geometry problem solving student

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Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the McGraw-Hill Mathematics program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 978-0-07-890523-0MHID: 0-07-890523-0

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 10 009 12 11 10 09 08

Illustrators: The Artifact Group, Greg Lawhun, Wayno, Scott Rolfs, Pat Lewis, Jim Callahan, Mark Ricketts

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TABLE of CONTENTS

Letter to the TeacherLetter to the Teacher ......................................................... ......................................................... iviv

Teaching Strategies and AnswersTeaching Strategies and Answers ................................... ................................... v

Reasoning and Proof 1 Series: What’s Shakin’? ........................................................................ 1

2 Reasoning: Money Mystery .................................................................. 4

3 Proof: King of the Learning Lab ........................................................... 5

Practice On Your Own .............................................................................. 6

Triangles and Quadrilaterals 1 Pythagorean Theorem: The Long Walk Home .................................... 7

2 Perpendicular Bisector: The Scavenger Hunt .................................... 10

3 Quadrilaterals: Sunshi Makes a Kite .................................................. 11

Practice On Your Own ............................................................................ 12

Similarity 1 Scale Factors: The Scale of Justice ................................................... 13

2 Ratios: Photo Paper Problem ............................................................. 16

3 Proportions: Radio Riddle .................................................................. 17


Transformations 1 Rotations: Fun By Design .................................................................. 19

2 Reflections: Bank On It ...................................................................... 22

3 Vectors: It’s Your Move ....................................................................... 23


Circles 1 Chords: The Mission .......................................................................... 25

2 Inscribed Angles: Circle Slicing .......................................................... 28

3 Semicircles: Fast Track ...................................................................... 29


Area, Surface Area, and Volume 1 Area: Julia Does Up the Gym ............................................................. 31

2 Surface Area: Problems in Pyramid Painting .................................... 34

3 Volume: What’s Your Volume? ........................................................... 35


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LETTER to the TEACHER

iv

Graphic novels represent a significant segment of the literary market for

adolescents and young adults. They are amazingly diverse, both in terms

of their content and usefulness. Graphic novels are exactly what teens are

looking for—they are motivating, engaging, challenging, and interesting.

They allow teachers to enter the youth culture and students to bring their

“out of school” experiences into the classroom.

Graphic novels have also been used effectively with students with

disabilities, struggling readers, and English learners. One of the theories

behind the use of graphic novels for struggling adolescents focuses on the

fact that the graphic novel presents complex ideas that are interesting and

engaging for adolescents, while reducing the text or reading demands.

However, graphic novels are motivating and engaging for all students.

They allow us to differentiate our instruction and provide universal

access to the curriculum. We hope you’ll find the graphic novels in this

book useful as you engage your students in the study of mathematics

and problem solving.

Sincerely,

Douglas Fisher & Nancy Frey

Douglas Fisher, Ph.D. Nancy Frey, Ph.D.

Professor Associate Professor

San Diego State University San Diego State University

USING GRAPHIC NOVELS:USING GRAPHIC NOVELS:Popular Culture and Mathematics InteractPopular Culture and Mathematics InteractPopular Culture and Mathematics Interact

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TEACHING STRATEGIES and ANSWERS

Graphic Novels in the ClassroomGraphic Novels in the ClassroomAs we have noted, graphic novels are an excellent adjunct text. While they

cannot and should not replace reading or the core, standards-based textbook,

they can be effectively used to build students’ background knowledge,

to motivate students, to provide a different access route to the content, and to

allow students to check and review their work.

Mathematical problem solving is presented in graphic novel format. The

novels contain real-world problems for each of the following mathematical

content strands: Reasoning and Proof, Triangles and Quadrilaterals, Similarity,

Transformations, Circles, and Area, Surface Area, and Volume.

• The first graphic novel that appears in each content strand describes a

real-world problem that is solved in graphic novel format.

• The second and third graphic novels that appear in each content strand

are left to the reader to formulate the solution.

• Finally, there are additional problems for students to practice on their own.

Teaching StrategiesTeaching Strategies

1. Previewing Content You can use a graphic novel as a lesson preview

to activate background and prior knowledge. For example, you may

display a graphic novel on the overhead projector and discuss it with the

class. By doing so, you may provide students with advance information

that they will read later in the book. Alternatively, you may display the

graphic novel and invite students, in pairs or groups, to share their

thinking with one another. Regardless of the approach, the goal is to

activate students’ interest and background knowledge in advance of the

reading.

