This course offers a broad introduction to advanced mathematics covering topics from right across the subject, with a particular emphasis on mathematical communication and problem solving skills. It is an essential course for students on degree plans involving mathematics, physics or statistics.

The course will be delivered in sections with lectures at different times to allow for a range of combinations with other subjects. Tutorial labs and long core skills labs will be offered at a variety of times throughout the week.

Mathematics 1C, 1G

Final exam (April / May): 50% 2 Hours

December exam (December): 20% 90 Mins

Set exercises: reading 5%. Weekly reading comprehension exercises before lectures (best 80% to count);

Set exercises: eAssignments and written feedback 15%. Weekly, alternating between eAssignment and written feedback (best 80% in each category to count).

Set exercises: tutorial group activities 10%. Weekly (best 80% to count). Group working skills will be explicitly assessed.

Additionally students must pass 5 `core skills' tests in order to receive a D3 grade for the course; otherwise the grade for the course will be capped at E1. This is to ensure students have a strong command of the essential skills needed to succeed all level 2 mathematics courses. These tests will take the form of randomly generated eAssessments, taken in supervised lab sessions.

The core skills tests will be offered regularly throughout the academic year, from freshers week through to the August resit diet, and students may repeat the tests on multiple occasions. The first attempt period is defined to last until the end of the April/May exam diet. Students who pass the 5 core skills test in this period will have their course grade returned as per the summative assessment schedule in 19. The second attempt period for the core skills is in the August resit diet.

It is the default expectation that all courses will offer opportunities for reassessment or deferred assessment. Where it is not possible to offer this in some assessment components, the grade achieved at the first attempt will be counted towards the final course grade, and any exceptions for this course are described below.

[No exceptions]

Mathematics 1 aims to transition students to university level mathematics through development of abstract structures and reasoning skills, the interplay between algebra and geometry, the underpinnings of calculus and its vast applications, and ensure students have a strong command of core skills crucial to further study. A strong focus throughout the course will be placed on developing mathematical communication skills.

By the end of this course students will be able to:

1 use the language of sets, functions, relations, number systems, and groups to communicate mathematical ideas and arguments.

2 analyse the structure of a mathematical proof or statement, identifying hypotheses, conclusions, contrapositives, converses, negations; produce valid mathematical proofs using methods such as direct argument, induction, proof by contradiction, counterexample.

3 use counting arguments and ideas from elementary number theory to solve problems with particular reference to topics including prime factorisation, congruence, cryptography, binomial expansions, permutations and cardinality.

4 solve equations and inequalities in a variety of settings including real and complex numbers, polynomial equations, systems of linear equations, lines and planes in 3 dimensional space, and differential equations using techniques from algebra, geometry and calculus.

5 define and identify groups and subgroups; perform group multiplications particularly in examples coming from permutation groups using cycle notation; compute orders of groups and elements; use Lagrange's theorem.

6 manipulate function limits, and use these to define and compute derivatives and integrals from first principles; relate differentiation and integration via the fundamental theorem of calculus.

7 compute derivatives (for both scalar and vector valued functions) and integrals using standard derivatives of polynomials, exponential, trigonometric, hyperbolic and logarithmic functions; chain, product and quotient rules; implicit differentiation; the fundamental theorem of calculus, substitution; integration by parts and other similar methods.

8 apply techniques from calculus to solve optimisation problems, compute area, arc lengths, tangent directions, volumes.

9 perform matrix and vector operations from both geometric and algebraic viewpoints particularly in the context of 3-dimensional space.

10 model situations using the mathematical tools developed in the course, and relate the mathematical solution back to the original problem.

11 read mathematics independently, extracting essential concepts, definitions, examples, methods and results.

12 present mathematical work in writing, using precise language and notation, providing clear conclusions and reasoning.

13 discuss and solve mathematical problems in small groups.