□ PROVIDES CHALLENGING INSTRUCTIONAL ACTIVITIES FOR 12 MATHEMATICS STRATEGIES

□ STRENGTHENS PROBLEM-SOLVING SKILLS AND IMPROVES MATH-RELATED WRITING SKILLS

□ FEATURES ASSESSMENT IN MATHEMATICS, INCLUDING SELECTED-RESPONSE AND CONSTRUCTED-RESPONSE PROBLEMS

EXTENSIONS INMATHEMATICS™ SERIES

G

EXTENSIONS IN MATHEMATICS

3

STRATEGY ONE Building Number Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

STRATEGY TWO Using Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

STRATEGY THREE Applying Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

STRATEGY FOUR Applying Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

STRATEGY FIVE Applying Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

STRATEGY SIX Applying Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54

STRATEGY SEVEN Converting Time and Money . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

STRATEGY EIGHT Converting Customary and Metric Measures . . . . . . . . . . . . . . . . . .74

STRATEGY NINE Using Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84

STRATEGY TEN Using Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .94

STRATEGY ELEVEN Determining Probability and Averages . . . . . . . . . . . . . . . . . . . . .104

STRATEGY TWELVE Interpreting Graphs and Charts . . . . . . . . . . . . . . . . . . . . . . . . .114

STRATEGIES ONE–TWELVE REVIEW . . . . . . . . . . . . . . . . . . . . . . . .124

Table of Contents

Learn About GeometryIn geometry, figures can be moved, or transformed, in different ways. The new figure is a same-size, same-shape image of the original figure.

You can use a coordinate grid to show a transformation. How has each figurebeen transformed?

Read the problem and the notes beside it.

Delia is making a design with this right triangle. She will reflect the triangle over side DF

—. She will rotate the triangle in a clockwise

direction around point F. The rotation will be a �� turn. Finally, Delia will draw a translation image that is 2 units to the right of the original image.

What will Delia’s design look like?

Studying the problem

Thinking about the strategy

94

Point E is how manyunits above the line of

reflection?

How much must the figure rotate to make

a �� turn?

How many units musteach point move to the

right for the translationimage?

How can Delia use a coordinate grid to solve the problem?

fountain spinner marble

clockwise

�� turn

�� turn

counterclockwise

Reflection: each point flipsacross a line of reflection.

Rotation: each point on theoriginal figure rotates the

same amount.

Translation: each pointmoves the same distance

to the right.

4 units

y

6

5

4

3

2

1

01 2 3 4 5 6

x

E

D F

line ofreflection

�

Using GeometrySTRATEGYTEN

95

A coordinate grid is a graphic organizer that you can use to draw transformationimages of geometric figures. Delia used this coordinate grid to draw thereflection, rotation, and translation images.

Delia’s design is shown on the coordinate grid above.

Read what Delia wrote to explain how she used a coordinate grid to solve the problem.

First, I drew the reflection image of triangle DEF. I used DF— as the line of reflection. Since point E is 3 units above DF—, I put a point 3 units below DF—. The other two points of the triangle are D and F. I drew a line to connect the point I plotted to points D and F. The rotation image is a

41 turn around point F. In a

41 rotation, the side of the image that

corresponds with DF— makes a right angle with DF— in the original triangle. Since DF— is 4 units long, I plotted a point 4 units above point F. From that point, I counted 3 units to the right and plotted another point. This side corresponds to DE—, which is 3 units long. From that point, I drew a line to connect to point F. This is a 41 clockwise rotation. To draw the translation image, I plotted 3 points. Each point is 2 units to the right of the corresponding point in the original figure. In the original figure, the points have the following coordinates: D (1, 1); E (1, 4); F (5, 1). Since the image is 2 units to the right, each x coordinate is increased by 2. The translation image has the coordinates (3, 1); (3, 4); (7, 1).

Understanding the solution

Studying the solution

y

8

7

6

5

4

3

2

1

–8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8x

–1–2–3–4–5–6–7–8

E

D F

•

• •

• •

•

96

Solve a ProblemRead the problem. As you read, think about how you could use a coordinate gridto solve the problem.

