ifymb002 mathematics business 2017-18 maths (business) syllabus 2017... · ifymb002 mathematics...

International Foundation Year (IFY)

IFYMB002 Mathematics Business

2017-18

Related Item:

Formula Booklet

© 2017 Northern Consortium UK Ltd. of 30

Amendment History

ReleaseDate

VersionNo.

Summary of Main Changes Author

May 2017 V3.1 Updated to 17/18 Minor formatting amendments Textbooks updated

Academic Team

June 2016 V3.0 Section B Coursework Guidanceamended.

Academic Team

May 2016 V2.1 Dates updated Section 7 updated as NCUK will

now write the IFY assessments.Some centres may set their ownassessments but must receivepermission from NCUK beforedoing so.

Academic Team

June 2015 V2.0 The content of this module has beenreviewed in full (Section 4).

The following changes have beenmade across all IFY modules: Final Examination duration

reduced from 3 hours (plus 10minutes reading time) to 2hours and 30 minutes (plus 10minutes reading time).

Module assessments are nowconsistent across allmodules. This has been done toremove assessment elementsthat contribute little to astudent’s performance in themodule and to reduceadministrative requirements ondelivery partners andNCUK. The change is notintended to reduce studentworkload.

An exemplar teaching has beenintroduced. The purpose of thisis allow judgements to be madeon the ability of the syllabus tobe delivered in the timeavailable and to provideteachers with a suggested, butnot mandated, delivery plan.

The duration of the End ofSemester One Test (EOS1 Test)is 2 hours.

Peter Davies(Module Leader),ProgrammeValidation Panel andAcademic Team.

Please note that the amendments previously introduced to this syllabus are detailedin the version that was released for 2014-15 teaching.

This syllabus is valid for the 2017-18 academic year.

IFYMB002 Maths (Business) Syllabus 2017-18


Contents

1. Module Specification 4

2. Aims 5

3. General Learning Outcomes 6

4. Module Content 7

5. Specific Learning Outcomes 10

6. Teaching and Learning Methods 14

7. Assessment 16

8. Resources 17

9. Core Text and Reading List 18

Appendix A Exemplar Teaching Plan 19

Appendix B Coursework Guidance 29

Appendix C Formulae Booklet 30


1. Module Specification

Module Code IFYMB002

Module Name Mathematics Business

Programme Name International Foundation Year

Percentage breakdown ofCoursework

30%

Percentage breakdown of Exam 70%

Delivery period The syllabus will usually be delivered over two15 week semesters

Recommended minimum teachinghours

4 hours per week

Recommended minimum hours(including independent study hours)

8 hours per week

Related documents NCUK IFY Programme Framework NCUK Academic Handbook



2. Aims

To develop student’s key knowledge, understanding, skills and application ofmathematics in subject-related contexts appropriate for entry to a degree course atany one of the NCUK Partner Universities.

2.1 General Aims

2.1.1 To develop abilities to think logically, to recognise incorrect reasoning and toexpress ideas clearly.

2.1.2 To develop an enthusiasm for the subject and the skills required to apply theknowledge to both the further study and application of mathematics.

2.1.3 To develop in students an understanding of how theory and application worktogether.

2.1.4 To develop students’ skills in modelling and the interpretation of results.

2.1.5 To develop the necessary English mathematics vocabulary and terminology touse their mathematics knowledge effectively in a UK/Western universitycontext.

2.1.6 To acquire the skills needed to use technology such as calculators andcomputers effectively, recognise when such use may be inappropriate and beaware of limitations.

2.1.7 To encourage students towards a level of independence in both the planningand organisation of their studies.

2.1.8 To assist the development of competence and confidence of the students aslearners, taking responsibility for their own learning through directed readingand study.

2.2 Specific Aims

2.2.1 To revise basic skills and develop further skills in algebra.

2.2.2 To demonstrate basic skills in trigonometry and coordinate geometry.

2.2.3 To differentiate and integrate, including the selection and use of appropriaterules and techniques, and the application of the calculus.

2.2.4 To develop concepts in probability and statistics relevant to business planning.

2.2.5 To be confident and competent with the operations of a scientific calculator andits use.

2.2.6 To apply mathematical techniques to simple “real life” problems.

2.2.7 To familiar with, and competent in, the use of computer software to solve pureand applied mathematical problems.



3. General Learning Outcomes

On successful completion of this module, a student will be able to:

Knowledge andunderstanding

Recognise, recall and apply specific mathematical facts,principles and techniques.

Select, organise and present relevant information clearlyand logically.

Select and apply appropriate mathematical andstatistical techniques to solving problems.

Intellectual skills Apply mathematical techniques to problems from avariety of relevant discipline areas.

Present and interpret data in tables, diagrams andgraphs, using generic and specific software packages.

Carry out appropriate calculations using a formulabooklet, a calculator and/or computer software whereappropriate.

Discuss and interpret results obtained, including anestimate of accuracy.

Practical skills Specify what data are required for a given task. Collect relevant data in an effective and efficient way.

Transferable skills Write mathematically-based reports that deliver both acogent argument and a neat and well-organisedpresentation style.

Study independently and make personal notes forproblem-solving and revision purposes.

