A Foundation in Pure Mathematics

Home Admissions Course Guide A Foundation in Pure Mathematics

This Course Guide has been taken from the most recent presentation of the course. It would be useful for reference purposes but please note that there may be updates for the following presentation.


A Foundation in Pure Mathematics

Welcome to MATH S121!

MATH S121 is one of the two mathematics foundation courses (along with MATH S122) designed for students beginning their studies with HKMU. As such it is structured so as to help new students develop the learning skills required for the distance learner. Studying in this type of education environment is challenging; however, I and my team of tutors are here to help you meet this challenge. By working together I am confident that you will be able to successfully complete the course and that you will enjoy the mathematics covered in MATH S121.


1.1 The purpose of this Course Guide

This booklet will help you become familiar with the overall shape and structure of MATH S121. You will be introduced to how the course is run, the different components of the course, what you have to do to pass the course, and what resources and help are available. Your first task in this course is to read through this Course Guide very carefully.


1.2 Course aims

This course aims to:

  • Provide you with experience of distance learning.
  • Develop your analytical skills to help you become an independent learner.
  • Lead you towards an appreciation of mathematics as a discipline.
  • Provide a transition between school and university-level mathematics and the associated learning environments.
  • Develop your skills in handling mathematics.
  • Develop your knowledge of mathematics in preparation for other courses in mathematics and computing.
  • Improve your confidence in studying in English.

1.3 Learning outcomes

When you have successfully completed this course, you should be able to:

  • Use correct notations to express mathematical ideas and processes.
  • Carry out logical and mathematical investigations using Boolean algebra and set language.
  • Explain the main features of relations, functions, and their applications.
  • Discuss the basic concepts and skills in combinatorial and recursive/iterative thinking.
  • Use counting principles to calculate simple probabilities.
  • Perform the basic operations of matrix algebra.

When studying any subject there will always be some ideas and skills that the authors will assume that you have. For example, in a mathematics course the most obvious skills to be assumed relate to basic arithmetic like addition and division. For MATH S121 the assumption is that you will have already experienced mathematics at school up to about Form 5 level.

It may have been some time since you used these skills and so we have provided you with some material to help you check on your level of understanding of these topics and to give you some practice to help you refresh your memory. In addition, we have provided some revision material to help you with any topics that you have forgotten, or perhaps you did not study at school.


2.1 Diagnostic Quiz

In this first package of course material you will find a booklet — Diagnostic Quiz. This contains a range of basic mathematics questions covering the kinds of topics assumed when writing the course material. You are strongly advised to work your way through all of these questions at the start of the course. Solutions are provided for you to check your answers.

If you come across a question that you find particularly difficult, then

  1. Check the Preparatory Unit to see whether it contains any related revision material.
  2. Contact your tutor to ask for help.

Note that you are not required to submit your answers to this quiz.


2.2 Preparatory Unit

This booklet introduces some basic strategies for studying MATH S121 and a review of some of the mathematics covered in the Diagnostic Quiz. You may find this section particularly useful if you have any problem with specific questions in the Quiz. Finally, the Preparatory Unit includes a hands-on introduction to Scientific Notebook — a software package that you will be using throughout the course (see below).

It is important to make sure that you are properly equipped for the challenges that lie ahead. Time and effort spent in preparing yourself at the start of the course will speed up your progress through the more difficult parts of the course.

Even if you already have experience with HKMU courses you may find that the approach adopted for this course is different because:

  • the Online Learning Environment (OLE) is given as a study and support tool; and
  • mathematical software is integrated into the course.

It is therefore very important that you set aside time to work carefully through the Preparatory Unit. This will ensure that you have the necessary familiarity with using Scientific Notebook, and you'll have many opportunities to identify your own readiness for study.


2.3 Calculator

You will need a calculator for this course. You should buy a calculator that has been approved for use in the examination. The list of approved calculators is available on the OLE — the University has strict rules about which type of calculators are permitted in the examination.

2.4 Computer

You will need access to a computer capable of running the mathematics software — Scientific Notebook — and connecting to the Internet.

To run Scientific Notebook you need a PC with 64 MB of RAM, and at least 250 MB of available hard disk space. Your system must run Windows 7 or higher.

You will need access to a computer throughout the course. If you have problems accessing a computer, then you can come to one of the University's PC labs.

3.1 The study units

The study units are the core course materials. They are grouped into five blocks. You are expected to complete one unit over a two week period including the associated assessment questions. You should refer to the course schedule on the OLE for details as to when each unit is to be studied.

