Musical Chairs

• No more than 2 people at one table who have been at the same table before.

• I suggest that one person remain at the table you have been at for the past few weeks and the rest move, making sure that there is no more than one person at your new table with whom you have previously shared a table.

Even and odd numbers

• What can you say about two numbers if their sum is even and their product is odd?

• Why?

Even and odd numbers

• The sum of three numbers is an even number. Does the product have to be an even number or an odd number or can it be either one?

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 Number

Divisible 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 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 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.

Exploration 4.3

• Do #1 yourself, compare answer with the others in your group.

• Do #2 with the following numbers

60

72

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 Factorization

60 Using a factor tree to do prime factorization

• 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.