Question and Exam Technique (8 Pages, 9/9/2013)

Question and Exam Technique(8 pages, 9/9/2013)

Part A: How to answer questions

Part B: Checking Methods

Part C: Exam Technique

Part A: How to answer questions

Tip 1: What is being tested here?

Look out for standard prompts to use a particular result. For example, any reference to a tangent to a circle usually suggests the result that the radius and the tangent are perpendicular.

Any request to maximise or minimise something should suggest the possibility of differentiating (assuming you have covered differentiation).

Consider what you know about the topic: If you know a topic well, the solution to any problem is bound to involve something you are already familiar with.

Note also how each part of the question leads on to the next. The usual convention is for question parts labelled (i), (ii), (iii) … to be related, whilst question parts labelled (a), (b), (c) … do not lead on to each other. It should be clear if your exam board follows this convention.

Tip 2: Creating equations

The general strategy for many questions is to convert the given information into mathematical form (usually one or more equations). This is especially true of Mechanics questions.

Example: Time, speed and distance questions

Set up a number of equations, using the relation Speed =

Equations can also be created as follows:

(a) Using a definition

Example: Definition of probability as

(provided that the outcomes are equally likely)

(b) Using a theorem, such as Pythagoras’ Theorem

Example: In order to prove some circle theorems, it is necessary to draw in a radius from a point given on the circumference. (Were this radius not drawn in, there would be nothing to distinguish this point from some other point, not on the circumference.)

Tip 3: How is the answer going?

At A Level (as opposed to more advanced activities, such as UKMT Maths Challenges or the Oxbridge entrance papers), a useful rule of thumb is: "if something is complicated, then you've probably gone wrong somewhere". However, don’t jump to the conclusion that the cause of the problem is a lack of understanding on your part: it is much more likely to be a simple algebraic or arithmetic slip.

Tip 4: Forcing an expression into the desired form

This technique is often useful in ‘Show that ...’ type questions, and has been mentioned earlier.

Example : Show that k(k+1)(2k+1) + (k+1)2 = (k+1)(k+2)(2k+3)

As we need the factors and (k+1) in the final expression, we can ‘force’ them into the left-hand side (LHS), as follows:

LHS = (k+1) { k(2k+1) + 6(k+1) }

= (k+1) { 2k2 + 7k + 6 }

= (k+1)(k+2)(2k+3)

Tip 5: Case by case approach

Example 1 : > 4

Notice, first of all, that we do not know whether x is +ve or –ve (though x cannot equal 0, as the left-hand side would then be undefined).

Case (i): x>0

Multiplying both sides by x gives 1 > 4x, so that x < ¼ (and x>0)

ie 0 < x < ¼ is one solution

Case (ii): x<0

Multiplying both sides by x gives 1 < 4x

[reversing the sign of the inequality, as we are multiplying by a –ve number]

so that x > ¼ ; which contradicts the assumption that x < 0

Hence the full solution is : 0 < x < ¼

Example 2: Demonstrate that – n is always even (where n is a +ve integer)

Case (i): n is even implies that – n = [even] – [even] = [even]

Case (ii): n is odd implies that – n = [odd] – [odd] = [even]

[In this example, it is in fact quicker to factorise – n as n(n-1), which will always be a product of an even number and an odd number, and will therefore always be even.]

Tip 6: Graph sketching

The first step will usually be to consider where the graph crosses the x and y axes (y = 0 and x = 0, respectively).

(A repeated root implies either a turning point on the x axis, if the factor appears an even number of times [eg y = (x-1)2, to give a very simple example], or a point of inflexion, if the if the factor appears an odd number of times [eg y = x3])

The next step will usually be to consider extreme values of x (ie to decide what happens to y). This will determine which quadrants the graph starts and ends in.

Part B: Checking Methods

Large numbers of marks are lost in exams through simple algebraic or arithmetic errors. It is important to have a checking routine that is second-nature (ie not one that is only applied in exams).

The approach should be along the lines of: “How do I know that my answer is correct?”

Possible checklist - SPARE Q (think of Scrabble, where you usually end up with an unwanted Q)

Substitution (S)

Substituting the answer into the original information, to see if it works.

eg solving equations (especially simultaneous equations)

Proofreading (P)

Reading over each line, as soon as you have written it, to see if there are any arithmetic or algebraic errors.

eg correct expanding of brackets (especially involving minus signs)

Alternative method (A)

Can you arrive at the same answer by another method?

Example: 0.05 x 0.3

(1) 15 gives 0.015 (counting the number of places after the decimal)

(2) 5/100 x 3/10 gives 15/1000 = 0.015

(3) we are expecting an answer just less than a third of 0.05 (and involving the digits 15)

(4) 0.5 x 0.3 is a half of 0.3; ie 0.15; we want a tenth of this

Reasonableness (R)

Is the answer sensible?

(A) Does that loaf of bread really cost £15? (Perhaps there has been an error with a decimal point; should it be £1.50?)

(B) Probabilities can’t be greater than 1

Estimation (E)

Work out an approximate answer (often by rounding).

23 x 19 ≈ 20 x 20 = 400

The exact answer will be a bit bigger than 400.

Read the Question again (Q)

Have you answered the question properly?

(A) Question asks for answer to eg 2dp

(B) Have you missed anything important?

Example

“What is the probability of drawing a Heartif the card is not replaced?”

It is often worth reading the question again just before you start your answer.

Part C: Exam Technique

Tip 1: Showing working

Putting down detailed working has the following advantages:

(a)It gives the examiner a good reason for awarding you one of the marks on the mark scheme.If you make a mistake, you can often still be given a ‘method’ mark.

(b) It makes it easier for you (as well as the examiner) to check through your work.

Tip 2: Explain what you are doing

Give a short written description of what you are doing at each stage

Example: “N2L implies that” [N2L = Newton’s 2nd Law]

This should help clarify matters if the method would otherwise be unclear to the examiner (eg if there is a mistake in the calculations).

Tip 3: Rounding

Examiners often like answers to be rounded to 3 significant figures. (Angles in degrees though are usually wanted to 1 dp.)

Ensure that sufficient decimal places are retained during the calculation. ‘Premature rounding’ is a common cause of lost marks.