Mathematical modelling: The steepest slopes on Baldwin St

This sequence explores how steepness can be measured and represented using Baldwin Street (the world’s steepest residential street) as a real-world context. Students model the slope using right-angled triangles and describe steepness in multiple ways, including ratios, percentages, and angles. They also consider how steepness is communicated in practice, analysing road signage and designing their own signs for different users. In doing so, students make decisions about clarity, precision, audience, and purpose, recognising that mathematical representations shape how information is interpreted.

The core mathematical goal is to link gradient, defined as $\frac{\text{rise}}{\text{run}}$, to the tangent ratio in trigonometry. Students make explicit that $\tan(\theta) = \frac{\text{rise}}{\text{run}}$ and therefore that gradient and angle describe the same geometric relationship in different forms.

The sequence assumes some familiarity with rise and run, but the context also provides an opportunity to introduce or revisit rise/run as a meaningful comparison of vertical and horizontal change.

The process of mathematical modelling in this sequence

Formulate

Students watch a video of the Jaffa race on Baldwin Street and generate their own questions about the street. They investigate how to measure steepness using photographs, finding angle and $\frac{\text{rise}}{\text{run}}$ values with geometry tools.

Solve

Students formalise gradient as $\frac{\text{rise}}{\text{run}}$ and establish that $\tan(\theta) = \frac{\text{rise}}{\text{run}}$, making making explicit that gradient and angle are different expressions of the same geometric relationship. They verify this using their own measurements and investigate why percentage gradient is the standard for road signage, finding that equal changes in angle do not represent equal changes in steepness.

Interpret

Students apply their understanding to a side-on elevation profile of Baldwin Street, calculating gradients for individual sections and for the street as a whole. They consider what a single number on a road sign should represent, then design signs for different audiences, making decisions about which value to display, which representation to use, and how much rounding is acceptable.

Evaluate

Students compare their road sign designs, examining how choice of value, representation, and rounding affects how steepness is perceived. An optional extension uses Baldwin Street's disputed claim to the world's steepest residential street title to explore how the definition of "steepest" is itself a modelling decision.