Maths in Action :: Exam Technique

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

www.drfrostmaths.com

Last modified: 15th November 2015

I used to do maths for robots!

where

A slide from my talk titled…“DUDE, WHERE’S MY CAT?”

(c) Simon’s Cat

But what the devil is that maths doing?

Each preposition in the English language, such as ‘behind’, ‘between’, ‘near’, ‘left of’, has a probabilistic model built for it. My algorithm generates descriptions of objects that disambiguates their location.

Algorithm output:“The sculpture is right of 101, just left of you and near the laboratory in front of me”

Robot

Things I really hate…

Generic TalksSo I’ll keep the advice practical!

TIP #1 :: Know Your Calculatrice* *French for calculator

First Tip: GET A SILVER CASIO CALCULATOR

There’s a remarkable number of A Level questions these will allow you to fully check answers.

B? S?Modules (Edexcel)Operation

Quadratic/cubic solver. All Solve 2/3 simultaneous eqns

Multiply/inverse matrices FP1,FP3

, , complex numbers FP1,FP2

Solve any equation.† C2 (e.g. log equations)

Calculate standard deviation, S1 Mean, PMCC (), line of bestfit.

† Using numerical methods. Thus only (at most) 1 solution will be found, and it won’t be exact.

Definite integration C2,C4

TIP #1 :: Know Your Calculatrice* *French for calculator

First Tip: GET A SILVER CASIO CALCULATOR

There’s a remarkable number of A Level questions these will allow you to fully check answers.

B? S?Modules (Edexcel)Operation

Find gradient for any -value Core for any equation.

Generate tables of values Core (e.g. for between -3 and +3)

Look up Z-table values S1-3 (but sadly not in reverse)

TIP #1 :: Know Your Calculatrice

Shameless Self Promotion Alert!

On my website you can find a ‘virtual calculator’ (both Silver and Black) which explains every single button and mode.

www.drfrostmaths.com KS5 Virtual Calculator

TIP #2 :: Document Your Errors

I once tutored a student who had previously failed to meet their Cambridge History offer, because of their maths A2 result, and reapplied whilst retaking maths.He was very bright and ‘got’ the questions, but regularly made silly errors.

We kept a log of his errors each time he made one, but while abroad for a few months (where he learnt Portuguese), he came up with various acronyms for Portuguese phrases, that he would write on relevant questions BEFORE he even started it!

NMSPN(Não misturar-se positiva e negative)

TIP #2 :: Document Your Errors

Proactively think about the types of errors you might make as you embark on a question, if you are liable to make them.e.g. Rounding errors or giving too little accuracy.

Common General Errors

Sign errors (e.g. )

Being clumsy/careless with your calculator (particular applies to S1)

,

Dividing an equation by a variable instead of factorising, e.g.

…

Topic Specific Errors

Doing for gradient instead of for gradient, or being inconsistent in the ‘to’ and ‘from’ point for and .

Forgetting /-intercepts on graph sketches.

Mixing up summation and nth term formulae for arithmetic/geometric series.

Forgetting the +c after integrating.

…

Click to Start Royal Boat Parade of

Maths Errorness

TIP #2 :: Document Your Errors

You are welcome to use my revision notes, which point out common errors.

Shameless Self Promotion Alert!

TIP #3 :: Know your Mark Schemes

You will more reliably answer an exam question correctly if you know exactly where the marks typically come from for that type of question.

“Find the mean time taken (2)”.

What you might be tempted to write:

But all you need to write: M1 A1You could exploit the ‘sum’ menu in STATS mode.

This is particularly important for common larger mark questions (e.g. hypothesis testing).

TIP #4 :: Don’t rely just on your textbook

Some textbooks have some drawbacks, e.g. the Edexcel ones:

1. Certain types of exam questions are underemphasised with limited practice, e.g.a. Forming an inequality from a discriminant (C1).

b. Solving more complex log equations (C2)

2. On the rare occasion the book prescribes a method that is not consistent with the main method in mark scheme.e.g. Linear interpolation (S1) mark schemes use idea of ‘fraction across the interval’ (much simpler!) whereas textbook uses similar triangles.

TIP #4 :: Don’t rely just on your textbook

3. Sometimes there’s an ‘alternative method’ (indicated as such in the mark scheme) that might work better, again not covered in the textbook.e.g. For showing divisibility in proof by induction (FP1).

• Practice past papers! (including Solomon, ‘Gold/Silver/Bronze’ papers, etc.HOWEVER: Don’t risk the opposite, revising just to the exam and hence lacking versatility to deal with less routine questions.

• Revision videos (e.g. www.examsolutions.net).

• (or slides on my website )

A simple solution…

TIP #5 :: Know your wordy questions

There’s sneaky last parts to questions (typically on applied modules), which are either wordy or require some form of explanation. It helps to know the standard response in advance.

“Explain whether the mean or median would be the more suitable average to use.”

