□ 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

D

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

14

Learn About EstimationWhen you figure out about how far you have to travel to get somewhere or about how much change you have in your pocket, you are estimating.To estimate, you round to find a number that is close to an exact number.

To round a decimal, you follow the same steps that you use to roundwhole numbers. You round up if a digit is 5 or more. You round down if a digit is 4 or less.

Francy is about the same height as her friend Rhonda, who is exactly 46.59 inches tall. Can you estimate to the nearest whole number how tallFrancy is? Need help? This estimation table shows you how to roundFrancy’s height. It also shows how to round other numbers.

Read the problem and the notes beside it.

How can Keary use an estimation table to solve the problem?

Keary plans to drive with her mom to the county fair onSaturday morning. According to a local tour-bus company, thedistance from the bus station to the fairgrounds is 182.63 miles.Keary and her mom live 3.875 miles from the bus station and must drive by the station on the way to the fairgrounds.

About how many miles, to the nearest hundred, do Keary and her mother have to drive from home to get to the fairgrounds?

Studying the problem

Thinking about the strategy

Exactly how far is itfrom the bus stationto the fairgrounds?

Exactly how far doesKeary live from the

bus station?

What operations canKeary use to figure

out about how far it isfrom her house to the

fairgrounds?

Numbers Nearest Nearest N e a rest Nearest Nearest Nearest Nearest0.01 0.1 Whole Number 10 100 1,000 10,000

46.59 46.59 46.6 47 50 100

881.391 881.39 881.4 881 880 900 1,000

37,928.717 37,928.72 37,928.7 37,929 37,930 37,900 38,000 40,000

Using EstimationSTRATEGYTWO

15

An estimation table is a graphic organizer that you can use to roundnumbers. Keary used this estimation table to estimate about how manymiles, to the nearest hundred, it is from her house to the county fair.

Keary estimated that it is about 200 miles to the fairgrounds.

Read what Keary wrote to explain how she used an estimation table to solve the problem.

I wrote the numbers 182.63 and 3.875 in the firstcolumn of the table. Then I rounded each decimal to thenearest whole number. I was careful to move from left toright across the table, rounding one place value at a time.I rounded up when a digit was 5 or more. I rounded downwhen a digit was less than 5.

Because 182.63 goes only to the hundredths place,I copied 182.63 in the hundredths place. Then I rounded to the nearest tenth. I rounded 182.63 down to 182.6. Then I rounded to the nearest whole number. I rounded 182.6 up to 183.

I used the same steps to round 3.875, except I start e dby rounding to the nearest hundredth, 3.88. Then I ro u n d e dup to the nearest tenth, 3.9. I rounded up again to then e a rest whole number, 4.

Because I wanted to estimate the total distance to thenearest hundred, I added 183 and 4 and got 187. I roundedup that sum to the nearest ten, 190. Then I rounded up 190 to the nearest hundred, 200. I estimated that it isabout 200 miles from my house to the fairgrounds.

Understanding the solution

Studying the solution

Numbers Nearest Nearest N e a rest Nearest Nearest Nearest Nearest0.01 0.1 Whole Number 10 100 1,000 10,000

182.63 182.63 182.6 183

3.875 3.88 3.9 + 4

187 190 200sum or difference of whole numbers→

16

Solve a ProblemRead the problem. As you read, think about how you could use an estimation table to solve the problem.

Use the information from the problem to complete this estimation table.Then write your answer below.

Answer:

Jason’s mom is training to run amarathon. She runs every weekdaymorning by herself. On weekendmornings, Jason runs with her. FromMonday through Friday, Jason’smom runs through the streets of herneighborhood. To keep track of hermileage, she wears a device thatmeasures exactly how far she runs.From Monday through Friday, sheruns a total of 32.348 miles. OnSaturdays and Sundays, when she runs with Jason, she takes adifferent, shorter path. The weekendruns total 6.742 miles.

About how far, to the nearest ten miles, does Jason’s mom run weekly?

