Mathematical modelling: The steepest slopes on Baldwin St
View Sequence overviewSteepness can be described by comparing vertical and horizontal change.
Gradient is calculated as rise divided by run.
The same slope can be represented using angle or ratio.
Whole class
Video: 2014 Cadbury Jaffa Race (1:02)
The steepest slopes on Baldwin St Slides
Each group
Question Quadrant Student sheet
Baldwin Street Student sheet, either in print or digital form
If using printed Student sheet
Rulers
Protractors
If using digital Student sheet
Access to a digital geometry tool such as Desmos Geometry or GeoGebra
Lesson
Play 2014 Cadbury Jaffa Race (1:02), showing a Jaffa race down Baldwin Street in New Zealand, the steepest residential street in the world.
Ask students to generate as many questions as possible about Baldwin Street and the Jaffa race and record their questions in the Question Quadrant Student sheet, grouping questions according to whether they are mathematical or non-mathematical and whether they are open or closed. Encourage students to write as many questions as they can and try to include at least one question in each section.
About the race/event:
- How many Jaffas are released?
- Has anyone ever been hit by a Jaffa?
- How often does the race happen?
- Who organises it?
- Do the Jaffas get eaten afterwards?
About movement/physics:
- Why do some Jaffas go faster than others?
- Do heavier Jaffas go faster?
- What happens if a Jaffa hits the kerb?
About the street/steepness:
- Could you ride a bike down it?
- Could a car drive up it?
- How do people who live there manage in wet weather?
- What’s the angle of the street?
Discuss the questions students have recorded. Once one or two questions about steepness or shape are identified, highlight them and explain that this will be the focus of the investigation.
The goal of this discussion is for the class to feel a sense of ownership over the investigation focus. Guide the conversation so that questions about the shape or steepness of the street come from students’ ideas, rather than being introduced as the teacher’s predetermined choice.
Invite students to step back and look at the full set of questions. You can ask:
- Which questions are about the race?
- Which are about movement or speed?
- Which are about numbers?
- Which are about the street?
If questions about the street appear, ask:
- What do these questions have in common?
- What are we wondering about the street itself?
If needed, shift attention by gently asking:
- If we paused the race and focused only on the street, what might we be curious about?
- What features of the street stand out visually?
Why invite students to generate their own questions?
Students are almost always asked to solve problems written by teachers or textbooks. They are rarely given opportunities to pose their own mathematical problems.
Problem posing is not separate from problem solving. It occurs before, during, and after solving a problem, as learners reformulate situations, identify relationships, and decide what is worth investigating (Silver, 1994). In this sense, generating questions is part of doing mathematics, not an extension task.
In this sequence, inviting students to ask their own questions at the start of the lesson positions them as authors of mathematical ideas. When students ask “How steep is it?”, the need for gradient emerges naturally from their own inquiry rather than being introduced as a formula to apply.
Silver, E. A. (1994). On mathematical problem posing. For the learning of mathematics, 14(1), 19-28.
Students are almost always asked to solve problems written by teachers or textbooks. They are rarely given opportunities to pose their own mathematical problems.
Problem posing is not separate from problem solving. It occurs before, during, and after solving a problem, as learners reformulate situations, identify relationships, and decide what is worth investigating (Silver, 1994). In this sense, generating questions is part of doing mathematics, not an extension task.
In this sequence, inviting students to ask their own questions at the start of the lesson positions them as authors of mathematical ideas. When students ask “How steep is it?”, the need for gradient emerges naturally from their own inquiry rather than being introduced as a formula to apply.
Silver, E. A. (1994). On mathematical problem posing. For the learning of mathematics, 14(1), 19-28.
Steep roads around the world
Baldwin Street in Dunedin, New Zealand is often described as one of the steepest residential streets in the world, with a gradient of about 35% at its steepest section and a total length of approximately 350 metres. Each year from 2002-2017 thousands of orange confectionery balls (‘Jaffas’) were released from the top of Baldwin Street and raced to the bottom as part of the Cadbury Jaffa Race. The video Jaffa Race 2015 captures one of these events.
To put Baldwin Street’s gradient in perspective: in engineering, road gradients are set based on what vehicles and people can safely manage. Motorways typically stay below 6%, residential streets below 12%, and driveways below about 15-20%. Wheelchair ramps are gentler still, often limited to around 7% (1:14). At 35%, Baldwin Street sits well beyond all of these.
If you prefer a different context, Alpe d’Huez is a well-known alternative. It is a mountain pass regularly featured in the Tour de France, with clearly marked percentage gradients on the road itself, which connects naturally to the road signage discussion later in the sequence. The video The most iconic Tour de France climb | Alpe D'Huez (3:01-5:49) provides useful footage.
When a gradient is too steep for a direct path, one common solution is to zig-zag the route. Switchbacks increase the horizontal distance travelled, which brings the gradient (rise divided by run) down even if the total height climbed stays the same.
Baldwin Street in Dunedin, New Zealand is often described as one of the steepest residential streets in the world, with a gradient of about 35% at its steepest section and a total length of approximately 350 metres. Each year from 2002-2017 thousands of orange confectionery balls (‘Jaffas’) were released from the top of Baldwin Street and raced to the bottom as part of the Cadbury Jaffa Race. The video Jaffa Race 2015 captures one of these events.