2. Narrative Writing Use the second and third graphic novels from each

content strand and ask students to solve the posed problem in graphic

novel format. Students should be encouraged to create character

dialogue and complete the story line detailing their solution. Another

alternative is to provide students with the first two pages of the first

graphic novel and ask students to complete the story line with the

solution to the problem posed. Not only does this engage students in

thinking about the content, but it also provides you with some assessment

information. Based on the dialogue that the students create of their

solution, you’ll understand what they already know, what they

misunderstand, and what they do not yet know.

v

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3. Reviewing Content In addition to narrative summaries, graphic novels

can also be used for content review. While there are many reasons to

review content—such as preparing for a test—graphic novels are

especially useful for providing students with a review of past chapters.

You can use a graphic novel from a previous chapter to review its major

concepts.

4. Analysis In the analysis approach, students read the graphic novel to try

to understand the main point the author is making. This approach is

particularly useful after students have covered the content in their

textbook. Encouraging students to pose questions about the text will help

to uncover the main points. For example:

• Why did the author choose this real-world situation to present this

concept we have studied? What are some other real-world situations

that can be used to present this concept?

• What does the graphic novel tell me about concepts we have studied?

Have students write a few sentences answering these questions. Then, have

them summarize what they believe is the main point of the graphic novel.

5. Visualizing Have your students skim over the exercises in the chapter

you are working on or the Practice On Your Own pages. The student

should then pick one exercise and create their own graphic representation

about it. Another option would be to use other forms of multimedia for

their topic. Students could take pictures, make a computer slide-show

presentation, make a video, or create a song.

These are just some of the many uses of graphic novels. As you introduce

them into your class, you may discover more ways to use them to engage

your students in a new method of learning while exercising the multiple

literacies that your students already possess. We welcome you to the world

of learning through graphic novels!

Cary, S. (2004). Going graphic: Comics at work in the multilingual classroom. Portsmouth, NH: Heinemann.

Fisher, D., & Frey, N. (2004). Improving adolescent literacy: Strategies at work. Upper Saddle River, NJ: Merrill

Education.

Frey, N., & Fisher, D. (2004). Using graphic novels, anime, and the Internet in an urban high school. English

Journal, 93(3), 19–25.

Gorman, M. (2002). What teens want: Thirty graphic novels you can’t live without. School Library Journal,

48(8), 42–47.

Schwarz, G. (2002a). Graphic novels for diverse needs: Engaging reluctant and curious readers. ALAN

Review, 30(1), 54–57.

Schwarz, G. (2002b). Graphic novels for multiple literacies. Journal of Adolescent & Adult Literacy, 46,

262–265.

Schwarz, G. (2004). Graphic novels: Multiple cultures and multiple literacies. Thinking Classroom, 5(4), 17–24.

ReferencesReferencesReferences

vi

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ANSWERSANSWERS

vii

Reasoning and ProofReasoning and ProofReasoning: Money Mystery, page 4

Set up a grid that contains the information Toshiro received. Use each piece

of information to eliminate as many boxes as possible. When the grid is

completed, you will know who put the money box in the wrong place.

long

hair

short

hair

curly

hair

black

hair

straight

hairjeans shorts skirt khakis

black

pants

Tina × × × × × × × ×George × × × × × × × ×Alexa × × × × × × × ×José × × × × × × × ×Paul × × × × × × × ×jeans × × × ×shorts × × × ×skirt × × × ×khakis × × × ×black

pants× × × ×

Alexa has curly hair and put the money box in the wrong place.

Proof: King of the Learning Lab, page 5To find the fallacy, justify each step with a property of real numbers.

1. a > 0, b > 0 Given

2. a = b Given

3. ab = b 2 Multiply each side by b.

4. ab - a 2 = b 2 - a 2 Subtract a 2 from each side.

5. a(b - a) = (b + a)(b - a) Factor each side.

6. a = b + a Divide each side by (b - a).

7. 0 = b Subtract a from each side.

8. b = 2b Add b to each side.

9. 1 = 2 Divide each side by b.

It appears that each step has a justification. However, in Step 6, each side of

the equation was divided by the quantity b - a. This can only be done if

b - a ≠ 0. However, it is given in Step 2 that a = b. By substituting b for a,

you find the quantity b - a = b - b = 0.

The fallacy happens in Step 6 because you cannot divide by zero.

Practice On Your Own, page 6

1. C 2. G 3. D 4. J 5. D 6. F

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viii

Triangles and QuadrilateralsTriangles and Quadrilaterals

Perpendicular Bisector: The Scavenger Hunt, page 10The point that is equidistant from the vertices of a triangle is the point

where the perpendicular bisectors of each side intersect.

