WI Elementary Math Problem Solving Report

300-Level Elementary Math Education (Writing-Intensive)

PROBLEM-SOLVING REPORT STUDENT SAMPLE

Sieve of Eratosthenes
(From: Hilton, Peter John and Jean Pederson. Fear No More: An Adult Approach to Mathematics. Menlo Park, California: Addison-Wesley, 1983).
The Sieve of Eratosthenes is a method that seeks out all the prime numbers in a given group of numbers. A prime number (p) is a number greater than 1 and whose multiples are "p" and 1. For example, 2 is a prime number because its multiples are 2 and 1, therefore, the Sieve of Eratosthenes says to cross out all numbers in a given group that are multiples of 2, except 2 itself. (See chart below.) As a result, the next smallest number is 3, and since 3 is prime, all the numbers that are multiples of 3 are crossed out. This method is continued until all the numbers remaining in a group are prime.
The Sieve of Eratosthenes "works" because it eliminates multiples of prime numbers by crossing them out and therefore the only remaining numbers are the prime numbers. A sieve is defined in Webster's New World Dictionary as "A utensil with many small holes for straining liquids or fine particles of matter." Thus, multiples of prime numbers are "strained" out of the group of numbers, leaving only the prime numbers.
Multiples of 6, 2, and 3 are not crossed out because they are prime numbers; their multiples are 2 x 1 and 3 x 1. On the other hand, 6 is crossed out because it is a composite number or a multiple of both 2 and 3 and its factors are not solely 6 x 1.
We stop crossing out number 7 because our number chart only goes up to number 102. Since 11 is the next prime number, the first multiple of 11 to get crossed out would be 121. Our number chart does not extend that far, therefore we stop crossing out numbers at 7. The same generalization would apply for 13, 83, and other prime numbers in our number chart because the first multiple of that number would be larger than the existing numbers on our chart.
After applying the Sieve, a generalization that I made from my observation of the chart was that all the even numbers, except 2 since it is a primary number, were crossed out because they were composite numbers. Of the prime numbers that remained (not crossed out), the last digit of the number after prime number 7 (since we stopped crossing out multiples after 7) was either 1,3, 5, 7, or 9. These numbers remained because the only multiple that could be multiplied with each number was 1, therefore making it a prime number. All the numbers that were crossed out had other multiples in addition to 1 and were multiples of a prime number.
One difficulty that I encountered was deciding whether to cross out a number that had multiples of only prime numbers, for example 6. At first I could not figure out why 6 would be crossed out and not its multiples, but then I realized that a prime number could only have multiples of "p" and 1.
One strategy that helped me determine if a number was a prime number was to recall the rules of division to see if a number on the chart had additional multiples other than "p" and 1. If it did, then I knew to cross out that number.
I felt that this was a good problem-solving activity because I had not heard of the Sieve of Eratosthenes before. As a result I found out what type of resource I had to refer to in order to become familiar with the term. It also challenged me because my recollection of prime numbers and multiples of numbers was fuzzy. I wonder if I solved the problem properly, but I did learn about the Sieve of Eratosthenes and how to apply it to the number chart.
WORKSHEET FOR THE SIEVE OF ERATOSTHENES

Go (back) to Professor Zilliox's Education class.

Mānoa Writing Program · 2545 McCarthy Mall, Bilger Hall 104 · Honolulu, HI 96822 · (808) 956-6660 · mwp@hawaii.edu

© 1997-2009 Mānoa Writing Program, University of Hawai'i

		300-Level Elementary Math Education (Writing-Intensive) PROBLEM-SOLVING REPORT STUDENT SAMPLE Sieve of Eratosthenes (From: Hilton, Peter John and Jean Pederson. Fear No More: An Adult Approach to Mathematics. Menlo Park, California: Addison-Wesley, 1983). The Sieve of Eratosthenes is a method that seeks out all the prime numbers in a given group of numbers. A prime number (p) is a number greater than 1 and whose multiples are "p" and 1. For example, 2 is a prime number because its multiples are 2 and 1, therefore, the Sieve of Eratosthenes says to cross out all numbers in a given group that are multiples of 2, except 2 itself. (See chart below.) As a result, the next smallest number is 3, and since 3 is prime, all the numbers that are multiples of 3 are crossed out. This method is continued until all the numbers remaining in a group are prime. The Sieve of Eratosthenes "works" because it eliminates multiples of prime numbers by crossing them out and therefore the only remaining numbers are the prime numbers. A sieve is defined in Webster's New World Dictionary as "A utensil with many small holes for straining liquids or fine particles of matter." Thus, multiples of prime numbers are "strained" out of the group of numbers, leaving only the prime numbers. Multiples of 6, 2, and 3 are not crossed out because they are prime numbers; their multiples are 2 x 1 and 3 x 1. On the other hand, 6 is crossed out because it is a composite number or a multiple of both 2 and 3 and its factors are not solely 6 x 1. We stop crossing out number 7 because our number chart only goes up to number 102. Since 11 is the next prime number, the first multiple of 11 to get crossed out would be 121. Our number chart does not extend that far, therefore we stop crossing out numbers at 7. The same generalization would apply for 13, 83, and other prime numbers in our number chart because the first multiple of that number would be larger than the existing numbers on our chart. After applying the Sieve, a generalization that I made from my observation of the chart was that all the even numbers, except 2 since it is a primary number, were crossed out because they were composite numbers. Of the prime numbers that remained (not crossed out), the last digit of the number after prime number 7 (since we stopped crossing out multiples after 7) was either 1,3, 5, 7, or 9. These numbers remained because the only multiple that could be multiplied with each number was 1, therefore making it a prime number. All the numbers that were crossed out had other multiples in addition to 1 and were multiples of a prime number. One difficulty that I encountered was deciding whether to cross out a number that had multiples of only prime numbers, for example 6. At first I could not figure out why 6 would be crossed out and not its multiples, but then I realized that a prime number could only have multiples of "p" and 1. One strategy that helped me determine if a number was a prime number was to recall the rules of division to see if a number on the chart had additional multiples other than "p" and 1. If it did, then I knew to cross out that number. I felt that this was a good problem-solving activity because I had not heard of the Sieve of Eratosthenes before. As a result I found out what type of resource I had to refer to in order to become familiar with the term. It also challenged me because my recollection of prime numbers and multiples of numbers was fuzzy. I wonder if I solved the problem properly, but I did learn about the Sieve of Eratosthenes and how to apply it to the number chart. WORKSHEET FOR THE SIEVE OF ERATOSTHENES Go (back) to Professor Zilliox's Education class.

Mānoa Writing Program · 2545 McCarthy Mall, Bilger Hall 104 · Honolulu, HI 96822 · (808) 956-6660 · mwp@hawaii.edu
© 1997-2009 Mānoa Writing Program, University of Hawai'i