Finding factors of a number

• Use the beans to find factors of 24Count out 24 beansWe know that products can be illustrated

using a rectangular modelMake a rectangle using the beansWhat are the numbers you multiply to get 24?Can you arrange the beans into a different

rectangle?What product does this represent?

How many different rectangles can you make?

Count out 11 beans.How many rectangles can you make with

11 beans?

Sieve of Eratosthenes• Eratosthenes was born in Cyrene which is now

in Libya in North Africa in 276 BC. He died in 194 BC.

• Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth.

• He was also fascinated with number theory, and he developed the idea of a sieve to illustrate prime numbers.

The Sieve of Eratosthenes

• Prime NumberDivisible only by 1 and itself

• Finding prime numbers using the sieve

Sieve of Eratosthenes• You will need many different colors. Use one

color for each factor.• Circle the number “1”. 1 is neither prime nor

composite, as we have seen earlier.• Now, circle 2. Every multiple of 2 is a composite

number, so put a dot of that color next to all of the multiples of 2.

• Use a new color. Now, circle 3. Every multiple of 3 is a composite number, so put a dot of this new color next to all multiples of 3.

Sieve of Eratosthenes• Now, 4 has a dot next to it--it is not prime. Skip

it and move on.• Use a new color. Circle 5, and then put a dot of

this new color next to all multiples of 5.• Now, 6 has a dot next to it--it is not prime. Skip

it and move on.• Continue until you know that only prime numbers

are left. When can you stop? How do you know?

Sieve of Eratosthenes• Questions to answer:• When you circled 11, were there any multiples of

11 that did not already have dots next to them?• Can you explain to a child why this was true?• What does this have to do with factors and

multiples?• What are the prime numbers that are between 1

and 100?• Is 1 a prime number?

Sieve of Eratosthenes1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100

Sieve of Erathosthenes

Names for these numbers

11 is an example of a

24 is an example of a

Factors of 24

• List

How should they be ordered?

• How do you know you have them all?

Factors of 24--How do we know when we have them all?

1 • 12

2 • 24

3 • 8

4 • 6

Exploration 4.2

• First, fill in the table on page 85, using the information on the sieve. It will help if you write them in pairs. For example, for 18: 1, 18; 2, 9; 3, 6. The order does not matter.

• Next, fill in the table on page 87. Use the table on page 85 to help.

Find all of the factors of 30

Now find all of the factors of 60

Factorization

• Factorization is writing a number as a product of factors.

24

60

Prime Factorization

• A factorization of the number in which all of the factors are prime numbers.

10

12

Prime Factorization

• 24

• 25

Prime FactorizationUsing a factor tree to do prime factorization.

60

112

• Exploration 4.3 is due on Thursday#1,2,6,7,8 along with some exercises from the textbook.

Please put the exploration on a separate paper than the textbook problems.