pre-activity reducing fractions preparation · in addition to the divisibility tests for the...

259

PRE-ACTIVITY

PREPARATION

You must often use numbers to communicate information to others. When the message includes a fraction whose components are large, it may not be easily understood. In that case, you might use a simplifi ed form of the fraction to most effectively convey the same information. The following example illustrates the practicality of the mathematical skill of reducing fractions to their simplest form.

Consider, for an example, the fi ve hundred twenty out of seven hundred eighty third graders in a certain school district who ride the bus to school (520/780) . The same fraction in its simplest form is 2/3—two thirds of the third graders ride the bus to school. This simplest form more effi ciently communicates the same information to prospective school parents.

• Reduce fractions to lowest terms by dividing or canceling out common factors.

• Use the meaning of equivalent fractions for validation.

Reducing Fractions

LLEARNINGEARNING OOBJECTIVESBJECTIVES

TTERMINOLOGYERMINOLOGY

NEW TERMS TO LEARN

cancel

common factor

cross-multiply

cross-product

does not equal sign ≠

equivalent fraction

lowest terms

reduce

simplify

PREVIOUSLY USED

denominator

divisibility tests

factor

numerator

prime factorization

Section 3.3

260 Chapter 3 — Fractions

BBUILDING UILDING MMATHEMATICAL ATHEMATICAL LLANGUAGEANGUAGE

Fractions are equivalent (equal) when they represent the same part of a whole or group, the same division, or the same ratio. For example, is equivalent to

One way to check whether two given fractions are equivalent is to apply the following test.

A cross-product is found by multiplying the numerator of one fraction by the denominator of the other. For example, to determine whether the fractions in the Introduction, and , are equivalent fractions, you could calculate the cross-products (cross-multiply):

Does 520 × 3 equal 2 × 780? Yes, 1,560 = 1,560 The two fractions are equivalent.

Now consider

These two fractions are not equivalent because they do not pass the Test for Equality.

That is, 2 × 9 ≠ 4 × 5 Read, “2 times 9 does not equal 4 times 5” 18 ≠ 20 Read, “18 does not equal 20.”

A common factor of two or more numbers is a factor that they share. That is, it divides evenly into each of the numbers. For example, 3 and 6 are the two common factors of 12, 18, and 36.

To reduce a fraction is to rewrite it as an equivalent fraction with a smaller numerator and smaller denominator. To reduce a fraction to its lowest terms or to simplify a fraction is to write the equivalent fraction whose numerator and denominator have no common factors.

Example:

1

22

4

VISUALIZE

Equality Test for Fractions

If two fractions are equal, their cross-products will be equal.

520

780

2

3

2

5

4

9 and .

52037802

=?

2

4

1

2= in reduced form.

261Section 3.3 — Reducing Fractions

MMETHODOLOGYETHODOLOGY

Reducing a Fraction

Steps in the Methodology Example 1 Example 2

Step 1

Prime factor— numerator

Determine the prime factorization of the numerator.

Step 2

Prime factor— denominator

Determine the prime factorization of the denominator.

Example 1: Reduce to its lowest terms.

Example 2: Reduce to its lowest terms.

►►

►► Try It!

This methodology breaks down the numerator and denominator of a fraction to their prime factorizations, in order to easily see their common factors. It is particularly useful to use when the common factors of the original numerator and denominator are not readily apparent to you.

Be sure to note its shortcut options!

30

3642

140

2 30

3 15

5 5

1

2 36

2 18

3 9

3 3

1

Quick reduction(see page 263, Model 2)

Shortcut:

Additional Divisibility Tests

• If the fi nal two digits of a number form a number divisible by 4, then the original number is divisible by 4.

• If a number is divisible by both 2 and 3, then it is divisible by 6.

• If the sum of the digits of a number is divisible by 9, then the original number is divisible by 9.

• If the number ends in 0, then it is divisible by 10; if it ends in 00, it is divisible by 100; in 000, it is divisible by 1000, and so on.

In addition to the divisibility tests for the numbers 2, 3, and 5 which you used to determine the prime factorization of a number, there are several other divisibility tests which you might use to determine common factors:


Steps in the Methodology Example 1 Example 2

Step 3

Write as prime factorization.

