Mathematical modelling: Playing with fire

In this sequence, students use mathematical modelling to determine when two small towns should evacuate in response to an approaching bushfire. Drawing on data from real and simulated fires in dry eucalypt forests, they investigate how weather conditions and topography influence the forward rate of spread. Using this data, students (re)create two commonly used rules of thumb:

• The forward rate of spread in dry eucalypt forests is approximately 10% of the average 10m open wind speed (the wind speed measured at 10 m above ground level in an open area) (Cruz & Alexander, 2019).
• The rate of spread of a bushfire doubles for every 10° of upward slope (McArthur, 1968).

Students are then introduced to mathematical equations used to model fire behaviour and compare the results to those generated by the rules of thumb. Through this comparison, they come to see that mathematical models offer greater accuracy in predicting the rate of spread of bushfires.

A key component of this sequence is the use of an Excel workbook. It enables students to efficiently represent data and perform calculations using more complex equations. By automating these processes, students can shift their focus from computation to critically analysing the data and comparing the accuracy of results generated by different methods.

This sequence has strong links to Science and can be used alongside the Science Connections sequence, Bushfire and ice.

The process of mathematical modelling in this sequence

Formulate

Students are introduced to the problem of predicting bushfire spread and deciding when to evacuate. By sorting fire simulation cards and observing how weather conditions affect fire behaviour, they identify the key variables at play and consider how mathematics might help them model the spread of a fire.

Solve

Students use real bushfire data to investigate the relationship between wind speed and rate of fire spread, fitting a trend line and developing a rule of thumb. They then apply a more complex mathematical formula, and repeat this process for topography, developing a second rule of thumb for the effect of slope. They use spreadsheets to calculate predicted rates of spread using both the rules of thumb and the formal equations.

Interpret

Students compare predicted rates of spread against observed data, evaluating how well each method performs. They apply their models to two specific town scenarios, calculating how quickly a fire would reach each town under given conditions.

Evaluate

Students determine when each town should evacuate, weighing their mathematical calculations against practical factors such as evacuation time and the unpredictability of fire behaviour. They present and justify their recommendations to the class and reflect on when a rule of thumb is adequate and when the formal equation is needed.ctors, such as the unpredictable nature of fire behaviour and the time required for a safe evacuation.

References

Cruz, M.G., Alexander, M.E. The 10% wind speed rule of thumb for estimating a wildfire’s forward rate of spread in forests and shrublands. Annals of Forest Science 76, 44 (2019). https://doi.org/10.1007/s13595-019-0829-8

McArthur, A.G. 1968. Fire Behaviour in Eucalypt Forests. Leaflet No. 107. Ninth Commonwealth Forestry Conference, India, 1968.