Do Now 1/4/10Do Now 1/4/10

Copy HW in your planner.Copy HW in your planner.– Text p. 214, #1-17 all (you need your text)Text p. 214, #1-17 all (you need your text)– Chapter 4 Test WednesdayChapter 4 Test Wednesday

Be ready to review sections 4.1 – 4.4.Be ready to review sections 4.1 – 4.4.

Be ready to copy POTW #6.Be ready to copy POTW #6.

ObjectiveObjective

SWBAT review prime factorization, SWBAT review prime factorization, greatest common factor, equivalent greatest common factor, equivalent fractions, and least common multiplefractions, and least common multiple

Section 4.1 “Factors and Prime Section 4.1 “Factors and Prime Factorization”Factorization”

Natural numbers are classified according to Natural numbers are classified according to how many factors they have:how many factors they have:

(1) Prime numbers: (1) Prime numbers: -a whole number that is greater than 1 and has -a whole number that is greater than 1 and has

exactly TWO factors, itself and 1.exactly TWO factors, itself and 1.

(2) Composite numbers:(2) Composite numbers:

-a whole number that is greater than 1 and has -a whole number that is greater than 1 and has more than two factors.more than two factors.

(3) The number 1:(3) The number 1: 1 is neither prime nor composite.1 is neither prime nor composite.

Factoring Natural NumbersFactoring Natural Numbers

You can use You can use TREE DIAGRAMSTREE DIAGRAMS to to factor a number until all factors factor a number until all factors are primes.are primes.3030

101033

33 22 55

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3030

151522

22 33 55

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3030

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55 22 33

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Prime FactorizationPrime Factorization**HINT**HINTWrite your factors Write your factors in increasing order.in increasing order.-writing a number as a product of prime factors.

MonomialMonomiala number, a number,

xx 3x3x³³yyz²z²

or the product of a or the product of a number and one or more variables with whole number and one or more variables with whole number exponentsnumber exponents

a variable,a variable,

77 To To FACTORFACTOR a monomial, write the monomial as a a monomial, write the monomial as a product of prime numbers and variables with product of prime numbers and variables with exponents of 1. exponents of 1.

3a3a³³ = = 3 3 · a · a · a· a · a · a

Greatest Common FactorGreatest Common Factor

The common factor of two The common factor of two numbers. numbers.

LARGESTLARGEST

8: 1, 2, 4, 8 8: 1, 2, 4, 8

12: 1, 2, 3, 4, 6, 12 12: 1, 2, 3, 4, 6, 12

List the factors of the two numbers and then compare.List the factors of the two numbers and then compare.

Find the GCF of 8 and 12.Find the GCF of 8 and 12.

From the list you can see that 1, 2, and 4 are common factors.From the list you can see that 1, 2, and 4 are common factors.Of these 4 is the Of these 4 is the greatest common factorgreatest common factor..

Section 4.2 “Greatest Common Factor”Section 4.2 “Greatest Common Factor”

Greatest Common FactorsGreatest Common Factors

Find the GCF of 39 and 50.Find the GCF of 39 and 50.

When the numbers are too large to list the factorsof the two numbers, find the prime factorization of eachand then compare.

39: 3 · 13

50: 2 · 5 · 5

From the list you can see that there are NO common prime factors. However, two numbers always have 1 as a common factor. So the GCF is 1 and the two numbers are relatively prime.

Relatively Prime:Relatively Prime:Two numbers are relatively Two numbers are relatively prime if their greatest common prime if their greatest common factor is 1.factor is 1.

GCF of Variable ExpressionsGCF of Variable Expressions

Find the GCF of 18xyFind the GCF of 18xy²² and 28x and 28x²²y³.y³.

Write the prime factorization of each and then compare.

18xy²: 2 · 3 · 3 · x · y · y

28x²y³: 2 · 2 · 7 · x · x · y · y · y

From the list you can see that 2, x, y, and y are common prime factors. The greatest common factor of is the product of 2 · x · y · y, which is 2xy².

EQUIVALENT FRACTIONS-EQUIVALENT FRACTIONS-

two fractions are equivalent if two fractions are equivalent if they represent the same they represent the same quantity.quantity.

4

1

8

2= 5

2

25

10=

A fraction is in SIMPLEST FORM if its numerator and denominator have a greatest common factor of 1.

Section 4.3 “Equivalent Fractions”Section 4.3 “Equivalent Fractions”

Simplifying Fractions Simplifying Fractions

18

12

63

62

=3

2

When simplifying fractions, look for the greatest common factor in the numerator and denominator. Then divide.