Source and retrieve information from a variety of originaland derived locations, such as textbooks, the internet,etc.

Select and employ problem-solving skills (description,formulation, solution/analysis, interpretation).

Manage and present data in a variety of formats. Use and apply information technology.



4. Module Content

All topics must be covered. Appendix A provides an Exemplar Teaching Plan whichindicates the proportion of time to be spent on each topic and activities to supportstudent learning.

SEMESTER ONETopic Content

A Linear Equations The equation of a line, parallel and perpendicularlines. Solving pairs of simultaneous equations usingelimination, substitution and graphical methods.

B Simple probability Define probability, use sample space diagrams tohelp calculate probabilities. Combining probabilitiesand using tree diagrams. (Knowledge of conditionalprobability is not expected in this module).

C Quadratic Equations,inequalities and RemainderTheorem

Quadratic Functions: Factorising, completing thesquare and using the quadratic formula.

Remainder Theorem: Simple algebraic division; useof the factor theorem and the remainder theorem.Graphs of quadratic and cubic functions.Geometrical interpretation of algebraic solutions ofequations.

Inequalities: Manipulating inequalities, solving linearand quadratic equations and inequalities.

D Binomial Expansions,Sequences and Series

Binomial expansions: Pascal’s triangle, factorials,binomial expansion (positive integer powers,binomial coefficient notation, evaluation of specificterms)

Sequences and series: Sequences, series, sigmanotation. Finite Arithmetic Progressions (AP) andseries including sum. Geometric Progressions (GP)and series including sum. Convergence anddivergence of geometric series.

E Indices, Exponential andLogarithmic Functions

Laws of Indices for all rational exponents.

Exponential function: Exponential function and itsgraph, introduction to rates of growth, solution ofequations involving exponential functions.

Logarithmic function: Rules and manipulation oflogarithms, logarithmic function and its graph,relationship between exponential/logarithmfunctions, solution of equations involving eitherexponential or logarithmic functions.



F Trigonometric Functions Angles (degree/radian measure). Trigonometricratios, trigonometric functions (sine, cosine, tangent)and their graphs. The identity cos + sin ≡ 1.Solutions of simple trigonometric equations.

G Calculus - differentiation Principles: Gradients of tangents and normals tocurves, limit form, polynomial rules (inc. FirstPrinciples). Derivatives of simple functions(exponential, log, trigonometric. The trigonometricfunctions are sin x, cos x and tan x only.) Use ofFormula Booklet (see Appendix C).

Generic applications: Using derivatives to help sketchcurves. Equations of tangents and normals. Maxima,minima and points of inflexion which are stationarypoints. Use of the second derivative.

H Calculus - integration Principles: Inverse of differentiation, standardintegrals (monomial, trigonometric, exponential),indefinite and definite integration. (The trigonometricfunctions are sin x and cos x only).Area under a curve.

SEMESTER TWOTopic Content

I Introduction to Statistics Data collection: Introduction to sampling andprobability for marketing research andexperimentation. Collection and presentation ofstatistical data. Histograms and the cumulativefrequency polygon and curve.

Data summaries: Mode, median and mean.Standard deviation. Quartiles and interquartilerange.

J Further Probability and SetTheory

Further Probability: Mutually exclusive events andindependent events. Laws of Probability. ConditionalProbability.

Set Theory: Sets, intersections, unions,complements. Venn diagrams, including their use tosolve probability problems.

K Correlation, LinearRegression and Time Series

Correlation: Scatter graphs. Calculation andinterpretation of the coefficient of correlation.

Linear Regression: Calculation of the equation of aleast-squares linear regression line.

Time Series: Trend-line, moving averages.

L Probability Distributions Discrete random variables: Probability distributionsgiven algebraically or in tables. Calculate the meanE( ) and the variance Var( ).



Distributions: Binomial distribution. Normaldistribution and confidence intervals.

M Financial Mathematics Percentage and percentage change. Interest.Appreciation and Depreciation.

N Further Differentiation Rules: Sum, product , quotient rules and the chainrule (composite functions)

Implicit differentiation.

O Further Integration Integration by substitution. Change of variable. Useof Formula Booklet (see Appendix C).

Partial fractions (linear factors, repeated linearfactors, improper fractions), integration by partialfractions.

Integration by parts.



5. Specific Learning Outcomes

On successful completion of the module, a student should be able to:

A Linear EquationsA1 Find the equation of a straight line using coordinate geometry.A2 Find parallel and perpendicular lines and sketch appropriate graphs.A3 Solve pairs of simultaneous equations using elimination, substitution and

graphical methods.B Simple probabilityB1 Find the probability of a single event.B2 Recognise that ( ) and ( ) mean the probabilities of event occurring and

event not occurring respectively.B3 Find, for two events and , the probabilities of both and occurring, and

the probabilities of either or occurring. (the use of the symbols ∩ and ∪will not be expected in this module).

