Nuffield Mathematics teaching resources are for use in secondary and further education

# FSMQ Level 3 (legacy) Using and applying decision mathematics scheme of work

This FSMQ requires a total of 60 guided learning hours that could be used in a variety of ways such as 2 hours per week for 30 weeks, 4 hours per week for 15 weeks, or 5 hours per week for 12 weeks. A suggested work scheme showing topics and methods to be covered is given below but the order and time allocations can be varied to suit different groups of students.

Note that you will also need to allow time for students to complete a Coursework Portfolio. The AQA Coursework Portfolio requirements are listed in the Advanced FSMQ specification for 'Using and applying decision mathematics' and in the AS 'Use of Mathematics' specification. The assignment below provides an example of the sort of work your students could include in their portfolios, but if possible you should use work that is more relevant to their other studies or interests.

Before starting the course students are expected to be able to:

• substitute values into formulae
• collect and organise data
• rearrange basic algebraic expressions.

Throughout the course students should carry out work that involves them in:

• defining areas of investigation and selecting appropriate data to use
• drawing appropriate networks and carrying out analyses using an algorithmic approach
• drawing conclusions and critically summarising findings.
 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Introduction to networks (2 hours) Represent situations using networks. Use terminology such as vertices (including the degree and odd/even), edges, edge weights, paths and cycles. Consider connectedness, directed and undirected edges and graphs. Store graphs as matrices (eg adjacency/distance matrices). Networks   Presentation introducing networks and matrices and the associated terminology. Hand-outs and examples for students to try. Trees and spanning trees  (8 hours) Understand that a tree is a connected graph with no cycles and that every connected graph contains at least one tree connecting all the vertices. Find minimum connectors using · Prim’s algorithm in graphical and tabular form · Kruskal’s algorithm. Appreciate the relative advantages of Prim’s and Kruskal’s algorithms. Understand when a situation requires a minimum spanning tree. Comment on the appropriateness of solutions. Cable TV   Presentation, data sheet and worksheets introducing and using Kruskal’s and Prim’s algorithms in real-life situations.

 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Shortest paths (6 hours) Appreciate that problems that involve finding paths of minimum time and cost can be considered to be shortest path problems. Apply Dijkstra’s algorithm using a labelling technique to identify the shortest path. Comment on the appropriateness of solutions. Shortest path  Presentation showing how to use Dijkstra’s Algorithm. Hand-outs and practice questions for students. Route inspection problem (8 hours) Appreciate the connection with the classical problem of finding an Eulerian trail. Understand the significance of odd vertices and solve problems with 0, 2 or 4 odd vertices. Apply the Chinese Postman algorithm and comment on the appropriateness of a solution. Chinese Postman problem  Two activities to introduce or revise the Chinese postman algorithm. Slide presentation helps class discussion and demonstrates possible solutions. Travelling salesperson problem  (8 hours) Appreciate the connection with the classical problem of finding a Hamiltonian cycle. Determine upper bounds by using the nearest neighbour algorithm. Determine lower bounds (finding the length of a minimum spanning tree for a network formed by deleting a given node and then adding the two shortest distances to the given node). Appreciate when a solution is sufficiently good (realising that a solution is not necessarily the best and commenting on its appropriateness). Sightseeing tour      assignment or classroom activity   Students plan the itinerary for a sightseeing tour in their local area. Critical path analysis  (10 hours) Construct activity networks. Find earliest and latest times (using forward and reverse passes).  Identify critical activities and find a critical path (calculation of floats). Construct and interpret cascade diagrams. Refurbishing a room  Presentation and activity to introduce critical path analysis. Revision (6 hours) Revise topics. Work through revision questions and practice papers. Discuss the Data Sheet - make up and work through questions based on it.

Page last updated on 25 January 2012