Use the information from the problem to complete this coordinate grid. Then write your answer below.

Finding the solution

Studying the problem

Answer:

Tam has the following homework for math class.

1. Copy this figure onto a coordinate grid.Label it with the number 1.

2. Draw a reflection image over RU—

. Labelthe reflection image with the number 2.

3. Draw an image that is a �� counterclockwiseturn around point U. Label the rotationimage with the number 3.

4. Draw a translation image that is 5 unitsup from the original figure. Label it withthe number 4.

How did Tam complete the homework assignment? What are the coordinates of the translation image?

y

6

5

4

3

2

1

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6x

–1–2–3–4–5–6

S

T

R

•

•

•

•

U

y

10

9

8

7

6

5

4

3

2

1

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10x

–1–2–3–4–5–6–7–8–9

–10

97

Reread “Understanding the solution” on page 95, which tells how Delia used acoordinate grid to solve a problem. Then write your own explanation of how youcompleted the coordinate grid on page 96 and found your solution.

Use your coordinate grid on page 96 to answer these questions.

1. What is true about the distances between corresponding points and the lineof reflection for the original figure and the reflection image.

2. What are the coordinates of the original figure?

3. Compare the corresponding coordinates of the original figure with thetranslation image. What do you notice?

4. What is true about a full clockwise rotation image of the original figure?

5. What are the coordinates of the reflection image?

Applying the solution

Explaining the solution

98

Learn More About GeometryYou may sometimes want to know how much space a solid figure takes up. Youcan find the volume of a sphere, cylinder, or cone using the appropriate formula.

A flowchart is a graphic organizer that you can use to find volume.

Dominic has a glass that holds about 16 ounces of water. To find the volume of the glass, he measured the height and diameter of the glass. How can Dominicfind the volume, to the nearest hundredth, in cubic centimeters?

Dominic used this flowchart along with the formula for the volume of a cylinderto find the answer.

Answer:

Read what Dominic wrote to explain how he used a flowchart to solve the problem.

I used the formula for the volume of a cylinder (πr2 h). I found thedistance across the circular top of the glass through the center; thismeasurement is the diameter. The diameter is 7 cm. The formula calls for the radius. Since the radius is half of the diameter, r = 3.5 cm. I foundthe height of the glass to be 16.5 cm. I know that π is equal to 3.14. I substituted the values into the formula. I multiplied 3.14 × 3.52 × 16.5 to find a product of 634.6725. I rounded to the nearest hundredth(634.67). Since volume is measured in cubic units, the volume of the glass is 634.67 cubic cm.

Understanding the solution

Dominic found the volume of the glass to be 634.67 cm3.

Thinking about the strategy

Find the radius, which ishalf of the diameter.

Measure the height.

Substitute theappropriate values into

the formula.

volume of a cylinder = πr2h

7 cm

π = 3.14diameter = 7 cmradius = 3.5 cmheight = 16.5 cm

1. Write the formula.

3.14 × 3.52 × 16.5 = 3.14 × 12.25 × 16.5 = 634.6725 cm3 � 634.67 cm3➧ ➧

2. Find the values. 3. Substitute the values into the formula,and multiply. Round to the nearesthundredth.

16.5cm

99

Solve a ProblemKevin plays baseball, and his sister plays softball. Kevin wanted to find the differencein volume between a baseball and a softball. He found that the diameter of abaseball is about 7.3 cm, and the diameter of a softball is about 10 cm. What is the difference in volume to the nearest hundredth?

Complete this flowchart to find the volume of each ball. Then write your answer below.

Answer:

Write an explanation of how you found the answer by completing the flowchart above.

Explaining the solution

Finding the solution

volume of a sphere = ��πr3

1. Write the formula.

➧ ➧

2. Find the values. 3. Substitute the values into the formula, andmultiply. Round to the nearest hundredth.

100

Numbers in ContextRead “The Junior Gym.” Think about the ways that numbers are used in the selection. Then answer items A–C on page 101.