B4 Construct and use a simple tree diagram.C Quadratic Equations, inequalities and Remainder TheoremC1 Carry out the process of completing the square to locate vertices (turning

points) of graphs.C2 Use the discriminant to determine the number of real roots.C3 Use surds to give exact solutions.C4 Use algebraic division by a monomial or quadratic function.C5 Sketch the graphs of quadratic and cubic functions.C6 Use the Remainder Theorem to determine the remainder when a polynomial

is divided by ( + ).C7 Solve by substitution a linear and quadratic pair of simultaneous

Equations: plot the functions using graph paper.C8 Recognise and solve linear/quadratic equalities and inequalities.C9 Use algebraic and graphical methods to solve inequalities.C10 Recognise and distinguish between open and closed intervals.D Binomial Expansions, Sequences and SeriesD1 Expand (1 + ) for small positive integer .D2 Use Pascal’s triangle to find binomial coefficients.D3 Expand ( + ) where is a small positive integer.D4 Understand idea of sequence of terms using general formulae and

Recurrence relations.D5 Use sigma notation for series representations.D6 Recognise and sum a finite arithmetic series (AP).D7 Recognise and sum a geometric series (GP).D8 Define, explain and test for convergence of a series.D9 Use an AP or a GP to solve certain practical problems.E Indices, Exponential and Logarithmic FunctionsE1 Know the equivalences e.g. × ≡ and ÷ ≡ .E2 Use a calculator to evaluate exponential and logarithmic expressions.E3 Sketch the graphs of = and = .E4 Apply exponential functions to problems.E5 Apply the rules of logarithms to problems.



E6 Change the base of a logarithm.E7 Solve equations involving exponential and logarithmic functions.F Trigonometric FunctionsF1 Convert from radians to degrees and vice versa.F2 Find sin, cos, and tan of any angle and plot their graphs.F3 Know the area of a triangle formula sin .

F4 Calculate inverse trigonometric functions.F5 Find particular solutions of simple trigonometric equations. (these equations

will take the form: sin = ; cos = ; tan = over any range).F6 Apply the sine and cosine rules to an arbitrary triangle.G Calculus - differentiationG1 Evaluate the gradient of a curve at a point.G2 Recognise and explain the notation and ( ).G3 Sketch the derivative graph .

G4 Apply the limit formula to simple functions (first principles). (this will beconfined to single integral powers of .)

G5 Use formula booklet to obtain derivatives of standard functions,Including where is a constant.

G6 Explain second-derivative notation.G7 Apply second derivatives to practical problems.G8 Find stationary points for a given function.G9 Distinguish between local maximum, local minimum and point of

Inflexion which are stationary points.G10 Apply G8 to practical optimisation problems.G11 Obtain the equation of tangent and normal of a curve at a specified point.H Calculus - integrationH1 Identify integration as the inverse of differentiation.H2 Use formula booklet to determine indefinite integrals including where

and are constants.H3 Form and explain the definite integral.H4 Evaluate definite integrals.H5 Calculate the area between a curve and the x -axis, including areas

Partly above and partly below the axis.I Introduction to StatisticsI1 Distinguish between continuous and discrete data.I2 Construct frequency distributions.I3 Draw a line graph using discrete data, a histogram using continuous data.I4 Evaluate mode, median and mean.I5 Understand what “standard deviation” means.I6 Evaluate the standard deviation (divisor ).I7 Evaluate the mean and standard deviation of data in a frequency distribution.I8 Know how to display and interpret a cumulative frequency distribution.I9 Use a cumulative frequency graph to estimate the median, the quartiles and

the interquartile range of a set of data.I10 Identify a distribution which appears to be skewed.J Further Probability and Set TheoryJ1 Explain set definitions.



J2 Compose two sets by union or intersection.J3 Illustrate sets by using Venn diagrams.J4 Use the laws of probabilityJ5 Distinguish between mutually exclusive and independent events.J6 Compute conditional probabilities.J7 Illustrate probabilities using Venn diagrams and more complicated tree

diagrams.J8 Calculate the probabilities of combined events (the understanding of set

notation will be expected).K Correlation, Linear Regression and Time SeriesK1 Explain the term “correlation” in relation to data sets where both variables

must be random.K2 Compute correlation coefficient and evaluate result in relation to appearance

of the scatter graph and with reference to values close to -1, 0 and 1.K3 Explain the principle of least-squares approximation.K4 Compute the equation of a least-squares linear regression line for random

variable and non-random variable .K5 Be able to interpret in context the uncertainties of estimating a valueK6 Construct time series chartsK7 Calculate moving averages of a time series.K8 Construct the trend-line of a time series.K9 Extrapolate a trend line but be aware of the dangers. (calculating seasonal

variations and residuals will not be expected.)L Probability DistributionsL1 Recognise and describe the nature of a “distribution”.L2 Construct a probability distribution relating to a given situation involving a

discrete random variable .L3 Evaluate the expected value and variance of a linear function of a random

variable.L4 Distinguish between discrete and continuous distributionsL5 Describe the binomial distributionL6 Perform calculations with the binomial distributionL7 Use the binomial distribution tables.L8 Use of standardised Normal Distribution tableL9 Convert general data into the standardised form.L10 Set up a confidence interval for a mean where the background distribution of

any samples will be Normal with a known standard deviation.M Financial MathematicsM1 Carry out any calculation involving direct percentage, reverse percentage,

percentage error and percentage change.M2 Perform calculations involving simple and compound interest.M3 Calculate appreciation and depreciation using knowledge of a geometric

progression gained in D7.M4 Estimate when a certain value is reached in an appreciation or depreciation

situation using knowledge of logarithms gained in E5.N Further DifferentiationN1 Apply the product rule, quotient rule and chain rule.N2 Find of an implicit function.