Studying the problem

Numbers Nearest Nearest N e a rest Nearest Nearest Nearest Nearest0.01 0.1 Whole Number 10 100 1,000 10,000

sum or difference of whole numbers→

Finding the solution

17

Reread “Understanding the solution” on page 15, which tells how Kearyused an estimation table to solve a problem. Then write your ownexplanation of how you completed the estimation table on page 16 andfound your solution.

Use your estimation table on page 16 to answer these questions.

1. Did you round up or down to find the nearest hundredth for 32.348?

2. What number did you write in the tenths column for 6.742?

3. If Jason runs with his mom for only three weekends in May, about howm a ny total miles, to the nearest ten miles, will he run on those we e k e n d s ?

4. Exactly how many miles does Jason’s mom run each week? Is this more

or less than the estimated sum?

5. Jason’s mom increased the distance of her weekday runs to 7.529 milesa day. What is 7.529 rounded to the nearest hundredth, tenth, and

whole number?

Applying the solution

Explaining the solution

18

Learn More About EstimationYou may sometimes want to estimate a quotient. You can do this by usingcompatible numbers. Compatible numbers are nearby numbers that youcan divide easily.

A flowchart is a graphic organizer that you can use to estimate quotients,using compatible numbers.

How can Dylan use compatible numbers to estimate 367 ÷ 52? Dylancompleted this flowchart to find the answer.

Answer:

Read what Dylan wrote to explain how he used a flowchart to solvethe problem.

First, I wrote the problem, as given, in the first section of theflowchart. Then I looked at the divisor, 52, and found a nearby numberthat divides easily into a number near 367, the dividend. I chose 50and then looked for a multiple of 50 that is near 367. I chose 350. I wrote the new problem, using the compatible numbers, in the secondsection of the flowchart. In the last section, I did the division. I foundan estimated quotient of 7.

Understanding the solution

Dylan estimated the quotient to be 7.

Thinking about the strategy

Compatible numbersare a factor and one

of its multiples.

Look at the divisor.Try to find a nearbynumber that divideseasily into a number

near the dividend.

367 ÷ 52

750 ) 350

–3500

350 ÷ 50

1. Write the problem.

2. Look at nearby numbers tofind compatible numbers.

3. Use compatible numbers tofind the estimated quotient.

19

Solve a ProblemInez offered to help set up folding chairs in the gym for the band concert.Mr. Winn told Inez and the other student helpers to arrange the 769 chairsinto rows with 18 chairs in each row. About how many rows should thestudents arrange?

Complete this flowchart to estimate, using compatible numbers, how manyr ows the students should arrange. 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

1. Write the problem.

2. Look at nearby numbers tofind compatible numbers.

3. Use compatible numbers tofind the estimated quotient.

20

Numbers in ContextRead “Reaching Out.” Think about the ways that numbers are used in the selection. Then answer items A–C on page 21.

Reaching Out

Two months ago, the Marston Family Shelterannounced that it wouldbe forced to close at theend of November ifdonations didn’t increase.The shelter serves needyfamilies in the towns ofAvery and Bentwood. Itoffers food, clothing, andemergency shelter. It alsohelps men and women find jobs and affordablehousing for their families. “Closing the doors to the Marston Family Shelter would be a great loss to the community,” the shelter’s director, Lin Han, said in a television news report.“Last Sunday, the shelter helped 35 families. It cost $578 just to feed those families.”

Adam Martin, who attends Avery Elementary School, saw Ms. Han on the news.The next day, Adam asked his principal if he could address students during the Fridaymorning assembly. Adam asked the students in his school to help him keep theMarston Family Shelter open.

“What can we do?” one student asked. “We’re just kids.”“Here’s my plan,” Adam said. “I want to challenge Bentwood Elementary School

to a contest to see which school can raise more money. And all the money can go to the shelter.”

The contest has been a great success. Since both towns have always enjoyedcompeting against each other, everyone has joined in, including parents, businessowners, and town officials. As of yesterday, Avery Elementary School had raised$3,776.62. Bentwood Elementary had raised $2,932.48.

Adam Martin and Jill Rowe, a student from Bentwood, will present the money to Ms. Han at a special ceremony on Sunday afternoon.

