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? 

An even bigger prime?

On the Radio 5 Live breakfast show this morning (6^th Feb 2013) Rachel Burden described the newly found biggest prime:  2^57,885,161 - 1    

Nicky Campbell, her co-host on the programme immediately said: “I know a bigger one: how about three to the power of … ”.

This is worth looking at with some classes. 

Nicky Campbell was joking, but how do we know that 3^57,885,161 - 1 isn’t prime?  I like this sort of question because it can be answered if you understand that powers of 3 must all be odd, or you could start with smaller powers: 3² - 1, 3³ - 1, 3⁴ - 1, and discover the answer is always even and could then try to explain it.

How can this be extended:  Well, what must happen to n^57,885,161 - 1 for any odd n (greater than 1)? 

More difficult is looking at some of the even values of n.  n=4 is nice, because we can rewrite it like this:

4^57,885,161 - 1 = (2^57,885,161)² - 1, which is the difference of two squares, so we have 

(2^57,885,161 + 1)(2^57,885,161 - 1) so we have a composite when n=4.

n=6 also works out neatly, because all powers of 6 end in 6, so when we subtract 1 from 6^57,885,161 we have a multiple of 5.

n=8 is harder for KS4 pupils to work on, but if we rewrite as a power of 2 we get something of the form x³ - 1, which factorises to give (x² + x + 1)(x - 1).

n=10 is good too.  We can tell by inspection that it is divisible by 9.  Is it also divisible by 11, though? 

On the inside the roses grow …

I don’t tend to buy The Times, so I haven’t seen their puzzle pages for years.  I was given a copy of Saturday’s edition and found they have some nice problems.  There are the expected crosswords and several variants of Su Doku (which they spell as two words).  And then there is something called ‘Cell Blocks’.

The puzzle itself is a nice one that can be used in class as a way of practising the recall of small primes and factors. 

The reason for mentioning this here, though is the instructions, which are a good talking point in their own right.

“Divide the grid into blocks.  Each block must be square or rectangular …”.  Um, so they must be “rectangular”, then!

This is reminiscent of The Independent who originally said about Sudoku (spelled like that): “There's no maths involved. You solve the puzzle with reasoning and logic.”   Um, so we are using mathematical skills, then!

[To be fair, as long as they are not meaning XOR the instructions in The Times are accurate, whereas the ones from The Independent were plain wrong.]

This is puzzle 1568 and if I assume that there is one published each day except for Sundays (figuring that the Sunday Times is a separate newspaper) then it has been over five years since Cell Blocks first appeared.  I wonder how many letters they have received in that time telling them that “rectangular” would cover all cases?

Using it in class

This will be a starter activity, with the following written on the board:

1]  Solve the problem.

2]  What is the unnecessary part of the instructions?

3]  Assuming there is one of these in The Times from Monday to Saturday, how long ago is it since they started being printed?