Six Columns

Some of the displays in my classroom serve several purposes.  Here is one that I think looks nice, provides useful mathematical facts for pupils to refer to, is intriguing and can be used to investigate and to explain.

The photograph shows the wall of my classroom.  There are 18 sheets of yellow A4 card (each sheet has four numbers on it).  The key features are that there are 6 numbers in each row and the prime numbers are picked out in orange.

It is handy having prime numbers for pupils to refer to, but displaying them in rows of six picks out a rather interesting result.  All of the prime numbers appear to be either in the first row, or the first column, or the fifth column. 

A natural question for pupils to ask (or for them to be asked) is: “will the rest of the prime numbers all be in the first/fifth columns too?”.

My classes have approached this question like this in the past: 

There can’t be any primes in the sixth column because all of those numbers are multiples of 6.

The numbers in the second and fourth columns are always even, so they aren’t prime (apart from the number 2).  The numbers in the third column are multiples of 3 so, apart from the number 3 itself, none of them can be prime either.  The only columns that we can’t find a reason for rejecting are columns one and five.   Some pupils then go on to talk about why the first row is an exception.

This means the pupils have essentially proved that all primes bigger than 3 are of the form 6n ± 1.

But the fun doesn’t stop there!  For some pupils this can then help them with the idea that x implies y does not necessarily mean that y implies x.  “All primes bigger than 3 appear in columns one and five” is not the same thing as saying “all of the numbers in columns one and five are prime”.

Then there are lots of other things we can do with the number patterns involved.  The sixth column is multiples of 6.  How can we describe the third column?  Are they the odd multiples of 3, or 6n-3, or “start at 3 and go up in 6s?”.

The multiples of 6 are in the sixth column.  Where are the multiples of 5 and the multiples of 7?  Why do they go diagonally?

In columns one and five every fifth number is a multiple of 5 (starting with 25 in column one and 5 in column five).  Does this pattern continue?  Why?  What about multiples of 7?  Or multiples of 11, etc?