P -A Rounding Whole Numbers P are the guidelines for rounding whole numbers when using a number line as a visual tool: ... Identify the place digit. ... Rounding to the largest place

27

PRE-ACTIVITY

PREPARATIONRounding Whole Numbers

“Dollar stores” have sprung up around the country as spin-offs of the “dollar days” promotions of major retailers. Their prices vary only slightly from customary dollars and cents pricing, so what is their appeal? Convenience and simplifi ed transactions, for both the retailer and you as the consumer, are two of the driving forces of this marketing tool.

Income tax forms permit you to round your entries to the nearest dollar amount. Is it reasonable for the I.R.S. to encourage this practice? Yes, because the I.R.S. has determined that the total revenue collected is insignifi cantly affected even when the taxpayers’ calculations have been simplifi ed.

How do the print and broadcast media present statistics such as the population of countries, national voter turnouts, or gross earnings for major motion pictures? Instead of exact counts, rounded fi gures are suffi cient to communicate the magnitude of these large numbers. On a smaller scale, consider the destination signs on U.S. freeways. Precise distances to upcoming cities measured in miles, feet, and inches would offer more information than you need to know, so distances are routinely rounded to the nearest mile.

These are just a few examples from the perspectives of business and mass communication. Think of the many other ways you encounter or use rounded numbers in your daily life. Even in casual conversation your friend might share with you, “By the time we paid for parking, ride tickets, food, and souvenirs for everyone, it cost the fi ve of us nearly $200 to go to the festival.” She may have actually spent $177, but from that rounded $200 fi gure you are able to get a sense of what the cost might be for your own family. Or you might share with your spouse that the car you test-drove sells for “about $19,000” when its actual price is $19,213.

For those times when it is unnecessarily complex or precise to use exact numbers, knowing how to round numbers to serve your purpose is an important mathematical tool and skill.

• Master the process of rounding whole numbers.

• Use consistent notation to present the process of rounding.

LLEARNINGEARNING OOBJECTIVESBJECTIVES

Section 1.2

28 Chapter 1 — Whole Numbers

NEW TERMS TO LEARNPREVIOUSLY USED

base ten system

decimal system

digit

place

place value

endpoint

greater than symbol >

inequality symbols

interval

largest place value

less than symbol <

methodology

midpoint

number line

place digit

round down

round up

rounding

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

Rounding is the mathematical process of re-stating a number as an approximate value to make it easier to read, write, and use for computation.

You can use an inequality symbol to state that one number is smaller than or greater than another.

< is the symbol for “less than” as in 4 < 7 “four is less than seven”

> is the symbol for “greater than” as in 7 > 4 “seven is greater than four”

Notice that both signs point to the smaller number and open to the larger number.

A horizontal number line is a visual representation of numbers as points on a line, with the numbers growing larger as you move from left to right. The number line below represents the whole numbers. The arrow indicates that the numbers go on without end.

The following number line zooms in on only the whole numbers in the interval from 20 through 30. The numbers 20 and 30 are marked as the endpoints of the interval and the midpoint 25 is halfway between them.

0 1 2 3 4 5 6 7 8 9 10 11 12

20 21 22 23 24 25 26 27 28 29 30

endpoint endpointmidpoint

TTERMINOLOGYERMINOLOGY

29Section 1.2 — Rounding Whole Numbers

For any given number you know that a digit represents each of its places according to the decimal system (see Section 1.1). For the rounding process, this book will refer to the digit in a specifi ed place as the place digit.

You round a whole number to a specifi ed place value (tens, hundreds, thousands, and so on) as it suits your purpose for using an approximate number. Once you specify that place value, its corresponding place digit is essential to the rounding process. Knowing the place digit establishes two numbers between which the original number falls. The lower number has the same place digit as the original and the higher number increases the place digit by 1.

You might draw a portion of a number line to illustrate the rounding process, as in the following example.

Round 1683 to its nearest hundreds place.

Focus in on the hundreds place digit and you know that 1683 falls somewhere in the interval between 1600 and 1700. The midpoint 1650 is half of one hundred, or fi fty, from each endpoint.