N3 Know the result = Ln .

O Further IntegrationO1 Integrate standard functions such as , , ( )( ) .O2 Clarify what is meant by the term ‘partial fraction’.O3 Find partial fractions for linear and repeated linear factors.O4 Find partial fractions for improper fractions.O5 Use substitution to evaluate indefinite and definite integrals.O6 Apply integration by substitution to practical problems.O7 Carry out integration by parts.



6. Teaching and Learning Methods

Teachers should use a range of different learning and delivery styles in order to givestudents experience of the types of approach they will encounter in a Western universitye.g. lectures and tutorials. Appendix A (Exemplar Teaching Plan) suggests a deliveryformat designed to facilitate the delivery of each topic. The centre is at liberty to divergefrom this Plan. Also, the centre may increase the minimum number of teaching hours(4 hours per week) to meet the needs and abilities of students.

The range of teaching and learning activities should be employed in lectures, tutorials,laboratories and directed self-study. The use of video clips and internet-based activityis to be encouraged where it might lead to enhanced learning (much higher-quality CALsoftware is now available).

A standard Formula Booklet (see Appendix C) should be used throughout the year, andeach student should be issued with a copy of this booklet at the start of the year.Students will require access to computers with both MS Excel and some mode ofelectronic/automated calculation platform installed.

A primary aim of the module is to develop in students an understanding of how theoryand application work together, and it is emphasised that teachers should develop andillustrate applications of relevant mathematics in order to encourage thisdevelopment in business and management contexts.

Lectures will be used to transmit, explain and demonstrate much of the factual material,and to develop and illustrate problem-solving methods. These sessions can beaugmented as appropriate with handout material, demonstration experiments and theuse of visual aids. These materials could be designed in collaboration with an EAPteacher.

It is important for the development of students’ English language and study skills thatthe delivery of the subject material is integrated with EAP, EAPPU or RCS. Regularcommunication between the subject module teacher(s) and the EAP, EAPPU or RCSteacher(s) will provide a basis on which to support and guide students. Students willbenefit from collaborative activities, where the subject module and EAP/EAPPU/RCSteachers jointly deliver classes in relation to activities such as essay writing style andusing academic sources.

As part of study for the EAP, EAPPU or RCS module, students will learn the Harvardreferencing system. Subject teachers will ensure that students carry these learning intothe work produced for this module; see Section 9 of the syllabus for details of thereferencing guide recommended by NCUK. For further information about referencingand citation, please consult the EAP or RCS syllabus (as relevant) for the texts andonline resources recommended by NCUK.

Students will have different backgrounds in the subject and it will be necessary to giveopportunities for directed self-study, and so allow each student to develop at their own



pace to reach the required level for the examinations. Activities (homework) for self-study should be set weekly.

Tutorials should not be managed in the same way as lectures. Tutorials should involveboth group and individual activities, with a strong emphasis on applying knowledge fromlectures and reading to problem-solving. It is important for all students to haveopportunities to speak in English during each tutorial. Suggested activities includestudents being encouraged to explain in English their answers jointly in pairs or smallgroups, students providing answers to the whole class whilst standing at the front ofthe group, and group activities that require discussion. These classes should be usedto verify that students are capable of using a scientific calculator correctly. Tutorialsmay be used to discuss practical applications of mathematics, particularly with respectto the content of the Business courses, including how to handle and analyse data.Teaching staff are advised to prepare examples for this purpose. The sessions can alsobe used for individual counselling of students and to assess student understanding ofthe subject.

Students should use Microsoft Office software in the analysis of data and preparationand presentation of reports.



7. Assessment

FormativeIt is important that students are given the opportunity to engage in and submitformative coursework assessments and receive feedback on this work. Formativeassessment should be designed to inform students of their progress and enable themto develop and practice coursework and examination skills.

SummativeSummative assessments contribute to the student’s final grade for the module. Thesummative assessment structure for the module is as follows:

COURSEWORK30%

FINAL EXAM70%

Semester 1Coursework

110%

Semester 2Coursework 2

10%

End of Semester 1Test10%

End of Module Exam(Set by NCUK)

Level 1(A – H)

Level 2(I – O)

Length of Test: 2hours plus 10 minutesreading time

Length of Exam: 2 hours 30mins plus 10 minutes readingtime

Section A: Compulsory40 marks

Section B: Choose 4out of 6 questions60 marks

Section A: Compulsory45 marks

Section B: Choose 4 out of 6questions80 marks

100 marks 100 marks 100 marks 100 marks

Unless the centre has been given permission by NCUK to write its own summativecoursework assessments, NCUK will produce all summative coursework assessments forthe module in accordance with the task rubric information presented in AppendixB. Where the centre has received permission from NCUK to write the summativecoursework, it will do this in accordance with the information and guidance given inAppendix B and the regulations set out in the NCUK Academic Handbook, IFYCoursework Writing and NCUK Approval.

