PRE-ACTIVITY Dividing Whole Numbers … 1.5 — Dividing Whole Numbers 91 Special Properties of...

89

PRE-ACTIVITY

PREPARATION

You and fi ve friends are enjoying your monthly get-together at a local restaurant. Your bill for appetizers, entrees, beverages, desserts, and gratuity totals $168. If you decide to simply split the bill equally, how much will you have to pitch in?

When you fi ll your gas tank with 12 gallons and the odometer on your car indicates that you have driven 276 miles since your last fi ll-up, you can fi gure out the gas mileage of your fi ve-year-old car by dividing miles by gallons.

So many everyday circumstances such as these require a basic knowledge of when and how to divide numbers. Learning the process of long division will strengthen your ability to think logically, help reduce your errors by carefully following a set procedure, and give you insight into the value of precision in your calculations. Furthermore, as you progress through this basic math course, you will appreciate your profi ciency with this fourth basic operation when you work through decimals, fractions, ratios, and percents.

Master the Long Division Methodology.

Dividing Whole Numbers

NEW TERMS TO LEARN

dividend

divisor

inverse operation

long division

partial dividend

quotient

remainder

undefined

PREVIOUSLY USED

operation

place value

product

validate

value

LLEARNINGEARNING OOBJECTIVEBJECTIVE

TTERMINOLOGYERMINOLOGY

Section 1.5

90 Chapter 1 — Whole Numbers

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

Division is the mathematical operation you use to fi nd out how many ways something can be distributed or split into equal portions.

Think about the simple example of four adults sharing a lunch of one large pizza cut into twelve slices. Each person takes one slice, then a second, then a third.

12 (slices) divided by 4 (adults) = 3 (slices per adult)

From another perspective, dividing the same pizza, one might ask, “If a lunch-size serving of pizza is three slices, how many adults can equally share a twelve-slice pizza?”

12 (slices) divided by 3 (slices per adult serving) = 4 (adult servings)

What happens if you try to divide the twelve-slice pizza equally among fi ve adults? In this case, each person receives one slice and then a second, but there would be only two slices left on the tray, not enough for everyone to have a third slice.

12 (slices) ÷ 5 (adults) = 2 (slices per adult) with 2 slices left over

In mathematical terms, this leftover amount is called the remainder.

Any of these phrases indicates division—“twelve divided by four equals three”“four goes into twelve three times”“four divides into twelve three times”“four divided into twelve equals three”

might be written symbolically in any of the following ways:

The second is the traditional set-up for a division problem with larger numbers.

The number being divided is called the dividend (12 in each example above), and the number you are dividing by is called the divisor (4). The result of the division process is called the quotient (3).

)12 4 3 4 123 12

43 12 4 3÷ = = = / 4 12

3

)divisor dividendquotient

91Section 1.5 — Dividing Whole Numbers

Special Properties of Division Involving One (1)

Any number (other than zero) divided by itself is equal to one (1).

Example: 8 ÷ 8 = 1

Any number divided by one (1) is the number itself.

Example: 9 ÷ 1 = 9

Special Properties of Division Involving Zero (0)

Zero (0) divided by any number is zero (0).

Example: 0 ÷ 2 = 0

Division by zero (0) is not possible (is undefined).

Example: 3 ÷ 0 is undefi ned

Division and multiplication are opposite operations (inverse operations), and you determine a quotient from your knowledge of the multiplication facts. For example, you can state with certainty that 30 ÷ 6 = 5 precisely because 5 × 6 = 30.

Following are the Special Properties of Division involving the numbers one (1) and zero (0). A simple example is given for each.

Steps in the Methodology Example 1 Example 2

Step 1

Set up the problem.

Set up the problem with the dividend and the divisor properly positioned. Allow enough workspace.

MMETHODOLOGYETHODOLOGY

Long division is the name given to the step-by-step process of breaking down a division problem into smaller manageable pieces and using division, multiplication, and subtraction to generate the answer. Specifi cally, the methodology breaks the dividend into parts (partial dividends) to simplify the calculation.

Long Division

Example 1: Divide: 47,658 by 18 Example 2: Divide: 39,598 by 21►►►► Try It!

)18 47658

)2 00

0 2 0 because × =

)0 3? because no number × =0 3



Step 2

Identify the fi rst partial dividend.

