Mathematics in nature: understanding bushfire
Exploring how mathematics can be used to understand and describe how bushfires spread across a landscape, and how different environmental factors such as wind and terrain influence bushfire behaviour.
Fire in culture
Fire plays a central role in Aboriginal and Torres Strait Islander culture. It is used for cooking, warmth and for making various tools, in ceremony (e.g. smoking ceremonies), in hunting and in agriculture. Fire is also used by Aboriginal and Torres Strait Islander people to manage the landscape and promote biodiversity – overall, fire plays an integral part in maintaining healthy country. Over thousands of years Aboriginal and Torres Strait Islander people have become highly skilled in fire management; for example, they learnt about ‘good fire’ and ‘bad fire’. Generally speaking, good fire burns cooler – this means that the fire spreads more slowly with smaller flames. Bad fire spreads more rapidly, with larger flames – these flames can often reach into the tops of the trees. This means that bad fires burn with an intensity that can damage ecosystems, causing certain plants and animal to disappear from an area. This can happen, for example, when really intense fires destroy logs and tree-hollows, which small mammals, reptiles and birds use as their homes. This impacts the whole ecosystem, and can jeopardise the viability of communities.
Aboriginal and Torres Strait Islander people developed an intimate understanding of fire and its relationship to the seasons, the plants and the animals, and the shape of the landscape. This was very important, because ‘good fire’ had to be very carefully introduced into the landscape – these fires were not allowed to spread too fast or burn too hot. As such, fire was only used in very specific seasons, under favourable weather conditions and when certain indicator species suggested it was the best time to do so. Fires would be ignited carefully to ensure that fires burnt in very specific patterns. Aboriginal and Torres Strait Islander people have a highly developed understanding of how fire behaves in country.
Understanding bushfire behaviour and spread
Bushfire behaviour is strongly influenced by the wind and the shape of the terrain. The wind determines how fast a fire will spread as well as the direction that it spreads – fires tend to propagate with the wind, and spread faster when winds are stronger. There are mathematical equations that can tell us how fast a fire will spread. Faster spreading fires tend to burn with greater intensity, particularly if fuels have been allowed to build up from lack of burning. These mathematical equations confirm the detailed knowledge of fire behaviour that has been passed from generation to generation of Aboriginal and Torres Strait Islander people over thousands of years.
The strength of the wind also determines the overall shape of a fire. For example, when there is no wind, and the ground is flat, a point ignition will tend to spread equally in all directions. The resulting shape that forms as the fire burns is roughly a circle. When the wind is blowing, however, the fire will spread faster in the direction the wind is blowing and slower against the wind. This means that the fire will make more of an oval shape as it burns, with the major axis aligned with the wind. The proper name for this type of oval shape is an ellipse. In general, the dimensions of the ellipse can be directly related to the strength of the wind.
Terrain also affects the rate of spread of a fire and its shape. Fires tend to spread faster when they spread up a hill, and slower when they burn downhill. In fact, it is known that a fire approximately doubles its rate of spread when a hill becomes 10 degrees steeper. So, for example, if a fire spreads at 10 km/h on flat ground, it will spread at about 20 km/h on a hill with a 10 degree slope, and about 40 km/h on a 20 degree slope.
Effects of bushfire
Aboriginal and Torres Strait Islander people understood that when country is not properly cared for, bad fires can become more likely. These fires burn with more intensity and can destroy entire forest stands, including the tree canopies, which many birds and animals rely upon for their survival. In recent decades there have been a number of destructive wildfires that have scorched vast areas of the country. These fires have killed hundreds of people and destroyed thousands of houses, and have also threatened the survival of certain plants and animals (e.g. Leadbeaters possum, corroboree frog, snow gum, sphagnum moss).
These very large bushfires are difficult and costly to fight against – some are not fully extinguished until the rains come to put them out. In some instances these fires can consume tens of thousands of hectares of forests and grasslands within a day, leaving very little of the landscape unburnt. This means that animals (humans included) have very limited time to escape, and only very limited unburnt refugia to escape to. These large fires are becoming more likely under climate change. In this context, understanding how much area is burnt by a fire is very important.
