Further Mathematics Support Programme

MEIModelling with Algorithms –Suggested Scheme of Work (2017-2018)

This template shows how Integral Resourcesand FMSP FM videoscan be used to support Further Mathematics students and teachers.

This is for the minor optional component Modelling with Algorithms – you will need to deliver another optional elementand Core Pure alongside this.

This content makes up 33⅓% of the MEI AS Further Mathematics content.

It is examined in AS level paper Y413 and in A level paper Y433.

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Teacher access to the Integral Resources (integralmaths.org/2017/) for Further Mathematics is available free of charge to all schools/colleges that register with the Further Mathematics Support Programme: furthermaths.org.uk.
This will include access to the FM videos. A single student login will also be included so that teachers can give students direct access to the FM videos.
/ Individual student access to the full range of Integral Resources andtheFM videos for Further Mathematics is available at a cost of £30 per student or via a full school/college subscription to Integral. Teachers will get access to the management system so they can monitor their students' progress: furthermaths.org.uk/fm-videos.

Integral Resourcesinclude a wide range of resources for both teacher and student use in learning and assessment. Interactive resources and ideas for using technology are featured throughout. Sample resources are available via:integralmaths.org/2017/.

FM videosare available for individual components of AS and A level Further Mathematics. There will be around 4-5 videos of 5-10 minutes in length for each section in Integral. The intention of these videos is that they are sufficient to introduce students to the concepts so that they can learn the material by working through appropriate examples. FM videosare ideal for schools/colleges teaching Further Mathematics with small groups and/or limited time allocation. They are also useful to support less experienced teachers of Further Mathematics. See furthermaths.org.uk/fm-videos.

Scheduling will depend on circumstances, but the template breaks the study intotopic sections.Each section corresponds to one set of videos and may be allocated approximately equal time – this would equate to approximately one week of teaching time for a single teacher delivering the complete AS course.

Further information on scheduling can be found at furthermaths.org.uk/offering-fm.FMSP Area Coordinators will be able to offer additional guidance if needed: furthermaths.org.uk/regions.

MEIModelling with Algorithms – Suggested Scheme of Work (2017-2018)

Date / Topic / Specification statements / Integral Resources / Exercises & Assessment
Integral Resources / FM videos / Notes / Other resources
Algorithms: Sorting, packing and complexity of algorithms /
  • Understand that an algorithm is a finite sequence of operations for carrying out a procedure or solving a problem. Understand that an algorithm can be the basis for a computer program.
  • Be able to interpret and apply algorithms presented in a variety of formats.
  • Be able to repair, develop and adapt simple algorithms.
  • Understand and be able to use the basic ideas of algorithmic complexity and be able to analyse the complexity of given algorithms. Know that complexity can be used, among other things, to compare algorithms.
  • Understand that algorithms can sometimes be proved correct or incorrect.
  • Understand and know the importance of heuristics.
  • Know and be able to use the quick sort algorithm. Be able to apply other sorting algorithms which are specified.
  • Be able to count the number of comparisons and/or swaps needed in particular applications of sorting algorithms, and relate this to complexity.
  • Be able to reason about a given sorting algorithm.
  • Know and be able to use first fit and first fit decreasing packing algorithms and full bin strategies.
  • Be able to count the number of comparisons needed in particular applications of packing algorithms, and relate this to complexity.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ 1.1 Introduction to algorithms and flowcharts
1.2 Algorithms using pseudocode
1.3 Bin packing algorithms
1.4 Sorting algorithms
1.5 Complexity of algorithms / TBC
e.g. UM, Nrich, ExamSolutions.
TBC
Graphs & networks 1: Modelling /
  • Understand and be able to use graphs and associated language.
  • Be able to model problems by using graphs.
  • Understand that a network is a graph with weighted arcs.
  • Be able to model problems by using networks.
  • Understand that network algorithms can be explored, understood and tested in cases in which the algorithm can be run by hand, but for practical problems the algorithm needs to be formulated in a way suitable for computing power to be applied.
  • Be able to use a network to model a transmission system.
  • Be able to specify a cut and calculate its capacity.
  • Understand and use the maximum flow/minimum cut theorem.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ 1.1 Introduction to graph theory
1.2 Modelling with graphs & networks
1.3 Further modelling with networks
TBC
Graphs & networks 2: Minimum spanning trees & shortest paths /
  • Be able to solve minimum connector problems using Kruskal’s and Prim’s algorithms.
  • Model shortest path problems and solve using Dijkstra’s algorithm.
  • Know and use the fact that Kruskal’s, Prim’s and Dijkstra’s algorithms have quadratic complexity.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ 2.1 Minimum spanning trees - Prim's algorithm
2.2 Minimum spanning trees - Prim's algorithm on a table
2.3 Minimum spanning trees - Kruskal's algorithm
2.4 Shortest paths
TBC
Critical path analysis /
  • Model precedence problems with an activity-on-arc network.
  • Use critical path analysis and be able to interpret outcomes, including implications for criticality. Be able to analyse float (total, independent and interfering), resourcing and scheduling.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ 1.1 Constructing an activity network from a precedence table
1.2 Using dummies in activity networks
1.3 Analysing an activity network
1.4 Finding critical activities and critical paths
1.5 Resourcing and scheduling
TBC
Linear programming 1: Introduction /
  • Understand and use the language associated with linear programming.
  • Be able to identify and define variables from a given problem. Be able to formulate a problem as a linear program.
  • Be able to recognise when an LP is in standard form.
  • Be able to use slack variables to convert an LP in standard form to augmented form.
  • Recognise when an LP requires an integer solution.
  • Be able to graph inequalities in 2-D and identify feasible regions. Be able to recognise infeasibility.
  • Be able to solve a 2-D LP graphically.
  • Be able to consider the effect of modifying constraints or the objective function.
  • Be able to use a visualisation of a 3-D LP to solve it. Be able to reduce a 3-D LP to a 2-D LP when one constraint is an equality.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ 1.1 Formulation of linear programming problems
1.2 Graphical solution - vertex method
1.3 Graphical solution - objective line method
1.4 Graphical solution - integer solutions
1.5 Visualising 3D LP problems
1.6 LP problems in augmented form
TBC
Linear programming 2: Simplex method /
  • Be able to use the simplex algorithm on an LP in augmented form.
  • Understand the geometric basis for the simplex method.
  • Recognise that if an LP includes >= constraints then the two-stage simplex method may be used; understand how this method works and be able to set up the initial tableau in such cases.
  • Be able to reformulate an equality constraint as a pair of inequality constraints.
  • Recognise that if an LP has variables which may take negative values or requires the objective function to be minimised then some initial reformulation is required before the simplex algorithm may be applied.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ TBC
TBC
Linear programming 3: Application to network problems /
  • Understand that some LPs can be solved using graphical techniques or the simplex method, but for practical problems computing power needs to be applied. Know that a spreadsheet LP solver routine, or other software, can solve an LP given in standard form or, in some cases, in non-standard form.
  • Be able to formulate a range of network problems as LPs.
  • Be able to interpret the output from a spreadsheet optimisation routine, or other software, for the simplex method or ILPs.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ TBC
TBC
Linear programming 4: Further network applications /
  • Be able to formulate a range of network problems as LPs.
  • Be able to interpret the output from a spreadsheet optimisation routine, or other software, for the simplex method or ILPs.
/ TBC /
  • Exercise level 1
  • Exercise level 2
  • Section test
/ TBC
TBC
Consolidation and revision

AS v1.027/09/2018