Use a compass and straightedge to construct the

perpendicular bisectors.

1. Set the compass opening to be slightly larger than

half the length of the line.

2. Draw an arc centered at each endpoint.

3. Draw the perpendicular bisector by connecting the

connecting the points where the two arcs intersect.

Repeat these steps to draw the perpendicular bisector

556 ft

428 ft

382 ft

Happiness Park

for each side of the triangle.

Extend all three perpendicular bisectors until they intersect.

This is the point where Jacob should look for the item.

Quadrilaterals: Sunshi Makes a Kite, page 11 Sketch a diagram of Sunshi’s kite and mark the given information.

Use the Pythagorean Theorem to find the lengths of a 1 and a 2 .

In the smaller triangles in the upper portion of the kite, b = 13

and c = 18.4.

a 2 2 + b 2 = c 2 Pythagorean Theorem

18.4 in.

29 in.

18.4 in.

29 in.

13 in. 13 in.

a 1

a 2

a 2 2 + 13 2 = 18.4 2 Substitute 13 for b and 18.4 for c.

a 2 2 + 169 = 338.56 Evaluate powers.

a 2 2 = 169.56 Subtract 169 from each side.

a 2 ≈ 13.02 Take the square root of each side.

In the larger triangles in the lower portion of the kite, b = 13 and c = 29.

a 1 2 + b 2 = c 2 Pythagorean Theorem

a 1 2 + 13 2 = 29 2 Substitute 13 for b and 29 for c.

a 1 2 + 169 = 841 Evaluate powers.

a 1 2 = 672 Subtract 169 from each side.

a 1 ≈ 25.92 Take the square root of each side.

The longer diagonal should be the length of a 1 + a 2 . a 1 + a 2 ≈ 13 + 26 = 39

Sunshi needs to cut the longer wood piece to be 39 inches.


1. C 2. H 3. A 4. H 5. A 6. G

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ix

SimilaritySimilarity

Ratios: Photo Paper Problem, page 16 Find the ratio of the areas of the papers.

The formula for the area of a rectangle is A = �w.

Write a ratio of the area of the smaller size paper to the area of the larger

size paper.

4 × 6

_ 8 × 12

= 24 _

96 or 1 _

4

The larger size paper is 4 times larger than the smaller size paper. The larger

paper should cost 4 times the cost of the smaller paper.

$0.25 × 4 = $1

The paper for an 8-inch by 12-inch photo should cost $1.

Proportions: Radio Riddle, page 17 Set up a proportion that equates Rosalyn’s height and the length of her

shadow to the tower’s height and the length of its shadow.

height of Rosalyn

__ length of her shadow

= height of the tower

___ length of the tower’s shadow

5.5

_ 3 = x

_ 273

Substitute.

1501.5 = 3x Cross multiply.

500.5 = x Divide each side by 3.

The tower is approximately 500.5 feet tall.


1. C 2. G 3. C 4. G 5. C 6. G

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x

TransformationsTransformations

Reflections: Bank On It, page 22 In order to line up the shot, Sandra considers the right side of the pool table

a line of reflection. She visualizes the location of the left side pocket if it were

reflected over that line. If she aims at the imaginary image of the pocket after

it is reflected over the line, the ball will bank off of the bumper on the side of

the table and into the left, side pocket.

This is because the ball will bounce off the right side bumper at the same

angle as it hits. Because a reflection preserves angle measure, the angle

from the ball to the image of the pocket is the same as the angle between

the point where the ball hits the side bumper and the target pocket.

Line of Reflection

Reflected Pocket

Back Angle

Vectors: It’s Your Move, page 23 Find the horizontal movement and the vertical movement of the knight.

The knight moves a horizontal distance from b8 to c8. This is a move of

1 square.

The knight moves a vertical distance from c8 to c6. This is a move of

2 squares.

The vector Kanya sends to Analiese to indicate his move is <1, 2>.


1. D 2. H 3. B 4. D 5. H 6. B

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xi

CirclesCircles

Inscribed Angles: Circle Slicing, page 28Madison can use properties of inscribed

angles to find the center of the circle.

When a right angle is inscribed in a

circle, the intercepted arc is 180°. Begin

by placing the vertex of a right angle

anywhere on a circle. Mark the points

where the sides of the angle intersect

the circle. Draw a line to connect these

points. This line is a diameter of the circle.

Place the right angle at another point

and draw a second diameter. The point

where the two diameters intersect is

the center of the circle.

The location of the fountain is the point

where the diameters intersect.

