Cardinal Newman School September 2020

A-Level Mathematics

Transition Booklet

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Contents

Folder Checklist and Organisation – Page 3

Calculator Information – Page 3

Course Overview – Page 4

Assessment Objectives and Weightings – Page 5

Resources and Support– Page 6

Independent Study – Page 7

Key Vocabulary – Pages 7-8

Greek Letters – Page 8

How to read an A-Level Maths Mark Scheme – Page 9

Exam tips – Page 9

Wider Reading List – Page 10

Year 12 complete PLC – Pages 11-13

Transition Unit PLC – Pages 14-15

Entry Assessment Practice – Pages 15 to 17

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Folder Checklist and Organisation

As this is a two year course, it is imperative you keep your notes organised and in one place so that you have everything needed for revision. You need to bring your folder with you to every lesson so that you can organise your re-sources and so that you have all the guidance with you to use within the les-son.

After each lesson ensure your notes and any resources provided are filed in the relevant section. Within your folder itself it is wise to have the following divided sections:

• Classwork • Homework • Pitstops + Assessments • Revision

You may also wish to divide this further into 3 sub-sections of Pure, Statis-tics and Mechanics.

Use this checklist to ensure your work is organised and so you can identify what you need:

Do I have… General Folder Requirements A4 Ring binder Folder Dividers for each unit/topic Transition Booklet Section 1 - Pure Classwork Homework Pitstops/Assessments Revision Paper 2 - Statistics Classwork Homework Pitstops/Assessments Revision Paper 3: Mechanics Classwork Homework Pitstops/Assessments Revision

Calculator

At A-level you need a calculator that can perform a number of statistical cal-culations. The one we recommend is :

• Scientific calculator (Casio fx-991EX Classwiz)

You may also wish to consider investing in a more powerful graphical calcula-tor but please check first whether the one you are thinking of buying is ap-propriate for the course.

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Course Overview:

From 2017 the A Level Mathematics course will follow the new specification published by the AQA exam board. The A Level Mathematics course is made up of 3 areas of study Pure(67%), Statistics(17%), Mechanics(17%). The table below describes the content of each of these units:

Assessment:

The course is assessed through three 2 hour exams worth 100 marks each.

Paper 1 Pure Paper 2 Pure and Mechanics Paper 3 Pure and Statistics

There is no coursework element and all formal examinations take place at the end of Year 13.

A-Level Mathematics

Topics Description

Pure This is the study of pure mathematics consisting of topics including proof, algebra

and functions, coordinate geometry, sequences and series, trigonometry, expo-

nentials and logarithms, differentiation, integration and numerical methods.

Mechanics An applied module consisting of vectors, quantities and units in mechanics, kine-

matics, forces and Newton’s laws and motion.

Statistics An applied module consisting of statistical sampling, data presentation and inter-

pretation, probability, statistical distributions and statistical hypothesis testing.

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Assessment Objectives and Weightings

The exams will measure your achievement against the following assessment objectives

AO1: Use and apply standard technique. Student should be able to:

• Select and correctly carry out routine Procedures

• Accurately recall facts, terminology and definitions

AO2: Reason interpret and communicate mathematically. Students should be able to:

• Construct rigorous mathematical arguments (including proofs)

• Make deductions and inferences

• Assess the validity of mathematical arguments

• Explain their reasoning

• Use mathematical language and notation correctly

AO3: Solve problems within mathematics and in other contexts. Students should be able to:

• Translate problems in mathematical and non-mathematical contexts into mathematical processes

• Interpret solutions to problems in their original context, and, where appropriate, evaluate their accuracy and limitations.

• Translate situations in context into mathematical models

• Use mathematical models

• Evaluate the outcomes of modelling in context, recognise the limita-tions of models and, where appropriate, explain how to refine them.

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The assessment weightings in each paper are shown below:

Resources

You will be issued with a textbook that covers the entire first year. As you near the end of the course you will be provided with exam practice papers.

Additional past papers and questions are available online. The following website are recommended:

• http://www.aqa.org.uk/subjects/mathematics

• https://www.examsolutions.net/a-level-maths

• https://www.kerboodle.com (Login details provided)

• http://www.physicsandmathstutor.com/

Support

You should never give up too easily on a mathematical problem but if you feel you have exhausted all possibilities then it is time to seek help. Try talking to other members of the class about the problem these discussions will help you learn. Alternatively find one of your teachers they will always e happy to help.