Each block is accompanied by a Problem Booklet for extra practice. The blocks are:


Block I - Foundations in discrete mathematics

Unit 1 Sets

  • Basic properties of sets
  • Subsets
  • Operations on sets
  • Set identities

Unit 2 Logic

  • Introduction to p-and-q game
  • Developing an algebra for p-and-q game
  • Logical operations
  • Equivalence of logical statements
  • Quantifiers

Unit 3 Relations

  • Properties of relations
  • Equivalence relations
  • Partial orders
  • Operations in relations

Block II - Functions

Unit 1 Introduction to functions

  • Domain, codomain, image
  • Graphs of functions
  • Linear, quadratic, exponential function
  • Trigonometric function
  • Flooring and ceiling functions

Unit 2 Properties of functions

  • Bijective, injective and surjective functions
  • Inverse function
  • Composition of functions
  • Sequences
  • Sum and product notations

Block III - Proofs and combinatorics

Unit 1 Proofs

  • Direct proof
  • Proof by contradiction
  • Proof by exhaustion
  • Mathematical induction

Unit 2 Counting

  • Fundamental principles of counting
  • Permutations and combinations
  • Elements of probability
  • Conditional probability

Unit 3 Binomial expansions

  • Binomial coefficients and combinatorial identities
  • Binomial theorem
  • Properties of binomial coefficients
  • Binomial expansion and approximations

Block IV - Recursion and iteration

Unit 1 Homogeneous recurrence relations with constant coefficients

  • Mathematical representations of recurrence relations
  • Solving recurrence relations by arithmetic and geometric progression
  • Second-order linear homogeneous recurrence relations with constant coefficients

Unit 2 Non-homogeneous recurrence relations with constant coefficients

  • Non-homogeneous linear recurrence relations with constant coefficients
  • Second-order non-homogeneous linear recurrence relations

Unit 3 Iterating functions

  • Iteration sequences
  • Exact and approximate solutions
  • Bisection method
  • Secant method

Block V - Algebraic structure

Unit 1 Matrices

  • Matrix property
  • Matrices addition and subtraction
  • Matrix multiplication and inverse
  • Transformation and matrices
  • Eigenvalues and eigenvectors

Unit 2 Complex numbers

  • Describe the complex number system
  • Manipulate calculations on complex numbers
  • Power of complex numbers
  • olving equations with complex numbers

All study units will be posted into the OLE.


3.2 Working through the study units

Each unit contains guidelines indicating what items you need to study the unit. Objectives at the beginning of each topic set out in detail what you should be able to do after completing it.

When studying a unit, you should remember to work through each exercise and problem within the unit, as well as those in the textbook as indicated in the unit. It is important to try your very best to solve problems on your own before looking at the solutions; don't just turn to the solution as the easy way out. On the other hand, don't ignore the solution — do use it if you are stuck, and read through the solution to compare it with your own.

You must not skip the sections that direct you to listen to audio recordings or watch the video clips; if you do, you'll miss some useful teaching points and explanations.

In the units you will come across examples, exercises and activities. At the end of your study of a unit, you should complete the corresponding assignment questions. Although it will be tempting to refer to the units while you do so, we think that you will gain more by trying to get as far as possible at your first attempt with any help other than referring to the handbook. You can, of course, refer back to the unit before you finalize your answers.


3.3 Scientific Notebook

In the same way that electronic calculators have replaced the abacus, computer mathematics software is increasingly being used by mathematicians to relieve them from tedious and lengthy calculations. It has now become essential for anyone using mathematics professionally to be able to make use of such software. To meet this aim we will be introducing you to the use of Scientific Notebook and at the same time using it to enable you to experiment with different mathematical techniques. This encourages discovery as well as allowing the analysis of more realistic problems without the burden of extensive calculations/manipulations.

The intention here is not to automate all calculations. The course material will require you to first develop mastery of all techniques through the use of pen-paper-calculator. Only when you are competent in handling this approach will you be allowed to switch to the software. This is important as you will not have access to the software in the examination.

Scientific Notebook combines the ease of entering text and mathematics in natural notation with the power of symbolic computation. This package, in fact, allows you to enter a mathematical expression and then to have the program manipulate it in a number of different ways. You can add explanatory text to your work and print it out for submission to your tutor.


3.4 Audio programmes*

Some units include teaching delivered on audio recordings. These audio programmes (as well as the video programmes) should provide you with some variety in your study. In most programmes you will be led by the speaker through an example or exercise displayed in the unit.

Each unit will indicate exactly when you should listen to the audio section. You will need to have your unit, a pencil and some paper to hand. You should pause the recording whenever you are instructed to do so, and work on the exercises as suggested. Of course, you may want to pause the recording at other times to gain a better understanding of some of the material.

A special feature of the audio programmes is that both English and Chinese versions are provided. You can choose the language of presentation with which you think you'll learn the most.

* Please note that all the audio and video recordings for this course have been uploaded to the OLE for easy access and no physical CDs will be provided. In the course materials, all references to the 'audio/video CD' should be taken to mean 'audio/video recordings on the OLE'.