Semicircles: Fast Track, page 29Because the width of the rectangle is 160 yards, the radius of each

semicircle is 160 ÷ 2 or 80 yards.

Because the radii of the semicircles are 80 yards, the length of the property

remaining for the straight sections is 300 - 2(80) or 140 yards.

300 yd

80 yd 80 yd

The length of the turns is the circumference of the semicircles. Because the

two semicircles make a whole circle, the total length of the turns is the

circumference of a circle with radius 80 yards.

C = 2πr Formula for circumference of circle

= 2π(80) or about 502 yd Substitute 80 for r.

The lengths of two

straight sectionsplus

the circumference

of the two turnsequals

the length

of track.

2(140) + 502 = 782

The maximum length Nate can make the track is 782 yards.


1. C 2. G 3. B 4. G 5. A 6. C

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xii

Area, Surface Area, and VolumeArea, Surface Area, and Volume

Surface Area: Problems in Pyramid Painting, page 34 Use the formula for the surface area of a pyramid: S = 1 _

2 P� + B, where

P is the perimeter of the base, � is the slant height, and B is the area of the

base. B = 6 × 6 or 36 ft2, and P = 6 × 4 or 24 ft.

SA = B + 1 _ 2 P� Formula for surface area of pyramid

= 36 + 1 _ 2 (24)(8) Substitute 36 for B and 24 for P.

= 36 + 96 or 132 ft 2 Multiply and add.

Alejandro needs paint to cover 132 square feet on the pyramid.

Volume: What’s Your Volume?, page 35 Find the volume of each of Della’s garbage cans. The formula for the

volume of a cylinder is V = πr 2 h, where r is the radius and h is the height.

r = 1 _ 2 d, where d is the diameter; 1 _

2 (34) or 17, and h = 32.

V = πr 2 h Formula for volume of cylinder

= π(17) 2 (32) Substitute 17 for r and 32 for h.

≈ 29,053.4 in 3 Evaluate the power and multiply.

Della collected 3 garbage cans, so she collected 3(29,053.4) or 87,160.2

cubic inches of aluminum.

Find the volume of each of Juanita’s boxes. The formula for the volume of a

rectangular prism is V = �wh, where � is the length, w is the width, and h is

the height.

For one box, � = 3, w = 4, and h = 5. For the other box, � = 2, w = 4, and h = 6.

V = �wh Formula for volume

of rectangular

prism

V = �wh Formula for volume

of rectangular

prism

= 3 × 4 × 5 or 60 ft3 Substitute. = 2 × 4 × 6 or 48 ft3 Substitute.

Juanita collected 60 + 48 or 108 cubic feet of aluminum.

The volume of the garbage cans is in cubic inches, and the volume of the

boxes is in cubic feet. Convert the volumes to the same units. There are

12 inches in 1 foot. Because volume is a cubic measurement, divide Della’s

volume by 123 to find the volume in cubic feet.

87,160.2

_ 123

≈ 50.44

Della collected about 50 cubic feet of aluminum. 108 cubic feet is more than

50 cubic feet, so Juanita collected more aluminum than Della.


1. C 2. F 3. D 4. H 5. B 6. G 7. C

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1

Reasoning and Proof 1: Series

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2

Reasoning and Proof 1: Series (continued)


3

Reasoning and Proof 1: Series (continued)

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4

Reasoning and Proof 2: Reasoning


5

Reasoning and Proof 3: Proof


PRACTICEPRACTICE

6

Reasoning and ProofReasoning and ProofRead each question. Then, fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

1. Which of the following is the inverse of

the statement, If it is Saturday, then

Jennifer is at work?

A If Jennifer is at work, then it is Saturday.

B If Jennifer is not at work, then it is not Saturday.

C If it is not Saturday, then Jennifer is not at work.

D If it is Saturday, then Jennifer is not at work.

2. Which of the following can you conclude

given the statement, Tara was not the

first person in line?

F Tara did have a person behind her in line.

G Tara did have a person in front of her in line.

H Tara did not have a person behind her in line.

J Tara did not have a person in front of her in line.

3. In the diagram below, �� AB is an angle

bisector of ∠DAC.

B

CA

D

Which of the following conclusions does

not have to be true?

A ∠DAC ∠BAC

B A and D are collinear.

C 2(m∠BAC) = m∠DAC

D ∠DAC is a right angle.

4. Which property justifies the following

statement?

If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C.

F Reflexive Property

G Substitution Property

H Symmetric Property

J Transitive Property

5. Which Venn diagram illustrates that all

reality TV shows are on Channel 10?