=

63

28

79

74

=9

4=

Simplify Variable ExpressionsSimplify Variable Expressions

To simplify a variable expression, you To simplify a variable expression, you can factor the numerator and can factor the numerator and denominator and then divide out any denominator and then divide out any common factors. common factors.

A variable expression is in A variable expression is in SIMPLEST SIMPLEST FORMFORM if the numerator and denominator if the numerator and denominator have no factors in common other than 1. have no factors in common other than 1.

34

12

x

xxxx

x

4

342

3

x

Section 4.4 “Least Common Section 4.4 “Least Common Multiple”Multiple”

Least Common MultipleLeast Common Multiple

Of all the common multiples of two Of all the common multiples of two numbers, the smallest is the numbers, the smallest is the least least common multiplecommon multiple. .

A A multiplemultiple of a number is the of a number is the product of the number and any other product of the number and any other number greater than zero.number greater than zero.

What are the multiples of 5? What are the multiples of 5? Think of counting by fives…Think of counting by fives…

5:5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 … 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 …

Find the LCM of 64 and 72.Find the LCM of 64 and 72.

When the numbers are too large to list the multiplesWhen the numbers are too large to list the multiplesof the two numbers, find the prime factorization of each of the two numbers, find the prime factorization of each number. number. The LCM is the product of the common prime The LCM is the product of the common prime factors and the factors that are not commonfactors and the factors that are not common..

6464: : 2 · 2 · 22 · 2 · 2 · 2 · 2 · 2 · 2 · 2· 2

7272: : 2 · 2 · 22 · 2 · 2 · 3 · 3· 3 · 3

From the list you can see that 2, 2, and 2 are common prime From the list you can see that 2, 2, and 2 are common prime factors, and the factors that are not common are 2, 2, 2, 3, and 3. factors, and the factors that are not common are 2, 2, 2, 3, and 3. The LCM is then The LCM is then 2 · 2 · 22 · 2 · 2 · 2 · 2 · 2 · 2 · 2· 2 · 3 · 3· 3 · 3 which is 576. which is 576.

Least Common MultipleLeast Common Multiple

Common Common FactorsFactors

Not Common Not Common FactorsFactors

2 · 2 · 22 · 2 · 2

2 · 2 2 · 2 · 2· 2 3 · 33 · 3

Find the LCM of 2Find the LCM of 2a³b and 3aba³b and 3ab..The LCM is the product of the common prime The LCM is the product of the common prime factors and the factors that are not commonfactors and the factors that are not common..

22a³ba³b : : 2 ·2 · a a · a · a · · a · a · b b

3ab 3ab :: 3 · 3 · aa · · bb · b · b · b · b · b · b · b · b

From the list you can see that From the list you can see that a and ba and b are common prime are common prime factors, and the factors that are not common are 2, 3, a, a, b,factors, and the factors that are not common are 2, 3, a, a, b,b, b, and b. The LCM is then b, b, and b. The LCM is then a · ba · b · 2 · a · 2 · a · a · 3· a · 3 · b · b· b · b · b · b· b · b

which is which is 6a³b. 6a³b.

Least Common MultipleLeast Common Multiple

Common FactorsCommon Factors Not Common FactorsNot Common Factors

a · b a · b 2 · a · a 2 · a · a 3 · b · b · b · b3 · b · b · b · b

55

55

55

B-I-N-G-OB-I-N-G-O Complete numbers #6, 8, 10, Complete numbers #6, 8, 10,

12 – 32 all on page 806.12 – 32 all on page 806.

Using your correct answers Using your correct answers we will play BINGO.we will play BINGO.

BINGO ReviewBINGO ReviewText p. 806, #6, 8, 10, 12-32 allText p. 806, #6, 8, 10, 12-32 all

6) prime6) prime 8) 8) 10) 1; yes10) 1; yes 12) 8; no12) 8; no 13) 4y13) 4y 14) 4p14) 4p 15) 2s15) 2s 16) 3x16) 3x 17) 3/1617) 3/16 18) 2/2118) 2/21 19) 7/1119) 7/11 20) 2/520) 2/5 21)21)

22)22)

23)23)

24)24)

25) 3025) 30 26) 8426) 84 27) 48027) 480 28) 9028) 90 29)29) 30) 30) 31) 31) 32) 32)

552222

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2

rs

6

c

ab

2

19

6

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HomeworkHomework

Text p. 214, #1-17 allText p. 214, #1-17 all