Do Now 1/31/11 Take out HW from Tuesday night. Take out HW from Tuesday night. –“The Sieve of...

Do Now 1/31/11Do Now 1/31/11

Take out HW from Tuesday night.Take out HW from Tuesday night.– ““The Sieve of Eratosthenes”The Sieve of Eratosthenes”

Copy HW in your planner.Copy HW in your planner.– Text page 175, #16-44 evensText page 175, #16-44 evens

Be ready to copy POTW #2Be ready to copy POTW #2

HomeworkHomeworkThe “Sieve of Eratosthenes”The “Sieve of Eratosthenes”

1) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 1) 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97; yes; these 73, 79, 83, 89, 97; yes; these numbers only have two factors 1 and numbers only have two factors 1 and itself.itself.

2) Removing multiples removes 2) Removing multiples removes composite numbers, which leaves composite numbers, which leaves only primes.only primes.

3) 10 numbers; 101, 103, 107, 109, 3) 10 numbers; 101, 103, 107, 109, 113, 127, 131, 137, 139, 149113, 127, 131, 137, 139, 149

Chapter 4 “Factors, Fractions, Chapter 4 “Factors, Fractions, and Exponents”and Exponents”

Section 4.1 “Factors and Prime Factorization”

Section 4.2 “Greatest Common Factor”

Section 4.3 “Equivalent Fractions”

Section 4.4 “Least Common Multiple”

ObjectiveObjective SWBAT write the prime factorization of a SWBAT write the prime factorization of a

numbernumber

SWBAT write the prime factorization of a SWBAT write the prime factorization of a monomialmonomial

In your journal, answer the question below using In your journal, answer the question below using the following definition. the following definition. A A factorfactor of a number is a of a number is a number that can be divided into the given number that can be divided into the given number without a remainder.number without a remainder. Which number do Which number do you think has more factors 24 or 29? Why?you think has more factors 24 or 29? Why?

(Hint: List all of the factors of the number 24 and 29)(Hint: List all of the factors of the number 24 and 29)

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.

““PrimePrime or or Composite?”Composite?”

66

4477

8822

55

PrimePrime CompositeComposite

33

11NeitherNeither

Factoring Natural NumbersFactoring Natural Numbers A natural number is A natural number is FACTOREDFACTORED when it when it

is written as the product of two or is written as the product of two or more numbers.more numbers.

– You can use divisibility tests to find factors You can use divisibility tests to find factors of numbers.of numbers.

– Example: Example: 3939 ÷ ÷ 33 = = 1313

3 · 133 · 13 = = 3939

dividend

divisor

quotient

factor factor

product

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

∙∙

∙∙ ∙∙

3030

151522

22 33 55

∙∙

∙∙ ∙∙

3030

6655

55 22 33

∙∙

∙∙ ∙∙

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.

Factoring Natural NumbersFactoring Natural Numbers Write the prime factorization of the following Write the prime factorization of the following

numbers using tree diagrams.numbers using tree diagrams.

4040

8855

55 22 44

∙∙

∙∙ ∙∙

225225

454555

55 33 1515

∙∙

∙∙ ∙∙**HINT**HINTWrite your factors Write your factors in increasing order.in increasing order.

55 22 22∙∙ ∙∙ 22∙∙

22 22 22∙∙ ∙∙ 55∙∙When you have repeatingWhen you have repeatingfactors use exponents.factors use exponents.³³22 55∙∙

33 55∙∙55 33∙∙ ∙∙

55 55∙∙33 33∙∙ ∙∙

²²33 55∙∙ ²²

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

Factoring Algebraic Factoring Algebraic ExpressionsExpressions

Factor each monomial. Factor each monomial.

63a63a³³

18x²y18x²y

= 3 · 3 = 3 · 3 · 7 · a · a · a· 7 · a · a · a

= 2 · 3 = 2 · 3 · 3 · x · x · y· 3 · x · x · y

6ab6ab

= 2 · 3 = 2 · 3 · a · b· a · b

36s36s³t²³t²

= 2 · 2 = 2 · 2 · 3 · 3 · s · s · s · t · t· 3 · 3 · s · s · s · t · t

What Did We Learn Today? What Did We Learn Today?

HomeworkHomework Text page 175, #16-44 multiples of Text page 175, #16-44 multiples of

4; #48-54 evens4; #48-54 evens

2424 -7-7

-4-4 -5-5

88