The examination paper will be provided by NCUK and it will contribute a maximum of70% to the final module grade. The paper will cover a broad range of the specificlearning outcomes.

It is essential that coursework and examinations are administered in accordance withNCUK regulations. Please refer to the following sections of the NCUK AcademicHandbook for details:

Coursework Administration and Regulations

Centre Marking and Recording Results

Academic Misconduct Policy



Examination Administration

Requirement for the End of Semester 1 Test and Final Examination

It is the centre’s responsibility to provide the following materials for the end of semester1 test and final examinations:

Calculator (refer to NCUK policy ‘Calculator Regulations’) Graph Paper Formulae Booklet ‘Data, Formulae and Relationships’ (refer to Appendix C of this

syllabus)

8. Resources

Microsoft Excel (or similar) processing package.



9. Core Text and Reading List

Please note that while resources are checked at the time of publication, materials maybe withdrawn from circulation and website locations may change at any time.

Core Text Rayner, D. & Williams, P. (2004). Pure Mathematics C1 C2.Elmwood PressISBN: 9781902214450

Rayner, D. & Williams, P. (2005). Pure Mathematics C3 C4.Elmwood PressISBN: 9781902214467

The above texts cover the common core (C1-C4) inmathematics.

Pledger, K. et al (2009). AS and A Level Modular Mathematics-Statistics 1. Pearson EducationISBN: 9780435519308

Further Reading Emanuel, R. & Wood, J. (2005). Advanced Mathematics AS Corefor Edexcel. LongmanISBN 0582842379This text book also includes a self-study CDISBN: 9780582842373

Useful Websites http://www.revision-notes.co.uk/A_Level/Maths/

http://maxima.sourceforge.net for a copy of the mathematicalprogram Maxima

http://www.geogebra.org for a copy of the mathematicalprogram GeoGebra

RecommendedReferencingGuide

Pears, R. and Shields, G. (2013). Cite them right: the essentialreferencing guide. 9th edn. Basingstoke: Palgrave Macmillan.ISBN: 9781137273116.There are online guides based on this publication. Centres mayconsider subscribing to the online version athttp://www.citethemrightonline.com. An older 2005 edition ofthe book can be downloaded (free) fromhttp://www.angelfire.com/planet/ibm2007/cite_them_right.pdfAdditional publications and online resources are listed in theEAP and RCS syllabuses



Appendix A Exemplar Teaching Plan

Method of delivery notation:L – by lectures; E – by doing exercises; T – use of tutorials; O – other (will bespecified)

Week Hour Topic + Lesson/activity/teacher guidance Further Notes1 1

1

1

1

A. Linear Equations. Identify the gradient andintercept in the equation of a straight line; calculategradients of the normal. (L/E)

Find equations of other lines which are parallel to orperpendicular to the equation of a particular line.Solve pairs of linear simultaneous equations usingelimination, substitution and graphical methods.(L/E)

Continue with simultaneous equations. Carry outpractice examples on linear equations. (Mostly E)

B. Simple Probability. Evaluate the probability ofa single event. Familiarity with the notation ( )and ( ) and realise that ( ) = − ( ). (L/E/T)

There may beconsiderablevariation in thealgebraic ability ofstudents.

Any practice in thisarea will probablybe good forindependent anddirected study.

2 1

1

1

1

Simple Probability. For two events and , workout the probability of both happening, and of eitherhappening. (Set notation is not required at thislevel – neither will candidates be expected to beaware of mutual exclusivity). Construct a treediagram and use it to work out combinedprobabilities. (Mostly L)

C. Quadratic Functions and Equations. Carryout a completing the square process and be able tosketch the graph of a quadratic function. (L/E)

Solve quadratic equations by factorising,completing the square and using the quadraticformula [if time runs short, there is scope forcontinuing the process of solving by factorisation inthe following week]. Candidates must be able topresent answers in surd form and understand thesignificance of the discriminant (but will not beexpected to evaluate the size of coefficients in theoriginal equation which give, for example, two realroots). (L/E)

Continuation of previous session. (T & mostly E)

Handling algebra isoften a weak spot,so what was saidabout independentand directed studyin the previousweek probablyapplies here.

e.g. questionsasking to find rangeof values of in+ + − =to give two realroots will not beset.



3 1

1

1

1

C. Further simultaneous equations. Solve bysubstitution two equations – one of which isquadratic and the other linear. (E & mostly L)

C. Inequalities. Solve linear and quadraticinequalities, either by using algebra or graphicalmethods. Recognise open and closed intervals andwhat an integer is. (L/E)

C. Remainder Theorem. Use the RemainderTheorem to show a given monomial is a factor of apolynomial, or to find its remainder upon division ifit is not a factor. Candidates will be expected todivide a polynomial by a monomial or quadraticexpression, and to be able to factorise anexpression completely. (L/E)

Continuation of previous session. (Mostly E)

Candidates shouldbe aware of theterm ‘factortheorem’ and relateits connection withthe remaindertheorem.

