Patterns
&
Sequences

Now of course,

just because numbers make a certain pattern when they are arranged in a ten by ten block

doesn't mean that is the only way you can arrange them.

Change the size of the block,

perhaps to twelve by twelve,

and the patterns look quite different,

and sometimes it's easier to see patterns

in one arrangement rather than another.

Apart from the simplest way of showing numbers,

on the number line

counting out from zero,

it's possible to find all sorts of different ways to write out the numbers

to look for patterns in their sequences.

They can be written out in a spiral,

with the zero in the middle,

or on a grid made of layers in 3D.

People have always loved the interesting ways that numbers could be set out on a grid,

so much so that they had special names for squares with the numbers arranged a certain way,

the arrangements they call magic squares.

They show patterns in certain sequences,

and some are hard to see,

and some you can see nearly everywhere.

One particular pattern can be seen

almost everywhere we look in nature.

It's a very simple sequence,

called the fibonacci sequence,

in the fibonacci sequence

each number is the sum of the two numbers that come before it,

0 1 1 2 3 5 8 13 21 34 55 89 144 and so on,

which is a pattern that things use to grow by.

The fibonacci sequence is one of the great number patterns of creation.

Now prime numbers are hard to find

because they can't be found by by multiplying other numbers together,

so that makes them the building blocks of numbers,

but the fibonacci sequence is

the mathematical building block of growing things.

The world is full of wonderful patterns,

from crystal structures in minerals

to the double helix of human DNA.

But there are also patterns in

the ways that numbers are written down when they are used.

For instance,

the way that brackets make what we are doing with numbers

look a certain way.

So 2(3+4) is the same as (2x3)+(2x4),

and because you know that the pattern works that way

whatever number you are using,

you can use a letter that means any number

and the maths works the same way.

So 2(a+b) is the same as 2a+2b.

It's easy to see a pattern when you know that 2a is the same as 2xa,

because in maths two of something is something multiplied by two,

when in words you say 'of'

in maths that means 'multiply'.

Doing maths using letters can be very useful

when you are trying to work out what a number should be

to fit in a certain place,

to find out what equals an answer.

Then patterns can be seen in equations,

the things that balance on either side of an equals sign.

Sometimes you might know the number that a letter stands for,

and sometimes you might need to work it out

from the numbers and signs and patterns that are there.

And sometimes it makes things easier to see a pattern

if you do something to each side.

If you add the same amount to each side they will still be equal.

Or you can simplify things by taking something away from each side.

a+b+c=b+c+3 is the same as a=3

because the b+c on each side cancel each other out.

Or sometimes you can find an easy way to see things

by switching things from one side to another,

but changing the sign of what you move to its opposite.

With equations you can make the numbers and letters

do a mathematical dance for you,

a dance that can be seen in the movements of the stars and planets,

and the life that is all around us.