Identify the part of the dividend to be used for the fi rst division (the fi rst partial dividend).

For the fi rst division, count the number of digits in the divisor. Count from the left digit in the dividend the same number of places.

• If the resulting partial dividend is greater than the divisor, you are ready for the fi rst division.

• If it is less than the divisor, pick up one more digit before beginning. (See Model 1 on page 95.)

To correctly align the quotient, underline the last digit of the fi rst partial dividend.

There are 2 digits in the divisor (18).

18 < 47

18 will divide into 47.

Underline the 7.

Step 3

Divide.

Divide the divisor into the partial dividend.

Place the answer in the quotient directly above the underlined digit.

Divide 18 into 47.

How many 18’s are in 47?

Best prediction: 2

Place the answer over the 7.

Step 4

Multiply.

Multiply the answer by the divisor and align the product correctly under the partial dividend. The product must be less than the partial dividend.

If your prediction results in a product greater than the partial dividend, lower your prediction, usually only by 1.

(See Model 2 on page 97.)

2 × 18 = 36

36 < 47

Step 5

Subtract.

Subtract the product from the partial dividend.

(To assure accuracy, mentally validate each subtraction as you proceed.)

Step 6

Verify the quotient digit.

Verify your prediction.

To verify that you predicted the correct quotient digit, the subtraction answer must always be less than the divisor.

11 < 18

OK to move on to the next step

)18 47658

? ? ?Why do you do this?

)18 476582

)18 476582

36

1

)18 4765836

2

11

1

−

THINK

THINK



Step 7

Bring down the next digit.

Bring down the next digit from the dividend and complete Steps 3–6 with the newly formed partial dividend.

That is,

Bring down,

Divide,

Multiply,

Subtract,

Verify. 8 < 18

Move to next step.

Step 8

Repeat the process.

Repeat the BDMSV process until there are no more digits to bring down. The fi nal operation in the long division process will always be a subtraction, resulting in 0 or a number between 0 and the divisor (the remainder).

13 < 18 Move on.

12 < 18

Division is complete.

Step 9

Present the answer.

Present your answer, with remainder, if there is one.

2647 R 12

? ? ? ?How and why do you do this?

)18 47 5836

11

21

0 1

4

6

6108

8

6

−

−

new partial

dividend

)18 47 6 836

116108

8

261

0 1

43

5

57213

4

−

−

−

new partial

dividend

)18 476536

116108

857213

4261

0 1

435

8

8126

7

−

−

−

−12

last partial

dividend

continued on the next page

BDMSV

Special Case:

Zero(s) in the quotient (see page 98, Model 3)

Special Case:

Zero remainder (see page 100, Model 5)

Zero (0) within the quotient (see page 99, Model 4)

Shortcut



Step 10

Validate your answer.

Validate your answer by multiplying the quotient times the divisor and adding the remainder.

The result should match the dividend.

quotient

divisor

remainder

dividend

264718

2117626470

476462

47658

5 3 5

1

×

+

+

1

? ? ?Why do you do Step 3?

It is important to identify the digits in the partial product. It is equally important to recognize the places these digits occupy in the original dividend. For it is their place values that will, in turn, determine the correct placement of the digits in the quotient.

In Example 1, the digits that make up the fi rst partial dividend, 47, occupy the ten-thousands and thousands places of the original dividend. Therefore, in Step 3, when you have divided the divisor 18 into the fi rst partial dividend 47, you have determined, in fact, how many thousands of 18’s are in 47,000.

)18 6 5 84 72

THO

USA

ND

S

HU

ND

RED

STEN

SO

NES

? ? ? ?How and why do you do Step 8?

The 2, by its thousands position in the quotient, indicates that there are 2000 18’s (= 36,000) in 47,000.

An effective and simplifi ed approach to solving a division problem with larger numbers is to determine the quotient digit by digit, beginning with the largest place value of the dividend.

The process of determining each digit in the quotient moves forward as you determine how many times the divisor goes into the next partial dividend until you can no longer divide into the number that remains.

In Example 1, the partial dividends 47, 116, 85, and 138 are divided, in order, by the divisor 18. Taking into account the values of the digits in the partial dividends and the corresponding value of each digit placed in the quotient, the actual long division process unfolds as follows:


By ignoring the ending zeros, this Methodology for Long Division greatly simplifi es this process while still assuring the same answer—with correct digits properly placed in the quotient, and with the same remainder.

