Embedded Files
Summary
Summary
Lesson Goals
Lesson Goals
By the end of the lesson, students should be able to:
By the end of the lesson, students should be able to:
Identify networks (and network features) from transport network maps
Define what weighted edges and paths are
Gather and use weight data from online to construct network diagrams
Convert two-way tables into weighted network diagrams
Calculate path lengths between two vertices
Discuss the possible benefits of analysing transport systems as weighted networks
Syllabus Content
Syllabus Content
N1.1 Networks
Students:
identify and use network terminology: vertices, edges, paths, the degree of a vertex, directed networks and weighted edges
solve problems involving network diagrams AAM
recognise circumstances in which networks could be used, e.g. the cost of connecting various locations on a university campus with computer cables
given a map, draw a network to represent the map, e.g. travel times for the stages of a planned journey
draw a network diagram to represent information given in a table
Working Mathematically
Working Mathematically
Recognise networks from real-world transport examples (Understanding)
Identify appropriate vertices and edges from descriptions of transport networks (Understanding, Fluency)
Represent networks in graphical form given real-world maps and two-way tables (Communicating, Fluency)
Gather data about the weight values of edges in a network using ICT (Understanding, Fluency)
Identify possible paths between two vertices (Fluency)
Calculate the total weight of a path by summing the weights of the included edges (Fluency, Communicating)
Explain why it may be inefficient to connect each vertex to every other vertex in a network (Understanding, Reasoning)
Resources
Resources
Transport Network Examples, Portable Whiteboard, Whiteboard Markers, Laptops (see Orientation)
A4-Sized Map of Australia, Laptop (see Body 1)
Weighted Edge Table (see Body 2)
Strategies Used
Strategies Used
Floorstorm
Group Work
ICT (Google Maps)
Class Discussion
Open-Ended Questions
New Vocabulary
New Vocabulary
Weighted Edges
Weighted Edges
"A weighted edge is an edge of a network diagram that has a number assigned to it which implies some numerical value such as cost, distance or time."
Path
Path
"A path in a network diagram is a walk in which all of the edges and all the vertices are different. A path that starts and finishes at different vertices is said to be open, while a path that starts and finishes at the same vertex is said to be closed. There may be multiple paths between the same two vertices."
"A shortest path in a network diagram is a path between two vertices in a network where the sum of the weights of its edges are minimised."
Image: shortest path shown using green vertices and black arrowheads
Lesson Details
Lesson Details
Orientation
Orientation
Working Mathematically: Fluency, Understanding
Floorstorm (5 min) [LIT]
Before the start of the lesson, print out several different transport networks (or have laptops with the websites open) and place them on each table group, along with a portable whiteboard and whiteboard markers. Students should go to their table but not sit down.
When ready, instruct students to use the coloured markers to brainstorm on their whiteboard anything that comes to mind when looking at the photo at their table. After 30 seconds, instruct the table groups to rotate clockwise and brainstorm the next photo, adding anything that the previous group did not write down. Repeat this process until each group has returned to their original photo. Give students a chance to read what was written on their photo, before discussing the following questions as a class (AFL):
What did these photos all have in common?
What do the vertices represent?
Is it possible to find out how far apart vertices are from the maps?
What information would we need/should we add to the maps?
Write students' answers on the board for the class to see.
Body 1
Body 1
Working Mathematically: Communicating, Understanding, Fluency
Vocabulary + Australian Highway Network (25 min) [LIT, ICT]
Explicitly introduce the new networks vocabulary for students to add to their Trip Log (AOL).
Weighted edge - an edge of a network diagram that has a number assigned to it which implies some numerical value such as cost, distance or time."
Path - a walk with no repeated vertices. Open paths are paths that start and finish at different vertices.
Students will be given a map of Australia and a series of highways that they need to find the length of using Google Maps (provide laptops). Using the data they collect, instruct groups to construct a network diagram on their map (the roads do not need to have the same shape; straight lines are fine). Encourage students to work together and use the portable whiteboard to record information. As students are constructing their network diagrams, ask the following questions (AFL):
What are the vertices and the weighted edges in this context?
What else could the weights represent in this context? (e.g. time, cost)
Is the network diagram connected (i.e. can I travel from one vertex to any other vertex)?
Australian Map - Details
Image from WikimediaAustralian Map - Details
Find the car travel distance between the following 7 cities (WORK IN GROUPS):
Perth to Darwin
Perth to Brisbane
Perth to Adelaide
Darwin to Adelaide
Darwin to Brisbane
Brisbane to Adelaide
Brisbane to Sydney
Brisbane to Canberra
Brisbane to Melbourne
Melbourne to Adelaide
Sydney to Adelaide
Sydney to Canberra
Sydney to Melbourne
Canberra to Melbourne
Canberra to Adelaide
Body 2
Body 2
Working Mathematically: Fluency, Communicating, Understanding
Network Drawing (20 min)
Briefly discuss with the class the following question (AFL):
Why might weighted edges be useful to understand for real-world networks?
Project the Weighted Edge Table on the board and show students how to interpret it (e.g. the distance between A and D is 30 kilometres). Instruct students to use the table to construct a network diagram in their workbook, with weighted edges labelled. Monitor students' work and check that they have successfully translated the weighted edge table into a weighted network diagram (AFL).
When students have completed drawing their network diagram, ask the following questions (AFL):
What could the vertices, edges and their weights represent?
What is the total weight of the path from A to E, via B, F and G (i.e. A > B > F > G > E)?
What would be the possible travel distances between vertices A and G (or any other pair)?
Suggest ways that these travel distances could be reduced
Conclusion
Conclusion
Working Mathematically: Understanding, Reasoning
Class Discussion (5 min)
As a class, discuss the following questions (AFL):
Would it be efficient to connect every vertex to every other vertex in a network? Why or why not?
Why might it be useful to redraw real-world networks as network diagrams?
How could the Department of Transport and Main Roads use network diagrams to improve current road or rail networks?
Homework/Follow-Up
Homework/Follow-Up
Remind students to update their Trip Logs (Weighted Edges, Path) (AOL).
Homework:
Students research a transport network online (or take a photo of a map) and offer a 1-3 sentence description of it, using the vocabulary introduced earlier: vertices, edges, degree, weights (if possible). Students should also respond to the following questions:
Are there particular stations (vertices) that are 'hubs', i.e. have a high vertex degree?
Are there particular stations (vertices) that are 'isolated', i.e. have a degree of 1?
Students should submit their response to the teacher at the start of the next lesson, or through Google Classroom (AFL).
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