# Geometry of water sources and landforms

# Geometry of water sources and landforms

A detailed knowledge of the availability of seasonal and permanent freshwater bodies is critical to the survival and culture of Aboriginal and Torres Strait Islander people.

Water sources are of special significance not only for drinking and survival, but also as places to meet and conduct ceremonies. Mathematical concepts of geometry and spatial positioning were applied for the use and management of these features.

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Aboriginal and Torres Strait Islander people applied several mathematical concepts, including geometry, time dependant variations and spatial location. The application of these concepts included understanding the capacity (volumes), water quality and seasonal variations in the levels of waterholes and springs. Many reliable water sources were spatially mapped using songlines to ensure the water source locations were remembered and this knowledge able to then be passed on to others.

### Waterbodies

There is a wide range of waterbody types across Australia and these can vary greatly in size, quality and permanence depending on local environmental conditions. Aboriginal and Torres Strait Islander people developed expertise in finding, maintaining and managing these resources for survival. Some typical waterbody types and their uses are described below. These descriptions are adapted from the Wetrocks resource^{[1]} and more detail regarding these, and many other waterbody environments, can be found in that document.

Desert waterholes – these features are often found within hill ranges in the arid interior of the country. They remain long after creek and river flows have subsided following flood events, and can be permanent to semi-permanent depending on climatic conditions. Because of their size they are able to sustain large populations of people for long periods of time, attract a range of fauna food sources as well as sustaining a variety of flora. These environments are therefore able to provide shelter and water, and support species used for food, medicine, tools and clothing.

Soaks and native wells – these can be located in many environments including ephemeral river beds, sand dunes and at the base of rocky outcrops. They can range in scale from less than a metre across to many tens of metres. Native wells describe the practise of accessing these resources, including digging to expose greater volumes of, and better quality, water. They could be filled with sticks, stones and even capped to reduce evaporation and prevent fouling.

Hanging swamps – are formed where rain percolating through rocky outcrops can discharge and form pools at altitude on exposed cliffs. In higher rainfall areas they could be a virtually constant year round source of water for Aboriginal and Torres Strait Islander people.

Subterranean cave systems – when the caves and caverns are eroded to a depth below the water table then these systems can allow access to usually reliable sources of good quality groundwater. The Nullarbor Plains and some areas in the southeast of SA are underlain by limestone cave system which contains various water bodies.

The locating and management of reliable water sources was critical to the survival and cultural expression of Aboriginal and Torres Strait Islander people. Waterbodies in various natural forms are able to provide almost everything required to not only survive but to flourish, including water and food, medicines, tools and clothing. It is not surprising then that these environments feature heavily in many songlines and dreaming stories. These oral records contain not only the locations and capacities of the resources, but in many cases also provide guidance on how to manage and protect the resources.

*Tjilbruke and the Coastal Springs* is a prominent example of a songline used for the mapping and marking of water sources (springs) along the coastal environment in South Australia. The springs mark the locations where Tjilbruke shed tears following the death of his nephew. There are many similar songlines within Aboriginal culture which relate the creation, location and nature of waterbodies^{[2]}.

Art works are commonly found on rock walls adjacent to or near waterbody environments, as these were often used as ceremonial gathering places and were therefore logical places to document information through the art work. The desert waterholes of the NT are a prominent example of this. Circles feature prominently in Aboriginal and Torres Strait Islander culture and art. Places of gathering and water holes are two examples of culturally significant features which are often depicted with circular symbols. Further detail on these cultural aspects of groundwater ecosystems can be found in the Wetrock resources.

## Classroom activity - Mathematics Year 10

Students will apply various techniques of geometry, trigonometry and algebra to investigate the nature of idealised water body shapes. The techniques can also be applied to describe angles and heights of landforms associated with water body features.

In this classroom activity, students will analyse several idealised waterbody and landform shapes to determine surface areas and volumes of regular and composite solids. Based on the calculations, students can also be asked to discuss the use of typical waterbody types as reliable water sources, and the seasonal variations. The calculations used for the example landforms can be used to analyse the most efficient ways to traverse difficult terrain.

