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Common Core State Standards

Lesson 10�3 805

�

Properties ofReflections

Objective To guide the discovery of basic properties

of reflections.o

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 193, 194, 197, 198

Key Concepts and Skills• Measure length. 

[Measurement and Reference Frames Goal 1]

• Draw and describe congruent figures. 

[Geometry Goal 2]

• Explore basic properties of reflections. 

[Geometry Goal 3]

• Solve problems involving spatial

visualization. 

[Geometry Goal 3]

Key ActivitiesStudents use a transparent mirror to

discover basic properties of reflections.

They draw reflected images.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 309. [Geometry Goal 3]

Key Vocabularyline of reflection

MaterialsStudy Link 10�2

Math Masters, p. 309

per partnership: 1 transparent mirror �

ruler � slate

World Tour Option: Visiting a Second Country in Region 4Math Journal 2, pp. 329–331, 338,

and 339

Student Reference Book, pp. 276,

277, 281, 288–291, 297, and 302–305

Math Masters, pp. 419–421 (optional)

Students resume the World Tour

in Region 4.

Growing PatternsMath Journal 2, pp. 277A and 277B

Students complete growing patterns.

Math Boxes 10�3Math Journal 2, p. 278

Students practice and maintain skills

through Math Box problems.

Study Link 10�3Math Masters, p. 310

Students practice and maintain skills

through Study Link activities.

READINESS

Creating Reflections with Pattern Blocks or Centimeter CubesMath Masters, pp. 389 and 403

centimeter cubes or pattern blocks � blank

paper or centimeter grid paper

Students create designs with pattern blocks

or centimeter cubes and reflect the designs.

ENRICHMENTExploring Reflections of 3-Dimensional FiguresMath Masters, p. 447

centimeter cubes

Students use centimeter cubes to construct

3-dimensional buildings and their reflections.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, p. 149

blank paper

Students solve problems involving reflections.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

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806 Unit 10 Reflections and Symmetry

�

LESSON

10�3

Name Date Time

Reflections

309

1. Use a transparent mirror to draw the reflected image of the head of the dog.

preimage imageline of reflection

2. On the back of this page, describe how the two drawings in Problem 1

are alike and how they are different.

3. Draw a picture on the left side of the line. Ask your partner to use

a transparent mirror to draw the reflected image of your picture.

preimage imageline of reflection

Answers vary.

106

Answers vary.

�

304-326_EMCS_B_MM_G4_U10_576965.indd 309 2/14/11 4:56 PM

Math Masters, p. 309

Teaching Master

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASS ACTIVITY

Have partners show their mirror-image poses to the class. Strike your own pose. Have students show the mirror image. Slowly change your pose and challenge students to mirror your changes.

Tell students that in this lesson they will investigate some of the basic properties of reflections.

� Examining Relationships WHOLE-CLASS ACTIVITY

between an Object and Its Reflected Image(Math Masters, p. 309)

Tell partnerships to put the recessed edge of the mirror on the line next to the dog’s head. When they look directly through the mirror, they will see the image of the dog’s head. (It is best to look through the mirror at eye level.) Ask students to carefully draw the image of the dog’s head on the paper where they see it through the mirror and describe, on the back of Math Masters, page 309, how the drawings are alike and how they are different.

When all students have completed the tasks, bring the class together and lead them in the following exploration:

� Point out that the picture of the dog’s head to the left of the line of reflection is the preimage. Remind students that pre- means “before,” so they can think of this as the “before image.”

� Ask: How are the two drawings alike? How are they different? Sample answers: The drawings are congruent; they are the same size and shape. They look exactly alike, except that the heads are facing in opposite directions.

Getting Started

Math Message Stand facing a partner. One partner poses. The other partner positions his or her body to be the mirror image of the partner. Then switch roles.

Study Link 10�2 Follow-Up Have students compare answers and share the preimages and images they created for Problem 7.