)18 6 5 83 6

5 81 0 8

87 2

1 2 6

1 2

0 0 0

0 0

0

4 72 64 7

1 1 6

8 5

1 3 8

−

−

−

−

18 × 2000

18 × 600

18 × 40

18 × 7

Remainder

MMODELSODELS

Model 1

Divide 38652 by 52.

Step 1

Step 2 2 digits in the divisor (52)

Underline the 6.

Step 3 Divide: Place the 7 over the underlined 6.

Step 4 Multiply: 7 × 52

)52 38652

52 will not divide into 38.

Pick up one more digit in the dividend for the fi rst partial dividend (386).

)52 386527

PREDICT

How many 52’s are in 386?

5 divides into 38 seven times.

Best predicition: 7

)52 38652364

71

)52 38652

THINK


Step 5 Subtract the product (364) from the partial dividend (386).

Step 6 Verify: 22 < 52 OK to go on to the next step.

Step 7 Bring down the next digit to form the the next partial dividend and repeat the process.

Divide 52 into 225.

Step 8 One fi nal digit to bring down and repeat the BDMSV process.

Step 9 Answer: 743 R 16

Step 10 Validate:

)52 3865236422

71

−For accuracy, validate this subtraction before proceeding!

)52 386 2364

2 2

71

1 1

5

5208

17

4

−

−17 < 52 OK

)52 38 65364

2 2 52 08

17

741

1 1

6 1

2

2156

16

3

−

−

−16 < 52 OK

5 divides into 22 four timesPREDICT

5 divides into 17 three timesPREDICT

74352

148637150

3863616

38652

2 1

1

1

×

+

+

quotient

divisor

remainder

matches the dividend


Divide 20,139 by 34.

Steps 1 & 2 The fi rst partial dividend is 201. Underline the 1.

Steps 3 & 4

Step 5

Step 6 31 < the divisor 34. OK to bring down the next digit from the dividend.

Steps 7 & 8

11 < 34 The division is complete.

Step 9 Answer: 592 R11

Step 10 Validate:

Model 2

)34 20139

)34 20139204

62

−

3 divides into 20 six times, so 34 divides into 201 approximately 6 times.PREDICT

THINK 204 > the partial dividend 201.Therefore, the predicted digit is incorrect. Try 5. )34 20139

170

52

−

)34 2 0139170

31

52 1 1

−

)34 2 01 391 70

3 1 33 0 6

7968

11

5 922 1 1

1

3

0

−

−

−

⇒

3 divides into 31 ten times. Since the digit in the quotient must be < 10, try 9.PREDICT

34 divides into 79 twice. PREDICT

59234

236817760

2012811

20139

3

1 1

1

2

×

+

+

remainder


7062

1404200

434027

4367

×

+

+

Divide: 4367 ÷ 62

Steps 1 & 2

Steps 3,4, 5, & 6

2 is less than the divisor (62). OK to bring down the next digit from the dividend.

Step 7

Step 8 Since there are no other digits to bring down, the division is complete. 27 is the remainder.

Step 9 Answer: 70 R 27

Step 10 Validate:

Model 3

6 divides into 43 seven times. The best prediction for 436 ÷ 62 is 7.

)62 4367

PREDICT

Special Case: Zero(s) in the Quotient

►►A

)62 4367434

2

71

−

Each time you bring down a digit from the dividend, you must place a single digit in the quotient—even if it is a zero—to occupy that place in the quotient.

Since 62 divides into 27 zero times, a zero (0) must be placed in the quotient above the 7 that was brought down from the dividend.)62 436

4342

27

717

70

0

−

− 0 × 62 = 0

27 – 0 = 27

27 < 62 OK

remainder


Model 4

►►B Divide : 122 405

3

,

)3 1224051202024240000532

4 8 1

−

−

−

−

−

0 0

3 divides into 2 zero times. Place a zero in the quotient.

3 divides into 0 zero times. Place a zero in the quotient.

Answer: 40,801 R2

408013

1224032

122405

2

×

+

Validate:

remainder

Shortcut: Zero Within the Quotient

Divide : 122 405

3

, Compare the following shortcut to the process demonstrated in Model 3B.

Answer: 40,801 R2

Shortcut (optional):After placing a zero (0) in the quotient, instead of multiplying by 0 and subtracting 0, simply bring down the next digit from the dividend and continue (DMSV).

