Gradient in real life

In this part of the course we are looking at the gradient of a line. But, how do we define a line?

What is a line?

A line is never ending. It extends forever beyond the edge of our page in both directions. A line segment, however, is finite, it has a length and can be thought as having two end points. A ray has one endpoint but extends in one direction without ending.

When we talk about the gradient of a line we will usually find two points which lie on the line. We can also find the gradient of a line segment. ‘Line’ and ‘line segment’ are often, incorrectly, used interchangeably.

As we saw in the last step, the gradient of a line is defined has how many units we have to travel in a vertical direction to get back onto the line when we have travelled one unit to the right in a horizontal direction. The gradient is a measure of steepness.

All of these lines have a positive gradient as they travel in an upwards direction from left to right. A line travelling in a downward direction from left to right has a negative gradient.

Where do we see gradients in real life?

In mathematics lessons gradients are usually expressed as a number. In the previous step the line in the example has a gradient of 2. This is in fact a ratio: travel two units upwards for every one unit we travel to the right, a ratio of 2 : 1. In real life, a gradient of 2 is very steep indeed. Most real life gradients are in fact relatively small and are less than 1.

Road signs in the UK used to use ratios to express steepness. In this example the road sign shows a ratio of 1 : 3. The first value represents the change in the vertical distance, and the second value corresponds to the horizontal distance travelled. A 1 : 3 ratio means that for every three units travelled horizontally, the road would be one unit higher up than when compared to a completely flat road. In a mathematics lesson we would express this as a gradient of 1/3.

This notation lead to a simple misconception which is a little counter-intuitive. A road with gradient of 1 : 3, is a lot steeper than a road with gradient 1 : 10. In general, steepness expressed as 1 : n the bigger the value of n the shallower the gradient.

Comment

How would you explain this misconception to students? Explain in the comments below.

Source: Wikimedia Commons

Task

Ski slopes are graded by colour, and the colour depends on steepness of the slope. In general the shallower the slope, the easier the route is. Black routes have steeper parts than red routes, which in turn have steeper parts than blue routes:

The blue ski slope is expressed as 25%. This means that for every one unit travelled to the right we have to travel one quarter of a unit in the vertical direction. We usually think of ski slopes as travelling downhill, so the blue slope will have a gradient of -1/4

• What is a gradient of 40% is expressed as a fraction?
• If a line has a gradient of 1, how would this be expressed as a percentage?

Give your answers in the comments below.

The steepness of a line could also be expressed as an angle, the greater the angle the steeper the slope