¿𝑘2+2𝑘+9

¿ (𝑘+1 )2+8

If real roots, .But and hence

C1

S1

Shameless Self Promotion Alert! e.g. From my S2 revision notes:

TIP #5 :: Know your wordy questions

TIP #6 :: Learn Formulae Effectively

Quick bit of maths! Try finding the gradient of the following two points in your head:

(−2 ,−3 ) ,(−8 ,9)

If you plugged directly into the formula:

you would have had to contend with various double negatives. It’s a faff and prone to errors.

𝑚=−2If however you just thought of the formula as:

Then we can clearly see without the need for subtraction that the value increases by 12 and the value decreases by 6.We get with much less effort.

TIP #6 :: Learn Formulae Effectively

Another example is the formula for calculating variance of data.

If you remember as:

Rather than the explicit formulae then the benefits are that:a. You will memorise the formula more easily.b. You can plug in numbers with more confidence because you

have a better appreciation of components of the formula.c. You get 3 different formulae for the price of one!

“the mean of the squares minus the square of the mean”

𝜎 2= Σ𝑥2

𝑛−𝜇2 𝜎 2= Σ 𝑓𝑥

2

Σ 𝑓−𝜇2 𝑉𝑎𝑟 ( 𝑋 )=𝐸 (𝑋 2 )−𝐸 ( 𝑋 )2

TIP #7 :: Reduce dependence on mark schemes

Sound familiar?

(While working on past paper)I’m a bit stuck with this

part of the problem.

Mar

kSch

emes

RUs.c

om

Cool beans. I know how to proceed with

this question now.

(Finishes practice paper)

Wicked cool. I got 85%. I feel confident on this material.

DOES REAL EXAM

FAILS EXAM Joins the

circus.

TIP #7 :: Reduce dependence on mark schemes

NEVER……have the full mark scheme next to you when you do a past paper. This will give you too much of a safety net and prevent opportunity to make (and hence learn from) mistakes, or build understanding.

BUT FEEL FREE TO……have numerical answers available.These provide the final answers only without any explanation. You can check whether you’re right as you go along without getting any assistance on the question.

Numerical answers can be found on various websites (including my own).

Shameless Self Promotion Alert!

TIP #8 :: Show clear structured workingYour working should tell a story, and it should be clear to the examiner what you’re calculating at any stage of your working.

Classic examples of unhelpful working:

𝑦=3 𝑥2+1

√𝑥∫ 3𝑥2−4𝑑𝑥

If , find

There should be a here, otherwise it’s not clear when they’ve just simplified and when they’ve differentiated.

They’ve already integrated, so why is the still there?

General tips:

• Work down the page.• If you have subsidiary results to show, ensure it’s clear when you’re

answering the original question again.• Use helpful variable names.

e.g. Use for the gradient function but for specific gradients. Distinguish between the gradient of a tangent (e.g. or just ) and the gradient of a normal ( or )

TIP #9 :: Plan for all types and subtypes of question

Often the mentality when preparing for exams is “prepare as much as I can and get a high as mark as possible”.

It’s the kind of mentality where you’re pleased if you’ve got the bulk of most of the questions right, and content you’ve perhaps got the odd last part of a question wrong or made a slip here and there.

The “0%+ mentality” The “100%- mentality”

It’s an approach where you meticulously plan in order to reduce the marks you lose, rather than thinking of it as maximising the marks you gain.

It requires a combination of what we’ve looked at, including:• Full appreciation of the exact types and

subtypes of questions, including wordy questions.

• A deep appreciation of the content rather than blindly applying processes.

• Algebraic dexterity.• Identifying mistakes you’ve made in

the past and proactively being mindful of avoiding them.

TIP #9 :: Plan for all types and subtypes of question

Without needing to do the questions, look through a large number of past papers and make a list for each chapter of what points come up.

Shameless Self Promotion Alert!

In some revision notes (on my website) I’ve already done this for you.

But there’s plenty of other notes out there…

TIP #10 :: Shallow understanding breeds misconceptions

Some errors are due to silly mistakes. Some are due to lack of preparation. But some are exposed when a superficial understanding of material is surfaced by non-routine questions.

What is ?

For anyone who has covered logarithms:

Solution: . Such an expression has never come up in an A Level exam. But if you understand and as inverse functions of each other, then you can appreciate that the two functions cancel each other out in this example.

The key to being a successful ‘Mathematician’ (rather than an ‘A Level mathematician’) is your capacity to internalise concepts successfully in such a way that techniques can be applied in unfamiliar settings. If you enjoy maths, MAT and STEP questions are perhaps the best way to enrich your skills of maths within the core A Level curriculum. https://www.maths.ox.ac.uk/study-here/undergraduate-study/maths-admissions-test

Fini

www.drfrostmaths.com VISIT IT – Everything is free(Go on, you know you want to)

http://www.examsolutions.netVideos explaining pretty much everything in the A Level syllabus.

http://www.mrbartonmaths.com/alevelcore.htm Links to various revision notes and websites.

Sites you might be interested in: