Section 3.4 Multiplying and Dividing PRE-ACTIVITY ... · PDF file271 PRE-ACTIVITY PREPARATION How much ﬂ our will you need to double a recipe that calls for 3⅛ cups of ﬂ our?

271

PRE-ACTIVITY

PREPARATION

How much fl our will you need to double a recipe that calls for 3⅛ cups of fl our? What is the surface area of your deck that measures 18¼ feet by 20⅝ feet? How many curtain panels can you cut from a length of fabric 6⅜ yards long if each panel is to be 1½ yards long? When the numbers in daily tasks such as cooking, carpentry, sewing, redecorating, and home repair are presented in fraction form, knowing how to multiply and divide such numbers is a practical skill to possess.

Beyond its relevance to these everyday contexts, having a thorough understanding of multiplying and dividing fractions is necessary for any further study of mathematics.

• Master the multiplication of fractions and mixed numbers.

• Master the division of fractions and mixed numbers.

Multiplying and Dividing Fractions and Mixed Numbers

LLEARNINGEARNING OOBJECTIVESBJECTIVES

TTERMINOLOGYERMINOLOGY

NEW TERMS TO LEARN

invert

of

reciprocal

PREVIOUSLY USED

cancel improper fraction

common factor mixed number

dividend product

divisor reduce

factor

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

To invert a fraction is to interchange the numerator and denominator of the fraction.

For example, to invert , write .3

88

3

Section 3.4

272 Chapter 3 — Fractions

The reciprocal of a fraction is the fraction that results from inverting it.

For example, is the reciprocal of .

When a given fraction is multiplied by its reciprocal, the product will always be 1.

The word of after a proper fraction indicates multiplication (read, “times”).

For example, to calculate of 52 acres, you would multiply

The product of two or more fractions is the product of the numerators over the product of their denominators, as illustrated by the following example.

Example: Find of .

First, shade in of a whole unit.

Then divide the shaded portion into thirds and mark of the shaded portion with ’s.

Now the whole has been divided into 21 parts, with 8 of them marked .

That is,

9

5

5

9

For example, , , and so on.3

8

8

31

5

9

9

51× = × =

3

4

3

452× to get 39 acres.

2

3

4

7

4

7

2

3

4

7

2 4

3 7

8

21 of ×( ) = ×

×=

8

21

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟

2

32

3

4

7

273Section 3.4 — Multiplying and Dividing Fractions and Mixed Numbers

MMETHODOLOGYETHODOLOGY

Multiplying Fractions and Mixed Numbers

Steps in the Methodology Example 1 Example 2

Step 1

Set up the problem.

Set up the problem horizontally for ease of calculation.

Step 2

Convert mixed numbers.

Convert the mixed numbers to improper fractions and rewrite the problem.

Step 3

Prime factor and cancel.

Simplify before multiplying.Determine the prime factorizations of both numerators and denominators; then cancel all common factors.

Step 4

Multiply across.

Multiply the remaining numerators and use the product as the new numerator. Multiply the remaining denominators, and use the product as the new denominator.

Example 1: Multiply by .

Example 2: Multipy:

►►

►► Try It!

Simply multiplying the numerators and denominators of two fractions to fi nd their product will often result in a fraction that must be reduced to lowest terms. The Methodology for Multiplication uses cancelingbefore fi nding the product so as not to end up with large numbers to reduce for the fi nal answer. It also addresses how to effi ciently multiply factors that are mixed numbers.

Be sure to note the shortcut for canceling in Step 3!

7

84

4

5

33

41

1

5×

? ? ? Why can you do this?

78

445

×

78

245

×

7

2 2 2

2 2 2 351 1 1

1 1 1

• •× • • •

7 35

215

• =

Whole number factor(s)(see page 277, Model 2)

Special Case:

Quick reduction(see page 275, Model 1)

Shortcut:

Product of more than two fractions (see page 277, Model 3)

Special Case:


The product of two or more fractions is the product of their numerators over the product of their denominators. It is the same whether you cancel before you multiply the numerators and denominators as indicated in Step 3, or after you fi nd their products and reduce the result to lowest terms.

For Example 1,

canceling before multiplying:

canceling after multiplying:

? ? ? Why can you do Step 3?