Finally there is an after school support session every Wednesday 3.10 to 4.10 in room 14, this is an opportunity to come and ask specific questions or get help with any topic you are finding challenging.

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Independent Study

It is vital that you spend as much time as possible practising mathematics, right from the start of the course. There should never be a time that you sit in private study and say you haven't been set any maths. You always have maths to do.

Sometimes you will be set an assignment - this must be completed by the deadline. If you are not set an assignment, you must do independent practice by: completing unfinished classwork / completing and marking extra exercis-es / doing practice exam papers.

Wording of Questions( Key Vocabulary)

Write down, state: answer should be fairly obvious – little or no work will be

needed

Find, Determine, Obtain, Calculate: Some work will be needed, and you will

need to show your method

Evaluate: Work out the value of an expression

Verify: You are given the answer – you just have to show that it works

Show that: You have to work out the answer as if it had not been given in the

question. You are given the answer because you may need it in the next part

of the question.

Prove: A formal proof is required – each line of you answer must follow on log-

ically from the previous line

Explain: This involves writing words (preferably in sentences!)

Interpret: Make a comment that refers to the context of the question

Exact: Not a decimal but a surd, fraction, etc.

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Hence / Hence or otherwise: Hence means you must use the last part of the

question to do the next part. Hence or otherwise means you can use the last

part of the question to do the next part (usually a good idea!) but don't have

to.

Sketch: Use pencil – don't use graph paper – label the axes – if there are two

graphs, label each one

In terms of: This usually means that your answer will not be a number but a mathematical expression including one or more letters.

Give your answer in simplified surd form: Simplified surd form means

. Bear in mind that c has to be as small as possible so take any square factors

Greek Letters

A guide to pronouncing Greek letters commonly used in A Level Maths

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How to Read an A Level Maths Mark Schemes If you do a past paper question and want to check your solutions, you might find the mark scheme confusing. This guide should help you make sense of it.

M marks: method marks are awarded for knowing a method and attempting to apply it. A marks: accuracy marks can only be awarded if the relevant method (M) marks have been earned. B marks are unconditional accuracy marks (independent of M marks). Abbreviations • bod – benefit of doubt • ft – follow through

• the symbol will be used for correct ft • cao – correct answer only • cso – correct solution only. There must be no errors in this part of the ques-tion to obtain this mark • isw – ignore subsequent working • awrt – answers which round to • SC: special case • oe – or equivalent (and appropriate) • dep – dependent • indep – independent • dp – decimal places • sf – significant figures

•＊ The answer is printed on the paper (eg in a show that question)

Exam Tips • State each formula you need before attempting to use it • Write each step on a new line • If a question has several parts (a) (b) etc:

• label each part clearly • leave gaps between your answers to each part • your answer to one part may well help you do the next part • if you can not do part (a) of a question, try part (b)

• Give answers to an appropriate accuracy (if final answer is to 3sf then do all intermediate working to at least 4sf)

• If you have two attempts at a question decide which one you want marked and cross out the other one with a single diagonal line

• If the final answer has a unit (eg cm) then include it • If you are stuck, try drawing a diagram or sketching a graph.

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A-Level Wider Reading List

Picking 1 or 2 of these books will help with your engagement and enjoyment of mathematics and will make you a better mathematician to.

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Year 12 PLC

Below is a complete personal learning checklist for what you will study in year 12. You will be given a copy of this to go in the front of your folder.

Topic Content ü

Pure Mathematics

Proof

Understand and use the structure of mathematical proof, proceeding from given assumptions

through a series of logical steps to a conclusion; use methods of proof, including proof by deduc-

tion, proof by exhaustion.

Disproof by counter example.

Algebra and

functions

Understand and use the laws of indices for all rational exponents.

Use and manipulate surds, including rationalising the denominator.

Work with quadratic functions and their graphs; the discriminant of a quadratic function, including

the conditions for real and repeated roots; completing the square; solution of quadratic equations

including solving quadratic equations in a function of the unknown.

Solve simultaneous equations in two variables by elimination and by substitution, including one

linear and one quadratic equation.