A

RealityTV Shows

Channel10

B Reality

TV Shows

Channel10

C

RealityTV Shows

Channel10

D

RealityTV Shows

Channel10

6. What is the hypothesis of the statement,

Any two Labradors are similar?

F if two dogs are Labradors

G if two Labradors are dogs

H if Labradors are similar

J if two dogs are similar


7

Triangles and Quadrilaterals 1: Pythagorean Theorem


8

Triangles and Quadrilaterals 1: Pythagorean Theorem (continued)


9

Triangles and Quadrilaterals 1: Pythagorean Theorem (continued)


10

Triangles and Quadrilaterals 2: Perpendicular Bisector

Are you ready for thescavenger hunt?

Here’s a mapwith all the details.

Read it over andbring back what you find.

Any questions? Yeah, it saysthe item is in

a spot equidistantfrom the swing,

the tree andkoi pond.

That’s what you needto figure out!

Hmmm?

Whereshould Jacob

look forthe item?

What doesthat mean?


11

Triangles and Quadrilaterals 3: Quadrilaterals

YOUR TURN!YOUR TURN!Help Sunshi makeHelp Sunshi make

her kite.her kite.

The shorterpiece needs to

bisect the longerpiece so that twothirds of the lengthof the longer piece

forms the lowerportion ofthe kite.

SUNSHI MAKES A KITE

Sunshiis making akite for the

spring parade.

Theshorter pieceof wood is 26inches long.

Sunshi uses aquadrilateral-shapedpiece of fabric with

two consecutive sideseach measuring18.4 inches andtwo consecutive

sides eachmeasuring29 inches.

YOUR TURN!Help Sunshi make

her kite.

?hi!

this isgoing to beawesome!

What lengthshould I cut the

longer pieceof woOd?

1 3

42


PRACTICEPRACTICE

12

Triangles and QuadrilateralsTriangles and QuadrilateralsRead each question. Then, fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

1. When Oscar looks at his roof on the side

of his house, he sees an isosceles

triangle. The legs of the triangle are 16

feet and the base is 10 feet. What is the

measurement of the altitude of the roof?

A 10 feet

B 11.6 feet

C 15.2 feet

D 16 feet

2. Cliff has cut an equilateral triangle out of

a sheet of notebook paper. He then

draws an angle bisector through one of

the angles and cuts along that line. Cliff

now has two triangles. Which word best

describes these two new triangles?

F hypotenuse

G equilateral

H congruent

J acute

3. Desiree is at a swimming pool with

her friends Katie and Michaela. Katie

and Michaela are at one corner of the

31 feet by 20 feet rectangular pool.

Desiree is at the opposite corner of the

pool. Katie swims along the diagonal

of the pool to reach Desiree. Michaela

walks around the sides of the pool to

reach Desiree. Estimate the distance

Katie saves by swimming to Desiree

rather than walking.

A 14 feet

B 23 feet

C 28 feet

D 37 feet

4. In the figure below, n is a whole number.

What is the least possible value for n?

25

n

2n

F 7 G 8 H 9 J 11

5. Delsin is constructing a triangular display

case in the shape of an isosceles

triangle. One of the angles is 40°. Which

of the following could be the measure of

one of the other angles?

A 70°

B 80°

C 110°

D 140°

6. Hallie cuts a hexagon from a piece of

poster board. She uses a protractor to

mark the first interior angle along the

bottom edge of the board as shown

below. What is the m∠1 in the piece of

poster board she cut off?

1

F 120° G 60° H 40° J 30°


13

Similarity 1: Scale Factors

The Scale of Justice

Thanks for helping mewith my civics project,Alex. I’m having a littletrouble getting started.

How so?

Well, I picked a court roomfor my project. And I have

to build a diorama.

Yeah, I even have thecourtroom plan showing

the actual dimensions. See…

Looks like you havewhat you need.

What’s the problem?

Brianna & Alex in

30 feet

20 fe

et

Plaintiff ‘s TableDefendant ‘s Table

Judge’s Bench

WitnessStand

Court clerk’sTable

CourtReporter

Table

Jury RoomJudge’s Chambers

Jury Box

5 ft × 2 ft

5 ft × 2 ft

7 ft × 5 ft

5 ft × 2 ft


14

Similarity 1: Scale Factors (continued)

I’m not sure what sizethings should be. My basefoam board is 40” x 60”

I get it, you need a scale factor.That will help you figure out

the size of the diorama comparedto the real courtroom.

Okay, how doI do that?