To put Baldwin Street’s gradient in perspective: in engineering, road gradients are set based on what vehicles and people can safely manage. Motorways typically stay below 6%, residential streets below 12%, and driveways below about 15-20%. Wheelchair ramps are gentler still, often limited to around 7% (1:14). At 35%, Baldwin Street sits well beyond all of these.
If you prefer a different context, Alpe d’Huez is a well-known alternative. It is a mountain pass regularly featured in the Tour de France, with clearly marked percentage gradients on the road itself, which connects naturally to the road signage discussion later in the sequence. The video The most iconic Tour de France climb | Alpe D'Huez (3:01-5:49) provides useful footage.
When a gradient is too steep for a direct path, one common solution is to zig-zag the route. Switchbacks increase the horizontal distance travelled, which brings the gradient (rise divided by run) down even if the total height climbed stays the same.
Provide students with Baldwin Street houses Student sheet, which contains four images of Baldwin Street:
- Two different houses photographed from typical street-level perspectives.
- One house photographed with the camera rotated so the house appears upright and the street appears flat.
- One wider view of the street taken from the top of the street.
If students are working with printed copies, provide rulers, protractors, and pencils for drawing over the images. If students are working digitally, provide access to a geometry tool such as Desmos Geometry or GeoGebra so they can insert the image, draw lines, construct triangles, and measure angles or distances.
Allow students time to investigate how steep Baldwin Street is. They will need to decide:
- what information is needed.
- which image (or combination of images) is most useful.
- how to obtain or estimate measurements.
- how to represent steepness.
Encourage students to compare the images before measuring.
To encourage the future connection between gradient and tangent, ask students to find as many different ways as they can think of to describe, represent, or measure steepness.
Students may describe steepness using:
- angle—defining steepness as the angle the road makes with the horizontal.
- rise compared to run—defining steepness as the ratio of vertical change to horizontal change, represented using a right-angled triangle.
- length of the road—defining a road as steeper if it is shorter for the same vertical height, or longer for the same horizontal distance.
- height gained—defining steepness in terms of total vertical rise, without considering horizontal distance.
- speed—defining steepness in terms of how quickly an object accelerates or how fast it moves down the slope.
- effort or fatigue—defining steepness in terms of how much effort is required to walk or climb the hill.
If students are unsure where to begin, prompt with:
- What exactly do we mean by “steep”?
- Could you draw over the image to make the steepness clearer?
- What measurements could help?
- Could we represent this steepness using a triangle?
Groups are likely to produce different answers. Rather than resolving differences immediately, invite comparison. You might ask:
- Where did you place your triangle?
- Which image did you use?
- Did you measure near the top, middle, or bottom of the street?
Students may notice that perspective affects measurements, that the foreground may give a different result than the background, or that the steepness of the street may change along its length. This creates productive discussion: Are we measuring the same thing? Is there one “correct” steepness?
Visual approach: Measuring the angle
Students may decide to measure the angle the road makes with the horizontal.
Working with a printed image
- Draw a straight line along the slope of the road.
- Draw a horizontal reference line (this can be aligned with the bottom edge of the page or drawn using a ruler).
- Use a protractor to measure the angle between the horizontal line and the line along the road.
Students might measure an angle of approximately 15$^\circ$ to 20$^\circ$.
Working with Desmos Geometry
- Import the image.
- Rotate the image so a known horizontal feature (such as the roofline or fence line) is level.
- Draw a horizontal line.
- Draw a line along the slope of the road.
- Use the angle tool to measure the angle between the two lines.
Ratio approach: Using rise and run
Students may decide to compare the vertical change to the horizontal change along the road.
Working with a printed image
- Choose two clear points along the road.
- Draw a right-angled triangle representing the slope.
- Measure the vertical distance (rise).
- Measure the horizontal distance (run).
Working with Desmos Geometry
- Import the image.
- Identify two points along the road.
- Construct a right-angled triangle by eye using vertical and horizontal segments.
- Use the distance tool to measure rise and run.
- Calculate the ratio.
For example, a student might measure:
- Rise = 1.39
- Run = 6.5
Then calculate $\frac{1.39}{6.5} \approx 0.21$. This gives a gradient of approximately 0.21.
Modelling approach: Constructing horizontal and vertical reference lines
Perspectives can slightly distort measurements if the photo is not taken directly side-on. This can lead to variation in answers. This variation is productive and reinforces that students are modelling a real situation and making assumptions.
One way to manage perspective distortion is to identify reliable vertical and horizontal references within the image before measuring the slope.
For example, the side of the house can be treated as an assumed vertical line. A perpendicular line then gives a horizontal reference. These two lines establish a coordinate frame within the image.
Students can then follow the below process:
- Draw a line along the slope of the road.
- Identify two points along this line.
- Construct a right-angled triangle using:
- a vertical line (rise).
- a horizontal line (run).
- a sloped road line (hypotenuse).
From this triangle, students can:
- measure the angle between the horizontal and the road.