3.5 Video programmes

Some study units include sections based on videos. These are used when a visual presentation better suits the topic being studied. You should try your best to watch these programmes when directed by the unit. The content of each one will be briefly described in the unit. The programmes are in English with Chinese subtitles.


3.6 Other materials

3.6.1 MATH S121 Handbook

Please read carefully the handbook regulation of this presentation as given below:

You will not be allowed to bring MATH S121 Course Handbook to the exam. Another copy of MATH S121 Handbook will be given to you together with the exam paper.


3.6.2 Course schedule

This sets out an overall timetable for the study weeks for each unit as well as when each assignment should be completed. It is available on the OLE.

Experience shows that spreading your study evenly over the presentation is very important. The submission dates of the assignments are designed to help you to do this. You are expected to have completed your study of the unit in time to be able to complete the assignment questions so that you can submit them before the associated due date.


3.6.3 Problem Booklet

Each block includes one of these booklets. Each one provides extra exercises that you can work through to help reinforce your understanding of the material. The Problem Booklet is divided into sections to match those in the units. You can pause your study at any point to work through relevant exercises.

You are not required to attempt every exercise in these booklets — only enough to give you confidence that you understand the associated technique/topic in the unit. You will also find these booklets a useful source of revision exercises for the examination.


3.6.4 Stop press

These act as a sort of course newsletter containing useful information or errata. They will be made available only through the OLE.

Your mastery of the materials in the course will be tested at regular intervals through assignments. Your marks for these and for your final examination will be used to determine your final grade.


4.1 Assessment

The course assessment contains two components: (1) continuous assessment, and (2) examination.

(1) Continuous assessment consists of two types of assignments (30%)

  • Assignments (multiple choice), which contain multiple choice questions and contribute 6% of the overall course score*
  • Assignments, which contain questions requiring you to prepare and submit written answers to your tutor for marking, and contribute 24% of the overall course score.

* Assignment (multiple choice) refers to assignment questions in multiple choice format and which are marked using computer software.

(2) Examination (70%)

There is a three-hour examination at the end of the course — this is based upon the whole course. A specimen examination paper with solutions will be provided and they are available on the OLE. You should work through this paper carefully some time before the examination.


  Number of assessments Weighting
(1)Continuous assessment:   
 - Assignments (multiple choice)2All are required6%
 - Assignments4All are required24%
(2)Examination (3 hours)1 70%
 Total  100%


You should refer to your Student Handbook for the method by which the University determines the final grade based on your continuous assessment score and examination mark.


4.1.1 Assignments (multiple choice)

Your answer script to each assignment (multiple choice) must be received by the University on/before the due date; e.g. if the due date is 15th September, then your solutions must be received before midnight (11:59 p.m.) of 15th September.

Each assignment (multiple choice) consists of short questions and assesses your understanding of the basic concepts/skills in the unit. Each question will include a list of possible options and you must choose the option that you consider represents the correct answer.

Your choice of option for each question must be submitted to the University through the OLE.

After the due date for each assignment (multiple choice) you will be able to find your mark on the OLE.


4.1.2 Assignments

There is an assignment question for each study unit. It is important that you understand that assignments are not simply a form of 'test'. All assignments fulfill a number of objectives:

  • the due dates encourage you to stay on schedule.
  • in preparing your answers you discover how much or little you understand of the unit. Hopefully, this will then encourage you to revise a particular section and to contact your tutor for help.
  • in marking your solutions your tutor will try to add comments that will identify where you have made an error and guide you on how this error can be corrected.
  • provide a mark representing your level of understanding for assessment purposes.

So, the assignments should be seen as an opportunity for both teaching and learning as well as for assessment. To maximize the benefits you need to start work on the assignment as early as possible so that you can ask for help if needed, and to work through the tutor's comments to correct any errors asking for further help if needed.

A common request from students is for model answers to be provided for all assessment questions. Experience shows that simply reading a model answer does not always help a student understand the cause of any errors. Attempting a similar question often results in the same error being repeated. It is only when a student discovers how to correct the error and then complete the solution that the student fully understands a technique.

All assignments must be submitted directly to your own tutor. There are two ways that you can do this:

  • write your solutions on paper and scan your answers to a pdf file. You can then upload your scanned answers via the OLE e-submission function.
  • use a package like Scientific Notebook or use Microsoft Office Word to prepare your answers and then upload the file to your account in the OLE. Your tutor will then access this file for marking. Instructions on how to do this are provided in the OLE User Guide.

Whichever method you use for submission your tutor will provide feedback on the script. The marks awarded will be entered into your account in the MATH S121 OLE together with any general comments. You will be able to view these as soon as your tutor has completed this entry. Again refer to the OLE User Guide for instructions.