Here. I’ll show you.First we’ll start with the

courtroom length and widthcompared to the foamboard

length and width and fillin the dimensions.

Yep, now you cantake the dimensionsin feet on your planand turn them into

inches in yourdiorama.

length of courtroomlength of foamboard

width of courtroomwidth of foamboard

=

=

30 ft60 inches

20 ft40 inches

1 ft2 inches

1 ft2 inches

or

So, since the ratio is1 ft./2 inches for both the

width and the height,my scale for both dimensions

is the same. Awesome!


15

Similarity 1: Scale Factors (continued)

So I can plan out mydiorama dimensionsand start building!

Thanks, Alex!

Let’s see… thecourt clerk’s tableis 5 feet x 2 feet.So in my diarama,

that’s 10” x 4”.

Hey sis, how’s thediorama coming?

Just putting onthe finishing touches.What do you think?

I think I’m getting anA on my Civics project!

One week later.

60 in.

40 in

.

Plaintiff ‘s TableDefendant ‘s Table

Judge’s Bench

WitnessStand

Court clerk’sTable

CourtReporter

Table

Jury RoomJudge’s Chambers

Jury Box14 in. × 10 in.

10 in. × 4 in.

10 in. × 4 in.

10 in. × 4 in.

Terra’s Diagram


16

Similarity 2: Ratios

I thought youhad plenty.

All I have is4” X 6” paper,and I wantedto print these

on 8” X 12”.

I wonder how muchmore that will cost.

How much isa single sheet

of 4” X 6”paper?

Only 25¢.

I wonder if thecost of paper

increasesproportionally.

Probably not,but if it did...

...a sheet of8” × 12”

photo papershould only

cost me...

?

Boy, the printeris getting low on

photo paper!


17

Similarity 3: Proportions


PRACTICEPRACTICE

18

1. The Eiffel Tower in Paris, France, stands

324 meters tall. The Paris Hotel in Las

Vegas has a 1 _ 2 scaled replica of the

tower. How tall, to the nearest foot, is the

tower in Las Vegas? Use the conversion

1 meter ≈ 3.3 feet.

A 99 ft

B 162 ft

C 535 ft

D 1063 ft

2. Each time a sheet of plain 8 1 _ 2 -inch

by 11-inch paper is folded in half, a

rectangle similar to the original rectangle

is formed. What are the dimensions of

the rectangle formed after the paper is

folded four times?

F 4 1 _ 4 in. by 5 1 _

2 in.

G 2 1 _ 8 in. by 2

3 _

4 in.

H 2 1 _ 4 in. by 2 1 _

2 in.

J 1 1 _ 16

in. by 1 3 _

8 in.

3. Travis, who is 5 feet 9 inches, measured

his shadow to be 2 feet 6 inches. At the

time, Taina measured the shadow of the

tree in their backyard to be 7 feet 3

inches. What is the estimated height of

the tree?

A 3 ft 2 in.

B 16 ft 6 in.

C 16 ft 8 in.

D 17 ft 10 in.

4. If you set a copy machine at 120%, what

will be the dimensions of the copy of a

6-inch by 8-inch image?

F 5 in. by 6 2 _ 3 in.

G 7.2 in. by 9.6 in.

H 8 in. by 10 in.

J 720 in. by 960 in.

5. Given that trapezoid BCDE is similar to

trapezoid KLMN, find the length of −−− MN .

15 cm 12 cm

10 cm 8 cm

6 cm

A 12 cm

B 8 cm

C 7.5 cm

D 4.8 cm

6. The dimensions of the home plate in a

professional baseball stadium are shown

in the diagram. An architect is creating a

model of a new baseball stadium that is

a 3 _

8 scale of the actual stadium. What is

the perimeter of a home plate he makes

for the model stadium?

F 14 in.

12 in.

8.5 in. 8.5 in.

12 in.

17 in.

G 21.75 in.

H 54.19 in.

J 58 in.

SimilaritySimilarityRead each question. Then, fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.


19

Transformations 1: Rotations


20

Transformations 1: Rotations (continued)


21

Transformations 1: Rotations (continued)


22

Transformations 2: Reflections


23

Transformations 3: Vectors


PRACTICEPRACTICE

24

TransformationsTransformationsRead each question. Then, fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

1. A reflection has been applied to the

letter N. Which of the following images

has had the same reflection applied?

A C

B D

2. In a board game, moves are made using

translations. Which translation will allow

the black chip to capture the white chip?

