Module FOUD0638: Calculus, Further Maths and Mechanics with Statistics and Strategic Maths

Department: Foundation Year (Durham)

FOUD0638: Calculus, Further Maths and Mechanics with Statistics and Strategic Maths

Prerequisites

• Any Core Foundation Maths or equivalent modules

Corequisites

• None

Excluded Combination of Modules

• FOUD0551, FOUD0651, FOUD0301.

Aims

• To extend knowledge of Cartesian coordinates in two and three dimensions to include equations of circles and lines.
• To introduce and develop a knowledge of matrices and applications.
• To develop a knowledge of vectors and their applications in two and three dimensions to include equations of lines.
• To improve confidence in algebraic and trigonometric manipulation.
• To introduce and develop understanding of trigonometric formulae and their uses.
• 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 introduce and develop knowledge of complex numbers and the concept of polar coordinates.
• To develop students' abilities to apply mathematics to problems based on physical situations.
• To provide the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.
• To introduce some statistical techniques.
• To use mathematics in a variety of applications

Content

• Trigonometrical functions of angles and graphs. Six ratios, inverse functions, use of identities, solution of equations, sine and cosine rule.
• Series expansions.
• Matrices (n x m): addition, subtraction, multiplication, determinant, transpose, inverse. Applications to simultaneous equations.
• Cartesian equations of straight lines, perpendicular lines and equation of circles.
• Complex numbers: +, -, x, /, complex conjugate, polar form, Argand diagrams, De Moivre's theorem.
• Vectors including: use of column and unit vectors, addition, subtraction and multiplication by scalar. Scalar (dot) and vector (cross) product and their applications.
• Consolidation of Integration by parts and integration by substitution.
• Differentiation of functions defined parametrically and implicitly.
• Applications of first order differential equations including seperating variables.
• Second order differential equations - Motion: velocity, acceleration, equations with constant acceleration, projectiles, variable acceleration/force, F=ma - Conservation of momentum, impulse - Hooke's law, SHM.
• Mean, Median and Mode and Standard Deviation.
• Normal Distribution and significance.
• Linear regression.
• Use of mathematics in a variety of applications appropriate to progression route. Appropriate content will be selected from Linear Programming, Spanning Trees, Applications to Finance, Induction, Functions, Hyperbolic Functions and Further Mechanics.

Learning Outcomes

• By the end of the module the student will be able to
• Solve a range of predictable problems in Discrete Mathematics. (SSK1)
• Give standard Cartesian equations for circles and lines (SSK2)
• Define the 6 trigonometrical functions (SSK3)
• State the Sine and Consines rules (SSK4)
• Understand vectors and rules of application in two and three dimensions. (SSK5)
• State the rules for addition, subtraction and multiplication of matrices and for finding inverses. (SSK6)
• Understand parametric and implicit functions and first order differential equations and second order differential equations (SSK7)
• Solve a range of physical problems in Mechanics. (SSK8)
• Solve statistics problems (SSK9)

• By the end of this module the student will have acquired the skills to be able to:
• Confidently manipulate a range of algebraic and trigonometric expressions as required in problems appropriate to the syllabus (SSS1)
• confidently manipulate a range of Cartesian and vector equations in two and three dimensions. (SSS2)
• use matrices in a number of mathematical situations. (SSS3)
• Select and use statistical techniques as required in problems appropriate to the syllabus. (SSS4)
• select and use trigonometric formulae and techniques as required in problems appropriate to the syllabus. (SSS5)
• understand and use complex numbers in a range of situations as appropriate to the syllabus. (SSS6)
• recall, select and use knowledge of appropriate integration and differentiation techniques as needed in a variety of contexts.(SSS7)
• understand and use first and second order differential equations in a range of situations as appropriate to the syllabus. (SSS8)
• apply mathematics to a variety of problems based on physical situations. (SSS9)
• apply mathematics to a variety of problems appropriate for progression routes (SSS10)
• apply descreet mathematics to a variety of problems (SSS11)

• By the end of the module students will be able to
• apply number in the tackling of numerical problems (KS1)
• demonstrate problem solving skills (KS2)
• Test 1 (Trigonometry and mechanics) covers SSK3, SSK4, SSK5, SSK8, SSS1, SSS2, SSS5
• Test 2 (Statistics) covers SSK2, SSK9, SSS4
• Test 3 (Mechanics) covers SSK8, SSS1, SSS7, SSS8, SSS9, KS1, KS2
• Test 4 (Strategic Maths) covers SSK1, SSK2, SSS1, SSS7, SSS10, SSS11, KS1, KS2
• End of module test (Further Maths and Calculus) covers SSK2, SSK6, SSK7, SSS1, SSS2, SSS3, SSS6, SSS7, SSS8

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, tutorials and students' own time.
• In class tests, developing or consolidating the previous weeks’ work will be set which will contribute towards final the module mark. These tests 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 end of module test.
• Ability to recall, select and use knowledge and manipulative skills will be tested by: tasks set by the tutor on a weekly basis, mid-module invigilated tests and an end of module test/
• 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 3, Test 4 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