Pretty straight forward once you get the hang of it!
Firstly radioactive decay is completely random, that means you cannot predict which nucleus will decay or when it will decay. There is however a set probability of radioactive decaying during any given second, when you get down to it Nuclear Physics does come down to probabilities, (that's Heisenburg's uncertainty principle and Schrodinger's Cat analogy is all about.) But I digress.
You have to remember that we are dealing with huge numbers of radioactive nuclei, for example in a 1kg sample of Uranium there are 2.56✕1024 nuclei. So although we can never know when or which nuclei will decay if we know how many there are then we can predict how many will decay in any given time. In other words there are so many, that by law of averages there will always be a certain ratio of them decaying each second.
There are lots of ways that we can express this statistical prediction of the decay rates of samples of radioactive isotopes but perhaps the most useful, and the one that we use in GCSE Physics is to say that no matter the number of unstable nuclei the average time for half of them to decay is a constant. It is this period of time that we call the half life of the isotope.
Some isotopes have very long half-lives, and others have very short ones, and that is one of the properties of radioactive substances that make them more or less useful for any given application. For example the 4 700 year half life of carbon-14 makes it excellent for accurately dating archaeological samples of organic matter. And the six hour half-life of technetium-99 means that it can be used as a medical tracer but will not leave a patient radioactive for long periods of time.
There are two type of common question in GCSE regarding half-life at GCSE, aside from the definition type, and the uses based on half-life. These either ask you to calculate a half-life from a graph, or use data about activity and a known half-life to work out a length of time something has been decaying.
So graphs, these should be easy marks. Simply look at the y-axis, take the first reading and half it, then interpolate to the line, and down to the x-axis, the time axis. This time is the half-life. Strictly speaking you should measure at least two and then average them, but there doesn’t seem to be consensus about whether you’ll be penalised for not doing this at GCSE level. Remember that half-life is the time taken to half, not the value of half the number. So if a sample starts at 300, and takes 50s to get to 150, the half-life is 50s not 150!
If you are asked to work out a time from activity data then the way around the problem is to work out how many half-lives have passed, i.e. how many times has the original activity halved to get to the final activity. Then calculate the number of half-lives multiplied by the half-life to get the total time that has passed.
Half life is a pretty straight forward topic… well as far as Nuclear Physics goes, and there are a few ways to explore it. Perhaps you've done a simulation with dice in the classroom? You start with, let's say a hundred dice and roll them, picking out the sixes. You repeat that and keep a table of how many dice you have left against turn number and plot number of dice against time (turn numbers). You get a pretty natty decay curve, from which you can work out the half-life of the dice.
The other way to get your head around these decay curves is to use a simulation, there are loads of good ones online. I've linked up a few just here:
Later in your study of Physics you find that there are many other phenomena which follow the same statistical trends. It's a really useful one to understand and at A Level we use this statistical analysis to explain a great deal more. In fact most natural processes at some point follow these curves, which we call exponentials.