Mathematical Modelling: Cornering

ACMMG197; ACMMG222; ACMMG272; ACMMG224
This unit Cornering is one of 5 units making up the Special Topic “Mathematical Modelling”.

This unit is about the way in which long vehicles such as buses turn corners, and the consequences for road safety for cyclists, drivers or pedestrians at a corner.  The main insight is that the rear wheels of a long vehicle do not follow the path of the front wheels, but ‘cut the corner’. Students start with paper scale models, experiment with a bike then use pre-made dynamic geometry programs that highlight the geometric features of cornering considering the vehicles’ width and front and back overhangs.  They prepare a road safety presentation and a report to council about road design.

Students discuss the meaning of the “Do not overtake turning vehicle” sign found on some trucks and buses, and recall instances when a truck or bus has had difficulty getting around a corner or roundabout. They use a simple paper scale model to visualise what happens when long vehicles turn corners, and discuss their findings, and the limitations of the model. The need for a model with steerable wheels is highlighted.

Lesson 2: A Better Model

Students use a bicycle or scooter to find the space covered when it turns a corner. They walk the bicycle in an arc, discovering that the rear wheel travels along an arc of smaller radius than the front wheel. They draw a diagram of the experiment (from above) to understand the geometry of the situation, and identify the centre of rotation. Students apply Pythagoras’ theorem to confirm that their diagram matches the results of the experiment.

Lesson 3: Software Models

Students use dynamic geometry models that match their bicycle experiments. They observe what happens to the rear wheel path as they vary the wheelbase or the tightness of the corner. Students then use the models to predict how much the rear wheels will cut in for a variety of real vehicles. The final products and human factors in vehicle safety are discussed.

Lesson 4: Modelling Four Wheels

Students use new software models, first to investigate the effect of front and rear overhangs on space required for turning, and then including variable width. They investigate the effect of changing these variables on the space required for a vehicle to turn, including the lane usage. Students use the models to gather data for their final products, and also sharpen their focus and scope. Clear communication of mathematical ideas is discussed.

Lesson 5: Presentations and Parking

After time to complete their work, students make their short road safety presentations, and give each other feedback. An optional activity uses the model in a new way to design a car park. The class reflects on the modelling process throughout the unit.

Last updated November 15 2018.