Your task is to determine whether 1683 is nearer 1600, or nearer 1700.

As the number line clearly shows, 1683 falls between the midpoint 1650 and 1700.

Conclusion: 1683 rounded to its nearest hundreds place is 1700.

These are the guidelines for rounding whole numbers when using a number line as a visual tool:

Once a place value has been specifi ed and the two endpoints of an interval have been determined,• A number that falls between the smaller endpoint and the midpoint rounds down to the smaller

endpoint number.• A number that falls between the midpoint and the higher endpoint rounds up to the higher endpoint

number.• Even though the midpoint number itself is no closer to one endpoint than the other, it rounds up to the

higher endpoint number.

1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700

1683

Methodologies

A methodology is an orderly arrangement of steps or procedures. A methodology serves as a model by listing a set of steps which describe how to best perform a process in an effective and effi cient manner. Throughout this book, methodologies will be presented as tools to help you learn key mathematical processes.


MMETHODOLOGYETHODOLOGY

Rounding a Whole Number

Example 1: Round 8,472 to the nearest thousand.

Example 2: Round 8,472 to the nearest hundreds place.►►►►

Steps in the Methodology Example 1 Example 2

Step 1

Identify the place digit.

Identify the digit in the specifi ed place value (the place digit) by marking it with an arrow.

8,472

8 is the thousands place digit

8,472

Step 2

Identify the digit to its right.

Identify the digit immediately to the right of the place digit by circling it.

4, in the hundreds place, is the digit to the right of the

place digit.

8, 4 7 2

Step 3

Compare to the number 5.

Determine whether the circled digit is less than, equal to, or greater than 5. 4 < 5

Step 4

Round up or round down.

If the circled digit is less than 5, do not change the place digit.

If the circled digit is 5 or greater, round up by adding 1 to the place digit.

The thousands place digit does not change.

8 , x x x

Step 5Present the answer.

To present your answer, replace all digits to the right of the place digit with zeros as placeholders.

8,000

? ? ?Why do you do this?

Try It!

The steps in this methodology come from establishing endpoint numbers and using the midpoint number as the basis for making the decision to round up or down. However, it does not require the use of a number line.

Rounding to the largest place value (see pages 32 & 33, Models 2 & 3)

Special Case:

Special Case:

Carry or carries required (see page 33, Model 3)


? ? ?Why do you do Step 3?

Half the specifi ed place value away from each of the two possible answers is their midpoint number. Because half of 10 is 5, half of 100 is 50, and half of 1000 is 500, and so on in the base ten system, the midpoint will always have 5 as the digit to the right of the place digit.

In Example 1, the two possible answers are 8000 or 9000. Half of one thousand is 500, so the midpoint number is 8000 + 500 or 8500.

The digit that you circle in Step 2 provides the key information for the rounding process because it is the indicator as to whether the original number is less than or greater than the midpoint number.

In Example 1, the circled digit 4 is in the hundreds place. You know that any number in the 8400’s (8400 through 8499) is less than 8500, the midpoint. (See the number line above.)

In fact, focusing only on the hundreds place digit in 8427 and 8500 (4 < 5), and not the digits in the tens and ones places, you can conclude that 8427 < 8500, the midpoint.

In general, to round any number to a specifi c place, it is suffi cient to compare only the digit to the right of the place digit in the original number to the digit 5 (always to the right of the place digit in the midpoint) before moving on to Step 4, rounding up or rounding down.

MMODELSODELS

Round 85,291 to the nearest hundred.

Step 1 2 is the hundreds place digit. 85,291

Step 2 9 is the digit to the right of 2. 8 5, 2 9 1

Step 3 9 is greater than 5 ( 9 > 5)

Step 4 Add 1 to the 2 in the hundreds place, making it 3. 85,3xx

Step 5 Answer: 85,300

Pictured on a number line:

85,291 is closer to 85,300 than to 85,200

THINKhundreds place—“Is 85,291 closer to 85,200 or closer to 85,300?” The Methodology confi rms the answer.

