9

1.2 roundIngWholenumbers

The Internal Revenue Service (IRS) instructions for filling out tax returns includes the following statement:

You can round off cents to whole dollars on your return. If you do round to whole dollars, you must round all amounts. To round, drop amounts under 50 cents and increase amounts from 50 to 99 cents to the next dollar. For example, $1.39 becomes $1 and $2.50 becomes $3.

Assume that your income for the year, as reported by your employer, is $13,498.57

Correctly rounding whole numbers to given place values

• accurate identification of the specified place value

• consistent and appropriate notation

• accuracy in the rounding process

1. Your yearly income: $

Fill in your reported income in the field atright, as you would on a tax form. Followthe IRS instructions about rounding.

Assess your readiness to complete this activity. Rate how well you understand: Not ready

Almost ready

Bringit on!

• the identification of place values

• the terminology and notation used when rounding whole numbers

• the significance of the digit to the right of the specified 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 specified place digit in the rounded answer

13,499.00

10

Chapter � — Whole Numbers

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

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

StepsintheMethodology Example1 Example2

Step1

Identifytheplacedigit.

Identify the digit in the specified place value (the placedigit) by marking it with an arrow.

Rounding to the largest place value (see Model 2)

Special Case:

8,472

8 is the thousands place digit

8,472

Step2

Identifythedigittoitsright.

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

Step3

Comparetothenumber5.

Determine whether the circled digit is less than, equal to, or greater than 5. 4 < 5 7 > 5

Step4

Rounduporrounddown.

If the circled digit is less than 5, donotchange the place digit.

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

Special Case:

Carry or carries required (see Model 3)

The thousands place digit does not change.

8 , x x x

The hundreds digit changes.

Add one to the hundred’s digit

8,5xx

Step5

Presenttheanswer.

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

8,000 8,500

Try It!

8,472

4 is in the hundreds place

8,472

7, in the ten’s place is the digit to the right

8, 4 7 2

11

Activity �.2 — Rounding Whole Numbers

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

hundreds place—“Is 85,291 closer to 85,200 or closer to 85,300?” The Methodology confirms the answer.

Pictured on a number line:

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

85,29�

midpoint85,30085,250

►AModel 1

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.

Step 5 Answer:90,000

On a number line:

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

midpoint80,000 90,00085,000

85,29�

►B

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

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”

1�

Chapter � — Whole Numbers

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

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

On a number line: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.

Model 3 Special Case: Carry or Carries Required

Make Your Own Model

Problem: _________________________________________________________________________

Step 1

Step 2

Step 3

Step 4

Step 5

Either individually or as a team exercise, create a model demonstrating how to solve the most difficult problem you can think of.

Answers will vary.

1�

Activity �.2 — Rounding Whole Numbers

1. Other than reporting income for the IRS, 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?

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

6. In Model 1A, why is 85,291 closer to 85,300 than it is to 85,200?

Answers might include: Determining a tip, Figuring an approximation of a grocery bill to see if you have enough cash, Validating that check book calculations are close, Estimating your budget, Figuring the amount of food needed for a party

Identify the place of the digit by placing an arrow above it. Circle the number that is to the right of the indicated place value.

If the number to the right of the digit’s place to be rounded is 5 or greater, then the indicated place digit would increase by 1. If the digit to the right is less than 5, then the indicated place digit remains the same. All digits to the right of the place to be rounded will change to zeros.

It is the farthest place to the left in the number (the leading digit).Example: 7,643 The largest place value is the 7 which is in thousands place.

Rounding up when the designated place value is a nine required adding one to the nine’s place, changing it to a zero and adding one to the place to the left of the nine. Using the number line and midpoint will validate the correctness of the “carrying.”

Since you are rounding to the nearest hundred, 85,291 is between 85,200 and 85,300. The midpoint is 85,250 and 85,291 falls in the interval from 85,250 and 85,300. Therefore it is closer to 85,300.

1�

Chapter � — Whole Numbers

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?

9. Which step of the model you created is the most difficult to explain to someone else? Explain why.

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

71,350,894 RoundingProcess Answer

a) hundreds 7 1 , 3 5 0 , 8 9 4 71,350,900

b) thousands 7 1 , 3 5 0 , 8 9 4 71,351,000

c) ten- thousands 7 1 , 3 5 0 , 8 9 4 71,350,000

d) hundred- thousands 7 1 , 3 5 0 , 8 9 4 71,400,000

e) millions 7 1 , 3 5 0 , 8 9 4 71,000,000

In general, it is better to use rounded estimates or rounded calculations when you do not need precise or exact numbers.

5 is the midpoint between 1 and 10. Also 15 is the midpoint between 10 and 20, and so on. You can then determine which value you are closest to by comparing to the midpoint, which is represented by 5.

Answers will vary.

1�

Activity �.2 — Rounding Whole Numbers

2. Round each number as specified.

RoundingProcess Answer

a) 2,197 to the nearest ten 2 , 1 9 7 2,200

b) 2,197 to the nearest hundred 2 , 1 9 7 2,200

c) 13,995 to the nearest hundred 1 3 , 9 9 5 14,000

d) 99,647 to the nearest thousand 9 9 , 6 4 7 100,000

3. Round each number to its largest place value:

RoundingProcess Answer

a) 234 2 3 4 200

b) 29,425 2 9 , 4 2 5 30,000

c) 5,678 5 , 6 7 8 6,000

1. Round 92,450,691 to each of the given places:

a) hundreds b) thousands c) ten-thousands

d) hundred-thousands e) millions

2. Round as specified:

a) 4,596 to the nearest ten 4,600b) 59,912 to the nearest thousand 60,000c) 59,912 to the nearest ten-thousand 60,000

92,450,700 92,451,000 92,450,000

92,500,000 92,000,000

1�

Chapter � — Whole Numbers

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.

WorkedSolutionWhat is Wrong Here? IdentifytheErrors CorrectProcess

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

Identified the wrong place.

4 is in the ten-thousands place.

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

The numbers to the right of the indicated place should be changed to zeros.

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

Correct process

Correct answer

4) Round 324,523 to the nearest hundred.

5 should stay 5.

Never round down.

5,246,3926>5

Answer: 5,250,000