Solve linear and quadratic inequalities in a single variable and interpret such inequalities graph-

ically, including inequalities with brackets and fractions.

Express solutions through correct use of ‘and’ and ‘or’, or through set notation.

Represent linear and quadratic inequalities such as y>x+1 and y>ax2 +bx + c graphically.

Manipulate polynomials algebraically, including expanding brackets and collecting like terms, fac-

torisation and simple algebraic division; use of the factor theorem.

Understand and use graphs of functions; sketch curves defined by simple equations including

polynomials, y = a/x and y = a/x2 (including their vertical and horizontal asymptotes); interpret

algebraic solution of equations graphically; use intersection points of graphs to solve equations.

Understand and use proportional relationships and their graphs.

Understand the effect of simple transformations on the graph of y = f(x) including sketching asso-

ciated graphs: y = af(x), y = f(x) + a, y =f(x+a) and y=f(ax).

Coordinate

geometry in

the (x,y) plane

Understand and use the equation of a straight line, including the forms y - y1 = m(x-x1) and ax +

by + c = 0; gradient conditions for two straight lines to be parallel or perpendicular.

Be able to use straight line models in a variety of contexts.

Understand and use the coordinate geometry of the circle including using the equation of a circle

in the form (x-a)2+(y-b)2 = r2; completing the square to find the centre and radius of a circle; use

of the following properties: the angle in a semicircle is a right angle, the perpendicular from the

centre to a chord bisects the chord & the radius of a circle at a given point on its circumference

is perpendicular to the tangent to the circle at that point.

Sequences and

series

Understand and use the binomial expansion of (a+bx)n for positive integer n; the notations n! and

nCr; link to binomial probabilities.

Trigonometry

Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and

cosine rules; the area of a triangle in the form (1/2)(a)(b)sinC

Understand and use the sine, cosine and tangent functions; their graphs, symmetries and perio-

dicity.

Understand and use tanθ = sinθ/cosθ

Understand and use (sinθ) 2 + (cosθ)2 = 1;

Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos

and tan and equations involving multiples of the unknown angle.

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Exponentials

and logarithms

Know and use the function ax and its graph, where a is positive.

Know and use the function ex and its graph.

Know that the gradient of ekx is equal to kekx and hence understand why the exponential model is

suitable in many applications.

Know and use the definition of logax as the inverse of a^x, where a is positive and x≥0.

Know and use the function lnx and its graph.

Know and use lnx as the inverse function of e^x.

Understand and use the laws of logarithms: loga(x) + loga(y) = loga(xy), loga(x) - loga(y) = loga(x/y),

kloga(x) = loga(xk) including k = -1 and k = -(1/2).

Solve equations of the form ax=b.

Use logarithmic graphs to estimate parameters in relationships of the form y=axn and y = kbx,

given data for x and y.

Understand and use exponential growth and decay; use in modelling (examples may include the

use of e in continuous compound interest, radioactive decay, drug concentration decay, exponen-

tial growth as a model for population growth); consideration of limitations and refinements of

exponential models.

Differentiation

Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y=f(x) at

a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change;

sketching the gradient function for a given curve; second derivatives; differentiation from first

principles for small positive integer powers of x.

Understand and use the second derivative as the rate of change of gradient;

Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary

points, points of inflection.

Identify where functions are increasing or decreasing.

Integration

Know and use the Fundamental Theorem of Calculus.

Integrate xn (excluding n = -1), and related sums, differences and constant multiples.

Evaluate definite integrals; use a definite integral to find the area under a curve.

Vectors

Use vectors in two dimensions.

Calculate the magnitude and direction of a vector and convert between component form and

magnitude/direction form.

Add vectors diagrammatically and perform the algebraic operations of vector addition and multi-

plication by scalars, and understand their geometrical interpretations.

Understand and use position vectors; calculate the distance between two points represented by

position vectors.

Use vectors to solve problems in pure mathematics and in context, including forces and kinemat-

ics.

Applied Mathematics

Statistical sam-

pling

Understand and use the terms ‘population’ and ‘sample’.

Use samples to make informal inferences about the population.

Understand and use sampling techniques, including simple random sampling and opportunity

sampling.

Select or critique sampling techniques in the context of solving a statistical problem, including

understanding that different samples can lead to different conclusions about the population.