21

A. To the nearest thousand, about how much money did students raise? Use the information from page 20 to complete this estimation table. Then write your answer below.

Answer:

B. About how much did the shelter spend to feed each family for a day? Use the informationfrom page 20 to complete this flowchart. Then write your answer below.

Answer:

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

1. Write the problem.

2. Look at nearby numbers tofind compatible numbers.

3. Use compatible numbers tofind the estimated quotient.

Numbers Nearest Nearest N e a rest Nearest Nearest Nearest Nearest0.01 0.1 Whole Number 10 100 1,000 10,000

sum or difference of whole numbers→

22

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

1. Together, Henry and his dad picked 27.376 pounds of apples. To the nearestwhole number, about how many poundsof apples did they pick?A 28 poundsB 30 poundsC 27 poundsD 29 pounds

2. What two whole numbers should Lilly use to estimate to the nearest foot the difference between 16,318.51 feet and 14,191.852 feet?A 16,320 and 14,190B 16,319 and 14,192C 16,318 and 14,191D 16,300 and 14,200

3. Anton and his brother have been savingsome of their allowance for two years.Each has exactly $219.67. To the nearestwhole number, about how much do the two boys have together?A $440B $400C $200D $438

4. Ella’s father was hired to paint 4 houses.He figured he would need about 122 gallons of paint to complete the whole job. What exact amount is closest to his estimate?A 121.409 gallonsB 121.115 gallonsC 122.814 gallonsD 122.185 gallons

5. What two compatible numbers shouldChico use to find the estimated quotient of 536 ÷ 98?A 530 and 94B 500 and 100C 532 and 96D 525 and 90

6. The city park department planted 315 treesalong 41 streets. If workers divided the trees among the streets as evenly as possible, about how many trees did they plant on each street? A 8 treesB 12 treesC 15 treesD 10 trees

7. Keesha’s older sister Yasmin plays on a college basketball team. This season,Yasmin played in 28 games and scored a total of 243 points. About how manypoints did Yasmin score in each game?A 6 pointsB 18 pointsC 15 pointsD 8 points

8. There are a total of 929 students and 34 teachers in the Norville Elementary and Middle Schools. About how manystudents is that per teacher?A 40 studentsB 30 studentsC 38 studentsD 35 students

9. What compatible numbers should be used to divide 635 by 78?

10. Tim’s dad is a truck driver. Last week, he drove a total of 681.508 miles.He drove about the same distance this week. To the nearest ten miles,about how many miles did Tim’s dad drive in two weeks? Explain howyou found your answer.

23

Extend Your Learning• Dividing up the Treasure

Imagine that you, your teacher, and 11 classmates are on a class picnic at thebeach when a treasure chest washes up on shore. Inside the chest, you find andcount 23,862 gold coins. You decide to divide the coins among all of you. Abouthow many coins will each of you get? Compare your answers in a group.

• Social Studies: Local Numbers

Working with a partner or in a small group, look in an encyclopedia or on theInternet to find the exact area in square miles of your city or town, of the capitalcity of your state, and of your state. Then round each of those numbers to thenearest whole number.

124

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

Snow Days

Hour after hour, snow fell, piling up on the streets and sidewalks of the city. For the third day in a row and the twelfth day in January, school had been canceledbecause of snow.

“Imagine, 12 out of 20 school days off!” Marcus’s next door neighbor, Ms. Berryexclaimed. “Why, you’ll be in school until the Fourth of July to make up for thesedays,” she said.

Ms. Berry is Marcus’s neighbor and had been his fourth-grade teacher. Usually,she was cheerful and pleasant, but the snow seemed to make her grumpy, eventhough it did mean a day off from work.

“You’re enjoying the free time now,” Ms. Berry warned Marcus, “but comesummer, you’re not going to like sitting in that hot classroom. I’ve been a fourth-gradeteacher for 24 years and taught 590 students over those years,” Ms. Berry toldMarcus. “I’m telling you, no one likes to go to school in the middle of July.”

“You’re probably right, Ms. Berry,” Marcus said politely. “But there’s nothing we can do about it now, except maybe go sledding in the park. Would you like to come with my mother and me?” Marcus asked on the spur of the moment.