Re-write the fraction using the prime factorizations.

Step 4

Cancel.

Cancel each common numerator factor with its matching denominator factor.

Step 5

Multiply remaining factors.

Multliply the remaining numerator factors to get the new numerator and the remaining denominator factors to get the new denominator.

Step 6

Present the answer.

Present your answer.

Step 7

Validate your answer.

Validate by using the Equality Test for Fractions. Compare the cross-products of the original fraction and the reduced fraction. The cross products must be equal.

Also, there should be no common factors between the numerator and denominator of the fi nal answer.

? ? ? Why can you do this?

Recall the Special Property of Division that states that any number divided by itself equals 1.

For any fraction, then, if a factor in the numerator is equal to a factor in the denominator, you can apply this property and replace the two factors with the number 1 (or 1/1), a procedure called canceling.

Example:

You will get the same result if you divide both the numerator and denominator by the same common factor.

OR, with canceling notation,

? ? ? Why can you do Step 4?

any number

that same number That is,

1

1,

2

2= = =1 1 1,,

3

3, and so on.= 1

6

10

3 2

5 2

3

51

1

1= ×

×× or or, by the Identity Property of Multiplication, simply

3

5

6

10

6 2

10 2

3

5= ÷

÷=

2 3 52 2 3 3× ×× × ×

1 1 51 2 1 3

56

× ×× × ×

=

56

56

52 3

=×

3036

56

=?

no common factors

?30×6 = 5×36

180 = 180

1

1

1

1

2 3 52 2 3 3× ×× × ×

3

5

6

10

3

5= .


Model 1

MMODELSODELS

Reduce to lowest terms:

Step 1 Step 2 Step 3 Step 4 Steps 5 & 6

Step 7 Validate:

84 × 19 = 12 × 133

1,596 = 1,596

84

133

2 84

2 42

3 21

7 7

1

7 133

19 19

1

2 2 3 77 19× × ××

2 2 3 77 19

1

1

× × ××

4 319× = 12

19

84133

1219

=?

1219

2 2 319

= × ×

no common factors

?

Model 2Shortcut: Quick Reduction

Shortcut: Before you do Steps 1 and 2, fi rst divide out the factor(s) you readily recognize as being common to both the numerator and denominator.

Simplify:

Step 1 Step 2

Step 3 Step 3

Step 4 Step 4

Steps 5 & 6 Answer: Steps 5 & 6 Answer:

Step 7 Validate:

2 440

2 220

2 110

5 55

11 11

1

2 1870

5 935

11 187

17 17

1

440

1870

2 2 2 5 112 5 11 17× × × ×× × ×

1 1 1

1 1 1

2 2 2 5 11

2 5 11 17

× × × ×

× × ×

417

►►A

2 2 1111 17× ××

2 2 11

11 17

1

1

× ×

×

4401870

417

=?4

172 217

= ×

no common factors

Step 1 Step 2

11 187

17 17

1

2 44

2 22

11 11

1

THINK

?440 × 17 = 4 × 1870 7,480 = 7,480

Shortcut Version (optional)

both divisible by 10

440 101870 10

44187

÷÷

=

417

Answer: 1219


AADDRESSING DDRESSING CCOMMON OMMON EERRORSRRORS

Issue Incorrect Process Resolution Correct

Process Validation

Not reducing all the way to lowest terms when simplifying a fraction

Simplify: For this issue, validating by cross-multiplying alone does not catch the error since both fractions are, in fact, equivalent.

Always do a prime factorization of your fi nal answer to assure that there are no remaining common factors to cancel.

Simplify:

Mismatching the factors when canceling

Reduce: Use effective notation.

Factors must be canceled in pairs: one numerator factor with only one denominator factor, with each pair equaling one (1).

Reduce:

PPREPARATION REPARATION IINVENTORYNVENTORY

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

the terminology and notation associated with reducing fractions

the meaning of equivalent fractions

what it means to reduce (simplify) a fraction to its lowest terms

why you can cancel common factors when reducing

the validation of the fi nal reduced fraction

►►B Reduce to lowest terms:

Use the shortcut to divide out the common factors.

