Mathematical modelling: The steepest slopes on Baldwin St
View Sequence overviewGradient is calculated as rise divided by run and can be expressed as a percentage.
The tangent ratio links angle and gradient in a right-angled triangle.
Angle and gradient describe the same steepness in different ways.
Different representations of steepness communicate information differently.
Each student
Why not degrees Student sheet
True or false Student sheet
Writing steepness as a ratio Student sheet (optional)
Task
In Lesson 1 students found a range of ways to describe steepness and discovered that road signs use gradient expressed as a percentage, not angles or decimals.
Refer back to a few values students found in Lesson 1.
Ask: We found angles and ratios to describe steepness. How do those connect to the percentage gradient we saw on road signs?
Take a few responses, then explain that this lesson will make that connection precise. Students will see exactly how gradient is defined, how it converts to a percentage, and how it links to the trigonometric ratios they have already used informally.
Explain that gradient is the formal mathematical measure of steepness. It compares vertical change to horizontal change and is defined as:
$$\text{gradient} = \frac{\text{rise}}{\text{run}}$$
A gradient can be expressed as a decimal (e.g. 0.35) or as a percentage (e.g. 35%). A road that rises 35 m for every 100 m of horizontal distance has a gradient of 35%.
Now connect gradient to the angle students likely measured in Lesson 1. Draw a right-angled triangle representing the road and label the rise, run, and angle of elevation.
Make explicit that:
$$\tan(\theta) = \frac{\text{rise}}{\text{run}}$$
Therefore $\text{gradient} = \tan(\theta)$, and the angle and gradient are numerically connected:
$$\theta = \tan^{-1}\!\left(\frac{\text{rise}}{\text{run}}\right)$$
The gradient and the angle describe the same geometric object. One expresses steepness as a ratio, the other as an angle. They are different representations of the same thing.
To consolidate this, ask students to verify the relationship using their own values from Lesson 1. Students who measured an angle can calculate $\tan(\theta)$ and compare it to their $\frac{\text{rise}}{\text{run}}$ value. Students who calculated a gradient can evaluate $\tan^{-1}(\text{gradient})$ and compare it to their measured angle.
Discuss:
- Are the results consistent?
- What might explain small differences?
Students complete the Why not degrees Student Sheet, which investigates the relationship between angle and gradient. Students calculate gradient values for a range of angles, convert those gradients to percentages, and compare what happens when the angle doubles and when $5^\circ$ is added.
Bring the class together to discuss findings. Surface these key ideas:
- Doubling the angle does not double the gradient.
- The increase in gradient becomes larger as the starting angle increases.
- Adding $5^\circ$ does not produce a constant increase in gradient.
- Equal changes in angle do not represent equal changes in steepness.
Then link explicitly to road signage: percentage gradient communicates changes in steepness more directly than degrees because equal changes in percentage represent equal changes in the rise/run ratio.
Communicating information: angles versus gradient
Angle and gradient describe the same slope, but equal numerical changes in each do not represent equal changes in steepness.
Consider an increase of $5^\circ$. Moving from $30^\circ$ to $35^\circ$ looks like a small numerical step. However:
$$\tan(30^\circ) \approx 0.577$$
$$\tan(35^\circ) \approx 0.700$$
This represents a change in gradient from approximately 0.577 to 0.700, an increase of 0.123, or more than 20%.
Now compare a change of 5 percentage points in gradient, for example, from 30% to 35%. This is a 5 percentage-point increase, and the change in steepness is directly visible in the ratio.
In other words, a $5^\circ$ change does not represent a fixed proportional change in steepness, but a 5% change in gradient does represent a fixed proportional change in the rise/run ratio.
On a road sign, both might appear as change of “5 units” ($5^\circ$ or 5%), yet the effect on steepness is not comparable. This is because angle and steepness have a nonlinear relationship. As the tangent graph shows, as angles approach $90^\circ$, even small increases in angle produce much larger increases in gradient.
Angle and gradient describe the same slope, but equal numerical changes in each do not represent equal changes in steepness.
Consider an increase of $5^\circ$. Moving from $30^\circ$ to $35^\circ$ looks like a small numerical step. However:
$$\tan(30^\circ) \approx 0.577$$
$$\tan(35^\circ) \approx 0.700$$
This represents a change in gradient from approximately 0.577 to 0.700, an increase of 0.123, or more than 20%.
Now compare a change of 5 percentage points in gradient, for example, from 30% to 35%. This is a 5 percentage-point increase, and the change in steepness is directly visible in the ratio.
In other words, a $5^\circ$ change does not represent a fixed proportional change in steepness, but a 5% change in gradient does represent a fixed proportional change in the rise/run ratio.
On a road sign, both might appear as change of “5 units” ($5^\circ$ or 5%), yet the effect on steepness is not comparable. This is because angle and steepness have a nonlinear relationship. As the tangent graph shows, as angles approach $90^\circ$, even small increases in angle produce much larger increases in gradient.
Provide students with True or false Student Sheet which contains a series of statements about gradient, angle, and road sign representation. For each statement, students decide whether it is true, false, or not sure, provide a written justification, and revise their thinking if new evidence emerges during discussion.
The statements are designed to draw attention to the following key ideas:
- 15% is not the same as $\mathit{15}\,^{\circ}$: Percentage gradient and angle describe the same slope in different ways. A $15^\circ$ slope is about 27%, while a 15% gradient is about $8.5^\circ$.
- Gradients can exceed 100%: A 100% gradient is equivalent to $45^\circ$. Gradients greater than 100% represent slopes with angles of elevation greater than $45^\circ$.
- A 100% gradient is not vertical: A vertical line would have an undefined (infinite) gradient. As the angle approaches $90^\circ$, the percentage gradient increases without bound.
- Road sign slope symbols are not drawn to scale: The triangle on a road sign is symbolic. It is designed for quick recognition, not geometric accuracy.
- Rounding can change how steep a road appears: For example, rounding 34.7% to 35% preserves meaning closely, but rounding to 30% may significantly alter perception.
- Representation affects perception: The same slope may feel steeper when expressed as a percentage than when expressed as an angle.
Students complete the Writing steepness as a ratio Student sheet, exploring how steepness can be written in the form $1:n$. Students convert rise/run values into this form, rescale triangles so the rise equals 1, and compare ratios such as $1:3$, $1:5$, and $1:2$, considering how these might be interpreted at a glance.
Bring the class together to discuss. Surface these key ideas:
- In the form $1:n$, the second number represents the horizontal distance.
- As the second number increases, the slope becomes flatter.
- A larger number in the ratio does not mean a steeper road.
- This may conflict with how people instinctively compare numbers.
Conclude by making explicit:
- Different representations describe the same steepness.
- Some formats communicate magnitude more intuitively than others.
- Public signage must consider how information is interpreted, not just whether it is mathematically correct.