7

8

24

5

7

2 2 2

2 2 2 3

5

7 3

5

21

51 1 1

1 1 1

× =× ×

× × × × = × =

7

8

24

5

7 24

8 5

168

40

2 2 2 3 7

2 2 2 5

3 7

5

21

5

1 1 1

1 1 1× = ××= = × × × ×

× × ×= × =


Step 5

Convert to a mixed number (if necessary).

If the product is an improper fraction, convert it to a mixed number.

Step 6

Verify that the fraction is reduced.

Verify that the fraction is fully reduced.

Note: If you canceled all common factors in Step 3, it will be fully reduced. If not, reduce fully now.

Step 7

Present the answer.

Present your answer.

Step 8

Validate your answer.

Validate the fi nal answer by division, using the original fractions and/or mixed numbers.

415

15

is fully reduced.

415

445

215

245

215

524

3 7

5

5

2 2 2 378

1

1

1

1

÷

= ÷

= ×

= • ×• • •

=

215

415

=


Model 1

MMODELSODELS

Multiply

Step 1 Step 1

Step 2 Step 2

Step 3 Step 3

Steps 4 & 5

Step 6

Step 7

Step 8 Validate: Step 8 Validate:

Shortcut: Quick Reduction

►►AShortcut version (optional)

17

92

5

8 by

Shortcut: Cancel the factors (not necessarily prime factors) you easily recognize as being common to both numerator and denominator.

THINK

179

258

× 179

258

×

169

218

×169

218

×

1 1 1

1

1

1 1 1

2 2 2 2

3 3

3 7

2 2 2

• • •

•× •

• •

2

3

7

1

16

9

21

8×

8 is a factor of both 8 and 16.

3 is a factor of both 9 and 21.

2 73

143

423

• = =

23

is fully reduced

423

258

143

218

143

821

2 73

2 2 2

3 7169

179

1

1

÷

= ÷

= ×

= • × • •

•

= =

423

258

143

218

143

8

21169

179

2

3

÷

= ÷

= ×

= =

Answer : 423

THINK

7 is a factor of 14 and 21.


►►B Multiply:

Step 1

Step 2

Step 3

Steps 4 & 5

Step 6

Step 7

Step 8 Validate:

11936

1045

×

119

3610

4

5×

5536

545

×

11

6

9

1

55

36

54

5×

Continue canceling until there are no more common factors to divide out.

THINK 5 is a factor of 5 and 55.

6 is a factor of 36 and 54.

THINK 3 is a factor of the “new” numerator 9 and the “new” denominator 6.

11

6

9

1

55

36

54

52

3

×

OR use this optional notation: When you recognize that you can cancel using “new” numerators and denominators, you may choose to rewrite the problem with its “new” factors so as not to lose track of them in your notation.

For example, 11

6

9

1 2

355

36

54

5

11

6

91

× ⇒ ×

11 32 1

332

1612

××= =

12

is fully reduced

Answer : 1612

1612

1045

332

545

332

5

5411 52 185536

11936

11

18

÷

= ÷

= ×

= ××

= =


Model 3 Special Case: Product of More than Two Fractions

Step 1

Step 2

Step 3

Steps 4 & 5

Step 6

Step 7

Step 8 Validate with two divisions:

Model 2

Multiply:

Step 1

Step 2

Step 3

Steps 4 & 5

Step 6

Step 7

Special Case: Whole Number Factor(s)

5 223

×

5 22

3×

51×8

3

In a fraction problem, if a factor is a whole number, write it in its improper form “the whole number” and proceed from there.

1

51

83× no common factors

5 83

403

1313

× = =

13

is fully reduced

Answer : 1313

Step 8 Validate: 1313

223

403

83

40

3

3

851

5

5

1

1

1

÷

= ÷

= ×

= =

Find the product of , , and .3

10

4

92

1

7

310

49

217

× ×

= × ×310

49

157

= × ×1

2 3

33

10

4

9

157

THINKThe common factor of 3 and 9 is 3.

The common factor of 15 and 10 is 5.

Rewrite and cancel again1

2

4

3

371

2

1

1

× ×

THINK 2 is a factor of 4 and 2. 3 and 3 cancel.