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Data presenta-

tion and inter-

pretation

Interpret diagrams for single-variable data, including understanding that area in a histogram rep-

resents frequency.

Connect to probability distributions.

Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter

diagrams which include distinct sections of the population (calculations involving regression lines

are excluded).

Understand informal interpretation of correlation.

Understand that correlation does not imply causation.

Interpret measures of central tendency and variation, extending to standard deviation.

Be able to calculate standard deviation, including from summary statistics.

Recognise and interpret possible outliers in data sets and statistical diagrams.

Select or critique data presentation techniques in the context of a statistical problem.

Be able to clean data, including dealing with missing data, errors and outliers.

Probability Understand and use mutually exclusive and independent events when calculating probabilities.

Link to discrete and continuous distributions.

Statistical dis-

tributions

Understand and use simple, discrete probability distributions (calculation of mean and variance of

discrete random variables is excluded), including the binomial distribution, as a model; calculate

probabilities using the binomial distribution.

Statistical hy-

pothesis test-

ing

Understand and apply the language of statistical hypothesis testing, developed through a binomi-

al model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail

test, critical value, critical region, acceptance region, p-value.

Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret

the results in context.

Understand that a sample is being used to make an inference about the population and appreci-

ate that the significance level is the probability of incorrectly rejecting the null hypothesis.

Quantities &

units

Understand and use fundamental quantities and units in the S.I. system: length, time, mass.

Understand and use derived quantities and units: velocity, acceleration, force, weight.

Kinematics

Understand and use the language of kinematics: position; displacement; distance travelled; veloci-

ty; speed; acceleration.

Understand, use and interpret graphs in kinematics for motion in a straight line: displacement

against time and interpretation of gradient; velocity against time and interpretation of gradient

and area under the graph.

Understand, use and derive the formulae for constant acceleration for motion in a straight line.

Use calculus in kinematics for motion in a straight line: v=dr/dt , a = dv/dt = d2r/dt2, r=∫v dt , v =

∫a dt.

Forces and

Newton’s laws

Understand the concept of a force; understand and use Newton’s first law.

Understand and use Newton’s second law for motion in a straight line (restricted to forces in two

perpendicular directions or simple cases of forces given as 2-D vectors).

Understand and use weight and motion in a straight line under gravity; gravitational acceleration,

g, and its value in S.I. units to varying degrees of accuracy.

Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a

straight line (restricted to forces in two perpendicular directions or simple cases of forces given as

2-D vectors); application to problems involving smooth pulleys and connected particles.

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Transition Unit

The transition unit will cover all the pure mathematics you study in the first four weeks. At the end of those 4 weeks you will sit a test which will indicate if it is likely you will be successful studying A-Level Mathematics.. Below is a personal learning checklist for the topics within the unit. There is some cross-over from GCSE.

Cardinal Newman Catholic School A-Level Mathematics

A Level Personal Learning Checklist

Transition Unit

Start of Unit End of Unit

Number Skills Red Amber Green Red Amber Green

Understand and use the laws of indices

Use and manipulate surds

Rationalise the denominator of a surd

Quadratic Functions Red Amber Green Red Amber Green

Work with quadratic functions and their graphs

Find the discriminant of a quad-ratic function.

Know the conditions for real and repeated roots

Completing the square

Solving quadratic equations

Simultaneous Equations Red Amber Green Red Amber Green

Solve 2 linear simultaneous equa-tions.

Solve 1 linear and I quadratic sim-ultaneous equation.

Interpret solutions to simul-taneous equations graphical-

ly.

Algebraic Manipulation(polynomials)

Red Amber Green Red Amber Green

Expand up to 3 brackets

Collect like terms

Factorise up to cubic equations

Know and use the factor the-orem

Perform simple algebraic long division

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Straight Line Graphs Red Amber Green Red Amber Green

Be able to use y=mx+c

Be able to use

Know the gradient condi-tions for two straight lines to be parallel or perpendicular

Be able to use straight line models in a variety of con-texts

Cirlces Red Amber Green Red Amber Green

Using the equation of a circle in the form

completing the square to find the centre and radius of a circle

Use the fact that the angle in a semicircle is a right angle

Use the fact that the perpendicular from the centre to a chord bisects the chord

Use the fact that the radius of a circle at a given point on its cir-cumference is perpendicular to the tangent to the circle at that point