The Junior Gym

Jess works at The Junior Gym, which is a gym for preschool children. Children takeclasses at the gym to get some exercise and to develop muscle coordination. They also go to the gym to have fun with friends.

The gym has lots of different equipment. Jess often uses cones to set up mini-obstaclecourses for the children. The kids run around the cones or zig-zag through them. For somegames, Jess uses small cones. The small cones each have a height of 9 inches and a basewith a diameter of 5 inches. For other games, Jess gets out the large cones. The large coneseach have a height of 17 inches and a base with a diameter of 7 inches.

The gym has stacks of rubber arrows that are used for treasure hunts. The arrows areplaced on the floor to give directions to the treasure. Jess made a diagram to show whereshe will place the arrows for a hunt that she is planning.

Jess will have two teamsof children. One team willfollow the arrows A, B, C.The other team will follow the arrows D, E, F. Teammembers take turns one-by-one bouncing a ball fromarrow to arrow. When theyget to the treasure bin,children throw their ball in.The first team to get their ball into their bin wins thetreasure.

Jess is curious to see whathappens with the game. It could be great fun . . . or it could be total chaos!

AB

bin bin

D E

C F

• Start

• •Point 1 Point 2

101

A. Look at the diagram Jess drew. Whichfigures show a reflection? Which onesshow a translation? Which ones show arotation? (Hint: There are two examples of translations and two examples of rotations.) Use the information from page 100 to copy the figures onto the grid. Then write your answer below.

Answer:

B. What is the difference in volume, to the nearest hundredth, between the small and the large cone? Use the information from page 100 and the flowchart to find each volume. Then write your answerbelow. (Hint: The B in the formula stands for the area of the base. Remember that the formula for the area of a circle is πr2.)

Answer:

C. Explain your solution to either item A or item B above.

volume of a cone = ��Bh

1. Write the formula.

➧ ➧

2. Find the values. 3. Substitute the values into the formula, andmultiply. Round to the nearest hundredth.

102

Check Your UnderstandingFill in the letter of the correct answers to questions 1–8. Write your answers to questions 9 and 10.

Use this coordinate grid to answer questions 1 and 2.

1. Which two figures show a reflection?� figures B and C� figures E and F� figures A and D� figures A and B

2. Which two figures show a translation?� figures E and F� figures A and E� figures A and D� figures D and C

3. The diameter of a cone is 6 cm. The height of the cone is 10 cm. What is the volume?� 847.8 cm3

� 282.6 cm3

� 94.2 cm3

� 9.42 cm3

4. What is the volume of this pipe?

� 3.14 in.3

� 50.24 in.3

� 100.48 in.3

� 200.96 in.3

5. A figure has a vertex at the point (2, 0) on a coordinate grid. If the figure is translated to the left 3 units, what is the new location of this vertex? � (-1, 0) � (-1, 3)� (3, 0) � (2, 3)

6. A sphere has a diameter of 8 inches. What isthe volume, rounded to the nearest hundredth? � 200.96 in.3 � 803.84 in.3

� 267.95 in.3 � 2, 143.57 in.3

7. A straw holds 16.485 cubic centimeters ofwater. The length of the straw is 21 centimeters.What is the diameter?� 1 cm � 0.785 cm� 0.8 cm � 0.5 cm

8. What pair of figures shows a reflection?

�

�

�

�

y

6

5

4

3

2

1

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6x

–2–3–4–5–6

D

•

•

•

•

A

• •

•

• B

•• C

•

•

•E

•

•

•F

2 in.

16 in.

•

103

Extend Your Learning• Transformation Drawing

Make designs on a coordinate grid. Draw a plane geometric figure; then show rotationimages (�� turn, �� turn, �� turn). Draw another figure, and show translation images. Create athird figure, and show reflection images. Exchange your work with a partner, and identifythe transformed images he or she has made.