Mental Math and Reflexes Dictate large numbers for students to write on their slates. Suggestions:

14,687,320 100,456,892 34,206,408,900

45,318,972 203,541,986 7,940,005,030

89,245,307 106,507,349 49,802,103,062

For each number, ask questions such as the following:

• Which digit is in the millions place?

• What is the value of the digit x?

• How many hundred millions are there?

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� Have students mark a point A anywhere on the preimage. Then ask them to look through the mirror, mark the image of point A, and label the image A′ (“A prime”). Write A′ on the board and explain that the little mark by the A is read “prime.” The prime mark shows that A and A′ are different, but related, points.

Check that all students have labeled the corresponding points A and A′ on the dogs. The points may be anywhere, but they must be in the same place on the image as on the preimage.

� Have students use a ruler to measure the distance from point A to the line of reflection and from point A′ to the line of reflection. Ask: What did you find out about the distances? They are the same.

� Have students mark and label several other points (B, C, and so on) on the preimage; use the mirror to mark and label the corresponding points (B′, C′, and so on) on the image; and check with a ruler that corresponding points are the same distance from the line of reflection.

� Remind students of the Pocket-Billiards Game. Ask: Where did you place the mirror in order to get the ball in the pocket? Halfway between the ball and the pocket

� Finally, have each student draw a picture on the left side of the reflection line in the bottom half of the page. Then ask partners to exchange papers and draw the image of their partner’s picture.

Ongoing Assessment:Recognizing Student Achievement

Math Masters

Page 309 �Problems 1 and 2

Use Math Masters, page 309, Problems 1 and 2 to assess students’ ability to

sketch and describe a reflection. Students are making adequate progress if they

are able to use a transparent mirror to accurately draw the reflected image of the

dog’s head and describe how the preimage and image are alike and different.

Some students may describe the relationship of the points on the drawings to the

line of reflection.

[Geometry Goal 3]

� Folding Paper to Observe WHOLE-CLASS ACTIVITY

Reflected Images(Math Masters, p. 309)

Lead the class through the following procedure:

1. Fold Math Masters, page 309 in half lengthwise along the lines of reflection.

2. Hold the folded sheet up to the light to check if the preimage and image in Problem 1 and the preimage and image in Problem 3 match.

Ask: For each problem, are the preimage and image congruent? yes How do you know? They are the same size and shape.

PROBLEMBBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEMMMLEBLELBLEBLELLLBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBLBLBBLBLBLBLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMEEEEEEEEEEEELLELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB ELEELEEMMMMMMMMMOOOOOOOOOBBBLBLBLBBLBBBLOOORORORORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRGGGGGLLLLLLLLLLLLLVINVINVINVINNNNVINVINVINNVINVINVINVINVV GGGGGGGGGGGOLOOOLOOLOLOLOO VVINVINLLLLLLLLLVINVINVINVINVINNVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLOLLOO VVVLLLLLLLLLLVVVVVVVVOSOSOSOOSOSOSOSOSOSOOSOSOSOSOOOSOSOOSOSOSOSOSOSOSOSOOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVLLLLLLLLVVVVVVVVVLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

Lesson 10�3 807

Date Time

5. a. Draw the next shape in the pattern.

b. Describe the rule used to extend the pattern.

6. a. Extend the number pattern: 1, 4, 10, 19, 31 , 46 , 64

b. Describe the rule used to extend the pattern.

The numbers in Problem 6 are called tetragonal numbers. The shapes in Problem 5

show tetragonal numbers as arrays of dots.

7. a. Make a list of the first 20 tetragonal numbers. Circle the ones digit in each number.

1, 4, 10, 19, 31 , 46 , 64 , 85 , 109 , 136 , 166 , 199 , 235 ,

274 , 316 , 361 , 409 , 460 , 514 , 571

b. Make a list of the ones digits in the first 20 tetragonal numbers.

