are not the same shape. Although their sides are the same length, their angles have different degree measures.

Mathematical PracticesSMP1, SMP2, SMP3, SMP4, SMP5, SMP6Content Standards6.NS.8, 6.G.3

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Constructing Congruent FiguresLESSON

5�6

Date Time

You may use any of your construction tools—ruler, compass, protractor, or GeometryTemplate—to complete the constructions below. Do not trace.

1. Draw a triangle that is congruent to triangle RST.

2. Draw a line segment BC so that BC�is congruent to AB� and the measure of angle ABC is 45�.

3. The plan below is for a paper cone with a glue tab. All dimensions and anglemeasures are shown. Draw a congruent copy of the plan on a separate sheet ofpaper. Cut out your copy and tape it on top of the plan below.

R

S

T

BA

C

110° 60°60°

60°3 in.

1 in.

1 in.

1 in.

1 in.

1 in.

190

Math Journal 1, p. 185

Student Page

Adjusting the Activity

Links to the Future

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

One way to determine whether two figures are congruent is to trace one of the figures, place the tracing on top of the other figure, and see whether the figures match exactly. One figure may have to be rotated or reflected to verify that it matches the other figure. Performing an isometry transformation is one way to decide if figures are congruent, especially when they appear to be the same shape and size.

To demonstrate the method described above, use the two congruent polygons you cut out. (See Advance Preparation.) Hold them up in different positions or place them on the overhead projector. Move one on top of the other so they match. Show how a combination of slides, flips, and turns can make one polygon fit exactly on top of the other. Mention that corresponding sides are the same length and corresponding angles have the same degree measure.

Use the demonstration polygons to show how to use slash marks and arcs to identify pairs of corresponding sides and angles. Also point out that when naming congruent polygons, the corresponding vertices of the polygons are listed in the same order for both polygons.

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

Go over the answers to Problems 2 and 3. Note that in these problems, some line segments and angles may be difficult to match. If there are disagreements, have students check answers by measuring with a ruler or a protractor.

In Unit 8, students will identify the corresponding sides and corresponding angles of figures that have the same shape but are not necessarily the same size. In their seventh- and eighth-grade mathematics courses, students will need to recognize congruent polygons when deriving and using formulas to calculate surface areas of polyhedrons.

▶ Constructing Congruent Figures PARTNER ACTIVITY

(Math Journal 1, p. 185)

The problems on journal page 185 provide practice with drawing and measuring tools. Students may use any combination of tools they wish. Then bring the class together and ask students which tools they used for their constructions. Compare approaches. Ask students how they know that their drawings are congruent to the given figures.

EM3cuG6TLG1_363-368_U05L06.indd 364 12/13/10 8:55 AM

Rough Sketches and Accurate DrawingsLESSON

5�6

Date Time

Roberta wants to build a bird feeder. She has found a plan for a feeder in a library book.

Roberta’s rough sketch for one side of the feeder is shown below. The labels for thelengths and angle measures are correct, but the scale of the drawing is not accurate.

In the space below, make an accurate drawing of Roberta’s rough sketch. Mark andlabel points A, B, C, D, E, F, G, and H on your drawing.

G

A CB

H

F

E

D2"

2"

1"

1 "12

90° 90° 120°

80°

1 "12

189194–200

Math Journal 1, p. 186

Student Page

Lesson 5�6 365

▶ Making Accurate Drawings PARTNER ACTIVITY

from Rough Sketches(Math Journal 1, p. 186)

To support English language learners, discuss the differences between the verbs draw, sketch, and construct. Introduce the distinction between rough sketches and accurate drawings.Explain that lengths and angle measures may be written on a rough sketch, but that does not mean that they are drawn correctly. Draw a rough sketch on the board or overhead and then make a careful drawing from the sketch. (For drawing on the board, multiply the given lengths by 10.)

Example:

A rough sketch of a trapezoid

An accurate drawing of a trapezoid

Caution students that the diagram on journal page 186 is only a rough sketch, so they should not try to measure the angles or line segments. They need to work from the written measurements. Again, they may use any of their tools to make an accurate drawing of the rough sketch.

Bring the class together and have students describe how they made their drawings.

Although many of the angle measures and segment lengths are unknown, students have sufficient information to make a precise drawing.

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187A

Date Time

Drawing a Carnival Map LESSON

5�6

Artie wants to draw a map of the school carnival grounds to give to visitors.