= 27

, proper fraction

27

is fully reduced

Answer : 27

27

217

49

27

157

49

2

7

7

15

9

4

310

1

1

1

5

3

2

÷ ÷ = ÷ ÷

= × × =

= × ×1

1

3

10

4

9

1571

1

2 3

32

The numerator and denominator in which you recognize a common factor do not have to be in adjacent fractions.

Continue canceling common factors.

OR


MMETHODOLOGYETHODOLOGY

Dividing Fractions and Mixed Numbers


Step 1

Set up the problem.

Set up the problem horizontally with the dividend fi rst.

Step 2

Convert mixed numbers.

Convert mixed numbers to improper fractions and rewrite the problem.

Step 3

Invert the divisor and multiply.

Invert the divisor (the second fraction) and change the operation to multiplication.

Step 4

Cancel.

Cancel the common factors by prime factoring fi rst or by using the quick reduction shortcut.

Step 5

Multiply across.

Multiply the remaining numerators and denominators.

Step 6

Convert to a mixed number.

Convert to a mixed number, if necessary.

Example 1: Divide

Example 2: Divide:

►►

►► Try It!

The methodology below converts a given division problem into a multiplication problem to solve.

? ? ? Why do you do this?

63

81

1

2 by .

83

41

7

8÷

638

112

÷

518

32÷

518

×23

1

1

1

1

17

4

1

1

3 17

2 2 2

2

3

51

8

2

3

•

• •×

×

or

172 2

174•

=

174

414

=

Whole number divisor or dividend(see page 281, Model 3)

Special Case:


? ? ? Why do you do Step 3?


Step 7

Verify the fraction is reduced.

Verify that the fraction is fully reduced.

Step 8

Present the answer.

Present your answer.

Step 9

Validate your answer.

Validate your fi nal answer by multiplication, using the original fractions and/or mixed numbers.

14

is fully reduced

414

no common factors to cancel

414×

= ×

= =

112

174

32

518

638

Consider Example 1, , the number divided by the number .

You know that you can also write this division as

The Identity Property of Multiplication tells you that if you multiply this fraction by (which equals 1), the value of your original number will not change.

Note that your denominator now equals 1 because times its reciprocal equals 1.

The same mathematical reasoning will hold for all dividends and divisors. That is why Step 3 simply says “invert the divisor and change the operation to multiplication.”

51

8

3

2÷

51

83

2

.

51832

2323

51832

51832

2323

= × 3

2

2

31

1

1

1

3

2

2

31× =

⎛

⎝⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

You are left with , or That is to say,

518

23

1

51

8

2

3

51

8

×× . ÷÷ = ×3

2

51

8

2

3.


Model 1 Model 2

Divide by 83

4

7

8. Divide:

3

8

1

14÷

MMODELSODELS

1 12 5

110

××=

78÷8

34

78÷ 35

4

78

×4

35

1

2

7

8×

1

5

4

35

110

is fully reduced

Answer : 1

10

110

834

110

354

72 4

78

2

7

×

= ×

=×=

Steps 1 & 2

Step 3

Steps 4 & 5

Step 6

Step 7

Step 8

Step 9 Validate:

38

114÷ no mixed numbers

to convert

38

×141

3

8

141

3 74

2144

7

× = × =

14

is fully reduced

Answer : 514

=514

514

114

214

1

143

4 238

3

2

×

= ×

=×=

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6 proper

Step 7

Step 8

Step 9 Validate:


Model 3 Special Case: Whole Number Divisor or Dividend

►►A

►►B

Divide 10 by 42

5.

Divide: 3 45

7÷

Step 1

Step 2

Step 3

Steps 4, 5 & 6

Step 7

Step 8

Step 9 Validate:

1025

÷4

525

÷41

In a fraction problem, if the divisor or dividend is a whole number, write it as “the whole number” and proceed from there.

