Mathematical Modelling A guidebook for teachers and teams Peter Galbraith and Derek Holton Contents Preparing students for mathematical modelling 3 What is mathematical modelling? 4 Mathematical modelling framework 6 Introductory problem: Adapting a recipe 9 Junior modelling problems for upper primary, junior or middle secondary 12 Pancakes 13 Hypothermia 14 Köchel numbers 16 Evacuating 18 Intermediate modelling problems for junior or middle secondary 19 Waste not, want not 20 Howzat! 24 How many trees make a newspaper? 27 Bushwalking 30 Farm dams I 32 Suggestions for student-generated modelling 36 Senior modelling problems for middle or senior secondary 37 Nuclear blast 38 Population growth 40 Fifteen-forty 44 Farm dams II 47 Super Size Me 51 Temporary traffic lights 55 Collaboration and communication 57 Modelling report illustrating use of criteria 60 Relating modelling qualities to judging criteria 66 The International Mathematical Modeling Challenge (IM2C) 67 Past IM2C questions 68 2015: Movie scheduling 68 2016: Record insurance 70 2017: Jet lag 78 Additional example problems from the IM2C 83 Published 2018 for ACER and IM²C This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC-BY-NC_SA 4.0). www.immchallenge.org.au Visit http://creativecommons.org/licenses/by-nc-sa/4.0/ Australian Council for Educational Research You are free to: share, copy and redistribute the material in any medium or format; adapt, remix, 19 Prospect Hill Road, Camberwell, VIC 3124 AUSTRALIA transform, and build upon the material. Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes Phone: +61 3 9277 5555 were made. You may do so in any reasonable manner, but not in any way that suggests the licensor www.acer.org endorses you or your use. ABN 19 004 398 145 NonCommercial — You may not use the material for commercial purposes. ShareAlike — If you remix, transform, or build upon the material, you must distribute your Design by ACER Creative Services contributions under the same license as the original. Australian Council for Educational Research 2 Preparing students for mathematical modelling Ultimately, only life educates, and the deeper that life, the real world, burrows into the school, the more dynamic and the more robust will be the educational process. That the school has been locked away and walled in as if by a tall fence from life itself has been its greatest failing. Education is just as meaningless outside the real world as is a fire without oxygen, or as is breathing in a vacuum. 1 Vygotsky The material here has been developed These considerations have influenced the audience. At times suggestions along with two purposes in mind. Firstly, it sets way in which the material is structured. these lines are included – but many others out to provide a basis for the development The mainstream development works are possible and it is the hope that the of workshops geared to those for whom systematically through, and illustrates contexts provide opportunity for individual mathematical modelling is a new by example, those features deemed initiative and development wherever these endeavour. Secondly, it provides support essential for the development of modelling are seen as relevant. material for teachers already working in the proficiency. This procedure gives autonomy This document aims to provide teachers field. The material may be of interest to: to the user in deciding if, when, and with material that will enable them to how much of the additional material is ■ individuals who want support prepare students to develop skills of appropriate for purposes of the moment. materials and ideas for teaching mathematical modelling. It is an espoused In particular it means that those already modelling up to senior secondary level goal of curricula that students are enabled working in the field can skip content, to apply the mathematics they learn to ■ pre-service and in-service education or indeed sections, according to their address problems in their personal lives, as courses featuring components in priorities and background. Ultimately it productive citizens, and in the workplace. mathematical modelling means that users are not forced to engage with everything that is included, while the In a specific international initiative to ■ mentors who intend to enter teams in 2 mainstream development does aim to support this ideal, an International the IM C 2 2 provide a complete package, beginning Mathematical Modeling Challenge (IM C) ■ teacher organisations, such as the from no assumed knowledge and building was instigated in 2014. A specific purpose Australian Association of Mathematics to sophisticated application of modelling of this material is to provide additional Teachers. techniques. Finally, problems can often be support to assist teachers and schools in adapted – simplified or extended according choosing and mentoring student teams to 2 to circumstance, intention, or target participate in the IM C. 1 Vygotsky, L.S. (1926/1997). Educational psychology. Boca Raton, FL: CRC Press LLC. 2 Note the American spelling. We use this for the IM²C title as this is used internationally. We have also used American spelling when we are using problems that have been sourced from America. Australian Council for Educational Research 3 What is mathematical modelling? It is the mark of an instructed mind to rest satisfied with the precision which the nature of the subject permits and not to seek an exactness where only an approximation to the truth is possible. Aristotle Mathematical models are used to help problem solving, the IM2C program begins In the field of education a variety of make decisions in a variety of new from an assumption that mathematics meanings have been attached to the developments. Governments consult is everywhere in the world around us – term mathematical modelling. At a very mathematical modellers to predict the a challenge is to identify its presence, simplistic level the term ‘mathematical consequences of new initiatives, such as access it, and apply it productively. The model’ has been used in the sense of a change of the tax system; engineers OECD, the American Common Core a formula. Another common notion is use mathematical models to build bridges Standards Initiative, the Australian associated with mathematical modelling and multi-storey buildings; and social and Curriculum, and other national curricula used as a vehicle to achieve other biological research relies on a great deal all avow that students should have a curricular goals. As such the purpose is of modelling that uses statistics. For a mathematical preparation which equips not the construction of mathematical good article on the importance and use of them to use their acquired knowledge models to solve problems per se, but mathematical modelling, see ‘5 Reasons in their personal lives, as citizens, and in rather to contrive the embedding of to Teach Mathematical Modeling’, by the workplace. some pre-determined mathematics in a mathematics professor Rachel Levy.3 This contextual setting as a mechanism for the Such a purpose implies two intersecting is well worth the read for both teachers learning of certain mathematical concepts, goals. Firstly, to develop a systematic and students. procedures and so on. and successful approach to addressing Children are natural modellers. individual problems located in real-world Mathematical modelling approached For example they know that often settings, and secondly, through this as content entails the construction of ‘1 = 2’ provides an equitable solution to the means, to enable students cumulatively mathematical models of natural and social problem of sharing lollies of different sizes to become real-world problem solvers. phenomena that are problem driven, and (one big lolly equals The latter means that they not only can where the choice of relevant mathematics two smaller ones). Yet formal schooling address productively problems set by is itself part of the solving process. It so often robs initiative in the name of others, but become able to identify and is this type of modelling that the IM2C conformity. As with all those who value address problems that matter to them. initiative seeks to promote and reinforce. mathematical modelling as real world 3 Levy, R. (2015, May 5). 5 reasons to teach mathematical modelling. American Scientist. https://www.americanscientist.org/blog/ macroscope/5-reasons-to-teach-mathematical-modeling Australian Council for Educational Research 4 Mathematical modelling is: Here a mathematical model is represented in three parts: ■ a process in which real-life situations and relations in these 1. an extra-mathematical domain (D) situations are expressed by using mathematics (Haines & 2. some mathematical domain (M) 4 Crouch ), or 3. mapping from the extra-mathematical domain (D) to the ■ a cyclical process in which real-life problems are translated mathematical domain (M), resulting in outcomes that are into mathematical language, solved within a symbolic system, translated back to the extra-mathematical domain (D). and the solutions tested back within the real-life system The extra-mathematical world is a useful way of representing 5 (Verschaffel, Greer, & De Corte ). that subset of the wider ‘real world’ that is relevant to a particular In both instances, mathematical models are seen to move problem. By real world we mean everything that is to do with beyond the physical characteristics of a real-life