1. The location of each attraction is below the grid on journal page 187B. Plot and label each location on the grid to make a map of the carnival.

2. There will be paths guiding visitors from one carnival attraction to another. Add paths to the map by connecting the sets of attractions given below in the order described. Connect the last attraction on each list to the first one. Then name the shape formed by each set of paths.

a. Entrance, cotton candy, Ferris wheel, ring toss square b. Bean bag toss, swings, balloon animals, dart game parallelogram c. Ferris wheel, slide, mini golf, cake walk trapezoid d. Dart game, petting zoo, fun house triangle e. Bean bag toss, elephant ears, cotton candy right triangle f. Finish the map by drawing one more path, from the mini golf to the dart game.

Suppose that one unit on the grid map represents one block. Use the completed map and the coordinates of the attractions to answer the questions below.

3. How long is the path from the cotton candy to the elephant ears? 4 blocks4. How far would you walk if you went straight from the entrance to the ring toss?

3 blocks5. Artie needs to make sure that the games have enough prizes. He walks from the

bean bag toss to the dart game, then to the mini golf. How far did he walk?

6 blocks6. a. Suppose you just got off the Ferris wheel. Are you closer to the cotton candy or the cake walk? The cotton candy b. How much closer? 3 blocks closer7. Explain how you used the coordinates to find the length of the path in Problem 3.

Sample answer: The y-coordinates of cotton candy andelephant ears are the same, so I did nothing with them. The

x-coordinate of elephant ears is 7 and the x-coordinate of

cotton candy is 3. 7 – 3 = 4, so the distance is 4 blocks.

EM3cuG6MJ1_U05_162-203.indd 187A 1/21/11 11:45 AM

Math Journal 1, p. 187A

Student Page

Date Time

Drawing a Carnival Map continuedLESSON

5�6

Entrance (0,0) Cotton candy (3,0) Ferris wheel (3,3)

Ring toss (0,3) Cake walk (3,9) Slide (5,5)

Elephant ears (7,0) Bean bag toss (7,3) Swings (10,1)

Fun house (6,10) Dart game (7,7) Mini golf (5,7)

Petting zoo (10,10) Balloon animals (10,5)

0

1

2

3

4

5

6

7

8

9

10

0 6 75432 10981

y

x

School Carnival

Entrance

Fun housePetting zoo

Cotton candy

Minigolf

Swings

Slide

Ferris wheel

Balloonanimals

Dart game

Bean bag toss

Cake walk

Elephant ears

Ring toss

EM3cuG6MJ1_U05_162-203.indd 187B 1/21/11 11:45 AM

Math Journal 1, p. 187B

Student Page

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

Here is one possible sequence of steps:

1. Draw a right angle with sides 2 inches and 3 1 _ 2 inches. Label the angle GAC.

2. Mark point B on ⎯⎯⎯ AC , 1 1 _ 2 inches from point A.

3. Draw a line segment perpendicular to ⎯⎯⎯ AC at point B. Mark point F on this segment, 2 1 _ 2 inches above point B. Mark point H on ⎯⎯⎯ BF , 1 1 _ 2 inches above point B.

4. Draw a 120° angle at point C. (The sketch shows how to position this angle.)

5. Draw an 80° angle at point H. (The sketch shows how to position this angle.)

6. Draw a line segment from point G through point F and extend the side of the angle C. Label the intersection E.

7. Extend the side of angle H to ⎯⎯⎯ CE . Label the intersection D.

2 Ongoing Learning & Practice

▶ Using a Grid Map

INDEPENDENT ACTIVITY

(Math Journal 1, pp. 187A and 187B)

Students plot and connect points on a grid map to represent the locations of attractions and paths at a carnival. They subtract coordinates to find the lengths of various routes through the carnival. Before students begin, review how to use coordinates and absolute value to find distances on a coordinate grid.

▶ Math Boxes 5�6

INDEPENDENT ACTIVITY

(Math Journal 1, p. 187)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 5-9. The skill in Problem 4 previews Unit 6 content.

Writing/Reasoning Have students write a response to the following: Identify another rotation in Problem 1 that will produce an image with vertices at the same location

as the image you got by performing a 90° counterclockwise rotation. If BDFH is rotated 270° clockwise, the vertices of the image will be in the same location as when BDFH is rotated 90° counterclockwise.

