Mathematics Developmental Continuum P-10

Networks: 4.0

Supporting materials

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Indicator of Progress

Students can create networks to show relationships, and can interpret networks to draw conclusions about those relationships. Success depends on students being able to identify clearly the relationship which a network is recording and to be clear what the network does and does not represent.

The simplest networks represent spatial information in a pictorial way. As students progress, they gradually come to deal with networks that depict more abstract relationships between more abstract entities.

The main goal for learning about networks is to develop students’ abilities to record and organise information. This is an important generic problem solving skill in adult life.

There are many examples of networks given below. In each case, the network consists of nodes joined by lines according to a clearly defined rule. See more information about networks.

 

Illustration 1: A simple network is like a picture

Nine Men’s Morris is a very old board game. Each player has 9 men (little pegs), who move around the board, from hole to hole capturing each other. The arrangement of holes and the allowable directions of the moves are shown here.

The Nine Men’s Morris is a network. The nodes are the holes and the lines show in which direction the men are allowed to move.

This network is simple to interpret, because it is just like a picture.

 

Illustration 2: Networks are not maps

This network shows the places that Mat can go to on his bike (the network nodes), with the riding times between places shown. Lines are drawn between nodes with a direct bike path between them.

It is not a map of the town. In exploring the differences, students can explain why proximity on the network does not indicate short times. They can use the network to find the quickest times between places (e.g. from school to river), but they cannot make conclusions that are not about time such as distance or location.

 

 

Illustration 3: Networks strip away unimportant information

Here is a plan of my house (showing the doors between the rooms) and a network that shows how the rooms are connected. The nodes are the rooms, and they are joined by a line if you can walk directly from one to the other. If I am only interested in how to go from one room to another, then this is all the information I need.

In a simple house plan like this, I can see from either the plan or the network that there are two ways to go from the laundry to the lounge room. However, if this were a very complicated space with hundreds of rooms, the network would be essential. Networks that simplify are an important tool for industry.

 

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Teaching Strategies

Networks record, organise and present information for analysis. Most learning about networks will occur in the context where the other information arises.

At each stage the central questions are:

  • What information do we want to represent (or what has been represented in a prepared network)?
  • What is the best display for representing this information? Possibilities include map, Venn diagram, Karnaugh map, or network (including tree diagrams etc).
  • If a network is chosen,
    • what will the nodes represent?
    • what relationship will the lines represent?

Activity 1: Transport suggests the types of networks that may arise in work on this theme.
Activity 2: Families suggests the types of networks that may arise in work on this theme.
Activity 3: Choosing the best representation presents some information that has been represented by a Venn diagram, a two way table and a network diagram.
Activity 4: Combinations and compound events introduces important uses of tree diagrams.
Activity 5: Mathematical examples suggests that a network approach to organising mathematical information leads to good visual learning and prompts investigations.

 

Activity 1: Transport (example of a theme)

A road map is a type of network, if we think of the towns as the nodes and the connecting roads as the lines. If it is only the connections that are important, then the positions and distances need not be geographically correct. This is usually the case for networks showing public transport.

Resources include many available on the internet, such as:

Students can use these ‘maps’ when working on a transport theme, e.g. in planning a trip. Discuss with students how these networks differ from a real map and how they are similar. The suburban train map, for example, shows which stations are directly connected by trains. The nodes are the stations and the lines represent train lines. It does not show the correct distance between stations, and they are only very approximately in the correct geographical position. It uses colour to show the fare zones.

Students can make their own transport networks (see Illustration 2).

 

Activity 2: Families (example of a theme)

Family trees are a special sort of network diagram. The nodes are the people in the family and the lines indicate the relationship ‘is a child of’. There are also different lines, which indicate the relationship ‘is married to’.

In the context of work on families, students can draw part of their own family tree (some sensitivity required here!) and use it to answer such questions as:

  • Who else is the grandchild of my grandmother? How can I see that on the family tree?
  • Who is my mother’s grandmother? How can I see that on the family tree?

Another network diagram that might arise from work on this theme could represent migration and movement around Australia. The network below accompanies this story: “My dad came from Italy and my mum came from Germany, and they got married in Shepparton. I was born there, and then we moved to Wodonga for 2 years, and came back to Shepparton. One of my older brothers has gone to live on the Gold Coast and my sister lives in Port Fairy.”

 

Activity 3: Choosing the best representation

Information can often be presented diagrammatically in several different ways, and so it is important that students learn to select good ways.

