Prime Numbers

If you asked your students to describe what makes a prime number prime, most of them will either shrug or tell you its a number whose factors are one and itself.  If you ask them to draw it, how would they react?  Mine simply would shut down because they have no idea how to do that.

I read a cool book called "The Joy of X" by Stephen Strogatz in which he presents prime numbers in a way I like.  Think of a prime number as a number than can be expressed as a square or rectangle no smaller than 2 by 2 with no left overs.

By this definition, it is easy to  draw prime and composite numbers and it makes it easy for students to see why one is easily factored while the other cannot.

If you look at this first picture you will notice that it is not until you get to four that you have a 2 by 2 shape - a square.  This means that 2 and 3 are prime numbers because they do not form the right shape.

 

Now look at the following illustrations for 5, 6, and 7.

 

The number 5 when drawn does not produce a square or a rectangle.  It produces a 2 by 2 square with one left over. On the other hand, the number 6 can be drawn as a perfect 2 by 3 rectangle while 7 is the 2 by 3 rectangle with one left over.

Notice that the drawings of composite numbers also illustrate at least one set of factors.  The factors of 4 are 2 because 2 x 2 is 4 or 4, 1 because 4 x 1 is 4.  Where as the 5 shows no nice factoring and that makes it easy to tell it is a prime.  For larger numbers, this is where the rules of divisibility come in or factoring trees.

I really like the way Stephen explained it because it gave me a way to help my students "see" a prime number as more than a definition.