Sieve of Eratosthenes By Tamika Allen, Frank Koegel, and Kathleen Sylla NSF Pedagogy Conference 2008 April 19, 2008.

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Presentation transcript:

Sieve of Eratosthenes By Tamika Allen, Frank Koegel, and Kathleen Sylla NSF Pedagogy Conference 2008 April 19, 2008

Eratosthenes of Cyrene (276 BC BC) was a Greek mathematician, geographer, and astronomer.

He is noted for devising a system of latitude and longitude, and for being the first known to have calculated the circumference of the Earth.

He also created a map of the world based on the available geographical knowledge of the era.

A sieve is a utensil used to separate solids from liquids or large particles from small particles

The Sieve of Eratosthenes can be used to filter out all numbers that are not prime. 5 What is a prime number? NumberFactors Prime Not Prime6 1,5 1,2,3,6

Start with a certain set of whole numbers, such as

Cross out 1 since it is not prime

Then begin with the next number not crossed out. Cross out all multiples of that number

Then go to the next number not crossed out. That would be 3. Cross out all multiples of that number

Continue through each number. What do you notice?

So the prime numbers between 1 and 20 are 2, 3, 5, 7, 11, 13, 17 and 19

What are the primes to 100?

National Library of Virtual Manipulatives Prime Number Sieve Sieve of Eratosthenes Applet

Can you apply the divisibility patterns? Key: multiples of 2 multiples of 3 multiples of 4 etc.

Lockers A school’s lockers are numbered 1 to 100. One hundred students enter the school one at a time. The first student opens the lockers. The second student closes the even-numbered lockers. The third student either closes or opens every third locker, and so on. After all the students have passed the lockers, which lockers are open? The perfect squares! 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Sieve of Squares

Multiple Representations Using the hundred chart, we can: – Filter the primes and composite numbers – Apply divisibility rules Using students and counters, we can: – Filter perfect squares We can even infuse technology… Sieve of Atkin Let’s play a game!

Sunshine State Standards MA.A.1.3 The student understands the different ways numbers are represented and used in the real world. MA.A.5.3 The student understands and applies theories related to numbers. MA.D.1.3 The student describes, analyzes, and generalizes a wide variety of patterns, relations, and functions.