)3 122405120240

32

4 8 1

−

−

−

2

0

4

5

0 0


For most division problems, you can easily estimate the quotient by fi rst rounding both divisor and dividend to their largest place values and then dividing.

In the following two examples, you can divide quickly into the rounded dividend with one simple mental calculation.

Example: 6,185 ÷ 28 Estimate: 6,000 ÷ 30, or 200 Actual answer: 220 R 25

Example: 92,397 ÷ 32 Estimate: 90,000 ÷ 30, or 3000 Actual answer: 2887 R 13

Some problems, however, require several computation steps, which you may or may not be able to quickly compute in your head, as in the next example:

Example: 8,974 ÷ 41 Estimate: 9,000 ÷ 40

For a quick one-step mental estimate of the answer to this kind of problem, you might predict only the fi rst digit of the quotient (and, of course, its corresponding place value).

Answer: 312

40 goes into 90 two times.

Estimated answer: 200Actual answer: 218 R 36

How Estimation Can Help

THINK

)30 600060000

200

−

)30 90000900000

3000

−

THINK

)40 90008010080200200

0

225

−

−

−

THINK

Model 5

When the fi nal remainder is zero (0), there is no need to indicate a zero remainder in the fi nal answer.

Divide 126,984 by 407.

)407 1269841221

488407814814

3122

−

−

−0

Validate: 312407

21840000

124800126984

1

×

Special Case: Zero Reminder


Remember, however, that estimation is not an exact process. The numbers you are working with will dictate the method of estimation you will use for your approximation of the answer.

Consider, for example, 2,490,015 ÷ 592 Rounding each number to its largest place:

Estimate: 2,000,000 ÷ 600, or 3,000

However, think about how far down you round 2,490,015 this way—by nearly half a million.A better estimation process might be to round the dividend to 2,500,000:

Estimate: 2,500,000 ÷ 600, or 4,000

Actual answer: 4206 R 63

There is another important use for estimation in long division, as you have seen demonstrated in the method-ology examples and models. Each time you predict the quotient digit as you divide the divisor into a partial dividend, you use an estimation process. It is usually easiest to round the divisor to its largest place value and the partial dividends accordingly. Remember that estimation serves you best when you round, correctly of course, to the places that allow for the easiest mental math computations.

Look again at the previous example in which you must divide 592 into the partial dividends 2490, 1220, 361, and 3615:

600 goes into 2000 three times

THINK 600 goes into 2500 four times

THINK

First, round the divisor 592 to 600.In this problem, it appears best to round each partial dividend as follows:

600 into 2500 6 goes into 25 four times

600 into 1200 6 goes into 12 two times

600 into 400 6 goes into 4 zero times or

600 into 3600 6 goes into 36 six times

592 2 4 9 0 0 1 5

2 3 6 8

1 2 2 0

1 1 8 4

3 6 1

0

3 6 1 5

3 5 5 2

6 3

4 23 8 1

1 1 1

1

15 1

1

5

−

−

−

−

)00 6

THINK

THINK

THINK

THINK

600 goes into 400 zero times


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

Issue Incorrect Process Resolution Correct

Process Validation

Misaligning the columns of numbers

Answer: 53 R 4

Alignment of place values is critical to success in long division problems.

Use down arrows for the “bring down” step.

Use graph or grid paper, lined paper, a straight edge, and/or different colored pencils.

Answer: 536

Not using zero as a placeholder in the quotient when necessary

Answer: 35 R 2

Alignment is critical.

Once the fi rst digit in the quotient has been positioned correctly, bring down the digits from the dividend one at a time. Each time you bring down a digit you must place a single digit in the quotient—even if it is a zero—to occupy that place in the quotient.

(If the partial dividend is smaller than the divisor, the quotient digit is zero (0) for that place.)

Answer: 305 R 2

Making subtraction errors

Answer: 126 R 9

Validate each subtraction step to ensure the accuracy of the next partial product.

Use borrowing notation.

