Chapter 1

Moving towards Mathematical Mastery

In This Chapter

Understanding the overlap with GCSE

Doing advanced algebra

Building on geometry

Diving into calculus

It's a big step up from GCSE to A level - especially if you're coming in with a B or a marginal A. The pace is pretty frenetic, and there's a fair amount of A and A* material from GCSE that's assumed knowledge at A level. If you're not especially happy about algebraic fractions or sketching curves, for example, you're likely to have a bit of catching up to do.

Luckily, this book has a whole part devoted to catching up with the top end of GCSE, as well as the stuff you'll need to learn completely fresh. In this chapter, I note where the content overlaps with GCSE and introduce you to A level algebra, trigonometry and calculus.

Reviewing GCSE

The good news is that if you've got a solid understanding of everything in your GCSE, quite a lot of Core 1 and a fair amount of Core 2 will be old news to you. Possibly less good news is that if you've got gaps in your knowledge, you need to fill them in pretty sharpish.

The four key areas where there's an overlap between the two qualifications are algebra, graphs and powers (all in Core 1) and trigonometry (which comes up a lot in Core 2). There are other bits and pieces, too - your arithmetic needs to be pretty decent in Core 1, where you don't have a calculator, and parts of Core 2 are likely to test your knowledge of shapes other than triangles - but generally speaking, these are the big four. The first part of this book is all about making sure you're up to speed with them.

Setting up for study success

Forgive me if you think it's patronising to tell you how to study - after all, you must have done pretty well with exams to get this far. I go into studying because A level is a much tougher beast than GCSE - it's possible for a reasonably smart student to coast through GCSE and get a good grade without needing to work too hard; by contrast, it's unusual to see someone glide through A level. And presumably, if you were finding it straightforward, you wouldn't be buying books like this to help you through it.

In this kind of scenario, everything you can do to optimise your working environment, your note-taking and your revision translates to quicker understanding and more marks in the exam.

Also, if you're studying at sixth form, there's likely to be a bit more going on socially than at secondary school. The more quickly you can absorb your studies, the sooner you can get out to absorb the odd lemonade with your friends. That is what you're drinking, isn't it?

All about the algebra

You've probably been manipulating algebraic expressions for years by now, and some of it will be second nature. However, just as a checklist, here are some of the topics you need to have under your belt:

• Solving linear equations (such as )
• Expanding quadratic brackets (for example, )
• Factorising and solving quadratics (such as )
• Solving linear and nonlinear simultaneous equations
• Simplifying algebraic fractions

I recap all of these in Chapter 3.

All this algebra isn't just for the sake of jumbling letters around and feeling super-smug when your answer matches the one in the mark scheme (although that can be a nice motivator). Algebraic competence underpins just about everything in A level. Even in places where you'd normally expect to use only numbers (for example, Pythagoras's theorem), you may be asked to work with named constants (such as k) instead of given numbers (such as 3).

Grabbing graphs by the horns

Somewhat related to algebra are graphs. You rarely need to draw an accurate graph at A level; it's far more common to be asked to sketch a graph.

That's good news: sketching is much quicker and more generously marked than plotting. However, you can no longer rely on painstakingly working out the coordinates and joining them up with a nice curve. Instead, you need to know the shapes of several kinds of graphs you've come across at GCSE: the straight line, the quadratic and the cubic graphs as well as the reciprocal and squared-reciprocal graphs.

You'll frequently be asked to work out where a graph crosses either of the coordinate axes (which is really an algebra question), and you'll be expected to be on top of curve transformations. I cover all these in Chapters 4 and 10.

Taming triangles and other shapes

Triangles, obviously, are the best shape of all, which is why you spend so much time on SOH CAH TOA, the sine and cosine rules, and finding areas at GCSE.

Oh, and Pythagoras's theorem. If there were a usefulness scale, Pythagoras's theorem would be way off it. I can't think of a more important equation at A level, and you can read about it in Chapter 4.

Those skills are extremely useful in A level maths. Pretty much every Core 2 paper I've ever seen has used a triangle somewhere, and triangles frequently crop up in other modules. (If you're doing Mechanics, having strong trigonometry skills is a massive help.)

It's not just triangles you need to know about, though. Be sure you know the areas and perimeters of basic two-dimensional shapes like rectangles, trapeziums and circles as well as the surface areas and volumes of three-dimensional shapes such as cuboids and prisms. I recap these shapes in Chapter 15.

Attacking Advanced Algebra

As you'd expect, the algebra you're expected to do at A level gets a bit more involved than what you did at GCSE. It comes down to learning some new techniques and linking some new notation to ideas you may already have a decent grasp of.

You start with powers and surds, a GCSE topic that sometimes gets glossed over. You'll need to be fairly solid on these, as they come up over and over again in A level. In later modules, you have a calculator that will happily tell you that the square root of 98 can be written as , but in Core 1, you need to be able to work that out on paper.

You also deal with sequences and series (extending the work you've done in the past), solidify the ideas of factorising polynomials, and do some work on functions - one of the most important ideas in maths.

Picking over powers and surds

Working out combinations of powers is one of the most critical skills for A level maths. I wouldn't say it's more important than topics like algebra, but a student's skill here is a strong indicator of how easy a student is going to find A level. If you're a bit rusty on the power laws, you're going to have to sort that out in fairly short order.

You also need to be pretty hot on your surds, especially in Core 1, when you're one calculator short of a pencil case. Throughout your course, you'll need to work square roots out in simplified surd form or, more generally, in exact form - examiners want to see things like rather than 4.443.

A step up from powers and surds are logarithms, which are handy functions for turning equations with unknown powers into equations with unknown multipliers. For example, without logarithms, is hard to solve (you know the answer is 4-and-a-bit but not necessarily what the bit is), but with logarithms, getting the answer is a simple bit of algebra: , where log(3) and log(100) are just numbers you can get from your calculator.

Lastly, under the 'powers' heading, you need to work with one of the most interesting numbers in all of maths, e. It's a constant (exactly , or roughly ) with the lovely property that if you work out the tangent line of at any point, you find its gradient is equal to .

Sorting out sequences and series

A sequence is simply a list of mathematical objects (often numbers, sometimes expressions). A series is what you get if you add them up.

You probably did some work on sequences in the past (finding the nth term, for instance, or deciding whether a term belonged to a sequence), and that will stand you in good stead. At A level, though, there's a lot more to it (who would have thought?).

As well as the arithmetic sequences you know and love, there are geometric sequences (where each term is a constant multiple of the one before). There are also explicitly and recursively defined functions, which sound like a horrible wild-card but in fact are quite nice because you're told precisely how they behave in the question.

And there are binomial expansions, which are a really neat way of expanding expressions like without needing to multiply out huge numbers of brackets. It's a particularly useful technique when you get to Core 4 and have to expand monsters like and use the result to approximate .

Of course, the binomial expansion is one of many things you do much more often in exams than you ever will in the outside world. (We have machines for that.) However, the idea of approximating things using polynomial series is a powerful tool for doing serious maths if you take the subject beyond A level. Oh, and...