STEP Preparation (10 Pages, 11/9/2013)

STEP Preparation(10 pages, 11/9/2013)

Contents

(A) Resources

(B) Possible Preparation Strategy

(D) Problem-solving Ideas

(E) Miscellaneous

(A) Resources

(a)Official STEP website:

This gives access to:

(i) STEP Specification ("About STEP" in the left-hand menu)

(ii) Formula booklet ("Preparing for STEP", then "STEP Documents")

(iii) past papers & solutions (1998-) ("Preparing for STEP" in the left-hand menu)

(iv) “Advanced Problems in Mathematics”by Stephen Siklos (based on past STEP questions (though not identified), with discussions & solutions; including STEP 3 questions; 98 pages) ("Preparing for STEP", then "STEP Documents")

(v)“Advanced Problems in Core Mathematics” by Stephen Siklos (as above, but STEP 1& 2 only; 164 pages) ("Preparing for STEP", then "STEP Documents")

(b) Other sources

(i) Peter Mitchell’s “Meikleriggs” website: solutions to past papers (2006-):

(ii)“Ask NRICH” option in NRICH website:

(iii) Notes (and a few solutions) for certain questions, on my website:

(B) PossiblePreparation Strategy

No candidate is expected to have studied all of the topics in the papers to sufficient depth, and it would seem to be a sensible strategy to specialise in a certain number of areas.

Also, given that there will be many rival candidates of a similar standard, one way of gaining the upper hand is by being better at particular topics. Another way is to be better at choosing questions!

Topics to specialise in

It probably pays to specialise in topics that have some of the following characteristics:

(i) they are likely to crop up

(ii) they are easily recognisable (eg Integration, Vectors)

(iii) they are limited in scope

(iv) they involve standard techniques

(v) they generally don’t involve long answers (eg Probability)

(vi) they can generally be checked in some way

Questions to target

(i) Q1 & (often) Q2 on each of the 3 papers are intended to be easier (or at least ‘accessible’ – ie not dependent on more advanced theory). However, nothing in STEP is guaranteed. Q1 in STEP 1 for 2007, for example, doesn’t fall into the category of an easy question – unless you happen to hit on a particular method for answering it.

(ii) ‘Show that ...’ (ie self-checking)

(iii) Questions with a clear topic and/or method (that you are happy with)

(iv) Short questions! (less time spent reading; especially if the question is not chosen)

Suggestions

Pick at least half a dozen topics, and arrange them in the order that you would attempt them in the exam.After attempting a paper under exam conditions, conduct a post-mortem to decide whether your selection of topics needs to be refined, and whether the order is still appropriate.

Whatever strategy you adopt, it is probably best to allow it to be refined gradually over the course of the preparation period, so that by the time you come to the exam the strategy has been tested.

(i) Simple Approach

Look for the simplest possible approach – even if it seems too obvious. The examiners are not trying to trick you.

(ii) Applying earlier results / methods

A recurrent theme of the STEP questions is the application of results or methods used earlier on in the question.

Possible variations:

(a) Precisely the same method is used for part (ii) as for part (i) – though it might be more awkward to apply.

(b) An expression or equation in part (ii) can be rearranged to be in the form of something appearing in part (i), so that the result of (i) can be applied. A substitution may be necessary.

Example: STEP 3, 2006, Q7

(iii) Refinements

Another recurrent theme is the presence of refinements in the questions for candidates to recognise and act upon.

Examples

(a) recognising the range of values for which a result is valid (eg if a Binomial expansion is involved)

(b) use of ln|x| (rather than just lnx)

(d) avoiding division by zero

A typical pattern is that the 1st part of a question is straightforward, but a modification introduced into the 2nd part means that a refinement has to be taken into account. The consequence of missing this refinement may just mean that full marks are not obtained, but it could result in the answer being worthless.

Even if a refinement cannot be acted on fully, it is probably worth at least showing an awareness of the problem. Borderline candidates (those who have just missed out on the required STEP grades) will have their scripts examined by the tutors of the college that made them the offer, and they will be looking for things that set them apart from other candidates with similar scores. Having said this, however, it is stated by the examiners that marked are only awarded for doing work, not for saying what you would do.

These refinements are often signalled by mentioning that eg <4c (STEP 1, 2008, Q2) – a fact that isn’t used until later on in the question.

(iv) Practice at algebraic and numerical manipulation

Examiners are always bemoaning candidates’ shortcomings regarding algebra.

(v) Checking

This needs to be an ingrained routine (ie not just saved for the exam).

For example, SPARE Q (think of Scrabble, where you generally have a Q left at the end):

S: substitution (does the answer satisfy the original equation(s)?)

P: proofreading (looking over each line after it has been written – to eliminate elementary slips; eg involving minus signs)

A: alternative method (eg for checking numerical work; 0.04x0.06 could be worked out by converting the numbers into fractions, as well as by counting decimal places)

R: reasonableness (is there anything suspicious about the answer?)

E: estimate (eg rough check on 243 x 47: 240 x 50)

Q: read the question again (eg before embarking on the solution, and at the end)[Note: zero marks are usually awarded for not using the specified method]

(vi) Explanations

Both to describe what you have done (to make it clearer), and to indicate the direction that you are pursuing. Both of these make it easier for the examiner to give you marks.

