Can be a tricky question, this.
Primary schoolchildren are taught that it is repeated addition, which makes a lot of sense.
4 x 4 =
4 + 4 + 4 + 4 =
16
Four lots of four/groups of/sets of.
But multiplication is also a scaling quantity. It ratchets things up massively quickly. If you type 2x2 into your calculator, then keep multiplying every answer by 2, the calculator will quickly run out of digits. To compare this with addition misses the point that multiplication is the centre of all geometric operations.
It does through the squaring and cubing and so on effect. If you take four centimetres, and for each of these four, you add another four, you get 4 x 4. It is known as squaring because the Greeks used to see it geometrically as the way to find the area of a square. For each centimetre across, there are four up (and vice versa).
x2 is a powerful, recurring idea, which has its role in pretty much the entire universe. (E = mc2)
It's also worth noting the effect of multiplication by a fractional quantity.
4 x 1/4 will obviously increase the fraction, but it will have a decreasing effect on the whole number. Unlike multiplication by two whole numbers, where the answer is greater than either, in this operation, the answer will always be smaller than one of the inputted values.
Thursday 28 January 2010
Tuesday 19 January 2010
Tips For Revising for a Module
It's exam season: in fact, it's just gone. So many of us will be turning our thoughts to retakes - or in my case, to reapplying for exams we had to drop out of.
So how do you study for an A Level maths exam?
1) Don't necessarily do all the questions. For some, like me, it's essential to do every question in the book and all the past papers. For many, that's just repetition. You need to do the first few and the last few. Past papers are good to study as well as to do, to work out which questions often come up.
2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years!
3) MAke sure you know the formulae which are NOT in the formula book for the module you are doing. There are usually several really common ones - like trig identities for example. Check each chapter in the textbook to see which formulae you need. It is easy to forget these, especially if you have been *ahem* working on the same module a while.
4) DON'T do anything the night before. The formulae will start to jump around in your head.
5) Revise the previous modules a little. You will find some identities and formulae you studied before that are not explicitly repeated come in handy. For example, the cosine and sine rules from C2 can be very handy when doing the questions on reciprocal trig functions in C3. Also the rules and techniques of radians are very useful indeed, though quite a bit of practice on this is done through C3.
6)If you do past papers and you can't get a tutor to mark them for you, you can do it yourself via the online answer papers which are published. It's not an exact science because you are not amazing at maths, unlike the markers, which is why you're studying it now. So BE HARSH on yourself and be conservative. Markers will often be generous with follow-through marks so you can at least be prepared. That's the pessimistic side of me coming through.
So how do you study for an A Level maths exam?
1) Don't necessarily do all the questions. For some, like me, it's essential to do every question in the book and all the past papers. For many, that's just repetition. You need to do the first few and the last few. Past papers are good to study as well as to do, to work out which questions often come up.
2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years!
3) MAke sure you know the formulae which are NOT in the formula book for the module you are doing. There are usually several really common ones - like trig identities for example. Check each chapter in the textbook to see which formulae you need. It is easy to forget these, especially if you have been *ahem* working on the same module a while.
4) DON'T do anything the night before. The formulae will start to jump around in your head.
5) Revise the previous modules a little. You will find some identities and formulae you studied before that are not explicitly repeated come in handy. For example, the cosine and sine rules from C2 can be very handy when doing the questions on reciprocal trig functions in C3. Also the rules and techniques of radians are very useful indeed, though quite a bit of practice on this is done through C3.
6)If you do past papers and you can't get a tutor to mark them for you, you can do it yourself via the online answer papers which are published. It's not an exact science because you are not amazing at maths, unlike the markers, which is why you're studying it now. So BE HARSH on yourself and be conservative. Markers will often be generous with follow-through marks so you can at least be prepared. That's the pessimistic side of me coming through.
Monday 18 January 2010
e (2.71828)
No, this is not a post for luvved-up early 90s teenagers.
e is a bizarrely cool number, sometimes known as the natural logarithm.
Like π it is an irrational number - ie it cannot be expressed exactly as the ratio of two whole numbers (and thus as a fraction).
How do we get it then?
Imagine an exponential curve. A curve that is nx. Say, y= 3x.
As usual, the gradient will generally differ as x differs.
Is there a curve which has a gradient of 1, where x=0? There is, and, coolly, this curve has a gradient of 1 where x=0 and ALSO passes through y=1, ie the value of the function is 1.
This curve is y = e x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.
As you can imagine, that makes differentiating with e easy...
There is an excellent page here which shows you some interesting examples of e in action.
e is a bizarrely cool number, sometimes known as the natural logarithm.
Like π it is an irrational number - ie it cannot be expressed exactly as the ratio of two whole numbers (and thus as a fraction).
How do we get it then?
Imagine an exponential curve. A curve that is nx. Say, y= 3x.
As usual, the gradient will generally differ as x differs.
Is there a curve which has a gradient of 1, where x=0? There is, and, coolly, this curve has a gradient of 1 where x=0 and ALSO passes through y=1, ie the value of the function is 1.
This curve is y = e x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.
As you can imagine, that makes differentiating with e easy...
There is an excellent page here which shows you some interesting examples of e in action.
Reciprocal Trig Functions
Sine, cosine and tangent are functions. You input a value - an angle in radians, or degrees, and for each angle there is a value. They are repeating functions.
They also have reciprocal functions.
The reciprocal of sine x is cosec x and is (1/sin x)
The reciprocal of cosine x is secant x and is (1/cos x)
The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).
They also have reciprocal functions.
The reciprocal of sine x is cosec x and is (1/sin x)
The reciprocal of cosine x is secant x and is (1/cos x)
The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).
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