4 1

1

1

1

D. Binomial Expansions. Revise expansion ofquadratic functions. Use Pascal’s Triangle to findbinomial coefficients, and be able to expandcompletely ( + ) and ( + ) where is a smallpositive integer (normally not more than 5). (L/E)


Consolidation of previous two sessions. Candidateswill be expected to pick out single terms in theexpansion of ( + ) where is much larger but stillan integer. Candidates must also know what acoefficient is. (L/E/T)

D. Progressions. Start on progressions andintroduce the difference between an ArithmeticProgression (AP) and a Geometric Progression (GP).Identify the first term and common difference in anAP. (E & mostly L)

Candidates couldbe asked to writedown the first fewterms in ascendingpowers of of anexpansion with ≫. The meaning of‘ascending’ or‘descending’ willalways be given.

5 1

1

D. Progressions. Use the relevant formulae tofind the term and the sum of the first termsof an AP and a GP. (L/E)

Candidates should be able to find the commondifference, common ratio and first term having beengiven the term or sum of the first terms. (L &mostly E)

Candidates couldbe asked to find thefirst term of a GP,or how many termsare needed forgeometric series, toexceed a certainvalue. This shouldonly be done oncelogs have beencovered.



1

1

Use sigma notation and be able to generate aprogression from a sigma expression. (L/E)

Understand the idea of convergence and theconditions needed. Find the sum to infinity of aconvergent series. (L/E)

6 1

1

1

1

E. Indices. Know that × ≡ and ÷≡ ; and that ( ) ≡ . Use fractionaland negative indices. (L/E)

Find the exact solutions of equations which useindices. (L & mostly E)

E. Exponential Functions. Understand what anexponential is and be able to sketch the graphs of= and = . (Mostly L)

Relate exponential change to a real-situation e.g.population growth and decomposition of a solid intoa liquid or a gas. (L/E/T)

Investigatingexponential growthand decay couldprovide goodmaterial forindependent anddirected study.

7 1

1

1

1

E. Logarithmic Functions. Understand how alogarithm behaves and compare it to real-lifesituations such as the Richter and pH scales.(Mostly L)

Establish the connection between logarithms andexponentials. (L/E)

Use the logarithmic laws and apply them to solveequations and simplify expressions. (L & mostly E)

Consolidation of previous work and practice in theuse of logarithms. (E/T)

Practice at usingand manipulatinglogs would beinvaluable timespent forindependent anddirected study.

8 1

1

F. Trigonometric Functions. Understand thedefinition of a radian; convert angles from degreesto radians and vice versa. Calculate the sin, cos andtan of any angle and be familiar with their graphs.(L/E)

Know how to find the inverse of a trigonometricexpression and solve simple trigonometricequations (These equations will take the form:= ; = ; = over any rangewhere can be positive or negative.) Candidateswill also need to be able to quote and recognise theexact values of the trigonometric functions of 0, 30,



1

1

45, 60 and 90 degrees and their radian equivalents.(L/E)

Consolidation of previous two sessions and practiceat examples – particularly solving the equations andidentifying all the angles in a given interval. (E/T)

Apply the sine and cosine formulae to a non right-angled triangle and use the formula for the area ofa triangle . Candidates should be familiar

with the identity + ≡ . (L/T)

It is probably a caseof the more timethat students spendat practisingexamples inindependent anddirected study, thebetter

9 1

1

1

1

G. Differentiation. Revision of finding thegradient of a straight line; understand the definitionof the gradient of a curve at a point and, using firstprinciples, find an expression for this gradient(using single integral powers of only). (Mostly L)

Using first principles, find gradient functions ofother small powers of . Introduce the idea ofdifferentiating any power of . (Mostly L)

Use and ( ) and be able to differentiate any

polynomial and any power of . (L/E)

Practice at differentiating polynomials andsubstituting in values of . (Mostly E)

A re-visit to thebinomialexpansions sectionmay be useful herefor

10 1

1

1

1

G. Differentiation. Extend differentiation to findthe equations of a tangent and a normal to a curveat a specified point. (L/E)

Extend the process to differentiation of exponentials(expressions of the form = ) and logarithms.(L/E)

Differentiation of trigonometric expressions (sin ,cos and tan only). Use of the formula booklet toobtain derivatives of standard functions. (L/E)

Consolidation of the previous two weeks’ work withplenty of practice at differentiation. (E/T)

Plenty of scopehere for practice inindependent anddirected study.

11 1

1

G. Differentiation. Find the second derivative of

a function and be familiar with the notation and( ). (L/E)

Find the stationary points of a function. (L/E)



1

1

Determine the nature of any stationary point. (L/E)

Sketch the graph of a function once the turningpoints are known. Sketch the graph of a derivative.(Mostly E)

Points of inflexionwill be identifiedonly in instances ofzero gradient.

12 1

1

1

1

G. Differentiation. Extend the process of findingstationary values to practical optimisationproblems. Typical cases could be to find themaximum volume of a solid which has a fixedsurface area, and to find the minimum surface areaof a solid with a fixed volume. (Mostly L)

Extension of previous session. (L/E)

More practice at optimisation problems and start ofa general consolidation of differentiation. (Mostly E)

Completion of the consolidation started theprevious session. (E/T)

Other examplescould bemaximising an areaof a shape withfixed perimeter.