85,200

85,291

midpoint85,30085,250

►►AModel 1

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 9000

8400–8499

midpoint8 4 2 7


Round 85,291 to the nearest thousand.

Step 1 5 is the thousands place digit. 85,291

Step 2 2 is in the hundreds place. Circle it. 8 5, 2 9 1

Step 3 2 < 5

Step 4 Do not change the place digit 5. 85,xxx


Pictured on a number line:

85,291 is closer to 85,000 than to 86,000

THINK thousands place—“Is 85,291 closer to 85,000 or to 86,000?”

85,291

midpoint85,000 86,00085,500

Round 238,450 to its largest place value.

Step 1 The largest place value in 238,450 is the hundred-thousands place. 238,450

Step 2 2 3 8, 4 5 0

Step 3 3 < 5

Step 4 The place digit 2 remains unchanged. 2xx,xxx

Step 5 238,450 rounded to its largest place value is 200,000. Answer: 200,000

On a number line:200,000 300,000250,000

238,450

►►B

Round 85,291 to the nearest ten thousands place.

Step 1 85,291

Step 2 8 5, 2 9 1

Step 3 5 = 5

Step 4 Round up. The 8 changes to 9.


On a number line:

THINK ten thousands place—“Is 85,291 closer to 80,000 or to 90,000?”

midpoint80,000 90,00085,000

85,291

►►C

The largest place value of a number is occupied by its leading digit (farthest to the left).Model 2

Special Case:

Rounding to the “Largest Place Value”


Round 297 to the nearest tens place.

Step 1 297

Step 2 2 9 7

Step 3 7 > 5

Step 4 9 becomes 10. Put the zero in the tens place and add 1 to the 2 in the hundreds place. 30x

Step 5 Answer: 300

On a number line:

THINKtens place—Is 297 closer to 290 or to 300?

290

297

300295

When the place digit is 9 and the circled digit is equal to 5 or greater than 5, adding 1 to the place digit 9 makes it 10. Put a zero in the place digit position. Carry the 1 to the next higher place value and add it to the digit in that place.


Step 1 49,953

Step 2 4 9, 9 5 3

Step 3 5 = 5

Step 4 Adding 1 to the 9 in the hundreds place makes it 10. Put a zero in the hundreds place. When you add 1 to the 9 in the thousands place, it also becomes 10. Put a zero in the thousands place and add 1 to the ten thousands place.

50,0xx

Step 5 Answer: 50,000On a number line:

Round 9,725 to its largest place value.

Step 1 9,725

Step 2 9, 7 2 5

Step 3 7 > 5

Step 4 10,xxx


49,900

49,953

50,00049,950

Model 3

►►A

►►B

►►C

Special Case: Carry or Carries Required


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

PPREPARATION REPARATION IINVENTORYNVENTORY

Before proceeding, you should have an understanding of each of the following:

the identifi cation of place values

the terminology and notation associated with rounding whole numbers

the signifi cance of the digit to the right of the specifi ed place value

why you round up when the digit to the right of the place digit is 5

why zeros are used as placeholders to the right of the specifi ed place digit in the rounded answer

Issue Incorrect Process Resolution Correct Process

Incorrectly identifying the specifi ed place for rounding

Round 76,345 to the nearest ten thousand.

7 6, 3 4 5

3 < 5

Answer: 76,000

Recall the place value chart to help identify the correct place for rounding.

Round 76,345 to the nearest ten thousand.

7 6, 3 4 5

6 > 5

Answer: 80,000

Lowering the place digit


7, 3 4 5

3 < 5

Answer: 6,000

Never subtract 1 from the original place digit.


7,394 is between 7,000 and 8,000.

7, 3 9 4 rounds down to 7,000

Not changing the digits to the right of the place digit to zeros


2 3, 8 2 4

8 > 5

Answer: 24,800

In a rounded answer, all digits to the right of the specifi ed place digit become zeros.


23,824 is between 23,000 and 24,000.