“Now that sounds like a really cool idea,” she said. Marcus groaned at Ms. Berry’s corny joke, but he was glad that he had found a way to cheer her up.

1. What fraction and decimal show the partof January school days that had beencanceled because of snow?

2. What two compatible numbers should be used to find about how many studentsMs. Berry has taught each year during her24 years as a teacher?A 35 and 700B 24 and 590C 25 and 600D 25 and 500

Re v i e wSTRATEGIESONE–TWELVE

125

Applying Addition; Applying Subtraction; Applying Multiplication; Applying DivisionRead “All About Snowy Owls.” Think about the ways that numbers are used in the selection. Then answer questions 3–6.

All About Snowy Owls

Enuka presented a report on snowy owls. Here are some facts she shared with her classmates.• Owls are birds of prey that usually sleep

during the day and hunt after dark. One special feature about owls is thatthe female is larger than the male. For example, one female snowy owlweighs 2.3 kilograms, but her malepartner weighs only 1.79 kilograms.

• Snowy owls must eat a lot to survive; for example, from 7 to 12 mice a day.

• Over roughly 27 days, a brood and their parents may eat as many as 750 lemmings.

• Snowy owls live in the arctic regions of North America and Eurasia. • Snowy owls often travel in pairs or small groups. During winter migration, you

might see 10 pairs of snowy owls within !s mile of one another among the trees.• Snowy owls dig their nests in the ground, usually on a ridge or other high spot.

A female snowy owl may lay up to 14 eggs in her nest over a week’s time.

3. In one month, owl A ate 297 mice; owl Bate 350 mice; and owl C ate 319 mice.How many mice did the 3 owls eat?

4. In the example that Enuka shared, howmuch heavier is the female snowy owl thanthe male snowy owl?A 0.51 kilogramB 1.49 kilogramsC 0.61 kilogramD 1.59 kilograms

5. Scientists kept track of 10 pairs of owls overa 3-year period. Each year, each pair ofowls produced an average of 9 chicks. Howmany chicks did all 10 pairs of owls hatchduring the 3 years?

6. How many lemmings may be eaten per dayby a brood and their parents during thefirst 27 days of the brood?A 27 R7B 27 R21C 25 R75D 28 R21

126

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

Setting Up Shop

To give her students a chance to use math in real-life situations, Ms. Stango hadthem plan and manage a used-goods shop. First, students passed out a flyer thatasked students and teachers in other classes to donate old toys and games, usedsports equipment, and used books. The flyer explained that part of the money earnedin the school’s “New to You” shop would go to school arts programs and part wouldgo to the City Center for Children.

Students spent several weekends turning an unused room in the school basementinto a store. A few generous people in the community provided paint for the wallsand wood for shelves and a sales counter. Ms. Stango’s father and brother, who arecarpenters, helped students build the 6-foot-long counter and measure, cut, and hangfourteen 48-inch-long shelves and five 36-inch-long shelves. Students arranged all thedonated goods on the shelves. They created a work schedule for everyone in theclass. Then they opened their shop for one hour two mornings a week before schooland one afternoon a week after school.

The first customer was the principal, Mr. Jung, who bought several things for hissmall children. He bought three picture books at $0.93 each; a Chinese checkers game for $1.17; and a junior hockey stick for $7.82. Larry, who was working at the counter, added up Mr. Jung’s purchases and gave him the correct change for a 20-dollar bill. There was no tax on the purchases. Mr. Jung congratulated Larry and all the classes for doing such a great job.

7. How much change should Larry have givenMr. Jung?

8. How many feet of wood did students useto build the counter and all the shelves?A 77 feet C 13 feetB 90 feet D 71 feet

127

Using Algebra; Using GeometryRead “Saving Summer.” Think about the ways that numbers are used in the selection. Then answer questions 9 and 10.

9. What fraction is equivalent to the fractionof daisy plants that Erica and hergrandfather dug up from the garden?A aUs C !s

B #f D @d

10. What is the area, in square feet, of thetabletop that Erica and her grandfathermade?

Saving Summer

Erica sighed as she looked at the garden that she and her grandfather had workedso hard on all summer. Fall had come, and a frost was forecast for the next night.

