Teacher Delivery Guide Pure Mathematics: Numerical Methods Specification Ref. Learning outcomes Notes Notation Exclusions PURE MATHEMATICS: NUMERICAL METHODS (2) Solution of equations Me1 Be able to locate the roots of by considering changes of sign of in an interval of x in which is sufficiently Finding an interval in which a root lies. This is often used as a preliminary step to find a starting value for the methods in Me3 and Me4. e2 Be aware of circumstances under which change of sign methods may fail. e.g. when the curve of touches the e.g. when the curve of has a vertical asymptote. e.g. there may be several roots in the interval. e3 Be able to carry out a fixed point iteration after rearranging an equation into the form and be able to draw associated staircase and cobweb diagrams. e.g. write as and use the iteration with an appropriate starting value. Includes use of ANS key on calculator. iteration, iterate e4 Be able to use the Newton-Raphson method to find a root of an equation and represent the process on a graph. e5 Understand that not all iterations converge to a particular root of an equation. Know how Newton-Raphson and fixed point iteration can fail and be able to show this graphically. Specification Ref. Learning outcomes Notes Notation Exclusions
Integration Mc34 Be able to find an approximate value of a definite integral using the trapezium rule, and decide whether it is an over- or an under-estimate. In an interval where the curve is either concave upwards or concave downwards. Number of strips. c35 Use the sum of a series of rectangles to find an upper and/or lower bound on the area under a curve. Problem solving Me6 Use numerical methods to solve problems in context.
Nrich Activity for students to do, finding the roots of a polynomial. e1-5 William & Mary Stretch material. Use of MATLAB with root finding algorithms. Extension work/reading for the most able. e1-5 Geogebra Demonstration of cobweb/staircase diagrams e3 CIMT Includes some real world scenarios linked with the maths of root finding. Covers most of the A Level numerical methods required content. e6 Underground Maths Investigation, using numerical methods, to evaluate the claim that the Serpentine covers an area of 40 acres. e6 Underground Maths Students are asked to decide whether using the trapezium rule would give an under or over estimate of the area between the curve and the x-axis for the sketch graphs from Gradient match. Rather than focusing on the numerical aspect of using the trapezium rule, in this resource students are only given sketch graphs and are prompted to think about geometric features of the curves. e6 Underground Maths Cambridge Colleges Examination for Entrance Scholarships and Exhibitions, Calculus (Group 3), 1952, Q2. e6 Underground Maths UCLES A level Mathematics 2, QP 840/2, 1970, Q17b. e6 MEI Notes on content, teaching ideas, suggestions on the effective use of technology, common errors and links across the curriculum. e6 Geogebra Demonstration of trapezium rule, using a slider to vary the number of trapezia and investigate the effect on the accuracy of the result. c34 Geogebra Demonstration of trapezium rule, using a slider to vary the number of trapezia and investigate the effect on the accuracy of the result. c34