- measure the vertical and horizontal side lengths and calculate the gradient using $\text{gradient} = \frac{\text{rise}}{\text{run}}$.
- use the measured rise and run to calculate the angle using inverse tangent $\theta = \tan^{-1}\!\left(\frac{\text{rise}}{\text{run}}\right)$.
How steep can our roads be?
In engineering, gradient limits are set so vehicles and people can safely use a path. Some useful reference points for Australian contexts:
Motorways and highways are typically designed at 3%-6% to preserve sight distance and allow heavy vehicles to maintain speed. Urban and residential streets commonly reach up to 8%-12% for short sections. Driveways are capped by most Australian states at around 15%-20%. Baldwin Street, at approximately 35% at its steepest, sits well beyond any standard road engineering limit.
When a gradient is too steep, one strategy is to change the path rather than the height. Switchbacks and zig-zag routes increase the horizontal distance travelled, which reduces the gradient (rise divided by run) even though the total vertical rise stays the same.
Why does direction matter?
A road that is safe to drive up is not necessarily safe to drive down. On inclines, gravity works against the vehicle: it slows naturally, but the engine must work harder. On declines, gravity accelerates the vehicle and the brakes must absorb the full energy of the descent. On long steep descents, repeated braking causes brakes to overheat and fail entirely, a phenomenon known as brake fade. This is why steep downhill roads feature emergency escape ramps: gravel-filled lanes where a vehicle with failed brakes can safely decelerate. These ramps appear on descents, not ascents, because the danger of losing control is specific to going downhill.
When is a sign required?
In Australia, the decision to install a steep gradient warning sign is guided by AS 1742, the national standard for traffic control devices. There is no single national percentage that automatically triggers a sign. Road agencies in each state apply engineering judgement, considering the gradient, the length of the grade, vehicle types, and crash history. Two roads with the same gradient might be treated differently depending on context, which is itself a modelling decision about what information matters most.
In engineering, gradient limits are set so vehicles and people can safely use a path. Some useful reference points for Australian contexts:
Motorways and highways are typically designed at 3%-6% to preserve sight distance and allow heavy vehicles to maintain speed. Urban and residential streets commonly reach up to 8%-12% for short sections. Driveways are capped by most Australian states at around 15%-20%. Baldwin Street, at approximately 35% at its steepest, sits well beyond any standard road engineering limit.
When a gradient is too steep, one strategy is to change the path rather than the height. Switchbacks and zig-zag routes increase the horizontal distance travelled, which reduces the gradient (rise divided by run) even though the total vertical rise stays the same.
Why does direction matter?
A road that is safe to drive up is not necessarily safe to drive down. On inclines, gravity works against the vehicle: it slows naturally, but the engine must work harder. On declines, gravity accelerates the vehicle and the brakes must absorb the full energy of the descent. On long steep descents, repeated braking causes brakes to overheat and fail entirely, a phenomenon known as brake fade. This is why steep downhill roads feature emergency escape ramps: gravel-filled lanes where a vehicle with failed brakes can safely decelerate. These ramps appear on descents, not ascents, because the danger of losing control is specific to going downhill.
When is a sign required?
In Australia, the decision to install a steep gradient warning sign is guided by AS 1742, the national standard for traffic control devices. There is no single national percentage that automatically triggers a sign. Road agencies in each state apply engineering judgement, considering the gradient, the length of the grade, vehicle types, and crash history. Two roads with the same gradient might be treated differently depending on context, which is itself a modelling decision about what information matters most.
Bring the class together to compare approaches. Invite several groups to briefly explain where they placed any lines or triangles, what measurements they took, how they represented steepness, and their final value. Record different results publicly.
Encourage students to respond to one another’s reasoning:
- Why might your answer be different?
- Did you measure the same section of road?
- Were your horizontal and vertical references the same?
- Does perspective affect the measurement?
By the end of the discussion, students are likely to have expressed the steepness of Baldwin Street either as an angle in degrees or as a ratio written as a decimal. Pause and ask whether angle and ratio describe the same steepness, and which representation might be more useful. Students will likely have a range of opinions. This tension is left unresolved here on purpose. The next step asks students to look at how steepness is communicated in the real world, which will reframe the question.
Display Slide 2 from The steepest slopes on Baldwin St Slides which includes images of real road gradient signs. Ask students what they notice. If needed, prompt with:
- What numbers or formats appear on these signs?
- What formats do you not see?
- Which sign would be easiest to understand quickly while driving?
Students are likely to notice that many signs use percentages, some use ratios, none use angles in degrees, and none use decimals.
Follow with:
- Why do road signs mostly use percentages?
- Why not angles?
- Why not a decimal like 0.35?
- What does a driver need to understand in a few seconds?
This is the key reveal of the lesson: the measurements of “steepness” students developed are likely mathematically valid, but they are not what engineers and road authorities use. The formal measure is called gradient, and it is expressed most often as a percentage (and sometimes as a ratio).
Close by letting students know: In the next lesson, we will learn exactly how gradient is defined, how it connects to the angle measures we have already found, and why it is the standard used by road engineers around the world.