4.1.3 Regulation for late submission application

You should always try to get your assignments to your tutor by the due date. This is to ensure that you keep to the schedule and that you are ready to start the next block on time. When unusual circumstances arise that prevent you from submitting on time, then the University allows you to apply through the OLE for permission to submit your solutions late, except for the last assignment. Who you apply to for permission depends on the amount of time that you require:

  • 1–7 days — your tutor
  • 8-21 days — the Course Coordinator
  • more than 21 days — the Dean of School

The OLE system presents you with these three options and will automatically forward your application to the appropriate individual for consideration. You will need to check with your account in the OLE for the decision. Note that requesting an extension is meant to be an exception — any delay for one assignment will make it harder for you to meet the due date for the following assignments. In some circumstances it may be better to omit one or two of the questions in an assignment so as to submit on time rather than make it harder for later assignments. In addition, this course consists of four assignments. They are all required to the course. All assignment marks will count towards your final grade.


4.1.4 Determining your overall continuous assessment score (OCAS)

  • Continuous assessment component: Each assignment will contribute 20% towards your overall continuous assessment score (OCAS).
  • Assignment (multiple choice) component: Each assignment (multiple choice) will count for 10% of OCAS.
 Number of assignmentsOCAS weightingCourse area covered
(multiple choice)
220%Each assignment (multiple choice) covers 3–4 units
Assignments480%Each assignment covers 3–4 units
Total 100% 


The combined marks from both assignments (multiple choice) and assignments contribute 30% of your overall score for the course.

5.1 Tutorials

These are face-to-face sessions run by the tutor. The main idea behind these sessions is to provide you with an opportunity to have any difficulties addressed by the tutor. This assumes that you will be working on the units according to the course schedule. Details of when and where the tutorials will be held are available in the course schedule on the OLE. Attendance at tutorials is optional although you are encouraged to attend.

The course schedule indicates which parts of the course are covered by each tutorial. To help your tutor design each session so that it meets your needs it would be helpful if you could inform him/her in advance about any specific topics that you would like covered. There will always be a tutorial scheduled before each assignment.

The first tutorial will introduce the use of Scientific Notebook.


5.2 Surgeries

Surgeries are different from tutorials, in that they are designed to offer a more personal level of consultation. A tutor will chair each surgery session. You will prepare and bring along your own individual study problems. Hopefully, you can then discuss these with the tutor and gain an insight into the problems or figure out an appropriate direction from which to tackle them.

Some tutors in charge might speak to the whole group so that the problem and solution of one student can benefit others. It may happen that the problem being discussed is the same one you have!

In tutorials and surgeries you will certainly have plenty of opportunity to get help from your tutor if you have any problems studying MATH S121. But students learning at a distance like you often need help at other times. There are a number of other ways for you to get the help you need to succeed in this course, no matter what point of the course you're at.


5.3 Your tutor

Your tutor is there to help you understand the ideas in the course. One of the best ways for him/her to do this is through comments on your assignments. When your tutor returns your assignment scripts to you after marking, you should always go through them and take note of the comments your tutor has written. This advice will help you avoid similar errors in later assignments and in the examination. Also, of course, you should try to attend tutorials because there you will have the opportunity to talk to your tutor directly and, just as important, to talk to other students.

Tutors are also available for you to telephone for help or advice. Contact details for your tutor will be sent to you at the start of the course. Your tutor should also let you know what hours he/she is available for telephone tutoring.

Email is another popular form of communication between tutor and student. Email allows you to attach a file containing a written description of your problem. This overcomes the problem that often arises in telephone contact — expressing a mathematical problem clearly.


5.4 The Online Learning Environment

The University provides the Online Learning Environment (OLE) for all courses. This provides a variety of features to support the presentation of a course:

  • Discussion board shared by all students and tutors on the course. Problems can be posted for anyone to offer help. In the past this has possibly been the most popular way of getting help. Often you can find that the answer to your own problem has already been supplied to another student with a similar problem.
  • Email. Each student and tutor has an email account for direct communication between individuals and groups. Most of the news and comments from the Course Coordinator will be sent to you through this email system.
  • Important news. Any stop press will be posted on the OLE. An alert will be displayed to indicate any new items posted.
  • Study units. Soft copies of the units will be available through the site.
  • Assessment questions and late submission application. All assignments are provided through the OLE. All assignments must be submitted through this site. Requests for assignment extension must be done through this site.
  • Check assignment score. You can check the marking score of any assignment and check the marking status.

The OLE should become a regular part of your study habits. Check the site regularly for updates or news as well as useful and interesting posting on the discussion board. If you are a new student, you will get a copy of the OLE User Guide, which explains how to use the system. You can also refer to the online OLE User Guide at http://ole.hkmu.edu.hk/help.html.


5.5 Your Course Coordinator (CC)

You will find an introduction to your Course Coordinator in the first mailing, which will include his/her contact details. Whilst your tutor should always be your first point of contact when seeking help, the CC is always available to provide additional assistance when needed.

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