UNIVERSITY OF KENT
MODULE SPECIFICATION TEMPLATE
SECTION 1: MODULE SPECIFICATIONS
1.
Title of the module: MA325: From geometry to algebra
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: First year intake in the
following BSc programmes: Mathematics, Mathematics & Statistics (optional),
Mathematics with Secondary Education (currently about 120 students).
6.
Modules to be withdrawn on the introduction of this proposed module and
consultation with other relevant Schools and Faculties regarding the withdrawal:
MA300, MA320. No consultation with other Departments or Faculties required.
7.
Level of the module: Certificate [C]
8.
The number of credits which the module represents 30, ECTS 15.
9.
Which term(s) the module is to be taught in Terms 1 and 2.
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.
11. The programme(s) of study to which the module contributes: Required for BSc
Mathematics and Mathematics with Secondary Education programmes, optional for
students on the BSc Mathematics and Statistics programme.
12. The intended subject specific learning outcomes and, as appropriate, their
relationship to programme learning outcomes
a. Be familiar with a variety of basic ideas from geometry and algebra.
(A4,B1,C1)
b. Understand the fundamentals of Euclidean and spherical geometry, the
geometry of the complex plane, and elementary notions in group theory.
(A4,A5,B1,C1)
c.
Be able to solve a variety of concrete problems in geometry and algebra.
(A4,A5,B1,B2,B3,B4,B5,C1,C2,C4,D1,D3,D7).
d. Be able to appreciate the unity of mathematics as demonstrated in the course
by
the
interplay
of
geometric
and
algebraic
ideas.
(A4,A5,B1,B2,B3,B4,C1,C2,C4,D7)
e. Be able to draw figures correctly to enhance geometric intuition.
(A4,B1,B2,B3,B5,C4,D2)
f.
Be able to give simple proofs in geometry and algebra, and be able to
understand slightly more complicated ones. (A4,A5,B1,B4,C1,C4,D1,D2)
g. Be able to manipulate combinatorial objects arising in geometry such as
permutations and graphs, and have learned how to solve simple problems
involving these objects. (A1,B2,B5,D1,D2)
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13. The intended generic learning outcomes and, as appropriate, their relationship to
programme learning outcomes:
On successful completion of the module students will:
a) have improved problem solving and communication skills (D1,D2,D3);
b) have learned how to argue logically (A5,B4,D1);
c) have shown a reasonable ability to revise the material in the module and answer
questions under time constraints (D6,D7);
d) have a better appreciation of the unity of mathematics (B3,B5).
14. A synopsis of the curriculum:
This module starts with an introduction to 2D and 3D geometry. Then it emphasises the
links between Geometry and Algebra. In particular, the geometric concept of symmetry
will lead to the algebraic concept of groups.
I.
Geometry of the Euclidean plane: Basic transformations,
classification of isometries, congruence, similarities, inscribed
and circumscribed circle, Sine and cosine rules, Ceva’s
Theorem.
II.
Spherical geometry: Spherical distance, great circles,
spherical triangles, area on the sphere. spherical excess
formula, tilings of the sphere by regular spherical polygons,
similarity and congruence on the sphere.
III.
Regular polyhedra: Definition by means of isometries, Euler’s
formula (proof using graphs and/or spherical excess formula),
classification of regular polyhedra.
IV.
Geometry of the complex plane: Complex numbers, polar
coordinates, lines and circles, isometries, roots of unity,
Fundamental Theorem of Algebra, algebraic numbers and
countable sets.
V.
Groups from geometry: Permutations, disjoint cycle
decomposition. cyclic and dihedral groups, infinite groups, an
examples led introduction to group theory, subgroups,
Lagrange Theorem, conjugacy classes. (Optional topics
include, examples of group homomorphisms, and an illustration
of group actions to describe the isometries of the cube or other
regular polyhedra.)
15. Indicative Reading List
1. A.F. Beardon, Algebra and Geometry, Cambridge Univ. Press, 2005.
2. H.S.M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1969.
3. G.A. Jennings, Modern Geometry with Applications, Universitext, Springer Verlag,
1994.
4. E.G. Rees, Notes on Geometry, Universitext, Springer Verlag, 1982.
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
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relate to achievement of the intended learning outcomes: The delivery is by means
of up to 72 lectures (up to 3 per week) and up to 24 exercise classes (up to 1 per week)
giving up to 96 contact hours in total. The total number of study hours is 300. Students
are expected to attempt exercises set by the lecturer, which will be discussed in the
exercise classes. Appropriate use of figures and devising simple proofs will contribute to
the communication skills and learning outcomes 12(a,b,c,e,f), Using methods and ideas
from geometry in algebra, and vice versa, will contribute towards a greater appreciation
of the unity of mathematics and contribute towards learning outcomes 12(d).
17. Assessment methods and how these relate to testing achievement of the intended
learning outcomes: Assessment will be 30% coursework (class test(s) and/or
assessed homework(s)) and 70% written exam. Coursework will test Learning Outcomes
[12(a-g), 13(a,b,d)] during the course. The exam in the Summer term will be 3 hours, and
will test Learning Outcomes [12(a-g), 13(a-d)].
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 A
convenor and one other lecturer. This is replacing existing modules, MA300 and
MA320. Timetable 4 lecture hours in a room with good blackboard facilities, which are
essential to demonstrate figure drawing skills.
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
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"
................................................................
..............................................
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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