(vii) Linking of statements

Example:

x y < xz [A]

y < z [B]

x > 0 [C]

As it stands, it is not clear whether [B] is supposed to lead on from [A] (which would be incorrect, as x could be negative), or whether [B] is a result established earlier, or perhaps something stated in the question. Likewise, is [C] being deduced, or brought in from somewhere else?

To avoid any uncertainty on the part of the examiner, each statement really needs something to say where it comes from.

A revised version of the above might be:

From (1), x y < xz

Also, the question states that y < z

Hence x > 0

(viii) Spotting shortcuts

These are often possible by the use of a symmetry argument, or a result established earlier in the question.

Be wary of any work that just repeats a method already applied; ie look for a shortcut (examiners will be reluctant to allocate marks for using an old idea).See STEP 2, 2008, Q10 (solution on this website).

Also, when you have discovered a suitable method, look for an improved version.

(ix) Checklist when answering questions

- should I be doing this question?

- observe the clues in the question

- is there an obvious approach to adopt (possibly involving standard tools)?

- where could I get inspiration from?

- what checks can be applied?

(x) For the end of the exam

Save some things until the end of the exam: it is useful to have a relatively straightforward task to complete in the last few minutes, rather than frantically looking through the paper for something sensible to do or check.

(D) Problem-Solving Ideas

Notes
Standard Approaches
Use information in the question to create equations / Create letters as necessary (eg r = radius)
Make a suitable substitution
Reformulate the problem / eg convert an equation into the intersection of 2 curves
Case by case / eg case(i): x<0, case (ii): x>0, case (iii): x=0
Listing (for “number of ways” questions) / Find a systematic way of listing the possibilities, and look for a pattern that will make it possible to count the number of items in the list.
Clues in the Question
Condition in the question / eg <q (could involve ); c0 (possible division by c at some stage)
Using a relevant theorem or definition
Special feature / Look out for a special feature of the problem that provides extra information. Example: A right-angled triangle has its corners on the circumference of a circle. This means that the hypotenuse of the triangle is a diameter of the circle.
Using an earlier result/method that appears in the question / eg by manipulating the result (perhaps by making a substitution), or applying the same idea to something similar
Presence of expressions in a question / (i) The presence of squares or pairs of brackets suggests the possible use of Difference of Two Squares.
(ii) The presence of a ± symbol strongly suggests the involvement of a square root.
(iii) division by n+1 could suggest integration
Looking for Inspiration
Try out some values / if only to get a feel for the problem
Critical points / Consider the point(s) at which the nature of a problem changes.
Simpler version of the problem / eg in order to sketch y = , start with y =
Penultimate Step / A result that would lead immediately to the required conclusion (eg in order to prove that m and n were never equal, we could show that m was always even and n was always odd).
Tools
Equating coefficients / Also applies to pairs of vectors in the form pa +qb (where ab are not parallel)
Greatest or least value / (i) Stationary point (ie = 0 )
(ii) ‘Completing the square’
0, 1 or 2 solutions of quadratic: b2 – 4ac / Example 1: If a straight line is to be a tangent to a circle, solve the simultaneous equations of the line and the circle, to give a quadratic, for which b2 – 4ac must equal zero (ie 1 solution).
Example 2: To show that is defined in a certain range, we want to show that f(x) = 0 has no solutions.

(E) Miscellaneous

(i) The first part of a STEP question may appear to be much too easy, and a trap may be suspected. Sometimes thereis a refinement to be taken into account, but often it is just intended as a gentle introduction, to point you in the right direction. It may, for example, have been added in as an afterthought – in order not to make the question too difficult. The examiners are generally keen for students to be able to at least start a question.

A result that is easy to derive may simply be needed for the next part of the question.

The last part of a STEP question isn’t necessarily any harder than the earlier parts - especially once you have got on the question-setter’s wavelength. Also, the last part might simply be the final (easy) stage in establishing an interesting result.

(ii) STEP 2 is intended to be harder than STEP 1. Both only involve minimal material from the Further Maths syllabuses (including proof by induction). Because STEP 3 opens up the possibility of questions on any Further Maths topic it may require more preparation of topics, but there is a school of thought which says that STEP 3 questions are generally easier than STEP 2 – once the topic in question has been studied - on the basis that STEP 2 questions have a smaller fund of suitable topics, and therefore tend to be hard questions on familiar themes.

Related to this point is the observation made by the examiners themselves that an unusual question, or one that quotes an obscure-looking result, often turns out to be simpler than a more standard question. Invariably no prior knowledge of the quoted result is in fact needed. The examiners are effectively rewarding candidates for coping on the spot with new territory.

(iii) At A Level, if the algebra becomes complicated it is usually the case that you have gone wrong somewhere. It is unfortunately the case with STEP that some questions simply involve a lot of messy algebra. You may or may not want to avoid these!

If trying to eliminate a variable X from two complicated equations, you may find it easiest to just make X the subject of each of the equations, and then equate the two results.

(iv) For “if and only if” proofs, it may be sufficient to indicate that the line of reasoning is reversible (assuming that this is the case).