1, 4, 0, 9, 1 , 6 , 4 , 5 , 9 , 6 , 6 , 9 , 5 , 4 , 6 , 1 , 9 , 0 , 4 , 1

c. Describe any patterns you see in the list of numbers in Part b.

d. Do you think this pattern will continue? Answers vary. Test your answer.

Sample answer: The dots switch back and forth between

closed and open. Each time a new set of dots is added, the

total number of dots increases by the next multiple of 3.

Sample answer: Start with 1. Add 3. Then keep adding the

next multiple of 3 to the sum. 1 + 3 = 4; 4 + 6 = 10; 10 + 9 = 19; 19 + 12 = 31; 31 + 15 = 46; 46 + 18 = 64

Try This

Growing Patterns continuedLESSON

10�3

The 10th and 11th numbers are 6s. The pattern of numbers

ends at the first 6 and then repeats in reverse. It’s like there’s

a line of reflection between the 10th and 11th numbers.

277A-277B_EMCS_S_MJ2_G4_U10_576426.indd 277B 3/16/11 11:13 AM

Math Journal 2, p. 277B

Student Page

Growing PatternsLESSON

10�3

Date Time

1. a. Draw the next shape in the pattern.

b. Describe the rule you used to extend the pattern.

2. a. Extend the number pattern: 2, 6, 12, 20, 30 , 42 , 56

b. Describe the rule you used to extend the pattern.

3. a. Draw the next shape in the pattern.

b. Describe the rule you used to extend the pattern.

4. a. Extend the number pattern: 1, 5, 12, 22, 35 , 51 , 70

b. Describe the rule you used to extend the pattern.

The number of rows and columns in each rectangle increases by 1 dot.

Start with 2. Add 4. Then keep adding the next multiple

of 2 to the sum. 2 + 4 = 6; 6 + 6 = 12; 12 + 8 = 20;

20 + 10 = 30; 30 + 12 = 42; 42 + 14 = 56

Each pentagon is made up of a square and a triangle. The

number of dots on each side of the square and on each side of

the triangle increases by 1.

Start with 1. Add 4. Then keep adding 3 more than the number

you added before. 1 + 4 = 5; 5 + 7 = 12; 12 + 10 = 22;

22 + 13 = 35; 35 + 16 = 51; 51 + 19 = 70

Sample answer:

Sample answer:

Sample answer:

Sample answer:

277A-277B_EMCS_S_MJ2_G4_U10_576426.indd 277A 3/30/11 12:56 PM

Math Journal 2, p. 277A

Student Page

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808 Unit 10 Reflections and Symmetry

STUDY LINK

10�3 Reflections 106

Name Date Time

Shade squares to create the reflected image of each preimage.

preimage image

line ofreflection

1. image preimage

line ofreflection

2.

preimage image

line ofreflection

3. image preimage

line ofreflection

4.

5. 54 º 6 = 324 6. 29 º 36 = 1,044

7. 2,025 = 45 º 45 8. 52,731 = 837 º 63

Practice

304-326_EMCS_B_MM_G4_U10_576965.indd 310 2/14/11 4:56 PM

Math Masters, p. 310

Study Link Master

2. Complete.

Rule: in out

3

_

8

7

_

8

1

_

6

9 _ 10

7 _ 20

1

_

4

3

_

4

Math Boxes LESSON

10 � 3

Date Time

3. Solve each open sentence.

a. 32.5 + y = 37.6 y =

b. 123.5 - k = 102.2 k =

c. b + 67.91 = 78.32 b =

d. 405.08 - w = 231.57 w =

5.1

21.3

10.41

173.51

1. a. If you spin this spinner 400 times,

about how many times would you

expect it to land on

red? 300 times

blue? 50 times

b. Explain how you figured out how many times

you would expect the spinner to land on blue.