Answer: 122 R 1

)7 37523525214

53

−

−

5367

3752

2 4

× )7 375235252142420

536

−

−

−

)3 9179017152

35

−

−

)3 917901017152

305

−

−

−

R

3053

9152

917

1

×

+

)23 2807235046147138

9

126

−

−

−

)23 280723

5 046

4746

1

122

4 1

−

−

− R

12223

36624402806

1

2807

1

×

+

22

−

21

3325

−

)7

917

−

1

5064647


PPREPARATION REPARATION IINVENTORYNVENTORY

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

the terminology and notation associated with dividing whole numbers

the repeated steps of the Methodology for Long Division (B D M S V)

when and why there would be a remainder in a long division problem

the validation process (including the remainder, if there is one)

Issue Incorrect Process Resolution Correct

Process Validation

Presenting an answer with a remainder greater than the divisor

Answer: 245 R 26

The remainder cannot be greater than the divisor.

Verify that the answer in each subtraction step, including the last, is less than the divisor.

2 < 24 OK

Answer: 246 R 2

)24 59 0648

11096

146120

26

2451

0 1

2

−

−

−

)24 59 0 64 8

1 1 09 6

14614 4

2

24 61

0 0 1

2

1

−

−

− R

24624

984

49205904

2

5906

1 2

1

1

1

×

+

−96

48

1116

01111

−

104

ACTIVITY Dividing Whole Numbers

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

Using long division to divide any two numbers • neatness of presentation • validation of the answer

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. What are four different ways to express “sixty divided by fi ve” in mathematical notation?

• •

• •

2. When will a quotient have a remainder?

3. How do you determine and signify the starting place of your quotient?

Section 1.5


4. In the Methodology for Long Division, when does a zero become part of the quotient?

5. What does it mean when the result of a subtraction step is a number greater than your divisor?

6. When dividing, what notational techniques can you use to improve your accuracy?


7. How can you be sure that you have validated your answer to a divison problem?

8. What strategies can you use to improve your prediction of a specifi c digit in the quotient?

TTIPS FOR IPS FOR SSUCCESSUCCESS

• Alignment is critical. Using graph paper can help with alignment.

• Show all of your work neatly and legibly, with proper notation.

• Know the methodology: DMSVB

• For each and every digit brought down from the dividend, place a single digit (0 through 9) above it in the quotient.

• Validate each subtraction as you proceed through the Long Division Methodology to assure accurate partial dividends.

• Always validate the fi nal answer!


DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

1. Estimate the answer by fi rst rounding each number to its largest place value and then dividing.

a) 839 ÷ 18 b) 4356 ÷ 52

c) 8,843,610 ÷ 2,920 d) 50,320 ÷ 39

2. Use long division to fi nd the quotient for each of the following problems.

Problem Worked Solution Validation

a) 585 ÷ 21



b) 32,475 ÷ 59

c) 23,218 ÷ 29

d) Divide 217,599 by 72.



e) 423,511 ÷ 39

f) How many times will 123 divide into 328,947?


TEAM EXERCISETEAM EXERCISE

Consider the following four division statements. What special property of division does each represent? Validate each example. Determine which property of multiplication is the basis for the validation.

Example Special Property of Division Validate the Example Property of Multiplication

a) 53 ÷ 1 = 53

b) 53 ÷ 53 = 1

c) 0 ÷ 53 = 0

d) 53 ÷ 0 = ? undefi ned

In the second column, identify the error(s) you fi nd in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column.

Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

1) 24,370 ÷ 12 Division process not completed.

The fourth digit of the quotient—a zero—is missing.

12 divides into 10 zero times.

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

203012

40602030024360

1024 370

×

+

,

)12 243702403

03736100

10

2030

−

−

−

−

Ans: 2030 R10


Worked Solution Identify Errors or Validate Correct Process Validation

2) 486,633 ÷ 221

3) 47,106 ÷ 23

4) 16,698 ÷ 33


Worked Solution Identify Errors or Validate Correct Process Validation

5) 5,302 ÷ 39

Perform the indicated operation in each of the following and validate your answers.

1. 4437 ÷ 11

2. 48,946 ÷ 57

3. Divide 35,598 by 79.

4. 893,590 ÷ 275

5. 321,147 ÷ 62

6. How many times will 12 divide into 48,082?

ADDITIONAL EXERCISESADDITIONAL EXERCISES

PRE-ACTIVITY Dividing Whole Numbers … 1.5 — Dividing Whole Numbers 91 Special Properties of...

Documents

Transcript of PRE-ACTIVITY Dividing Whole Numbers … 1.5 — Dividing Whole Numbers 91 Special Properties of...