Decision Mathematics – Teaching Resources

Linear Programming Grapher

Trying to make things easy to find, I have added two new pages to the Further Mathematics series. This series can be found on the A Level (16+) Tab.

GraphsDecision Mathematics 1Mathematical PreliminariesAlgorithms
NetworksAlgorithms and Graph TheoryGraphs and NetworksNetworks
Network FlowsAlgorithms on GraphsAlgorithmsNetworks and graphs
Linear ProgrammingCritical Path AnalysisNetwork AlgorithmsNetwork Flows
Critical Path AnalysisLinear ProgrammingDecision Making in Project ManagementCritical path analysis
Game theory for zero-sum gamesDecision Mathematics 2Graphical Linear ProgrammingSolving network problems using technology
Binary OperationsTransportation ProblemsThe Simplex AlgorithmLinear Programming
Allocation (assignment) problems Game TheoryThe Simplex Algorithm
Flows in networks
Dynamic Programming
Game Theory
Recurrence relations
Decision Analysis
Decision Mathematics Subject Content
Teaching Resources – GeoGebra Resource from the OR Society

Mathematical Miscellany #44

Some updates this week…

mathsgrader Interactive Assessments

mathsgrader interactive example

MATHSgrader Wk1 Assessment Tier C

From Ross Blair on MATHSgrader comes a complete set of Interactive assessments to complement the Maths Genie homeschool program. The resources are provided in three tiers, Tier A roughly accords with grades 1-2. Tier B with grades 3-5 and Tier C is 6-9. Check the menus for Teachers and Pupils for easy site navigation.

For A Level Owen (Owen134866 on TES) has a library of Mathematics teaching resources, these are really clearly structured with step by step examples. Recently added, is A-level Core Pure Mathematics Year 2/A2 for the second year of Further Mathematics. These resources have been added to the Teaching Resources page for Further Mathematics. This Calculus content is part of the core Pure Mathematics specification common to all examining boards. For more on Further Calculus in Further Mathematics, see my post here.

Also note for Further Mathematics, Jack Brown who has created thousands of videos covering the complete A Level specification has been very busy with his Further Maths collection of teaching videos and exam paper walkthroughs;  the easiest way to navigate the videos is through his website, A recent addition to the exam paper walkthroughs is the specimen paper for OCR MEI Modelling with Algorithms.

My post on Graspable Math proved popular last week and included a canvas of the problems from one of Dave Taylor’s wonderful Increasingly Difficult Question sets. I decided I liked the idea of having a canvas ready for his Simplifying Expressions, starting with IDQ-Simplifying Expressions 1, I opened it on my own canvas and adapted it slightly. I do like to keep all steps of the working displayed, so I have put the exercises on the left, creating a good space on the right.

One Incorrect - Don Stward - first problemTaking a canvas and adapting can help you learn to use the interface if you are unfamiliar with it. Using this One Incorrect canvas by Eric Weitnauer which is based on Don Steward’s one incorrect simplification I made a copy of Eric’s canvas and created one for the first problem from Don Steward’s blog post, discovering along the way that if you enter text for a web link, then Graspable Math inserts that as a link.

Note formulas on the menu.
Graspable Math Formulae

For a clear example of this feature in action, have a look at this video.

HELM Notes

I have referred to the HELM (Helping Engineers Learn Mathematics) notes many times over the years on this blog. They are referred to on the Notes and Examples page for Further Mathematics as well as in many individual blog posts, for example, Mechanics – Dimensional Analysis, Differential Equations, and Further Calculus.

HELM Notes - Basic Algebra

HELM (2008): Workbook 1: Basic Algebra

If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. Whilst the workbooks cover the basic engineering mathematics and statistics teaching for first and second year students in a typical UK undergraduate engineering degree many of the workbooks include content appropriate for A Level Mathematics and particularly, Further Mathematics. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics Learning Support Centre. Alternatively, the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

Looking at Loughborough University’s Mathematics Learning Support Centre as well as providing access to the workbooks, we can additionally read about the past, present and future of the resources and see the details of the HELM consortium members and their roles.

You will see from Loughborough’s description that each workbook varies in length from 25 pages to 75 pages (average 50 pages), and includes Mathematics (and Statistics) for engineering simply explained, worked examples, tasks and exercises with answers provided. Note the last workbooks in the series, Workbook 49 is a Student’s Guide and Workbook 50 a Tutor’s Guide.

The Student guide includes a description of the format of the workbooks and a comprehensive list of contents.

HELM - Student Guide

HELM Project – Workbook 49, Student’s Guide



Technology in Mathematics

An update has been made to the Use of Technology page.

In particular note the GeoGebra resources from Edexcel (and also OCR).

