Our Networks and Paths Unit of Work begins with the famous Königsberg bridge problem (Ferrarello & Mammana, 2018), whose solution (by Leonhard Euler in the 18^th century) is widely recognised as the beginning of Graph Theory (Harris, Hirst, & Mossinghoff, 2008; Niman, 1975; Wilson, 2003). The problem itself demonstrates several concepts of Networks & Paths, including vertices (land), edges (bridges), Eulerian trail and the degree of a vertex. Initially depicting the problem using a picture of the bridges also reinforces the visual connection between tracing out a journey across the bridges and network-related ‘walks’ (Ferrarello & Mammana, 2018). From this concrete depiction, students can then simplify it by reducing the landmasses to points and the bridges as lines connecting the points (Watson, 2018). The bridge problem serves as an introduction to explore the general concept of networks: interconnecting systems of objects or places (Watts, 2004a).

Researching the topic of Graph Theory revealed the wide applicability of many of the Network and Paths concepts in different fields, including:

• Biology (e.g. food chains and webs, metabolic networks, neurons, trees and plant structures, universal tree of life, the spread of disease, ant colonies, slime moulds) (Cozzens & Koirala, 2018; Niman, 1975; van Steen, 2010; Watt, 2004a, 2004b)

• Chemistry (e.g. covalent network solids, molecular structures) (Niman, 1975; Wilson, 2003)

• Computer Science (e.g. Internet and hyperlinks, email and instant messaging, routers and cables) (Ferrarello & Mammana, 2018; Watt, 2004a, 2004b)

• Council Planning (e.g. connecting homes with roads, electrical and internet cables, water & gas pipes) (Niman, 1975; Watt, 2004a, 2004b)

• Public Transport Design (e.g. trains and train stations, buses and depots, ferries and ports, airports and flight connections, highways) (Ferrarello & Mammana, 2018; Quinn, 2015; van Steen, 2010; Watt, 2004a)

• Sociology (e.g. social networks, criminal networks, six degrees of separation, family trees) (Watt, 2004a, 2004b)

From this, we dedicated our lessons towards focusing on these areas in order to demonstrate how Graph Theory can provide a new perspective and means of analysing everyday (and technical) systems (van Steen, 2010; Watts, 2004a, 2004b). For example, applying Graph Theory to food webs (Cozzens & Koirala, 2018) reveals the importance of particular species and the effects of their removal from ecosystems. Graph Theory also enables the efficient design and usage of real-world transport, power and communication networks (Watts, 2004b). Within these focused lessons, we include a mixture of practical (Cozzens & Koirala, 2018; Ferrarello & Mammana, 2018; Science in School, 2014; Trim, 2009) and ICT-based (Quinn, 1997, 2015) activities that allow students to both create representations of and analyse networks.

Our emphasis on teaching Networks and Paths within the context of real-world applications reflects the ever-growing belief that at all levels of mathematical study, students should be studying real-world problems (Ferrarello & Mammana, 2018). Broadly speaking, students will be provided opportunities to translate real-world problems into problems solvable using mathematical knowledge and skills, and also learn to interpret solutions in terms of the context of the problem.

The Networks and Paths unit contains a significant amount of new technical terminology (Gross & Yellen, 2003; NESA, 2019), with some of the key terms (such as vertex, edges, degree and weight) having separate meanings in different mathematics topics, while others being unique to Networks and Paths. Additionally, these new terms have numerous modes of representation, including worded descriptions, images, tables (Tammadge, 1966) and matrices (Quinn, 2015). To navigate these literacy and numeracy demands, students will be asked to create and regularly update a 'Trip Log' (Ferrarello & Mammana, 2018) with new definitions, example diagrams, and evidence of their research into real-world network examples.

What do Facebook, Sydney Trains, the Internet, food webs, molecular structures, and the human brain all have in common? They are all examples of networks: systems of interconnecting objects or places.

The topic of Networks & Paths provides students with a unique perspective from which they can mathematically analyse the human and natural world. Students will learn new terminology and algorithms from the field of Graph Theory, and apply them in analysing or solving problems related to networks. In doing so, students are given opportunities to develop their skills in determining and communicating solutions applied to real-world contexts, e.g. connecting homes to a water supply whilst minimising the cost of installing pipes. Additionally, students will develop their geometric intuition, visualisation and problem-solving skills and their ability to negotiate different representation forms.

By studying Networks and Paths, students can also learn to recognise and appreciate the commonalities between distinctly separate real-world networks, e.g. rail networks and friendship connections. The language and tools given to students allow them to analyse networks from many fields of study, including physics and biology, economics, sociology, computer science and engineering. Additionally, developing students' understanding of Networks and Paths provides them with a foundation to explore the related topic of Critical Path Analysis or other fields of study such as computer science.

Additional Resources Worth Checking:

• Hart, E. W., Maltas, J., & Rich, B. (1990). Teaching discrete mathematics in grades 7-12. The Mathematics Teacher, 83(5), 362-367.

• Levenkova, N. (2014). Applications of graph theory to real-world networks (Masters thesis, University of NSW, Australia). Retrieved from http://unsworks.unsw.edu.au/fapi/datastream/unsworks:12628/SOURCE02?view=true

• McDuffie, A. R. (2001). Flying through graphs: An introduction to graph theory. The Mathematics Teacher, 94(8), 680-688.

• Peters, J., & Metz, D. (2015). Using Graph Theory to Understand First Nations Connections. Mathematics Teacher, 109(4), 311-313.

• Picker, S. H. (2008). The Paris bridge sweepers. Mathematics in School, 37(2), 30.

• Smithers, D. B. (2005). Graph theory for the secondary school classroom (Masters thesis, East Tennessee State University, USA). Retrieved from https://dc.etsu.edu/cgi/viewcontent.cgi?article=2172&context=etd

• The Royal Society. (2013). Webs. Retrieved https://www.stem.org.uk/system/files/elibrary-resources/legacy_files_migrated/35236-LifesHiddenOrder_Webs_Introandactivity.pdf