1525

×14

13525

135

235

×1

41 = =

35

is fully reduced

Answer : 235

235

4

135

41

525

1035

×

= × = =

Steps 1 & 2

Step 3

Steps 4 & 5

Steps 6 & 7 proper fraction, fully reduced

Step 8

Step 9 Validate:

3 457

31

337

÷ = ÷

= 31

×733

= × =1

11

31

7

33

711

Answer : 7

11

711

457

7

11

33

7

31

31

1

3

1

×

= × = =


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

Issue Incorrect Process Resolution Correct

Process Validation

Multiplying or dividing without fi rst changing to improper fractions

Converting mixed numbers to improper fractions (Step 2) must be done prior to multiplying or dividing. Using the equivalent improper fraction is the most effi cient way to multiply or divide mixed numbers.

Not inverting the divisor before multiplying in a division problem

Division is performed by multiplying the dividend by the reciprocal of the divisor.

In other words, invert the divisor (the second fraction) before multiplying.

Not inverting the divisor before canceling common factors in a division problem

Canceling common factors can only be done when the operation is multiplication or when reducing a fraction.

Not dividing out common factors fi rst and ending up with large numerators and denominators to reduce, thus making the processes of multiplication and reduction more diffi cult

If you cancel as many common factors as possible, your product will be much more easily reduced.

In fact, if all common factors are divided out before multiplying, there is no need to reduce the fraction. It is already in lowest terms.

so it can be reduced further; therefore, it is not the correct lowest terms answer.

345

614

19

5

254

954

2334

1

5

×

= ×

= =

2334÷ = ÷

= ×

= =

614

954

254

95

4

4

25195

345

19

51

1

49

15

49

51

2 23 3

51

209

229

÷ = ×

= ••×

= =

229×

= × =

15

209

1

5

49

4

1

35

57

35

75

2125

÷ = × =2125×

= × =

57

21

25

5

7

35

3

5

1

1

21147

3 73 7 7

= •• •

27

1415

34

57

2

7

14

15

3

4

57

17

1

1

2

3

1

2

11

1 1

× × ×

= × × ×

=

17

57

34

1415

1

7

7

5

4

3

15

1427

1

1

1

2

1

1 3

2

÷ ÷ ÷

= × × ×

=

345

614

18420

1815

× =

=

49

15

445

÷ =

3

5

57

371

1

÷ =

27

1415

34

57

4202940420 102940 104229442 2294 221

147

× × ×

=

= ÷÷

=

= ÷÷

=

× 1614

49 4

15÷

3

55

551

37

÷

242

=

2

94÷÷

4940


PPREPARATION REPARATION IINVENTORYNVENTORY

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

the terminology and notation associated with multiplying and dividing fractions

the value and process of reducing before multiplying

how to deal with mixed numbers in multiplication and division

the division process—how and why division is turned into multiplication

how to present a fi nal answer

how to validate the answer to a multiplication problem involving fractions

how to validate the answer to a division problem involving fractions

Issue Incorrect Process Resolution Correct

Process Validation

Improperly dividing out common factors (improper cancellation)

Cancel any one factor in the numerator with only one matching denominator factor.

Incorrectlyrepresenting a whole number as a fraction

In a fraction problem, write a whole number as

and proceed.

Not fully reducing the fi nal answer

Before presenting your fi nal answer, always verify that the proper fraction portion is fully reduced, in case you missed a possible cancellation.

any numberthat same number

=1

the whole number1

37

29

1415

3

7

2

9

1415

2 23 15

445

1

1 3

2

× ×

= × ×

= ••

=

445

1415

29

4

45

15

14

9

2

37

1 2

1 3

1

7

3

1

÷ ÷

= × ×

=

413

25

41

13

25

815

× ×

= × ×

=

815

25

13

8

15

5

2

31

41

4

4

1 3

1

1

1

÷ ÷

= × ×

= =

412

216

92

136

11712

99

12

× = ×

= =

Is reduced?

Not yet. 9 312 3

Final answer:

912

34

934

÷÷=

934

216

394

136

39

4

6

1392

412

3

2

3

1

÷ = ÷

= ×

= =(There was a common factor of 3 that could have been canceled before the multiplication.)