From Edexcel you can use A Guide to using GeoGebra when teaching AS and A Level Mathematics which links to numerous GeoGebra files clearly mapped to the specification content. You can also access an excellent collection relating to the Pearson books (see table below). Whilst linked to the books many of these GeoGebra resources do stand alone, for example check this equations and inequalities example from the Pearson Maths A Level Pure 1 collection.
Edexcel Geogebra Equations and Inequalities

Pearson Maths A Level Year 1
Pearson Maths A Level Year 2
Pearson Maths Statistics and Mechanics Year 1
Pearson Maths Statistics and Mechanics Year 2
Further Mathematics
Pearson Maths A level Core Pure 1
Pearson Maths A level Core Pure 2
Pearson Maths A level Further Pure 2
Pearson Maths A Level Further Mechanics 1
Pearson Maths A Level Further Mechanics 2
Pearson Maths A Level Decision 1
Pearson Maths A Level  Decision 2
Pearson Maths A Level Further Statistics 1
Pearson Maths A Level Further Statistics 2

It is good to see all the GeoGebra resources for Further Mathematics, for example, check Explore toppling and sliding using GeoGebra.

Pearson Maths A Level Further Mechanics 2 Toppling and Sliding

For exploring the Large Data, on AQA’s All About Maths see AQA Large Data Set – Guidance and Worksheets including a tool to help support teaching the statistics content of the specification. This is an amended version of the large data set spreadsheet, the first three sheets are as on the AQA large data set but there are many additional sheets which allow students to explore the large data set. The tools are all provided for students, they do not need knowledge of Excel, meaning time can be spent on interpreting the data rather than learning about Excel. The worksheet on Exploring the Large Data Set could I think be usefully used lower down the school to get students used to working with more data.

A separate page is available on Statistics.

Mechanics – Dimensional Analysis

For UK Examination Boards AQA, OCR A, OCR B MEI Further Mathematics.
This posts gives details of the specifications and teaching resources for Dimensional Analysis.

First a look at the specifications and we see a similarity between all three specifications. For AQA see the first image below. OCR A and B (MEI) additionally mention the relationship between the units of a quantity and its dimensions, with OCR B MEI adding ‘be able to change the units in which a quantity is given’.

AQA: Optional application 1 – mechanics  Assessed at AS and A Level
AQA Dimensions Specifiication
AQA have a very helpful Teaching Guidance document for Mechanics, explaining how AQA have interpreted the content of the specification, the guide provides examples of how the content of the specification may be assessed.

The guidance is available on AQA’s free Maths portal for teachers, All about Maths. The resources are for teachers who offer, or are considering offering AQA maths qualifications. See this page for how to register.

OCR A Mechanics (Optional paper Y543) Assessed at AS and A Level
OCR A Dimensions Specifiication
OCR B MEI Dimensions Specifiication
Note that mapping documents from the legacy specifications to the new specifications are available on this page in the Further Maths series.

Examination Questions
New Specifications
AQA Specimen
questions paper and mark scheme. The only question on Dimensional Analysis is question 2, a multiple choice question on dimensional consistency.
AS Practice Papers and A Level are available on All About Maths

For some Legacy questions and mark schemes see the paragraph on MathedUp below.
The Legacy Mechanics 3 papers which included Dimensional Analysis are available on All About Maths.

OCR A Sample Question paper and mark scheme. Question 3 is on Dimensional Analysis.
OCR B Mechanics Minor (question 3) and Mechanics Major (question 6)
OCR also have practice papers – these require an Interchange log-in.

The Legacy M3 OCR MEI questions are very useful indeed for this topic. (No log-in required.

A Level Maths Revision very usefully provides Further Maths exam questions by topic (scroll down). Legacy questions from OCR MEI and mark schemes are available for Dimensional Analysis.

Further Resources

From amsp this brilliant collection of short videos produced by the legacy Further Mathematics Support Program supports the Further Maths Specification. I have used many of these successfully in class and recommended them to students to support their studies. Look at any of the examination boards to see the coverage for the course. Check the list for your examination board and you can find the videos on Dimensional Analysis for AQA, OCR A and OCR B MEI.

OCR Section Check In Tests and Delivery Guides
OCR have very useful check in tests on Dimensional Analysis with fully worked solutions for OCR A and OCR B MEI. These both seem to use the same questions.

Each check in has 10 questions, the first four questions are are routine procedural questions which primarily assess ‘Use and apply standard techniques (AO1)’ The remaining questions primarily assess ‘Reason, interpret and communicate mathematically (AO2)’ and ‘Solve problems within maths and in other contexts (AO3)’

At least half of the questions are appropriate for the AS course.

OCR A Delivery Guide Dimensional Analysis
OCR B MEI Delivery Guide Dimensional Analysis

These Delivery Guides discuss general approaches to the delivery of the topic, common difficulties and common misconceptions. A good approach, whenever a formula is taught, a dimensional analysis can be performed to check the validity. Both guides have the same suggested list of Further Resources including some good Nrich resources such as this article. The Springer notes are helpful, with clearly explained examples. The University of Guelph resource seems to have an unwanted full stop at the end of the address – the resource is here.

MathedUp – Mohammed LadakMathedupOn Mohammed Ladak’s ‘MathedUp’ see his A Level Further Maths Takeaway, a wonderful source of exam questions by topic with mark schemes. AQA questions have been used here.
A video of a legacy AQA question is provided, also four challenge questions with mark schemes.
AQA Legacy question

Helm notes

For a very comprehensive set of notes with many fully worked examples and exercises, have a look at Section 47 of the HELM Project which was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. For easy access to these resources, the HELM Project Workbooks are hosted by Loughborough University’s Mathematics Learning Support Centre. Alternatively, the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.