2 3, 8 2 4 rounded to the nearest thousand is 24,000

Forgetting to carry 1 to the next higher place value when the place digit is 9 and must be increased by 1


7, 9 7 4

7 > 5 (9 becomes 10)

Answer: 7,000

When the place digit is 9, and it rounds up to 10 because the circled number to the right is 5 or higher, the 1 is always carried to the next higher place value.


7,974 is between 7900 and 8000.

7, 9 7 4 rounds up to 8,000

7 is in the ten-thousands place

THINK

THINK

THINK

THINK

6

7

6, 3 4 6,

< 5

6 0

,

wer: 6

, 3 4 537,

<

6,0

2

2

3, 8 2 2 3,

> 5

4

7

(9

wer:

om

77 7,

bec

7

35

ACTIVITY Rounding Whole Numbers

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

Correctly rounding whole numbers to given place values• accurate identifi cation of the specifi ed place value• consistent and appropriate notation• accuracy in the rounding process

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. Other than those mentioned in the Pre-Activity introduction, what are two additional situations where you have observed the use of rounding?

2. What notation will you consistently use to show your work for the rounding process?

3. What are the decision rules for rounding?

4. What is the position of the largest place value in a given number?

•

•

Section 1.2


5. When rounding up, what must you do when the specifi ed place digit is a 9?

6. In Model 1A (on page 31), why is 85,291 closer to 85,300 than it is to 85,200?

7. Why is the number 5 the key comparison number in the rounding process?

8. In general, what sort of circumstances lend themselves to the use of rounded estimates or rounded calculations?


TTIPS FOR IPS FOR SSUCCESSUCCESS

• Use a place value chart to improve your skills at identifying the specifi ed place value for rounding.

• Use a consistent notation for the rounding process.

• Drawing a number line may help visualize the comparison to the halfway point in the rounding process.

DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

1. Round 71,350,894 to each of the given places.

71,350,894 Rounding Process Answer

a) hundreds

b) thousands

c) ten- thousands

d) hundred- thousands

e) millions

2. Round each number as specifi ed.

Rounding Process Answer

a) 2,197 to the nearest ten

b) 2,197 to the nearest hundred

c) 13,995 to the nearest hundred

d) 99,647 to the nearest thousand


3. Round each number to its largest place value:

Rounding Process Answer

a) 234

b) 29,425

c) 5,678

TEAM EXERCISETEAM EXERCISE

Complete the following statements and fi ll in the missing numbers on the accompanying number lines. Indicate the original number on the number line as well.

1) The number __________ is midway between 630 and 640. 639 is closer to 640 than it is to 630 because____________________________________

_______________________________________________________________________.

2) The number __________ is midway between 600 and 700. 639 is closer to__________ than it is to 700 because_____________________________

_______________________________________________________________________.

3) The number __________ is midway between 2,900 and 3,000. 2,979 is closer to 3,000 than it is to 2,900 because_______________________________

_______________________________________________________________________.

630 640

600 700

2,900 3,000

____?

____?

____?


4) The number ________ is midway between 2,000 and 3,000. 2,979 is closer to 3,000 than it is to ________ because____________________________

_______________________________________________________________________.

____?

Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the second column. If the worked solution is incorrect, solve the problem correctly in the third column.

Worked SolutionWhat is Wrong Here? Identify the Errors Correct Process

1) Round 5,246,392 to the nearest ten thousand.

Identified the wrong place.

4 is in the ten-thousands place.

5,246,3926>5

2) Round 8,267 to the nearest tens place.

3) Round 99,909,990 to the nearest hundred.

4) Round 324,523 to the nearest hundred.

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

____?

____?

Answer:

5,250,000


1. Round 92,450,691 to each of the given places:a) hundredsb) thousandsc) ten-thousandsd) hundred-thousandse) millions

2. Round as specifi ed:a) 4,596 to the nearest tenb) 59,912 to the nearest thousandc) 59,912 to the nearest ten-thousand

3. Round each to its largest place value:a) 977b) 439,148c) 15,960

ADDITIONAL EXERCISESADDITIONAL EXERCISES

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P -A Rounding Whole Numbers P are the guidelines for rounding whole numbers when using a number line as a visual tool: ... Identify the place digit. ... Rounding to the largest place