Download MA322 - University of Kent

UNIVERSITY OF KENT
MODULE SPECIFICATION TEMPLATE
SECTION 1: MODULE SPECIFICATIONS
1.
Title of the module
MA322: Proofs and Numbers
2.
School which will be responsible for management of the module
School of Mathematics, Statistics and Actuarial Science
3.
Start date of the module
Autumn term 2011
4.
The cohort of students (onwards) to which the module will be applicable
2011 entry
5.
The number of students expected to take the module
(currently about 300 students)
SMSAS Stage 1 class
6.
Modules to be withdrawn on the introduction of this proposed module and
consultation with other relevant Schools and Faculties regarding the withdrawal
MA303: Algebra. MA303 was available as a wild card module. MA322 is not available as
a wild card module.
7.
Level of the module (e.g. Certificate [C], Intermediate [I], Honours [H] or
Postgraduate [M]) Certificate [C]
8.
The number of credits which the module represents
9.
Which term(s) the module is to be taught in (or other teaching pattern) Term 1
15 (7.5 ECTS)
10. Prerequisite and co-requisite modules Prerequisites are an ‘A’ level in Mathematics or
in Mathematics and Statistics or Pure Mathematics or equivalent. In particular, students
starting the course are expected to be familiar with the Pure Mathematics core of ‘A’
level. Co-requisite module: MA321 (Calculus and Mathematical Modelling)
11. The programme(s) of study to which the module contributes. Compulsory for
students registered for a BSc in Actuarial Science, Financial Mathematics, Mathematics,
Mathematics with a Foundation Year, Mathematics and Accounting & Finance,
Mathematics and Computer Science, Mathematics & Statistics, Mathematics with
Secondary Education.
12. The intended subject specific learning outcomes and, as appropriate, their
relationship to programme learning outcomes
On successful completion of this module students will:
a)
have acquired precision in logical argument and have gained a core mathematical
understanding in discrete mathematics (A1, A5, C1);
b)
have been introduced to fundamental mathematical notation and have practiced its
use (A1,C1);
c)
have learned and practiced basic concepts of mathematical proof (direct proof, proof
by contradiction, mathematical induction) (A1,C2);
d)
have learned elementary combinatorics and counting techniques (A2);
e)
have learned how to solve simple matching and combinatorial problems (A5);
f)
be able to solve combinatorial problems in discrete structures (B2,D1,D3);
g)
have learned how to state precisely and prove elementary mathematical statements
and solve problems (C1,C2);
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h)
13.
be able to handle simple mathematical symbols with some confidence (B2,C2);
i)
be able to simplify complex mathematical expressions and apply general formulae
to specific contexts (B2,C2);
j)
be able to perform calculations involving: simplification of polynomial expressions,
simple factorizations, expression of rational functions as sums of “partial fractions”,
complex numbers, integers, elementary arithmetic (A1,A4,B2,B3,B5,B7,C1);
k)
have a basic understanding of the nature of the algebraic concepts covered in the
module: the system of complex numbers and its geometrical interpretation,
polynomial identities, polynomial equations, greatest common divisors, modular
(congruence) arithmetic (B1,B4,C2);
l)
have reasonable ability to make simple deductions from algebraic equations and
identities, and be able to state them with reasonable clarity (A5,B4,C4).
The intended generic learning outcomes and, as appropriate, their relationship to
programme learning outcomes
On successful completion of the module students will:
a) have developed a basic knowledge of the logical approach to solving problems on the
subject matter in this module and be able to present the solution in a reasonably
coherent way (B1,D1,D2,D3);
b) have learned how to argue logically and will have gained a higher understanding and
appreciation for the objective nature of mathematical knowledge (B4,D1);
c) have developed higher numeracy and computational skills (D3);
d) have shown a reasonable ability to revise the material in the module and answer
questions under time constraints (D6,D7);
e) have improved their skills in written communication, numeracy and problem solving
(D1,D2,D3).
14. A synopsis of the curriculum
Numbers and proofs are key notions in modern Mathematics that have found
applications in many other sciences, but also in our every day life. For instance, the
security of our mobile phones relies on Fermat Little Theorem. This course will introduce
the students to these fundamental concepts, as well as the notion of proof.
Topics covered may include:
 Logic and Proofs: Propositions, Logical operators (not, or, and, imply, if and only if).
 Quantifiers and other mathematical symbols (sums, product), Introduction to Proofs.
 Basic Sets: introduction to sets, operations on sets (union, intersection, Cartesian
product, complement), basic counting (inclusion-exclusion for 2 sets), countable sets.
 Proof by Induction
 Selection models.
 Integers: divisibility, greatest common divisor and Euclidean algorithm, unique
factorisation, base representation.
 Modular arithmetic: congruence classes including Chinese remainder theorem and
Fermat Little Theorem. (Time permitting: RSA)
 Polynomials: long division algorithm, rational functions in one variable, and partial
fractions.
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 Complex numbers: definition, addition, multiplication. Inverse. Geometrical
interpretation. Modulus, conjugate. Triangle inequality, polar coordinates, arguments,
De Moivre Theorem. Lines and circles in the complex planes.
 Roots of polynomials: Fundamental Theorem of Algebra. Roots of unity, and their
geometric interpretation. Roots of polynomials with real coefficients.
 Functions: injective, surjective, bijective. The Pigeonhole Principle.
 Relations and partitions: Equivalence relations. Equivalence classes. Ordered and
unordered partitions. Link with equivalence relations.
15. Indicative Reading List

