The artificial intelligence (AI) platforms of today are largely based on a technology known as artificial neural networks. These neural networks mimic the way that the human brain works. They consist of neurons which takes some inputs and fires a signal to the next neuron if the inputs pass some threshold value. This diagram shows a sample neural network used to classify handwritten numbers. The nodes are arranged in layers with the output of one layer becoming the input to the next layer until the final output layer is reached. The diagram shows a neural network that has been trained to recognise handwritten numbers recognising the number 3.

Image source: IEEE Spectrum Oct 2021

3 - Solving Problems Involving Network Diagrams

Summary

This lesson applies networks to real-life situation. Students learn to recognise and critically think of the application of network in workforce. They also learn to use ICT to build networks from geographic maps. Directed and weighted diagrams are examined throughout the lesson. Learning resources and teaching process are highly differentiated. Famous mathematical problems such as Konigsberg Bridge Problem and Travelling Salesman Problem are also introduced as extension tasks for talented students.

Context

Sequence of Lessons

In Lesson Two, the students were introduced to real-life application of network. This lesson will enhance students' problem-solving ability using different representations with differentiation and reasonable adjustment.

In Lesson Four, students will be introduced to minimum spanning tree in weighted networks.

Prior Learning

Students were introduced to network terminologies and learnt how to translate information from tables and network diagrams. They were also exposed to network problem-solving questions. This lesson enhances students' problem-solving abilities with the use of geographic map, and allows students to gain deeper understanding of how network is applied in future work.

Future Learning

This lesson will introduce students to the notion of shortest path, which will lay foundation of students' learning about shortest paths in lesson Six.

Lesson outcomes

Students learn to:

applies network techniques to solve network problems MS1-12-8
chooses and uses appropriate technology effectively and recognises appropriate times for such use MS1-12-9

Syllabus Content

Students solve problems involving network diagrams

– recognise circumstances in which networks could be used, eg the cost of connecting various locations on a university campus with computer cables

– given a map, draw a network to represent the map, eg travel times for the stages of a planned journey

– draw a network diagram to represent information given in a table

Working Mathematically indicators

Communicating, Problem solving, Understanding

Lesson plan

Time
(min)

0 - 5

Teacher

The teacher asks students to complete glossary worksheet [LIT] with network diagrams, in order to elicit their prior knowledge.

Students

Students independently complete the glossary worksheet and check their answers.

5 - 15

The teacher gives students 3 real-life scenarios, and asks students to draw up corresponding network diagrams using pen and paper. 3 scenarios are ordered by different difficulty levels, and the teacher allow students to do think-pair-share when needed.

3 star real-life scenario (edge representing distance): the internet underseas world, airline route maps

4 star real-life scenario (edge representing cost/time): travel time between stations for inner city trains

5 star real-life scenario (using network to represent geographic map): Konigsberg Bridge Problem

[Differentiation & reasonable adjustment]

The teacher asks prompt questions such as what are the vertices and what are the edges to promote students' thinking.

The teacher encourages students to re-examine each other's diagram and work together to identify erros and come up with the solutions.

The classroom projector need to be set up prior to the lesson to make sure the speakers are able to successfully deliver ideas on their diagrams to everyone in the classroom, including students with vision impairment.

Students choose 1-3 scenarios from the scenario list that teacher provided, and draw up the diagrams independently or in pairs, then share their ideas in front of the class.

15 - 20

The teacher raises the following reflection questions [AFL]:

Q1: Why do we need network diagrams to represent these situations?

Q2: In what other situations we can use network diagrams? Who may need to use network diagrams in their work?

Q3: In Konigsberg Bridge Problem, can you visit each landmass by crossing each bridge exactly once?

Q4: Examine whether the diagrams in the worksheet are traversable, what are the conditions for a network diagram to be traversable?

Extension task: students are given Konigsberg Bridge Problem video to view after class to verify their thinking

Answers:

Q1: Network is a clearer way to represent connection, it helps us better allocate recourses and increase efficiency.

Q2: Local area network (LAN) and wide area network (WAN), representing workflow. IT personnel, project manager, delivery man.

Q3: No.

Q4: All vertices must be of an even degree or there must be exactly two odd vertices.

Students listen and participate in the conversation.

Students review the worksheet.

20 - 25

The teacher then summarises the activity by showing students a real-life application video of how network concept is applied in Information Technology.

Students are required to share interesting points they found by viewing the video.

25 - 40

The teacher asks the class to pick 6 local places on Six Maps [ICT], and to measure distances between each of them. The teacher asks students to work in pairs to determine the shortest path visiting these 6 places (students can pick any points as the starting and ending point), and draw up the network diagrams.

The techer then invites students to share their work and investigation experience, reflection questions include:

Q1: How did you find your shortest path?

Q2: Are there other ways to find the shortest path, with or without using the distance table?

Q3: What are challenges you encounter when looking for the shortest path?

Q4: Imagine a postman use shortest path to deliver mails to different places, are there any pros and cons?

Extension task: Students watch Eddie Woo's classroom demonstration on the Travelling Salesman Problem after class. They retry the problem and submit their work and justification by the next lesson.

Students use worksheet to record data and draw up a directed network diagram with distances on each edges, and calculate the total distance.

40 - 55

The teacher introduces students to the activity: Olympic Torch Tour. The teacher models how to interpret tables into network diagrams for students, then asks students to construct their own diagrams using a different table.

Students investigate the activity and construct their own diagrams, and then check the answers.

55 - 60

The teacher asks students what they've learnt during the lesson [AFL], and conclude the learning content and outcomes for students.

Students do oral reflection to review the classroom content.