Validate:?

48 × 4 = 3 × 64 192 = 192

4864

34

=? no common factors34

32 2

=×

48 864 8

68

6 28 2

÷÷

= ⇒ ÷÷=

34

Answer

48

64

150240

150240

150 10240 10

1524

÷÷

=

1524

3 5

2 2 2 358

1

1= ×

× × ×

=

150240

58

150 8 5 2401200 1200

=

× = ×=

58

52 2 2

=× ×

60126

60126

60126

=

60126

2 2 3 5

2 3 3 71021

1 1

1 1= × × ×

× × ×

=

no common factors

1021

2 53 7

= ××

no common factors

60 21 10 261260 1260× = ×

=

?

?

?

?

150 10240 10

1524

150 24 15 2403600 3600

÷÷

=

× = ×=

= × × ×

× × ×

=

1 1 1

1 1 1

2 2 3 5

2 3 3 757

24

÷

×244 =4

10

1?

1

1

÷

=

×

× ×

1

1

2 ×1

×1

×

33

265

ACTIVITY Reducing Fractions

PPERFORMANCE ERFORMANCE CCRITERIARITERIA

• Reducing a fraction to its lowest terms – correct reducing techniques – validation of the fi nal answer

CCRITICAL RITICAL TTHINKING HINKING QQUESTIONSUESTIONS

1. What is a fully reduced fraction?

2. How do you validate that fractions are equivalent?

3. When reducing to lowest terms, what is the result when all the factors in the numerator cancel out?

Section 3.3

Fractions are in its lowest terms if there are no factors common to both the numerator and denominator.

Find the cross-products by multiplying the numerator of the fi rst fraction times denominator of the second fraction, then, denominator of the fi rst fraction times numerator of the other fraction, then, comparing to make sure the products are equal.

The result is that the numerator (or denominator) becomes 1.


4. When reducing to lowest terms, what is the result when all the factors in the denominator cancel out?

5. How can you tell if a fraction is in lowest terms?

6. How can you be sure that your reduced fraction answer is correct?

• Know and use Divisibility Tests to quickly cancel common factors.

• When you cannot readily recognize common factors, use the prime factorization of the numerator and denominator to reduce (by using the Methodology for Reducing a Fraction).

• Cross multiply to test the equivalency of your reduced fraction.

• Even when you do quick reduction by common factors, always prime factor your fi nal answer to assure that there are no remaining common factors.

TTIPS FOR IPS FOR SSUCCESSUCCESS

Set the original fraction equal to the reduced answer. If the products of the cross multiplication are the same, then the reduced answer is correct.

The result will be the product of the factors left in the numerator over 1. But we know that by the Division Property of One that any number divided by one is equal to the beginning number.

For example:

You can tell a fraction is in lowest terms, if you cannot fi nd a common factor that will divide into both the numerator and the denominator. If the numerator is larger than the denominator, you must change to a mixed number.

2 3 3

3

2 3

1

6

16

× × = × = =


DDEMONSTRATE EMONSTRATE YYOUR OUR UUNDERSTANDINGNDERSTANDING

Fraction Factorization Reduced Fraction Validation

1)

2)

3)

4)

5)

Reduce each of the following to lowest terms. If improper, write as a mixed number with its fraction in lowest terms.

2496

2842

6450

7801820

68102


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) Reduce to lowest terms:

2) Reduce to lowest terms: Can be reduced further.

3) Reduce to lowest terms: Cannot cancel the 5 on the bottom (denominator) twice.

IDENTIFY AND CORRECT THE ERRORSIDENTIFY AND CORRECT THE ERRORS

25100

130260

20150

Reduce to lowest terms and validate your answers.

1) 2) 3) 4) 5)

ADDITIONAL EXERCISESADDITIONAL EXERCISES

42

1085400

750075

165

27

56

120

162

25100

14

25 4 1 100100 100

=

× = ×=

Correct

fully reduced

14

12 2

=×

?

?

718

1825

511

2756

2027

already reduced

pre-activity reducing fractions preparation · in addition to the divisibility tests for the...

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