About 1 _ 8 of the board is blue,

+ 1 __ 2

4

__ 10

4

_ 6

17

__

20

34––3755 57

4. Angle ART is an obtuse

(acute or obtuse) angle.

Measure of ∠ART = 125 °

140

82–86

93142 143

A

R T

red

redwhiteblue

red so the spinner should land

on blue about 1 _ 8 of the 400

spins. 1 _ 8 of 400 is 50.

5. Alberto and Lara estimated the weight

of their adult cat. What is the most

reasonable estimate? Fill in the circle

next to the best answer.

A. 2 pounds

B. 10 pounds

C. 50 pounds

D. 100 pounds

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Math Journal 2, p. 278

Student Page

2 Ongoing Learning & Practice

� World Tour Option: Visiting SMALL-GROUP ACTIVITY

a Second Country in Region 4(Math Journal 2, pp. 329–331, 338, and 339; Student Reference Book,

pp. 276, 277, 281, 288–291, 297, and 302–305; Math Masters,

pp. 419–421)

Social Studies Link If you have chosen to extend the scope of the World Tour for your class, have small groups visit

a second country in Region 4. You may let them choose which countries to visit, or you may assign a country to each group. Remind students to update their Route Log (if they are keeping one) and Route Map and to complete Country Notes for the country they visit.

� Growing Patterns PARTNER ACTIVITY

(Math Journal 2, pp. 277A and 277B)

Students complete growing patterns of numbers and shapes. Encourage students to discuss the patterns.

After students complete the journal pages, point out that the numbers in Problem 2 are called rectangular numbers and the numbers in Problem 4 are called pentagonal numbers. The shapes in Problem 1 show the rectangular numbers as arrays of dots. The shapes in Problem 3 are nonregular pentagons.

� Math Boxes 10�3 INDEPENDENTACTIVITY

(Math Journal 2, p. 278)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-6. The skill in Problem 5 previews Unit 11 content.

Writing/Reasoning Have students write a response to the following: How did you choose the weight for the cat in Problem 5? Sample answer: I eliminated two of the choices. I knew the cat weighed more than 2 pounds of butter, and some of my friends weigh about 50 pounds. That made 10 pounds the best choice.

� Study Link 10�3 INDEPENDENTACTIVITY

(Math Masters, p. 310)

Home Connection Students shade grid squares to create images of given preimages. A transparent mirror is not required to do this page.

Adjusting the Activity Have students determine the latitude and longitude of the capital city of the country they visit.

AUDITORY � KINESTHETIC � TACTILE � VISUAL

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3 Differentiation Options

READINESS PARTNER ACTIVITY

� Creating Reflections with Pattern 5–15 Min

Blocks or Centimeter Cubes(Math Masters, pp. 389 and 403)

To explore reflections using concrete materials, have students use pattern blocks or centimeter cubes to create preimages for a partner to reflect.

� For centimeter cubes, have students fold a sheet of centimeter grid paper in half on a grid line. One partner creates a design on one side of the fold for the other partner to reflect.

� To use pattern blocks, have students fold a blank piece of paper in half to create a line of reflection. One student creates a design on one side of the fold for a partner to reflect on the other side of the fold.

On an Exit Slip, have students explain how they made their reflections.

ENRICHMENT PARTNER ACTIVITY

� Exploring Reflections of 15–30 Min

3-Dimensional Figures(Math Masters, p. 447)

To apply students’ understanding of reflections, have partnerships use centimeter cubes to construct a 3-dimensional building, then trade buildings and construct the reflections. Each point in the first building should correspond to a point in the reflected building. Have students draw the original cube building and the reflection of the building on isometric dot paper.

EXTRA PRACTICE SMALL-GROUP ACTIVITY

� 5-Minute Math 5–15 Min

To offer students more experience with reflections, see 5-Minute Math, page 149.

Lesson 10�3 809

Cube structure with reflection

Cube structure and reflection drawn

on isometric dot paper

Centimeter cube design with reflection

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