There are three representations below relating to the same information. Two train lines, LongLine and ShortLine, start in different suburbs (A and P). They both go to the city (stations D, E and F) and then they continue in different directions to H and S. Information about this situation can be presented in a network diagram, a Venn diagram or a two way table (karnaugh map). In this case, for most purposes, the most useful is the network diagram, which shows the way that the stations are connected by trains, not just how many stations are connected or which is on each line.

However, the network diagram only tells us how the stations are connected by trains. It does not tell us how long the lines really are, for example.

Students can take a network (e.g. a transport network) and reconstruct the information as a Venn diagram and as a table. They can then report on the usefulness of each representation for different purposes.

 

Activity 4: Combinations and compound events

Students can use networks to describe compound events and combinations, as in the examples below.

Showing all combinations
A tree diagram is very useful for working out all the possibilities in situations of multiple choices. For example, they can produce networks which show the all the possible combinations in situations such as:

  • making salad sandwiches (with or without tomato etc)
  • choosing teams of 2 boys and 1 girl from 3 boys and 2 girls
  • select a sandwich, a drink and a piece of fruit from a small menu.

Example: If a shop offers 3 sorts of ice-cream and 2 toppings, then there are 6 different possibilities. All of the combinations can be found by reading along each path of the network. The top path is for vanilla ice-cream with nuts, the next is for vanilla ice-cream with sprinkles and the 6 th is for chocolate ice-cream with sprinkles. The set of all 6 possibilities is called the Cartesian product of the set of all combinations of the set of three ice-creams and the set of two toppings: Cartesian product = {(vanilla, nuts), (vanilla, sprinkles), (strawberry, nuts), (strawberry, sprinkles), (chocolate, nuts), (chocolate, sprinkles)}

Showing compound events
A tree diagram is a type of network that is very useful for keeping track of possibilities and choices. It is the sort of network diagram that will be very important for students when studying probability. Here is a tree diagram made to represent the possible events that can happen on a student’s way to school. There are two buses (the earlier and the later) that he can catch to the station and two trains (the earlier and the later) for the rest of the journey

At this level, the intention is to display the possible events in an organised way. At a more advanced level, the tree diagram is annotated with the probability of each event. The diagram helps work out the probability that he arrives at school on time or not.

 

Activity 5: Mathematical examples

There are many ways in which mathematical information can be presented graphically. They help organise concepts and information. Here are some examples using networks.

Factor Networks
Students can see the differences in the factor structures of numbers by presenting them in a network as shown below. This also highlights the importance of prime factors.
In these networks the nodes are numbers and a line is drawn if a number is a factor of another. In this example, the network lines are really arrows, because each line has a direction (6 is a factor of 12, but 12 is not a factor of 6).
Some of the lines have been omitted in these networks (e.g. 1 is a factor of every number but there is no direct line drawn). Looking at the factor networks gives rise to many questions, with important mathematical answers relating to prime numbers.

Sample Investigations on Factor Networks

  • What numbers have factor networks that are just in one line, like 32? (Answer: powers of prime numbers).
  • What numbers have factor networks that are diamonds, like 15? (Answer: a prime multiplied by a prime)
  • The factor network for 24 has some ‘cross-overs’, as does the factor network for 18. Which numbers have factor networks with and without cross-overs? (Answer: they all have cross-overs except for the diamonds

Sets and subsets
Relationships between sets of geometric figures can be represented as a network. Students can make their own networks, to clarify their understanding of how geometric shapes are related. It is also important that they interpret networks, according to the meaning of the connections between nodes.

For example, the network below shows which sets of triangles are subsets of other sets.
A set is joined to another if it is one of its subsets.
One of the difficulties that students will encounter in making or interpreting this network is to appreciate how it is different to the Venn diagram for the same relationships (given below).
In situations where there is potential confusion such as this, teachers need to develop each idea separately and strongly, and then bring them into contrast so students see the differences between them.

 

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Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn object *

Journey Planner: quickest route 1 – students help two children travel around town. They look at a map, then check bus and train timetables. They choose the fastest route.
(https://www.eduweb.vic.gov.au/dlr/_layouts/DLR/Details.aspx?ID=3184)

Journey Planner: quickest route 2 – students help two children travel around town. They look at a map, then check bus and train timetables. They choose the fastest route.
(https://www.eduweb.vic.gov.au/dlr/_layouts/DLR/Details.aspx?ID=3183)

* Note that Digilearn is a secure site; DEECD login required.