Named after the Greek Mathematician Erastosthenes, the sieveprovides a very efficient method for finding prime numbers.
We start with a large grid of whole numbers. If we use thesimple definition that a prime number is any number that has exactly 2factors. Then we can eliminate 1 as not prime. The next number 2 isthe first prime number, it also is uniquely the only even primenumber.
Whenever a prime is found, we choose a color for example red andpaint all of its multiples, so for 2 this would be 2,4,6,8...
Anything other than 2 that is painted red cannot be a primenumber as it has more than two factors.(1,2 and the number itself). Wenow look for the next number that has not be colored, in this case 3.This is are next prime, You can use a different color to mark off itsmultiples 3,6,9,12...
The process is continued, look for the next number that has notbeen colored and paint its multiples. Eventually you will see only theprime numbers remaining.
Grid Sizes
Use the slider to change size from 2x2 upto 30x30, the really large squares can show how the method works alsofor larger numbers. Small grids can be used to examine the factors ofa particular number (discussed below).
Colors
To select a particular color, click on it and to remove colorsfrom the grid click on the white brush
The split color control when turnedon it allows for a square to show more than one color.
Click the trash button to remove allcolors from the grid
Modes
The activity has three different modes. Inmanual mode choose a color, then click and drag over thenumber squares to color them. In multiples mode, click asquare and all it subsequent multiples will be automatically colored.The final automatic mode will run through the sieve,automatically allocating colors for each prime and its multiples. Thiscan be useful for large numbers of squares, note press the startbutton to begin the process.
Animation Speed
Use the animation speed slider tochange how fast multiples are highlighted
Using the Sieve
This activity has many different uses and can help explain manymathematical concepts apart from finding prime numbers.
Finding the Lowest/Least Common Multiple (LCM)
This example will work fine with the 100 squares or less, press trash to clear the grid, select multiples mode and make sure splitcolor is on. So an example problem would be find the LCM of 4,6. Todo this click the red paint icon, and click number 4 in the grid, allthe multiples of 4 are now red. Click the yellow paint and then clickthe number 6, all the muliples of 6 will be shown as yellow. Youshould see the common multiples are colored boy red and yellow, theseare 12,24,36... the LCM the least/lowest of the these is 12.
You can do this for more than two numbers, just make sure youuse a different color for each one of the multiples you are using.
Finding the factors of a number
Like the LCM example, the default settings are used, multiplesmode and split colors.
It is important to understand the relationship between factorsand multiples. For example if 12 is one of the multiples of 4 then weknow that 4 must be a factor of 12. So using the grid if I wanted totest if 5 is a factor of 15, then I can click trash to clear the gridand then choose a color and click on 5 to show all its multiples. Ifone of them is 15 then we know that 5 is indeed a factor of 15. Totest if 4 is a factor of 15 you can either clear the grid or select adifferent color and then click 4, in this case the color does not hit15 and so we know it is not a factor.
If you are only interested in the number upto 16, it is possibleto select the 4x4 grid and assign a different color for each number1-16. In this way the number of colors on each number, gives itsnumber of factors
Prime factors class exercise
This is a whole class activity, as it can help studentsunderstand how each number can be expressed as a product of primenumbers. Start with a 100 square, split colors and automatic mode,click start to run through the sieve. When complete the primes have asingle border color, and the composite numbers have one or more solidcolors. These colors tell which prime factors the number has. So forexample number 60 has 3 colors: red, yellow and lime. Look at theprimes that represent these colors, red is 2, yellow is 3 and lime is5. In this case 60=2×2×3×5
A question for the class would be what is special about the numbersthat are not prime but only have a single color. Hopefully they canwork out that these numbers only have one prime factor, so exampleswould be all the square numbers, all the cube numbers or in fact anynumber that can be expressed as anwhere a is a prime number and n is apositive integer.
Divisibility Tests
These grids can also be a good starting place for a discussionon division tests
So for example, any number that is divisibly by 2, must be amultiple of 2. So look at these numbers on the grid and hopefully thestudents can identify these numbers always end in 0,2,4,6 or 8. Othertests can also be investigated.
Related activities
The number explorer is also auseful teaching tool for factors, multiples and primes.