• Science: Density

Materials can have the same volume, but different mass. For example, a cubic liter of gashas less mass than a cubic liter of water. Density is a comparison of mass to a givenvolume. Compare the density of a tennis ball and a baseball. Find the volume and massof each ball. Which, do you think, has a greater density? Why?

9. Draw a geometric figure. Then draw a reflection, translation, and rotation image ofthis figure. Label each image.

10. Stephanie has two paper cups. One is a cone with a diameter of 8 cm and a height of 12 cm. The other cup is a cylinder with a diameter of 6 cm and a height of 8 cm.What cup will hold more water? Explain how you found your answer.

Building Number Sense; Using EstimationRead “Protecting Land and Sea.” Think about the ways that numbers are used in the selection. Then answer questions 1 and 2.

Protecting Land and Sea

“I’m doomed!” Rena thought. Rena and Clarence had gone to school together since firstgrade. As far as Rena could remember, she and Clarence hadn’t agreed on one single thingin all that time. Now Mr. Jones had made them project partners.

When Rena walked into the library that afternoon to meet with Clarence, she was bracedfor an argument. Clarence presented his project ideas first. “I’d like to do something onconservation and the National Park Service,” he said. “I’m sure you know that PresidentTheodore Roosevelt established the first national park in 1872. That, of course, wasYellowstone. Today, our national park system protects 378 areas that encompass more thanthree trillion, six hundred twenty-eight billion, five hundred forty-eight million square feet ofland, which, by the way, is equivalent to 83.3 million acres.”

“That’s a good idea,” Rena said, “but I was hoping we could focus on marine sanctuaries.In 1972, the Marine Protection, Research and Sanctuaries Act established the NationalMarine Sanctuary Program. Since then, 13 national marine sanctuaries have been created.These sanctuaries protect 18,618 square miles of our seas and coasts. One of the largestsanctuaries in the program is right here in Monterey Bay, California. The Monterey BaySanctuary protects over 5,300 square miles of ocean habitats and coastlines.”

“I have an idea,” Clarence said. “Why don’t we call our project ’Protecting Land andSea.’ We could focus on national parks and national marine sanctuaries. We could do ourresearch independently, and then work together to show how both programs help protectand preserve our lands and seas. What do you think?”

Rena couldn’t believe it as she heard herself say, “I agree. It’s a great idea!”

1. Which of these correctly expresses the wordnumber in the story?� 3,628,548,000,000� 362,854,800,000� 3,628,500,480,000� 3,000,628,000,548

2. Rena divided 5,300 by 18,618 to find that thearea protected by the Monterey Bay Sanctuary isabout 0.284670748 of the area protected by theentire Marine Sanctuary Program. If shecorrectly rounds the decimal to four decimalplaces, what will be the rounded value?

124

REVIEWSTRATEGIESONE–TWELVE

125

Applying Addition; Applying Subtraction; Applying Multiplication; Applying DivisionRead “The Charity Walk.” Think about the ways that numbers are used in the selection. Then answer questions 3–6.

The Charity Walk

For the last two years, Troy and his sister Robin have participated in a charity walk to benefit the homeless. Last year, the walk from start to finish, was 18 mi 3,109 ft 6 in.Robin walked that distance. Troy had walked exactly 16 mi 4,070 ft 9 in. when hedeveloped a blister on his right foot; the injury made it impossible for him to go on.

This year, the walk had been extended to 20 �� miles. Troy was determined to go thedistance. He had also promised the editor of the school newspaper that he would write an article describing the walk from a participant’s point of view. Troy decided to divide the walk into 2 �� mile segments. He would wear a pedometer to keep track of his mileage, and he’d carry a small tape recorder. At the end of each segment, he would record how he felt. He planned to use his recorded notes to write the article.

The day of the walk started out sunny and warm. Troy had completed 3 �� segments and was feeling great when he noticed dark clouds gathering. He took a sweater and a rain poncho out of his backpack, put them on, and kept going. A few moments later, rain began falling, softly at first and then harder and harder. By the time Troy reached the finish line, he was cold and his feet were wet. However, he had completed the whole walk and felt great, which is exactly what he said as he recorded the last of his notes.