Classroom activity - Mathematics Year 6
In these classroom activities students will be introduced to the notion that mathematics is inherent in nature, using bushfire as an example. They will be encouraged to consider how factors like the wind and the slope of the terrain affect the way fires spread, and make the connection between fire and geometry through discussion of the various shapes that fires make when they spread across the landscape. The activities will allow students to develop numerical and algebraic concepts using fire as a motivating example. Throughout classroom activities emphasis will also be placed on measurement and area, and interpretation of graphical representations of mathematical relationships.
This resource addresses the following content descriptions from the Australian Curriculum:
- Convert between common metric units of length, mass and capacity (ACMMG136)
- Solve problems involving the comparison of lengths and areas using appropriate units (ACMMG137)
- Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)
- Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles (ACMMG141)
This resource addresses the following excerpts from the achievement standard for Year 6 in Mathematics:
- describe the use of integers in everyday contexts
- solve problems involving length and area
- solve problems using the properties of angles
- locate an ordered pair in any one of the four quadrants on the Cartesian plane
Activity 1 – The shape of fire
Suggested timing for activity: 20-25 minute class discussion (including 5 minute video), followed by 10 minutes looking at handout and group discussion of the questions
Required resources: handout with discussion (see below)
- Introduce students to the idea of fire intensity (how hot a fire burns) and link this to ‘good fire’ versus ‘bad fire’ Watch the “Fighting carbon with fire” video: https://www.youtube.com/watch?v=Qfjw5Vts8hQ
- Follow with a brief discussion of what factors can make fires more intense (bad fire). Focus on wind in particular.
- Inquiry-based learning question: if a fire starts from a small point ignition, what shape would a fire make if there was no wind? How would this shape change if there was a wind blowing?
- Introduce concept of oval or ellipse.
Homework task: find examples of where shapes like ellipses are found in the real world. E.g. the shape of a stadium, cutting a sausage, cucumber or carrot at an angle, shining a torch on the ground at an angle, planets going around the sun, etc.
Activity 2 – Linking fire shape to wind speed
This builds on Activity 1 and the concept that wind-driven fires have a roughly elliptical shape.
Suggested timing for activity: 20-30 minute session for students to work on assigned problems. 15 minute follow up class discussion about whether they think the weather outside means it would be a suitable day for ‘good fire’. Mainly consider how strong the wind is (could go outside, or look at how much tree branches are swaying).
Required resources: handout with problems (see below)
- Students refer to handout showing elliptical fire shapes of various dimensions and are required to decide which shapes correspond to fires burning under strong winds or light winds. Which fire shapes correspond to the most intense fires?
- Students are introduced to the concept of length-to-breadth ratio (L/B) of an ellipse and are required to measure the length and breadth of a number of ellipses, and convert then to length-to-breadth ratios using decimal representation.
- Introduce the relationship between L/B and wind speed via graphical handout and get students to assign wind speeds to the elliptical shapes previously considered.
Activity 3 – Fire spreading on hills
Suggested timing for activity: One 15-20 minute session to outline concepts and one 30 minute session for students to work on assigned problems.
Required resources: handout with problems (see below)
- Students are introduced to the notion that fires spread faster uphill. They are given a hand-out that explains that a fire doubles its rate of spread for every additional 10 degrees of slope, and presented with an example or two.
- Students answer worksheet questions that requires: calculation of up to four terms in a geometric sequence (doubling from a flat ground rate of spread), calculating flat ground rate of spread (via successive halving) from a given rate of spread on a 20 degree slope, measurement of angles using a protractor and then using the measured angle to answer questions about rate of fire spread on a hill.
Activity 4 – Fire area
Suggested timing for activity: one 15-20 minute session to outline concepts and one 20 minute session for students to work on assigned problems.
Required resources: handout with problems (see below)
- Fire area is usually measured in hectares. Introduce hectares (relate it to the area of a football field, for example). Give examples that illustrate conversion from square metres or square kilometres to hectares.
- Students answer worksheet questions that require them to approximate the shapes of real fire scars (the fire shapes in the example worksheet are traces of actual fire perimeters, including the Murrundindi fire on Black Saturday 2009) using familiar shapes (i.e. rectangles and triangles) and use these to approximate the area of real fires (converting to hectares).
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.