Mathematics, Computing and Technology/Science MST210 Mathematical methods, models and modelling MST210 TMA 07 Contents Cut-off date 2 TMA MST210 07 (modelling activity assignment) 27 April 2016 The module website gives details of how to submit assignments for this module. In order to encourage you to present your report in a good mathematical style, your tutor will comment on how you: • use correct mathematical notation • define any symbols that you introduce in formulating and solving a problem • give references for standard formulas and derivations • include comments and explanations within your mathematics • explicitly state results and conclusions, giving answers to an appropriate degree of accuracy and interpreting answers in the context of the question • draw diagrams and graphs. These features are seen as being essential to complementing your mathematical skills. Your tutor will make comments on how well you achieve these objectives and give you guidance on how to satisfy them. Five of the marks for this TMA will be allocated to the way you write your report. It is expected that most students will receive the majority of these presentation marks; such marks are included in TMAs to encourage, and emphasise the need for, thinking about how you present your mathematics. Copyright ! c 2015 The Open University WEB 04386 2 2.1 TMA MST210 07 Cut-off date 27 April 2016 This assignment covers mathematical modelling. Question 1 – 10 marks Briefly describe how your modelling group worked together and your role within the group. The marks awarded for this question relate to the group activity. Question 2 – 85 marks You should extend and complete your modelling work on one of the two problems listed on page 3, and write up your work in the form of a report of up to 2000 words. In no circumstances should your report exceed 3000 words; you may be penalised by up to 10 marks if it does. If a report obviously exceeds that limit, then your tutor will not mark any material beyond 3000 words. (These word limits may be taken as excluding the captions for diagrams or graphs, linking words between equations, and any appendices.) You should normally base your report on the group work of the modelling activity, but you are under no obligation to do this. The mark scheme for this report is given on page 4. This mark scheme shows a typical outline for the report. Your report will actually be read (and marked) by your tutor, of course, but in writing your report, you may find it helpful to imagine that it is to be read by another student. This should allow you to make sensible judgements about the amount of mathematical detail to include. Your report should explain to your tutor clearly and fully what results you obtained and how you obtained them. Remember that your tutor cannot award marks for work that he or she cannot comprehend or for steps that are missing. It is your responsibility to explain your work satisfactorily. This report is in some ways different to other assignments in that you have a great deal of freedom as to how to express yourself. Take care to be consistent in your choice of words and symbols. You do not need to aspire to exacting standards of literary style. Your tutor will not worry unduly about occasional spelling mistakes. On the other hand, you should be concerned with those aspects of writing that are closest to mathematics: being systematic; laying out your work in a logical fashion; making sure that each step of an argument or calculation follows from the previous one; avoiding any obscurity or ambiguity in the use of symbols and technical terms; using appropriate diagrams and tables. Moreover, no piece of writing is going to make much sense if it contains lots of meaningless or ambiguous sentences, so you should also ensure that what you write is clearly expressed. Your report does not have to be drafted in the order in which it will be finally presented and read. Most people find it quite difficult to write anything longer than a postcard by starting at the beginning and keeping steadily on until the end is reached. It is usually necessary to make at least one rough draft, and to work backwards and forwards through the draft as the ideas develop. If you proceed in this way, then you will find that the resulting account of your model is more comprehensive and better structured than if you just start at the beginning and carry on to the end. page 2 of 5 Take every opportunity to include figures and illustrations in your report. You can give sketches of graphs, drawings of situations, diagrams of processes and relationships, and computer output. Explanations of complicated derivations are often easier to understand if they are accompanied by appropriate diagrams. Data, numerical results and lists of variables are usually best displayed in tables. You may find it helpful during your later work on this report to find one or two (non-mathematical) friends who might be interested in the problem that you are going to solve. Persuade them to be your audience, and ask them to read your report when first drafted. Your audience will be useful for discussion when you become stuck, and should help you to produce a report that is clear and easy to understand. Your report should have the following section headings, which are based on the stages of the modelling cycle: Specify the purpose Create the model Do the mathematics Interpret the results Evaluate the model Revise the model Conclusions You should choose one of the following problems to address in the modelling activity. Problem 1: Safety zone around playground swings Children or objects that children have with them (e.g. sweets, trainers, purse) may well fall off a playground swing when it is in motion. In fear of possible litigation, a town council is proposing to establish a soft landing zone around a swing, so that if a child or an object does fall off it, then any consequences of such an accident are reduced. Advise the council on the minimum area that should be established as a soft landing zone around a playground swing. Problem 2: Yellow lines When one is travelling by bus or car along a major