3. How many miles combined did Troy andRobin walk in last year’s charity walk?� 34 mi 1,899 ft 5 in.� 35 mi 1,900 ft 3 in.� 34 mi 961 ft 3 in.� 24 mi 2,300 ft

4. Last year, Troy had to stop walking because of a blister. How much farther would he havehad to walk to get to the finish?

5. How many miles had Troy gone when henoticed dark clouds gathering?� 3 miles

� 7 �� miles

� 9 �� miles

� 6 �� miles

6. How many times did Troy plan to record notesduring the walk, not including his final notes?

126

Converting Time and Money; Converting Customary and Metric MeasuresRead “Lunch Talk.” Think about the ways that numbers are used in the selection. Then answer questions 7 and 8.

Lunch Talk

“Kayla!” Tito said. “I’m glad you’re here. We need your help.”“What’s going on?” Kayla asked as she placed her lunch tray on the table.“We’re trying to figure out how many miles per day you’d have to travel to go from

New York City to Los Angeles in about 4 days,” Dave said.“Why?” Kayla asked.“Why what?” Janet said.“Why do you want to know?” Kayla asked.“Know what?” Janet said.“Janet!” Kayla, Tito, and Dave yelled together.“You don’t have to yell,” Janet said. Tito turned back to Kayla. “Dave’s brother Rob is going to college in Los Angeles,”

Tito said. “Dave’s dad is planning to drive Rob to school, spend a couple of days getting Rob settled, and then drive back home in 4.5 days. We were just wondering how far he’d have to drive each day to make it in 4.5 days.”

“How far is it from New York City to Los Angeles?” Kayla asked.“It’s 2,820 miles,” Janet said, “or would you rather know in kilometers?”Kayla, Tito, and Dave stared

at Janet.“How can you know the exact

distance from New York toCalifornia and yet not be able to keep track of a simpleconversation?” Kayla asked her friend.

“I don’t know,” Janet said.“Now, what is it that we weretalking about?”

7. To the nearest hundredth, how many miles perday will Dave’s dad have to drive to go fromLos Angeles to New York City in 4.5 days? � 125.33 miles per day� 62.67 miles per day� 705.56 miles per day� 626.67 miles per day

8. According to Janet’s figures, what is the totaldistance in kilometers that Dave’s dad willdrive if he drives from New York to Los Angelesand then back to New York?

1 mi = 1.609 km

127

Using Algebra; Using GeometryRead “The Man’s Glasses.” Think about the ways that numbers are used in the selection. Then answer questions 9 and 10.

The Man’s Glasses

Once, there was an old man who collected beverage glasses. This man had the largestcollection of glasses ever seen. He kept all of his glasses on shelves that lined every inch ofevery wall in every room of his house. There were glasses in the living room. There wereglasses in the dining room. There were glasses in the kitchen. There were glasses in the hall,up the stairs, in the bathroom, and in all 3 bedrooms of his house.

If that wasn’t odd enough, all the glasses in the man’s collection were exactly the sameshape and same size. Each glass was 11 cm in diameter and 18.5 cm high. Furthermore,each glass was kept filled to the brim with water. Of course, water evaporates, which meantthat every day, the man spent hours checking and refilling his glasses to the brim.

The man’s neighbors thought the man and his collection were quite strange. However,since the man caused no trouble whatsoever, no one ever felt inclined to criticize. As a result,the man and his neighbors lived in peace and harmony.

One day, a fire broke out in the center of the village. A gusty wind caused the fire tospread rapidly. Soon, flames threatened the entire village. To make matters worse, theweather had been dry of late, and villagers quickly ran out of water to fight the fire.

Women, children, and even men began to weep as they realized that without water, their village would be devoured by the angry flames.

And it might have been if the man hadn’t come to their rescue. For if you have beenpaying attention, then you know that the village was not out of water as long as the man’sglasses were filled to the brim. In fact, there was more than enough water to put out theflames and save the village.

