Module FOUD0551: Calculus and Further Maths with Statistics and Strategic Maths

Department: Foundation Year (Durham)

FOUD0551: Calculus and Further Maths with Statistics and Strategic Maths

Prerequisites

• Any Core Foundation Maths modules or equivalent modules

Corequisites

• None

Excluded Combination of Modules

• FOUD0651, FOUD0638, FOUD0301

Aims

• To provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications
• To extend knowledge of the six trigonometrical functions and inverses.
• To introduce and develop a knowledge of matrices and their applications.
• To extend understanding of a range of standard techniques for differentiation and integration.
• To introduce and develop a knowledge of first and second order differential equations and their applications
• To extend knowledge of Cartesian coordinates in two dimensions to include equations of circles and lines.
• To introduce and develop knowledge of complex numbers and polar coordinates.
• To introduce the concept of linear programming
• To introduce some statistical techniques
• To introduce the use of mathematics in financial applications

Content

• Spanning trees (Prim's and Kruska's algorithm)
• Matrices (nxm): addition, subtraction, multiplication, determinant, transpose, inverse. Applications to simultaneous equations.
• Trigonometrical functions of angles and graphs.
• Cartesian equations of straight lines, perpendicular lines and circles.
• Complex numbers: +, -, x, /, complex conjugate, polar form, Argand diagrams, De Moivre's theorem.
• Series expansions
• Consolidation of Integration by parts and by substitution
• First Order Differential equations and applications including separating variables.
• Second Order Differential equations.
• Implicit and parametric differentiation
• Linear Programming
• mean, mode, median and standard deviation
• Normal Distribution and significance
• Linear Regression
• Use of mathematics in financial applications

Learning Outcomes

• By the end of the module students will be able to
• Define the 6 trigonometrical functions (SSK 1)
• State the rules for addition, subtraction and multiplication of matrices and for finding inverses. (SSK2)
• State the standard forms for general solutions of second order differential equations (SSK3)
• Give standard Cartesian equations for circles and lines (SSK4)
• Solve a range of predictable problems in Discrete Mathematics. (SSK5)

• By the end of the module the student will have acquired the skills to be able to:
• select and use statistical techniques as required in problems appropriate to the syllabus. (SSS1)
• confidently manipulate a range of Cartesian equations. (SSS2)
• understand and use complex numbers in a range of situations as appropriate to the syllabus. (SSS3)
• use matrices in a number of mathematical situations. (SSS4)
• recall, select and use knowledge of appropriate integration and differentiation techniques as needed in a variety of contexts.(SSS5)
• understand and use first and second order differential equations in a range of situations as appropriate to the syllabus (SSS6)
• reduce problems to a series of equations and inequalities and solve using linear programming techniques. (SSS7)
• apply mathematics to a variety of problems (SSS8)

• By the end of the module students will be able to:
• apply number in the tackling of numerical problems
• demonstrate problem solving skills.
• Test 1 covers SSK4, SSS1, SSS2, SSS8, KS1, KS2
• Test 2 covers SSK4, SSK5, SSS2, SSS5, SSS7, SSS8, KS1, KS2
• End of module test covers SSK2, SSK3, SSK4, SSS2, SSS3, SSS4, SSS5, SSS6, SSS8 KS1, KS2
• Portfolio covers SSK1-5, SSS 1-8, KS1, KS2,

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Theory, initial concepts and techniques will be introduced during lectures.
• Much of the learning, understanding and consolidation will take place through the use of structured worksheets during seminars and students' own time.
• Small coursework tasks testing, developing or consolidating the previous week’s work will be set usually on a weekly basis. These tutor marked tasks allow rapid feedback and build confidence. Whilst the marks accumulate towards the overall portfolio mark, the tasks also perform a formative role enabling students to reflect on their own performance, identify areas of weakness, and practice some of the skills and techniques which will be required in the longer in-class tests and end of module test. Additionally, they ensure that students master specific skills to an appropriate level prior to their requirement in more complex tasks. As an example, a task on first order differentiation might require students to solve four different first order equations with certain boundary conditions. Tutor feedback from this task ensures that students are ready to build on these skills when solving second order equations with boundary conditions.
• Ability to recall, select and use knowledge will be tested by two class tests and an end of module invigilated test in addition to the portfolio of tasks. The two class tests will focus on a selected subset of the content. In addition to their summative role, these tests also serve a formative function helping to prepare students for the end of module test which will test a wider area of content.
• In addition to obtaining an overall weighted mark of 50% or above, students must obtain a mark of 50% or above for the following element/s - Test 2 and end of module test

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Students will be given self-testing units on a weekly basis in the form of worksheets with answers and/or DUO quizzes. Portfolio tasks with a rapid marking turnaround fulfill a formative as well as summative role (See Section 14). Students have access to two or more mock papers and answers to help prepare for the class tests and the end of module test.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University