“I hate that summer is over, Grampa,” Erica said. “It’s so sad that all our beautifulflowers and plants have to die. Let’s dig them up and bring them inside. I’ll help takecare of them. Then next spring, we can replant them.”

Erica’s grandfather shook his head. “There’s no spot in my house that gets enoughlight to keep these plants healthy.”

“We get lots of light in our kitchen,” Erica said. “Let’s talk to Mom.”“The kitchen is bright,” Erica’s mom said, “but where will we put the plants?”“Grampa can build a table,” Erica said. “Okay,” Erica’s mom said. “You can take a few of the plants home.”Erica and her grandfather got right to work. First, they dug up plants. They dug

up 7 of the 12 geranium plants, 2 of the 4 daisy plants, and 3 other plants that they thought might survive indoors. They replanted all the plants in small pots.

Next, they measured wood for a tabletop 24 inches long and 36 inches wide. Then they sanded and varnished the wood. Erica’s grandfather planned to mount the tabletop onto 4 old table legs that he had stored in his workshop.

The next afternoon, Erica’s grandfather brought the table and plants to Erica’s house and placed them in a sunny corner of the kitchen. “Now summer never has to end,” Erica said as she looked at the colorful flowering plants. “Well, at least not for these lucky plants.”

128

Determining Probability and Averages; Interpreting Graphs and ChartsRead “Play Time.” 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.”

Play Time

Every spring, each class at Weaver Elementary School presents a play thatstudents write themselves. This year, Mrs. Gomez’s class decided to write a funny playbased on familiar fairy tales and Mother Goose rhymes.

Mrs. Gomez put students into small groups. Each group picked a different fairytale or rhyme to rewrite as a play.

Students had fun writing their plays and practicing their lines. Meanwhile, Mrs. Gomez asked each group to time its play so that she would know how long the whole presentation would take. If the combined time was too long, she would have to ask some groups to cut their play by a few minutes. Mrs. Gomez recorded each group’s time in a stem-and-leaf plot.

Students made their own costumes. These were simple capes with felt decorationssewn onto them. Students were given a choice of red, blue, or black velvet cloth to maketheir capes; and a choice of purple, green, yellow, or white felt cloth for the decorations.To fasten their capes, they were given a choice of buttons, hooks, snaps, or ribbon.

The play was a great success. Everyone loved the costumes too.

11. What is the total number of differentcombinations of velvet, felt, and fastenersthat students could use to make theircostumes?A 11 C 19B 16 D 48

12. Mrs. Gomez asked students to shorten all the plays that were longer than 10 minutes. How many groups had to shorten their play?

Stem Leaf0 6, 91 0, 1, 3, 62 2, 4

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 toeach. Use a separate sheet of paper.

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□ PROVIDES CHALLENGING IN

□ STRENGTHENS PROBLEM-S

□ FEATURES ASSESSMENT INCONSTRUCTED-RESPONSE

EXTENSIONS INMATHEMATICS™ SERIES

DMathematics 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□ Della and Her Calculator

□ Reaching Out

□ Spring Plants

□ The Costume

□ Ernesto’s Garden

□ Computer Time

□ The Amazing Brush

□ Fun at the Fair

□ School Pride Month

□ Construct a Character

□ The School Newsletter

□ Snow Days

□ All About Snowy Owls

□ Setting Up Shop

□ Saving Summer

□ Play Time

Graphic Organizers□ Grids

□ Flowcharts

□ Estimation Table

□ Place-value Chart

□ Pie Chart

□ Multiplication Pyramid

□ Partial-products Grid

□ Time Table

□ Calendar

□ Schedule

□ Tables

□ Money Table

□ Conversion Table

□ Function Machine

□ Fraction Strips

□ Dot Paper

□ Tree Diagram

□ Bar Graph

□ 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 social studies, geography, science, technology, art, health, and physical education

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EXTENSIONS INMATHEMATICS™ SERIES

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