Visualizing Prime and Composite Numbers.

 I know in high school we always define a prime number as having only two factors, one and itself while composite has at least three factors but visually how does that play out?  How do we make pictures that show that idea.

I read something in the "Joy of X" by Steven Strogatz that flipped a switch in my brain to give me a pictorial way of understanding the difference.  I'll go through the explanation step by step here with the appropriate pictures.  I think it makes more sense to students than using just the standard explanation.

Lets start with the numbers one, two, and three:

If you look at the picture there is only one way you can arrange the blocks so you have a rectangular shape.  You cannot rearrange the blocks to change the measurements.  The only way these can be arranged is so it is 1 by either 1, 2, or 3.

Yes, students will say for two that it is  1 x 2 or 2 x 1 so they are different but you can show that the 1 x 2 is rotated to become 2 x1 which brings transformations into visualizing prime and composite rather than applying transformations only to geometry.


Now for the number four.

Four can be drawn as 4 x 1 or 1 x 4 rectangle but it can also be drawn as a 2 x 2 square.  This shows that four times one and  two times two both equal four.

This means that four is composite because there is more than one way to represent the number four .  This shows all the factors for four visually and makes it much easier to understand.



Lets look at five.
When you draw five in the normal rectangular arrangement, it works but if you try to rearrange it into any other configuration, it won't give you a rectangle or square.  It gives you something that is missing part of the shape.

This indicates five is a prime because the only arrangement you can do is a five by one, nothing else gives a full rectangle or square.  Any other arrangement, and something will be missing.  That missing part is what indicates it is a prime.

Furthermore, the five by one shape represents the 5 times 1 multiplication that works.

Finally six.

If you look at the picture, I was able to represent six in two different ways.  The first is in the six by one configuration which is the standard way of showing all numbers but it is the three by two arrangement that makes it a composite number.

So basically, prime numbers can only be arranged in one configuration while composite numbers can be arranged in two or more shapes.  The larger the number, the more arrangements are possible.

If you look at 100, you should be able to draw a 1 by 100 rectangle, a 2 by 50 rectangle, a 4 by 25 rectangle, a 5 by 20 and a 10 by 10 rectangle so it is definitely a composite.

I've found this explanation is easier for students to understand the difference between prime and composite.  The pictures make turn an abstract definition into something more concrete.

I admit that until I read the explanation in the book, I'd always used the verbal definition.  I'd never seen any visual explanation because I'd never connected the number as representing area and the factors represented the lengths of the sides of either a square or rectangle.

Let me now what you think, I'd love to hear.  Have a great day.