In view of the sizeof this topic, a longconsolidation willprobably benecessary.

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H. Integration. Identify integration as the inverseof differentiation. Integrate any power of . (MostlyL)

Integrate where is a constant; integratesin and cos . (L/E)

Realise the integration of = ln + . Use the

formula booklet to determine indefinite integralsincluding integrals of the form where and

are constants. (L/E)

Find the indefinite integral of any of the functionsabove. (Mostly E)

Practice atintegration wouldprobably be timeusefully spent inindependent anddirected study.

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H. Integration. Evaluate definite integrals. (L/E)

Apply definite integrals to finding the area betweenthe curve and the − axis. (L/E)

Find more difficult areas, including those below the−axis. (L & mostly E)

Extend integration to finding an area which is boundby two or more curves. (E/T)

Students shouldrealise that areasbelow the − axiswill give negativevalues andappreciate themeaning of anintegral giving zero.

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End of Semester Test

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I. Introduction to Statistics. Distinguishbetween, and give examples of, continuous anddiscrete data. Evaluate mean, mode and median.(L/E)

Calculate a standard deviation and understand whatit means. Construct a frequency table and use it tofind the mean and standard deviation. (L/E)

Use grouped frequencies to estimate the mean andstandard deviation. (L/E)

Draw a line graph to illustrate discrete data and ahistogram to illustrate continuous data. (L/E)

Students shouldtake care to labelaxes and will beexpected to scalethem sensibly.

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I. Introduction to Statistics. Evaluate cumulativefrequency and construct a cumulative frequencygraph. (L/E)

Use the graph to estimate the median, quartiles andinterquartile range. Identify possible skew in adistribution, giving a reason. (L & mostly E)

J. Further Probability and Set Theory. Befamiliar with set notation (knowledge of thefollowing symbols will be expected: ∩, ∪, ∅, ∈ ′).(L/E)

Construct, and interpret, Venn diagrams. Use themto find probabilities. (Mostly E)

If students arequestioned aboutskew, they will notbe asked to identifyif it is positive ornegative.

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J. Further Probability and Set Theory.Construct, and interpret, more complicated treediagrams. (L/E)

Combine probabilities using tree diagrams and Venndiagrams. (L/E)

Use the laws of probability (these are in the formulabooklet). (L/E)

Distinguish between independent and mutuallyexclusive events. Calculate conditionalprobabilities. (L/E/T)

Tree diagrams willnot have more thanthree branchesfrom any givenpoint.Candidates willneed to be aware of( ∩ ) = formutually exclusiveevents and ( ∩) = ( )× ( ) forindependentevents.



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K. Correlation. Draw a scatter graph and the lineof best fit. (L/E)

Obtain some idea of sign and strength ofcorrelation. (L/E)

Calculate the correlation coefficient and relate itsvalue to the strength and type of correlation, withparticular reference to values close to -1, 0 and 1.Appreciation that correlation does not implycausation. (L & mostly E)

Continuation of previous session and furtherpractice at calculating and interpreting correlation.(Mostly E)

Candidates will nothave to carry outsignificance tests.

Unless there arevery few pairs ofreadings, the datawill normally bepresented insummary form inan examination. Ina coursework taskstudents may havemore readings butwill be expected touse Excel or othersimilar package.

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K. Linear Regression. Explain the principle ofleast-squares approximation, and compute theequation of a least-squares linear regression line forrandom variable and non-random variable .(L/E)

Use the equation to estimate values but be awareof the reliability of estimates. The equation willnormally be in the form = + and candidateswill be expected to know what and represent,and to recognise if their values make sense. (L/E)

K. Time Series. Calculate the moving averages ofa time series. (L/E)

Construct a time series chart and draw a trend-line.Candidates may be asked to extrapolate the trendline for a short distance but may also be asked toexplain why extrapolation cannot be relied on.(Calculating seasonal variations and residuals willnot be expected.) (L/E)

The above alsoapplies concerningthe numbers ofpairs of readings.Students may haveto draw theequation on ascatter graph.Questions askingfor an equation ofon in the form =+ will not beset.

Any number ofpoints could beasked, but they willnormally be 3 or 4point.

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L. Probability Distributions. Understand what adistribution is and be able to recognise and describeone. Distinguish between discrete and continuousdistributions. (Mostly L)

Construct a probability distribution relating to agiven situation involving a discrete random variable. Evaluate the expected value of a randomvariable. Appreciate the connection betweenexpected value and mean. (L/E)

Students canusefully use



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Evaluate the variance and standard deviation. Findthe expected value and variance for expressions like− . (L/E)

Describe the binomial distribution and performsimple calculations involving small values of .(L/E)

independent anddirected study timepractising typicalexaminationquestions on thistopic.

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L. Probability Distributions. Performcalculations using the binomial distribution andlarger values of . Use the cumulative binomialdistribution tables. Appreciate importantassumptions when using the binomial distribution.(L/E)

Describe the Normal distribution and use thestandardised Normal distribution table. Convertdata into standardised form. Be aware of thelimitations of the Normal distribution (e.g. theassumption that there is no restriction on the valuestaken by but this is not usually the case in reality).(Mostly L and some E)


Set up a confidence interval (any samples will betaken from a background distribution which isassumed to be Normal with a known standarddeviation.) Explain what is meant by a confidenceinterval. (L/E/T)

Students canusefully spend timein independent anddirected studypractising examplesin the binomial andNormaldistributions.