The suggested activity utilising State / Federal water resource databases will enable students to search for, obtain and analyse time series monitoring data for known water sources (e.g. springs, wetlands). Ideally the water sources would be located along or close to publicly listed songlines. The data can be statistically analysed to determine mean, minimum and maximum values and to explore relationships between water quality and water quantity variables, against time and against each other. Conclusions can then be made regarding the suitability of selected water resources as reliable sources of potable water.

### Curriculum connections

#### This resource addresses the following content descriptions from the Australian Curriculum :

**Factorise algebraic expressions by taking out a common algebraic factor (ACMNA230)****Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids (ACMMG242)****Solve right-angled triangle problems including those involving direction and angles of elevation and depression (ACMMG245)**

#### This resource addresses the following excerpts from the achievement standard for Year 10 in Mathematics:

- solve surface area and volume problems relating to composite solids
- apply deductive reasoning to proofs and numerical exercises involving plane shapes

### Activity 1 - Calculate areas and volumes for typical water body shapes (ACMMG242)

#### A – small seasonally fragile rock hole, single solid shape

- This type of rock hole is usually of limited size and can be especially sensitive to climatic conditions. They can be idealised as a cylindrical shape.

- Calculate the total surface area and volume of the idealised water hole shape assuming the circumference is 7 metres and average depth is 0.5 metres.

#### B – semi-permanent and seasonally variable waterhole, single solid shape (ACMMG242, ACMMG245)

*Images by N. Watkins *

The example of this type of waterhole can be idealised as a triangular prism.

- Calculate the volume of the water hole assuming the dimensions shown on the idealised shape above.
- Calculate the slope angle A of the water hole bottom.
- Calculate the total surface area of the idealised shape above. (Note the length of the sloping base of the water body is equal to the hypotenuse of the end triangle shape).
- What would the volume be if the water depth reduced to 2 metres during dry periods ? (Note both the depth and the width of the water body will reduce).

#### C – large permanent waterhole, composite solid shape (ACMMG242, ACMMG245)

*Images by N. Watkins *

Assume that this type of waterhole can be idealised by a composite triangular prism / rectangular prism water body.

- Calculate the volume of the water hole with the dimensions shown on the idealised shape above.
- Calculate the slope angle A of the water hole bottom.
- Calculate the total surface of the idealised shape above.

#### D – water volume use

If each person requires 20 litres of water per day, for how long could each of the water bodies sustain a group of ten people? (Note 1 m^{3} = 1,000 litres). Comment on the likely use of these waterbodies and how this information might be passed on.

### Activity 2 - Calculate angles and distances of ascent and descent (ACMMG242, ACMMG245)

Significant water features are often found in steep terrain which is difficult to traverse. The exercise below shows an idealised traverse across a range of hills to access permanent waterholes. The hills rise at an angle of 30^{°} to a height of 150 m above the flat ground level. The range of hills is then flat for a horizontal distance of 200 metres and then falls at an angle of 60^{°} back to flat ground.

*Images by N. Watkins *

- Calculate the horizontal distance AB.
- Calculate the total walking distance from A to B over the range of hills.
- Can you comment on the reality of traversing down the 60° slope. Estimate a traverse distance down this side of the hills if you were to traverse across it at a less steep angle of 20°.
- Calculate the area of the composite shape shown.

#### Activity 3 - Interesting attribute of a circle (ACMMG245)

*For any position of point B around the circle the angle ABC is always 90 ^{°}. This property has many useful applications in maths, science, engineering and design. *

Construct 3 circles of different radiuses. Mark 4 random points around the circle and construct the ABC triangles. Using a protractor measure the angle ABC.

#### Extension *(ACMNA230**,* *ACMMG245)*

Prove the angle ABC is always 90^{°} using Pythagoras Theory:

(length AC)^{2} = (length AB)^{2} + (length BC)^{2}

### Notes

^{2} Moggridge, B.J. (2005). *Aboriginal People and Groundwater*. Master of Science in Hydrogeology and Groundwater Management. University of Technology, Sydney.

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

*The development of these resources was funded through an Australian Government initiative delivered by the University of Melbourne's Indigenous Studies Unit. The resources include the views, opinions and representations of third parties, and do not represent the views of the Australian Government. They have been developed as a proof of concept to progress the inclusion of Aboriginal and Torres Strait Islander content in Australian classrooms. In drawing on the material, users should consider the relevance and suitability to their particular circumstances and purposes.*