F ⟨6, -3⟩ H ⟨5, -3⟩G ⟨-3, 5⟩ J ⟨-4, 2⟩

3. What is the order of rotation for the yard

ornament shown here?

A 2

B 4

C 8

D 16

4. What are the coordinates of the image

of vertex C after a reflection over the

x-axis?

O

y

x-6 -4-8 -2 2

-3

-1

1

3

5

7

B

A

C

A (-2, -4) C (0, -1)

B (2, 4) D (0, 1)

5. A series of transformations are shown.

What is a single transformation to get

from Step 1 to Step 7?

Step 1

Step 7

F vertical reflection

G 90° counterclockwise rotation

H 90° clockwise rotation

J 180° rotation

6. What angle of rotation does Riley use to

completely surround the circle with his

name?

A 30° C 120°

B 60° D 360°

NN

QQ

KK

RR

FF

RileyRile

yR

ileyRileyRileyR

ileyR

iley


25

Circles 1: Chords

There is a distress beaconburied somewhere in the woods.

It’s my mission to locate it.

Marcos in

THE MIsSION

The pressureis on! |'m upagainst the

clock.

If | want to become apart of my community

rescue team, |’ll have toretrieve the beacon...

...before mytime runs out.

blip! blip! blip! blip! blip! blip! blip! blip! blip! blip! blip! blip!blip! blip! blip! blip! blip! blip! blip! blip! blip! blip! blip! blip!blip! blip! blip! blip! blip! blip! blip! blip! blip! blip! blip! blip!




26

Circles 1: Chords (continued)

At this point, | need to walk at a rightangle, away from the sound, until | canno longer hear the signal--point C.

The beacon emits a signalthat can be heard within a

30-meter radius.

To locate the center of a circle with a 30-meter radius, | must use the properties of chords

and diameters.

| can hear the signal at point A, so | shouldwalk in a straight line until | can no longer

hear the signal--point B.

If a radius of a circle intersects a chordat a right angle, then the diameter bisectsthe chord. | should find the midpoint of thepath AB. That is where the sound should

be the loudest.

A B

A B

C

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

| haveto hurRy. the

clock isticking!


27

Circles 1: Chords (continued)

Because it is perpendicular toAB at its midpoint, this path is part of

a radius of the circle that has thebeacon at its center.

thedistresSbeaconis buriedhere!

To find the rest of the radius,| need to turn 180˚ and walkthe opposite direction until| can no longer hear the

signal--point D.

The midpoint of the path, CD, isthe center of the circle.

Okay,this is

the centerof thecircle.

D

A B

C

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

blip!blip!blip!

blip!blip!blip! blip!blip!blip!

CONGRaTULATIONS,MARCOS.

youmade the

team!

END

almostthere, and I thinkI’m making goOd

time, toO.


28

Circles 2: Inscribed Angles

What’chadrawing, Sis?

with Madison and her little brother

I need to come upwith a sketch

for my communityservice project.

Is it a horse?

No, silly!I’m drawing acircular water

garden forthe park.

I can use my paper cupto make a circle.

This will be the total area of my garden.

And thehorse will bein the middleof the garden?

No horse!I want to place a fountain

directly in the center,but all I have with me is this

cup and a fewpieces of paper

to use as right anglesand a straightedge.

If I keep talking insteadof drawing, I’m gonna be

a little hoarse!

That wouldmake

you a pony!

What canMadison do

to findthe center

of hercircle?


29

Circles 3: Semicircles


PRACTICEPRACTICE

30

1. The diameter of Earth at the equator is

about 7926 miles. An airplane flies at

600 mph about 5.5 miles above Earth in

a path that follows the equator. About

how long will it take for this plane to

travel all the way around Earth?

A 42 days C 42 hours

B 21 days D 21 hours

2. In a circular theater, Laura wants to sit

along the edge of the room, as close to

the center of the theater as possible. In

the diagram below, she is seated at

Point L. What is the minimum angle of

vision that Laura needs to be able to see

the entire stage?

stage

118°110°

L

F 33° H 76°

G 66° J 152°

3. A guest that wants the largest portion

should select a slice from which of the

following pizzas?

A a 10-inch pizza cut into four equal-sized pieces

B a 14-inch pizza cut into six equal-sized pieces

C a 16-inch pizza cut into eight equal-sized pieces

D an 18-inch pizza cut into ten equal-sized pieces

4. Circle C has radius r and ABCD is a

rectangle. Find DB.

F r √ � 3

G r

H r √ � 2 _

2

J r √ � 3 _

2

5. James bakes an apple pie in an

8-inch pie plate. He cuts the pie twice

through the center to make 4 equal

pieces. What is the length of the arc in

each piece that the outermost crust

makes?

