Exercises on core topics

The pre-16 syllabi contain a bewildering number of different topics, but it is important to understand that all of these topics are not created equal. There are a number of core topics, without a full understand of which, your mathematical education will be hampered and frustrated. On this page, I have collected together a number of resources which I have written over the past few years, which are designed to give structured practice on a few particularly important topics. These files are quite large and dense with no clip art or other distraction. They are simply exercises to practise important techniques. My students have found them useful either, as part of a structured teaching schedule, and also as revision aids.

A word of warning: Doing exercises alone will not help you to improve your mathematical skill. When you do an exercise, it is vitally important to check your answer after every single example you do. If your answer is correct, move on, but if not examine your error, trying to find out where you went wrong. Get help if you can't deal with the problem yourself. Only move on when you know you have correctly completed and understood the prior example. As with music practice, maths practice is great if you do it right, but if you don't, you will become supremely talented in making the same error over and over again. Once this error is fixed in your mind, you will need to do a great deal of un-learning before you can proceed. Doing endless, unchecked examples is worse than a complete waste of time. You will get worse! Most students find this hard to believe, but it is true.

Let me reiterate; doing exercises without checking your answers after each example is very likely to make you worse at the skill, not better.

In the list below, you will see topics which will help you to become good at maths. Sometimes, these topics are not the most exciting for many but like learning your scales in music practice, these topics form the backbone of pre-16 mathematics.

  1. Multiplication tables.

    If you know your multiplication tables by heart, you will progress more quickly than if you do not. Knowing multiplication tables enables your brain to make connections which will help you in all of the maths you do. If you do not know your tables, print off some of the tables grids on this page and do them. If you run out of sheets, do them again. (N.B. Repeating exercises can be very beneficial!)
  2. Fractions.

    There is no getting away from the fact that if you can easily manipulate fractions, you have a major advantage, as fractions pop up almost everywhere in mathematics.

    These two sets of exercises go quite slowly through the various techniques. Take your time and ask if you get stuck.


      Fraction arithmetic.
  3. Ratios.

    Ratios are another really important idea.

    This set of exercises go through the simpler concepts and methods. Each exercise is preceded by examples showing a complete method.


  4. Algebraic manipulation.

    Most mathematics requires the solving of equations at some point, and you cannot solve equations effectively unless you can manipulate algebraic expressions, so this is where to start to get your algebra up to scratch.

      Algebraic manipulation.
  5. Equations.

    You have to be able to solve equations if you are going to any good at mathematics. Sorry, but it is true. Inverst the time and you will not be disappointed.


These are not the only vital topics but if you can conquer these, then all other mathematics will make more sense and will be easier to acquire.