Students should beaware of theCentral LimitTheorem but willnot have to apply itin an examination.

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M. Financial Mathematics. Be thoroughlyfamiliar with all types of percentage. Calculate anydirect and reverse percentage, and evaluate anypercentage change and error. (L/E/T)

Identify what type of percentage is required in amiscellany of examples. (L/E/T)

Appreciate what interest on an investment is.Calculate simple interest and compound interest.(L/T)

Compare two types of investment to see whichgives the better interest. (Mostly E)

Students can usethe compoundinterest formulaand may need todraw on theirknowledge of logs.



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M. Financial Mathematics. Consolidate onprevious week and extend the ideas to finding, forexample, the total interest as a percentage of whatwas originally invested. (E/T)

Calculate appreciation. (L/E)

Calculate depreciation. In both this session and theprevious one, students will be expected to work outthe value of an article after any number of years byapplying the idea of a geometric progression(specific learning outcome D7). They will also beexpected to work out the expected time that acertain value is reached by applying logarithms(specific learning outcome E5). (L/E)

Consolidate on appreciation and depreciation. Atask could ask students to investigate if the benefitsof appreciation in one part of a project outweigh thedrawbacks of depreciation in another part of thesame project. (E/T)

Examples may bethe price ofproperty or a car.

There must be anawareness of whyMathematicalmodels may not bereliable in the longrun (e.g. interestrates may change).

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N. Further Differentiation. Know that thedifferentiation of . Apply the productrule. (L/E)

Apply the quotient rule (L/E)

Apply the chain rule. (L/E)

Practise examples using a mixture of the aboverules. (L/T)

Candidates may beasked to derivecertain results (e.g.showing that thederivative of secis sec tan .)Practice atdifferentiation inindependent anddirected study timewill be invaluable.

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N. Further Differentiation. Understand what animplicit function is and how to apply differentiation.(Mostly L)

Find of an implicit function which could involve

use of the product rule. (L/E)

Find the equations of a tangent and a normal at apoint. (L/E)

Find stationary values. (L/E)

In this module,candidates will beexpected to drawon concepts learntin module G.

Students will nothave to find thenature of turningpoints.

Again, time spentpractising inindependent anddirected study willbe invaluable.

27 1 O. Further Integration. Integrate standard

functions such as , , ( )( ) . (L/E)



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Resolve expressions into partial fractions. (L/E)

Resolve improper fractions into partial fractions.(L/E)

Integrate expressions which have been resolvedinto partial fractions. (L/E)

Expressions willhave linear andrepeated factorsonly, with noquadratic factors.

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O. Further Integration. Use substitution toevaluate indefinite and definite integrals. (L/E)

Use integration by parts. (L/E)

Consolidation of integration by substitution and byparts. (Mostly E)

Practice at recognising and performing any integral.(E/T)

The substitution willnormally be given.

Integration by partswill need to beapplied not morethan twice.

With regard toindependent anddirected study, themore practice thebetter.

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30 Final Exam Week



Appendix B Coursework Guidance

Continuous assessment makes up 30% of the final grade and tasks are set locally bythe teacher(s) delivering the module. Assessments should comprise substantive tasksrequiring students to demonstrate different skills and knowledge identified in themodule learning outcomes.

Students should complete three assignments, which must be undertaken in advance ofthe end of module examination.

These assignments are:

A. Two pieces of coursework.One of these is based on Maths Semester 1 (modules A – H) and the other onSemester 2 (modules I - O).

The coursework is made of three tasks. Each should be based on an applicationexercise to investigate a substantial multi-part problem. The project reportsshould include narrative and models/calculations. An outline structure of thecoursework reports should include:

Introduction and background (approximately 100 words) Description of how the task was carried out, outlining any methods used and

any assumptions made. Results and findings, example relevant calculations/solutions, charts and tables Case study and/or validation with narrative to describe the study. Summary and conclusions. Approximately 100 words which should include any

limitations to any relevant results. Suggestions for further investigation orimprovements to the work.

Summary including key points and conclusion, any limitation and suggestion forfurther investigation or improvements.

Each piece of coursework will contribute 10% towards the final module grade.

Students should be given 2 weeks to complete the coursework task.

Students will be expected to make use of ICT in at least one of the courseworktasks.

The first piece of coursework will normally be carried out in the second half ofSemester 1 and the second piece during the second half of Semester 2.

B. End of Semester 1 TestHeld in class during the 15th week. The test will be two hours long (with ten minutes’reading time) and will contribute a maximum of 10% towards the final module grade.

Section A is compulsory and carries a total of 40 marks.Section B carries 60 marks and students are required to choose 4 out of 6questions. The total for this paper is 100 marks.



Appendix C Formulae Booklet

Refer to separate formula booklet.

It is the centre’s responsibility to print “clean” (new) copies of the Formula Booklet forthe end-of-semester 1 test and final examination. (Refer to Section 7 Assessment)

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