37

29

1415

3

7

2

9

14

152 23 5

415

1

1 3

2

5

× ×

= × ×

= ••=

413

25

4

4

13

25

215

2

12

1

× ×

= × ×

=

412

216

92

136

11712

99

12

×

= ×

= =

33

77 9×

9

993

44

2

22 3

5

3×

3

443×

172

69

127 =

284

ACTIVITY

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

• Multiplying any given combinations of fractions and mixed numbers correctly – presentation of the fi nal answer in lowest terms – validation of the answer

• Dividing any given combinations of fractions and mixed numbers correctly – presentation of the fi nal answer in lowest terms – validation of the answer

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. What is the fi rst critical step when multiplying or dividing mixed numbers?

2. How are whole numbers converted to fractions for multiplying and dividing?

3. How do you convert a division of fractions into a multiplication of fractions?

4. What can you do to simplify a multiplication of fractions problem before computing the fi nal answer?

Multiplying and Dividing Fractions and Mixed Numbers

Section 3.4


5. What property(ies) permit you to cancel a factor in a numerator with a factor in another denominator when multiplying fractions?

6. What is the result when all factors in the numerators cancel out?

7. What is the result when all factors in the denominators cancel out?

8. How do you validate that your fi nal answer is both properly presented and correct?


TTIPS FOR IPS FOR SSUCCESSUCCESS

DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

Problem Worked Solution Validation

1)

2)

3)

Solve each problem and validate your answer.

1235

730×

23

18

45

35

× × ×

38

27

÷

• Write fractions using a horizontal fraction bar rather than a slash ( rather than 2/3). Using a slash can interfere with proper alignment of the problem.

• Use neat and consistent notation when dividing out common factors so that you do not cancel too many or too few of them.

• Replace each completely canceled factor with a 1.• For multiplication, if you cancel all common factors within the problem, your result will be a fully reduced

answer.• Always verify that your fi nal answer is fully reduced by prime factoring your answer.• Because a fraction problem has intermediate steps, it is especially important to validate the fi nal answer

using the original fractions and/or mixed numbers.

2

3



4)

5)

6)

7)

59

637

×

634

849

×

312

514

÷

613

÷



8)

9)

10) Bruno’s share of the profi ts from a land sale is to be 2/7 of $280,000. Calculate his share.

Try to do these “in your head.”

a) What is of 42? b) What is of 80?

c) of 90 is what number? d) 12 is what part of 36?

e) of what number is 20? f) of what number is 12?

558

3÷

516

425

6× ×

1

2

2

3

1

4

1

4

1

5

MENTAL MATHMENTAL MATH


TEAM EXERCISESTEAM EXERCISES

Discuss and circle the correct answer to each of the following.

1. When you multiply a proper fraction by a proper fraction, your answer will always be: a) a mixed number b) an even smaller proper fraction c) a larger proper fraction

2. When you multiply a proper fraction and a mixed number, your answer will always be: a) less than the mixed number b) greater than the mixed number

3. When you divide a mixed number by a proper fraction, your answer will always be: a) less than the mixed number b) greater than the mixed number

4. When you divide a mixed number by a larger mixed number, your answer will always be: a) less than one b) greater than one


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) The final answer must be reduced to lowest terms.

is equivalent to

the correct answer, but it is not fully reduced:

2)

3)

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

78

45×

535

318

÷

56

310

89

34

• • •

1420

7

8

45

7102

1

× =

78

45

2840

28 240 2

14 220 2710

× =

÷÷

= ÷÷

=

OR

Answer:710

710

45

7

10

54

78

2

1

÷

= ×

=

1420

2 72 2 5

= ×× ×


Worked SolutionWhat is Wrong Here?

Identify Errors or Validate Correct Process Validation

4)

5)

6)

117

514

×

Find the product of

, and 35

415

58

, .

1216

623

×


ADDITIONAL EXERCISESADDITIONAL EXERCISES

Perform the indicated operations and validate your answers.

1.

2.

3.

4.

5

32

9

3

5

27

81

1

4

3

7

2

5

14

15

10

11

21

64

7

8

×

×

× × ×

÷

..

6.

7.

8. 1

9.

31

91

1

2

2 11

2

12 11

24

1

3

21

2

5

7

÷

÷

× ×

÷

1183

420÷

Documents

Section 3.4 Multiplying and Dividing PRE-ACTIVITY ... · PDF file271 PRE-ACTIVITY PREPARATION How much ﬂ our will you need to double a recipe that calls for 3⅛ cups of ﬂ our?