A 2π inches

B 3π inches

C 4π inches

D 8π inches

6. In geometry class, Callie was given a

piece of grid paper with the graph of the

circle with equation

(x + 2) 2 + (y + 2) 2 = 25.

She must write the equation of

another circle that can be graphed

on the same piece of paper and

completely fit into the circle she was

given. Which equation could be the

one Callie wrote?

A (x + 1) 2 + (y - 1) 2 = 9

B (x - 1) 2 + (y + 1) 2 = 9

C (x + 1) 2 + (y + 1) 2 = 9

D (x - 1) 2 + (y - 1) 2 = 9

CirclesCirclesRead each question. Then, fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.


31

Area, Surface Area, and Volume 1: Area


32

Area, Surface Area, and Volume 1: Area (continued)


33

Area, Surface Area, and Volume 1: Area (continued)


34

Area, Surface Area, and Volume 2: Surface Area

Your minature golf course is gonna be cool

Alejandro and Tyler in...

Yeah... all we needto do is paint this

pyramid!

when we’re done!I want it to look like sandstone

when it’s finished.

That will besweet!

It would be so mucheasier to figure if the sides were

square.

Like you?

How wide is the base,Alejandro?

I come upwith 6 feet

wide.

And what is theslant height?

Looks like8 feet.

What is thesurface area

that Alejandroand his

buddy needto paint?


35

Area, Surface Area, and Volume 3: Volume

Juanita & Della in

What’s yourVolume?

You FigureIt Out!

Who HasMore

Volume?

Okay. All 3 ofmy trash cans are34 inches high and

have a 32 inchdiameter.

Whatever! Let’s figureout the volume to see

who has more.

How do youfigure? I think I’ve got more!

I think it’sobvious. My boxeshave more volume.

Hi, Juanita! Are youready to see who has the

most aluminum cans?

RECYCLINGRECYCLINGCENTER Hey, Della!

Take a look at my boxes. Ithink I’ve got you beat!

You’re on! Whoever hasless can pay for movie

tickets tonight.

RECYCLINGRREECCYYCCLLIINNGGCENTERSpr ing C leanup C o n t e s t

My first box is 3feet by 4 feet by5 feet. The otherbox is 2 feet by4 feet by 6 feet.


PRACTICEPRACTICE

36

Area, Surface Area, and VolumeArea, Surface Area, and VolumeRead each question. Then, fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.

1. Hinto is wrapping a box that is 14 inches

long, 8 inches wide, and 2 inches tall. At

the very minimum, how much wrapping

paper will he need?

A 224 in 2 C 312 in 2

B 224 in 3 D 312 in 3

2. During the week, Evita drinks 6 glasses

of water each day using the glass shown

below on the left. On Saturday and

Sunday, she drinks 5 glasses of water

using the glass shown below on the

right. How much more water does she

drink each weekday than each day on

the weekend?

6 in.

3 in.

4 in.

2 in.

F about 3 in 3

G about 8 in 3

H about 250 in 3

J about 670 in 3

3. Nestor wants to know the volume of the

Great Pyramid of Giza in Egypt. The

height of the pyramid is 455 feet and the

length of each side of the base is 756

feet. What is the approximate volume of

the Great Pyramid of Giza?

A 38,278,280 ft 3

B 52,170,300 ft 3

C 65,012,220 ft 3

D 86,682,960 ft 3

4. Roberto pulled the pages out of a

catalogue and laid them side by side.

The catalogue had 750 8-inch by 10-inch

pages. What was the total area

covered by the pages?

F 750 in 2

G 6,000 in 2

H 30,000 in 2

J 60,000 in 2

5. Cleveland wants to build a fence around

the circular field where his horses graze.

The diameter of the field is 500 feet.

Approximately how many feet of fencing

does Cleveland need?

A 786 feet C 196,350 feet

B 1570 feet D 785,399 feet

6. Ella finds an artist that will paint an

ornamental garden ball with any design

she wants, but she charges $0.06 per

square inch of surface area. The ball

she wants painted has a diameter of

12 inches. About how much will it

cost her for a design that covers the

entire ball?

F $4.50 H $54.00

G $27.00 J $90.00

7. Bianca has a rectangular fish tank. Its

dimensions are 4 feet by 3 feet by 2 feet.

How much water does she need to fill

the tank if 6 cubic feet are taken up by

coral and sand?

A 144 ft 3 C 18 ft 3

B 24 ft 3 D 3 ft 3
