Mathematical modelling: The steepest slopes on Baldwin St
View Sequence overviewMathematical information can be communicated in different ways.
Designing for an audience requires decisions about clarity and precision.
A mathematically accurate representation is not always the most useful one.
Each student
How steep is Baldwin Street Student sheet
Access to the article Baldwin Street in New Zealand reinstated as the world’s steepest street | Guinness World Records (either digital or print) (optional)
Lesson
Provide students with How steep is Baldwin Street Student sheet which shows a side-on profile of Baldwin Street.
Students draw and calculate three triangles:
- a triangle over the steepest-looking section.
- a triangle over a gentler section.
- a triangle from the bottom to the top of the street (representing the average steepness).
For each triangle, students calculate the gradient expressed as a percentage and the angle using $\theta = \tan^{-1}\!\left(\frac{\text{rise}}{\text{run}}\right)$.
Ask:
- Do the different sections have the same steepness?
- Which section is steepest?
- If a road sign could only display one number, what should it show?
- Should that number represent the steepest section or the whole street?
Students may suggest averaging the three gradients. Record this method alongside the gradient calculated using the single triangle from bottom to top.
Compare the results and ask:
- Why might these two methods give different answers?
- Do all sections contribute equally to the overall steepness?
- Does the horizontal length of each section mater?
Conclude by making explicit: The gradient calculated from the single triangle from bottom to top represents the overall change in height compared to the overall horizontal distance. This is not necessarily the same as averaging the gradients of individual sections.
Street profile as an approximation
The side-on profile of Baldwin Street is shown as a series of straight-line segments. This is a modelling decision. In reality, the street is unlikely to change slope in perfectly straight sections. The straight segments approximate how the steepness varies along the road.
Each segment represents a section where the gradient is treated as a constant and provides an opportunity to highlight that mathematical models simplify reality. The representation is useful for calculation, but it is not a perfect copy of the real street.
The profile is based on data from myCols.
The side-on profile of Baldwin Street is shown as a series of straight-line segments. This is a modelling decision. In reality, the street is unlikely to change slope in perfectly straight sections. The straight segments approximate how the steepness varies along the road.
Each segment represents a section where the gradient is treated as a constant and provides an opportunity to highlight that mathematical models simplify reality. The representation is useful for calculation, but it is not a perfect copy of the real street.
The profile is based on data from myCols.
Local and overall gradient
When students calculate the gradient of each section of the street, they are finding the average gradient of that subsection (the rise divided by the run for that segment alone).
When students calculate $\frac{\text{total rise}}{\text{total horizontal distance}}$ they are finding the average gradient of the whole street. This value summarises the entire climb using a single ratio.
The gradient of a subsection describes one part of the street. The average gradient of the whole street describes the entire interval. These are not necessarily the same.
When students calculate a single value for their road sign, they may average several subsection gradients. This provides an opportunity to highlight an important idea: averaging individual subsection gradients does not necessarily produce the overall average gradient unless the subsections have equal horizontal length.
These ideas are developed formally in senior mathematics when students study rates of change in calculus. Here, the focus is on building an intuitive understanding that the interval chosen matters, and that a single number displayed on a sign represents a modelling decision about what aspect of the street is being summarised.
When students calculate the gradient of each section of the street, they are finding the average gradient of that subsection (the rise divided by the run for that segment alone).
When students calculate $\frac{\text{total rise}}{\text{total horizontal distance}}$ they are finding the average gradient of the whole street. This value summarises the entire climb using a single ratio.
The gradient of a subsection describes one part of the street. The average gradient of the whole street describes the entire interval. These are not necessarily the same.
When students calculate a single value for their road sign, they may average several subsection gradients. This provides an opportunity to highlight an important idea: averaging individual subsection gradients does not necessarily produce the overall average gradient unless the subsections have equal horizontal length.
These ideas are developed formally in senior mathematics when students study rates of change in calculus. Here, the focus is on building an intuitive understanding that the interval chosen matters, and that a single number displayed on a sign represents a modelling decision about what aspect of the street is being summarised.
Challenge students to design two road signs for Baldwin Street, using any of the gradient or angle measurements they calculated from the side-on profile (or choose new measurements).
For each sign, students should:
- decide who the sign is for (e.g. driver, tourist, cyclist, Jaffa ball).
- make design choices that suit that user.
The two signs should include:
- a recommended sign—a sign you believe should genuinely be used on the street.
- a mathematically precise sign—a sign that prioritises mathematical and real-world accuracy.
At least one sign must include a numerical representation of the steepness.
Students should be prepared to explain:
- who each sign is for.
- why they chose that representation.
- what trade-offs they made between clarity and precision.
Encourage students to refer to the side-on profile and choose or construct measurements that support their design, considering the following:
- Which measurement best represents the street?
- Should the sign show the steepest section or the average rate of change?
- Is rounding acceptable? If so, how much?
- Is percentage clearer than angle or ratio?
- Would including both angle and percentage be confusing?
- Does the diagram (if included) need to be to scale?
Students are not just calculating steepness. They are making decisions about how information should be communicated to real people in a real setting. The focus is on usefulness, clarity, and audience.
When introducing the activity, you might say:
A road sign is not a maths worksheet. It has to work quickly, clearly, and safely. Your job is to decide what information matters most and how best to present it.
Emphasise that there is no single correct design, that good design depends on who will use the sign and what they need in the moment, and that clarity and safety might matter more than mathematical precision.
If students default to simply putting the largest number on the sign, redirect with questions like:
- Who is the sign for?
- Drivers need to know whether to slow down before the steepest section.
- Tourists might just want a sense of how extreme the slope is.
- How much time will different users have to read the sign, and how does that affect how information should be presented?
- Would different users need different information?
You might even assign roles:
- Design this sign for a cautious tourist visiting the Jaffa race.
- Design this sign for a confident local driver.
- Design this sign for someone towing a trailer.
As students share their work, listen for attention to the user’s perspective, trade-offs between precision and usability, awareness that representation affects perception, and recognition that rounding or choice of interval affects meaning.
Have students share their designs and compare:
- Which values appeared the most often? Were these maximum values, average rate of change values, or something else?
- Which representations appeared the most often (percentage, angle, ratio, symbol-only)?
- What made some signs misleading or hard to interpret (choice of value, rounding, representation, diagram)?
- How did representation affect perception (e.g. did the same slope feel “steeper” when written as a percentage versus an angle)?
Explain that Baldwin Street was temporarily stripped of its title as the world’s steepest residential street after another street in Wales was measured differently. Baldwin Street's title was later reinstated. Provide students with the article at Baldwin Street in New Zealand reinstated as the world’s steepest street | Guinness World Records.
Ask students to read the article, identify any information about how steepness was measured, and consider whether the measurement refers to the steepest section, an average over the whole street, or another definition.
Bring the class together and ask:
- What rule did Guinness use to decide which street was steepest?
- Did the definition change?
- Why did that change affect the result?
Then ask students to compare the maximum gradient of each street, consider whether the steepest section or the overall gradient should determine the title, and discuss how the length of the measured section affects the result.
Prompt:
- If one street is steeper but only for a very short distance, is it fair to call it the steepest?
- Does your answer depend on purpose?
Ask students: If you were writing the rule for Guinness World Records, how would you define “steepest street”?
Make explicit that:
- “steepest” depends on how steepness is defined.
- different definitions produce different outcomes.
- mathematical rules are modelling decisions.
- public claims rely on agreed definitions.