road, one often sees on the approach to a roundabout a succession of yellow lines painted across the carriageway. These warn the driver to slow down before reaching the roundabout. However, the lines are intended to do more than just warn: they are designed to force the driver to decelerate by creating the impression that the vehicle is going too fast otherwise. In order to do this, the lines are positioned progressively closer together as they get nearer to the roundabout. Crossing the lines provides a very strong visual clue to a vehicle's speed (and also an auditory clue, as the lines are usually painted so as to produce a click as the vehicle passes over each one). If a vehicle is approaching the roundabout at a constant speed, then the lines come past more and more quickly, giving the driver the feeling that the vehicle is accelerating. To compensate, the driver will tend to drive so that the lines come past at a constant rate. By careful spacing of the lines, therefore, drivers can be made to slow down on the approach to the roundabout. Work out exactly how the lines should be spaced to take the greatest advantage of this effect. page 3 of 5 Mark scheme The following mark scheme assigns a number of marks to each of the section headings mentioned previously. Under each section heading, some instructions are given as to what the section should contain. You are not expected to follow these instructions to the letter; how you write your report will depend to some extent on the problem that you have chosen and the model that you have developed. Specify the purpose 5 marks • Define the specific problem to be solved. Write a clear, succinct statement of the specific problem addressed in your report, in your own words (do not just repeat the specification on page 3). • Describe the features that you are going to investigate. Give some indication of the approach used to create the model. Create the model 30 marks • Outline the mathematics to be used in the model. Give a qualitative description of the approach to be used in the first model, to explain why and how the first model will be formulated. • State assumptions. Create a numbered list of clearly stated assumptions used in the model (take care not to miss assumptions or include assumptions that are never used). Do not attempt to justify assumptions here. Data values should not appear in assumptions, so, for example, 'the width of the road is 10 m' should be replaced by 'the width of the road is constant'. • Choose variables and parameters. Create a table of all symbols used in the model. For each symbol, state a clear definition and its associated units. It is not necessary to distinguish between variables and parameters. • Formulate mathematical relationships. Derive relationships between your variables and parameters. You should explain how the equations follow from your assumptions, which should be referenced. Do the mathematics 10 marks • Derive a first model. Solve your first model to find the variable of interest (as specified in the purpose of the model) in terms of other variables and parameters. Clearly state the mathematical model derived. It is not necessary to have one overall explicit equation; it is possible to have a series of equations, which may aid clarity, or an implicit equation (that will be solved numerically). Your solution at this stage should not include particular data values. • Draw graphs showing typical relationships. Sketch graphs to show the expected variation of variables predicted by your model. Use typical values for any parameters. • Check your model using dimensional analysis. Interpret the results 10 marks • Collect relevant data for parameter values. Usually relevant data is available on the internet (or in the library), in which case the source should be referenced. For some models it is easier to perform a simple experiment, in which case the deduced parameter values should be stated here and the experiment should be described in an appendix. • Describe the mathematical solution. Substitute data into your model to find a solution. Clearly state in words this solution and how it relates to the purpose of the model. This should be written in a form that could be understood by a lay-person, by presenting it, for example, as a set of instructions, a graph or a table. page 4 of 5 • Find predictions to compare with reality. Look for any predictions of your model, or part of your model, or a corollary of your model that may be tested. Evaluate the model 15 marks • Collect data to compare with the model. Collect additional data to test your model. Do not use the data used to define parameter values. As before, the additional data can be from the internet, the library or an experiment. • Test your first model. Compare model predictions with the additional data that you collected. Some models may be impossible to test in this way, in which case you should explain why it is not possible to test your model. Some marks are available for describing an experiment without actually being able to perform it. • Criticise your first model. Criticise your model based on the tests that you performed. • Review your assumptions. Consider each assumption in turn, and explain what would be the effect of changing it – would it improve the model to fit better with the evaluation? Revise the model 10 marks • Decide whether to revise your first model. Decide whether a revision of your first model is justified. Explain why you made your decision, referring to the evaluation of the first model and your review of the assumptions. If your first model fits your data well, then consider if a simpler model might be better. • Describe your intended revision. Include a clear statement of any assumptions that are being revised and the new assumption(s) that will replace them. Note that a change of a parameter value does not constitute a revision of the model. Conclusions 5 marks • Summarise your modelling. Include the performance of your first model, any attempts to improve on it, and any comments on the modelling process. This short summary should not introduce any new considerations. page 5 of 5