Chetwynd & Diggle, Discrete Mathematics, Butterworth Heinemann, 1995

Childs, A concrete introduction to higher algebra, 3rd edition, 2009
16. Learning and Teaching Methods, including the nature and number of contact
hours and the total study hours which will be expected of students, and how these
relate to achievement of the intended learning outcomes
The delivery is by means of up to 36 lectures (up to 3 per week during Term 1) and up to 12
exercise classes (up to 1 per week during Term 1) giving up to 48 contact hours in total. The
total number of study hours is 150. Students are expected to attempt exercises set by the
lecturer, which will be discussed in the exercise classes.
Lectures address Learning Outcomes 12(a-e,g-l) 13(a,b);
Classes and/or workshops address 12(a,b,c,e-l) 13(a,b,c,e);
Personal study addresses 12(a-l) 13(a-e).
17. Assessment methods and how these relate to testing achievement of the intended
learning outcomes
A variety of assessments (class tests and/or assessed homework) will test Learning
Outcomes [12(a-l), 13(a-c,e)] during the course and will provide assessment for 20% of total
credit. A 2-hour written exam in the Summer term testing Learning Outcomes [12(a-l),13(a-e)]
will provide assessment for 80%.
In each case, questions may involve elements of bookwork (that can be derived both from
lectures and the literature) and solving mathematical problems.
18.
Implications for learning resources, including staff, library, IT and space
This is replacing an existing module, MA303.
19. The School recognises and has embedded the expectations of current disability
equality legislation, and supports students with a declared disability or special
educational need in its teaching. Within this module we will make reasonable
adjustments wherever necessary, including additional or substitute materials,
teaching modes or assessment methods for students who have declared and
discussed their learning support needs. Arrangements for students with declared
disabilities will be made on an individual basis, in consultation with the
University’s disability/dyslexia support service, and specialist support will be
provided where needed.
SECTION 2: MODULE IS PART OF A PROGRAMME OF STUDY IN A UNIVERSITY
SCHOOL
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Statement by the School Director of Learning and Teaching/School Director of
Graduate Studies (as appropriate): "I confirm I have been consulted on the above module
proposal and have given advice on the correct procedures and required content of module
proposals"
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Director of Learning and Teaching/Director of Graduate
Studies (delete as applicable)
Date
…………………………………………………
Print Name
Statement by the Head of School: "I confirm that the School has approved the introduction
of the module and, where the module is proposed by School staff, will be responsible for its
resourcing"
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Head of School
Date
…………………………………………………….
Print Name
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