March 24, 2009 The road to success is dotted with many tempting parking places. ~Author Unknown.

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March 24, 2009

The road to success is dotted with many tempting parking places. 

~Author Unknown

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March 24, 2009

Sections 4.1 & 4.2:

• Divisibility

• Odd/Even

• Prime/Composite

• Factor/Multiple

• Sieve of Eratosthenes

• Begin Exploration 4.2

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4.1 – Divisibility & Related Concepts4.2 – Prime & Composite NumbersUse the manipulatives to make arrays for the

following numbers:

7 12 23 22

Can you make an array for each? Can you make more than one array for any?

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4.1 & 4.2 (Cont’d)

• If you can make two rows of the same length, then your number is even.

• If there is one left over after making even rows, then the number is odd.

• Is the number 0 even or odd?

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4.1 & 4.2 (Cont’d)

• If you can only arrange the manipulatives into a rectangle in just one way, then the number is prime.

• If you can arrange the manipulatives into two or more different rectangles, then the number is composite.

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4.1 & 4.2 (Cont’d)

Definition: A number is prime if and only if it has exactly two factors (1 and itself).

A composite number has more than two factors.

Is the number 1 prime? Composite? Neither?

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4.1 & 4.2 (Cont’d)

Use the blocks to show why…

1. The sum of two odd numbers is an even number.

2. The sum of two even numbers is an even number.

3. The product of two odd numbers is not even.

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4.1 & 4.2 (Cont’d)

Factors & Multiples

A factor of a number n is a divisor of n.

Ex: 3 is a factor of 12 since 12 ÷ 3 = 4 R0

A multiple of a number n is a product of n with another number.

Ex: 12 is a multiple of 3 since 4 × 3 = 12

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4.1 & 4.2 (Cont’d)

Factors & Multiples• One way to visualize factors and multiples is to

make rectangles or rectangular arrays. In a rectangle, the length times the width is equal to the area.

• Can you see that the sides are factors and the area is a multiple?

• Use the blocks. How many different rectangles can you make using 24 blocks? Be systematic.

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4.1 & 4.2 (Cont’d)

Factors of 24 (how can you tell when you have them all?)

1 • 24

2 • 12

3 • 8

4 • 6

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4.1 & 4.2 (Cont’d)

You try: Find all factors of 36

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4.1 & 4.2 (Cont’d)You try: Find all factors of 36

1 • 36

2 • 18

3 • 12

4 • 9

6 • 6

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4.1 & 4.2 (Cont’d)

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.

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4.1 & 4.2 (Cont’d)

Sieve of Eratosthenes*This is an algorithm for finding prime numbers*

Use a new color for each prime number.

On the handout, cross out the number 1 (it is neither prime nor composite).

Next, circle 2 in red (2 is prime), then cross out all multiples of 2 using the red pen.

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1 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

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

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4.1 & 4.2 (Cont’d)

Sieve of Eratosthenes• Once you’ve crossed out all multiples of 2, the

next number remaining is prime (circle it in a new color).

• Cross out all multiples of this prime.• Continue the process until you reach a number

that is larger than the square root of 140 (the largest number in the list). Once you reach this number, all remaining numbers that are not crossed out are prime.

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1 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

101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130

131 132 133 134 135 136 137 138 139 140

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4.1 & 4.2 (Cont’d)

Sieve of EratosthenesQuestions to answer:• When you circled 13, were there any multiples of

13 that were not already crossed out?• 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 140?• Is 1 a prime number?

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4.1 & 4.2 (Cont’d)

Sieve of Eratosthenes • How can you tell that a number is divisible

by 2?

• By 5?

• By 10?

• By 3?

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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.

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Factor game• Player 1 picks a number from 1 - 30, and gets that

many points. However, there must be an available “proper” factor for that number. (Be careful with primes!) If you pick a number with no available factors, you lose your turn, and get no points.

• Player 2 then selects as many factors as possible for that number, and scores the sum of those factors.

• Roles switch, and play continues.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=12

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Taxman game (if time) Exploration 4.1

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HomeworkLink to online homework list:

http://math.arizona.edu/~varecka/302AhomeworkS09.htm