9. There are a total of 88 shelves of glasses in the 3 bedrooms. The largest bedroom has 8 fewershelves than twice the number of shelves in eachof the other bedrooms. How many shelves ofglasses are in the largest bedroom? How manyshelves are in each of the other two bedrooms?

10. To the nearest hundredth, what is the volumeof each of the glasses in the man’s collection? � 638.99 cm3

� 1,757.22 cm3

� 319.50 cm3

� 2,238.50 cm3

128

Determining Probability and Averages; Interpreting Graphs and ChartsRead “Up and Down, Round and Round.” Think about the ways that numbers are used in the selection. Then answer questions 11 and 12. After completing question 12, do “Explaining the Solution.”

Up and Down, Round and Round

Sally lives in a condominium complex adjacent to one of the most popular theme parks in the country. Sally’s father is the general manager of the park. All the students in Sally’sschool live in the same complex, and all of their parents also work at the park.

Because of their relationship with the park, Sally’s math teacher often creates problemsrelated to the park. Here are two problems that appeared on Sally’s last math quiz.

Problem 1: Imagine that visitors to the park had to spin a wheel to determine what ridethey could take. Look at the wheel shown here. What is the probability that a visitor willget to ride a roller coaster or an adventure ride?

Problem 2: For a few hours each day, park officials track the number of people who ride Storm Alley. Riders range from a low of 56 people per hour to a high of 99 people. The results for the last 5 days are displayed in this stem-and-leaf plot. What was the median number of people per hour who rode Storm Alley?

Storm-Alley Riders (per hour)

5 6 7 8 96 5 6 6 97 0 3 5 78 0 0 6 89 0 1 3 9

11. What is the correct answer to problem 1 in the story?

12. What is the correct answer to problem 2 in the story?� 66 � 79� 74 � 80

Explaining the solution Look at your solutions to questions 1–12 on pages 124–128. Choose twoquestions, and write an explanation of how you found the solution to each. Use a separate sheet of paper.

Twis

t an

d Sh

out

Anxiety D

rop

Sleepy Hollow

Adventure Ride

Whirl Wind

Storm

Alle

y

Wild Cave

Adventure Ride

Mighty

Roller Coaster

Fanta

sy F

ores

t

Rock’n Roller Coaster

Sea FloorAdventure Ride

Mathematics Strategies□ Building Number Sense

□ Using Estimation

□ Applying Addition

□ Applying Subtraction

□ Applying Multiplication

□ Applying Division

□ Converting Time and Money

□ Converting Customary and Metric Measures

□ Using Algebra

□ Using Geometry

□ Determining Probability and Averages

□ Interpreting Graphs and Charts

Numbers-in-Context Reading Selections□ Viewing Power

□ Five Nights for the Round Table

□ Good Friends

□ The Dream

□ A Rush of Water

□ Journey into the Past

□ Kayla’s Journal

□ Changing Waves

□ The Basketball Game

□ The Junior Gym

□ Northstar Hockey Camp

□ The Traffi c Study

□ Protecting Land and Sea

□ The Charity Walk

□ Lunch Talk

□ The Man’s Glasses

□ Up and Down, Round and Round

Graphic Organizers□ Place-value Chart

□ Flowcharts

□ Number Lines

□ Tables

□ Table of Signs

□ Conversion Tables

□ Coordinate Grid

□ Grid

□ Spinner

□ Venn Diagram

□ Line Plot

□ Histogram

□ Stem-and-leaf Plot

Math-related Writing□ Short-answer responses

to math problems

□ Extended-answer responses to math problems

Assessments□ Assessment of each

mathematics strategy

□ Final Review of all mathematics strategies

□ Selected-response problems

□ Constructed-response problems, including short-answer responses and extended-answer responses

□ Extended-answer responses require students to explain their problem-solving process

Extend Learning□ Challenge problems

□ Class projects

□ Practical applications

□ Cross-curricular activities in economics, social studies, home economics, language, science, technology, and physical education

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EXTENSIONS INMATHEMATICS™ SERIES

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