Ongoing Assessment: Math Boxes Problem 1 �

Recognizing Student Achievement

Use Math Boxes, Problem 1 to assess students’ ability to rotate a figure and name points on a coordinate grid. Students are making adequate progress if they can correctly plot the image and accurately name the coordinates of each vertex.

[Measurement and Reference Frames Goal 3; Geometry Goal 3]

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Study Link Master

STUDY LINK

5�6 Congruent Figures and Copying

178

Name Date Time

Column 1 below shows paths with the Start points marked. Complete each path in Column 2 so that it is congruent to the path in Column 1. Use the Start points marked in Column 2. In Problems 2 and 3, the copy will not be in the same position as the original path.

(Hint: If you have trouble, try tracing the path in Column 1 and then slide, flip,or rotate it so that its starting point matches the starting point in Column 2.)

Example: These two paths are congruent, but they are not in the same position.

Start

Start

Start

Start

Start

Start

Start

Start

1.

Column 1 Column 2

2.

3.

Math Masters, p. 163

187

Math Boxes LESSON

5�6

Date Time

4. Write a number sentence for each word sentence. Then tell whether the number sentence is true or false.

1. Rotate rhombus BDFH 90° counterclockwise about point F (0,0). Then plot and label the vertices of the image that results from that rotation.

B�� ( , ) D� � ( , )

F� � ( , ) H� � ( , )001�20�4

2. Use the partial-quotients algorithm todivide the numerator by the denominator.Round the result to the nearest hundredthand rename the result as a percent.

�196� � 0. � % 5656

3. Add, subtract, or multiply.

a. 2 �36� � 3 �

12� �

b. 4 � 2 �35� �

c. 2 �14� � 1�

23� �

d. �23� � �

45� �

6

55–57 84–8689

241–243

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1

2

4

3

5

–1

–2

–3

–4

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y

x

D'B'

H'F'F

H

B

D

180 234

1�25�

�172�

�185�

false5 � 8 � 4515 � 10 � 272 / (32) � 8

truetrue

True or Word Sentence Number Sentence

False?

Five times eight is equal to 45.

15 is greater than 2 less than 10.

If 72 is divided by the square of 3,the result is 8.

�2 �1

�

Math Journal 1, p. 187

Student Page

Lesson 5�6 367

▶ Study Link 5�6

INDEPENDENT ACTIVITY

(Math Masters, p. 163)

Home Connection Students copy figures made of line segments, using a grid as a guide. In Problems 2 and 3, the copies have different orientations than the originals.

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Using Congruence to 5–15 Min

Sort Shapes(Math Masters, pp. 164 and 165)

To provide experience applying vocabulary and properties of congruence, have students cut out the 16 quadrangles and 6 set labels on Math Masters, page 164. They place the quadrangles in the rings of a Venn diagram on Math Masters, page 165 according to each pair of labels. Students can use a ruler to compare side lengths and a protractor to compare angle measures.

NOTE Explain to students that the label Adjacent Sides Congruent for Pair C means that at least two adjacent sides must be congruent.

NOTE Consider collecting and saving the quadrangles for the optional Readiness activity in Lesson 5-10.

3 6

7

8

11

13

All Right AnglesA All Sides CongruentA

2

3

4

67

8

9

11 10

1514

13

All Pairs Opposite Sides CongruentB At Least 2 Acute AnglesB

1

3

4

5

6

7

811

12

13

16

Adjacent Sides CongruentC At Least 1 Right AngleC

EM3cuG6TLG1_363-368_U05L06.indd 367 12/18/10 1:48 PM

Name Date Time

Graph Paper (1 in.)

Math Masters, p. 418

Teaching Aid Master

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

ENRICHMENT

SMALL-GROUP ACTIVITY

▶ Exploring Isometry 15–30 Min

Transformations and Congruence(Math Masters, p. 418)

A pentomino is an arrangement of five unit squares that are joined along their edges.

This is a pentomino: This is not a pentomino:

To further explore isometry transformations and congruence, students use five same-size square tiles to form as many different pentominoes as possible. They draw each pentomino on Math Masters, page 418, cut them out, and tape/glue them onto a large sheet of paper. Students apply their knowledge of isometry transformations (translations, reflections, rotations) to check that each pentomino is different.

The 12 possible pentominoes are shown below.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer more practice using geometry vocabulary terms to describe polygons and other shapes, see 5-Minute Math, pages 57, 61, 64, 220, and 229.

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