A gradient is not an angle

1 in 20

0.05gradient

2.9°angle

gentleverdict

The run is the same every time. Only the rise changes, and the gradient is the rise divided by it.

The figure needs JavaScript. It draws a ramp at 1 in 20 (2.9°), 1 in 12 (4.8°), 1 in 10 (5.7°) and 1 in 5 (11.3°), then puts a 1 in 10 ramp beside a 10° ramp, where the 10° one has a gradient of 0.18 and is nearly twice as steep. Everything below stands on its own.

What a gradient actually is

Gradient is rise over run, and it is a ratio, not a number of degrees. The two are connected by the tangent:

gradient = tan(angle), so angle = tan⁻¹(gradient)
1 in 10 means a gradient of 0.1, so the angle is tan⁻¹(0.1) = 5.7°
A 10° slope has gradient tan 10° = 0.18, which is a rise of about 1 in 5.5

So reading "1 in 10" as ten degrees understates the angle by a factor of nearly two in gradient terms. For small slopes the two numbers happen to look similar, which is exactly why nobody notices.

You will not be asked to convert a gradient to an angle in this sub-topic. It is here because the words are used interchangeably outside mathematics and are not interchangeable inside it. When a question says "find the gradient of the road", it wants the ratio.

Three forms, one line

Form	Looks like	Use it when
Gradient-intercept	y = mx + c	You want to read the gradient and the y-intercept straight off.
General	ax + by + d = 0	The question gives it this way, or you want whole numbers with no fractions.
Point-gradient	y − y₁ = m(x − x₁)	You have one point and a gradient, which is most of the time.

Take 2x + 3y = 12 and get all three out of it.

Rearrange: 3y = −2x + 12, so y = −⅔x + 4
So the gradient is −0.67 to 2 decimal places, and the y-intercept is 4
The x-intercept comes from y = 0: 2x = 12, so x = 6
Through (3, 2) with that gradient: y − 2 = −⅔(x − 3), which tidies back to the same line

The gradient of 2x + 3y = 12 is not 2, and it is not ⅔. Those are the two answers people give. It is −⅔: you have to divide by the coefficient of y and carry the sign. If you ever read a gradient off the general form without rearranging, write the rearrangement out. It is one line and it is the difference between the mark and nothing.

Parallel and perpendicular

For the same line, gradient −⅔:

Parallel: same gradient, −⅔. A parallel line differs only in its intercept.
Perpendicular: the negative reciprocal, 3⁄2, that is 1.5. Check it: −⅔ × 1.5 = −1.

The product being −1 is the test, and it is quicker than remembering which way round to flip. If your two gradients do not multiply to −1, one of them is wrong.

On the GDC: a line through two points

The machine will fit a straight line to two points and hand you the gradient and the intercept, which saves the arithmetic and none of the thinking.

When you may use it. Applications allows a calculator in every paper. Analysis does not allow one in Paper 1, where a gradient from two points is a one-line subtraction anyway.

TI-Nspire CX II

ctrl doc → Add Lists & Spreadsheet, and type the two x values into one column and the two y values into the next
menu → Statistics → Stat Calculations → Linear Regression (mx+b)
Read m for the gradient and b for the y-intercept
For two points it is exact, so r² comes back as 1. That tells you nothing: any two points lie on a line.

Casio fx-CG50

MENU → Statistics, x values into List 1 and y values into List 2
F2 CALC → F3 REG → F1 X for a linear fit
Read a for the gradient and b for the y-intercept, in y = ax + b
Note the letters: the Casio's a is the gradient, where the Nspire calls it m

The mark people lose. Giving the gradient as an angle, or an angle as a gradient. If a question asks for the gradient of an access ramp and you write 4.8°, you have answered a question nobody asked. The other one is the Casio's letters: in y = ax + b the gradient is a, and students who learned y = mx + c reach for b and write the intercept down as the gradient.

Your turn

1. A road sign says 1 in 5. What angle is that, in degrees, to 1 decimal place?

2. What is the gradient of 2x + 3y = 12? Give it to 2 decimal places.

3. What is the gradient of a line perpendicular to it? