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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?
Baldwin Street - a house on a road with a steep slope
© STEM Learning
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.
On the left above the word Gradient in black text and the number 4 are four squares half a centimetre by half a centimetre, stacked on top of each other as a grid, this shape is cut in half by a black line diagonally, the area cut in half is filled with pink to the right and is white on the left, the number 1 appears at the bottom of the grid and the number 4 appears next to the good on the right, in the centre is another square, dissected by a black diagonal line, the area dissected is pink on the right, the area above the line is white, the word Gradient and the number 1 appear at the bottom, a 1 appears next to the square at the bottom and to the right, on the right three half centimetre squares appear attached left to right to form a grid, this grid is dissected by a black line, the area to the right of the line is pink, the area above the line is white, the word Gradient appears below this grid and 1/3, directly next to the squares at the bottom is a 3 and to the right there is a 1
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.
On the left there is a British road sign for steepness, this is a red equal sided triangle, with a white triangle centre, this has a ratio 1:3 in black text above a black triangle which fits within the white centre and fit half of the centre on the right and slopes down to the bottom of the white centre on the left. To the right of this is a another triangle of black grid lines the mimics the black triangle in the road sign and has a black, pink and purple image of a bicycle on top of the slope of the triangle grid
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.
Most UK road signs have now been converted to express steepness as a percentage. This has the advantage of the greater the percentage, the steeper the slope.
On the left there is a British road sign for steepness, this is a red equal sided triangle, with a white triangle centre, this has 10% in black text above a black triangle which fits within the white centre and fit half of the centre on the right and slopes down to the bottom of the white centre on the left. To the right of this is a another triangle of black grid lines which is half centimetre high on the left and slope down to the left and has a black, pink and purple image of a bicycle on top of the slope of the triangle grid
These road signs still need to be interpreted. A gradient of 10% is the same as the ratio of 10 : 100. This can be simplified to 1 : 10 and interpreted as for every one unit we travel up we have to travel ten units horizontally.
Baldwin Street, in Dunedin, New Zealand, is recognised as the steepest street in the world by Guinness World Records. At its steepest point, the road has a gradient of 35% Expressed as a ratio this is about 1 : 2.86 and as a fraction as 7/20. Even the steepest road in the world has a gradient of much less than one.

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: On the left there is a triangle which slopes upwards left to right and is split into black, red and blue triangles, on the left there is a black filled quarter centimetre circle with .40% next to it, below this is another red filled circle with 40% 22º next to it, then a blue filled circle with 25% and 14º, this forms a key for the
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  
© STEM Learning
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Maths Subject Knowledge: Graphs, Functions and Solving Equations Graphically

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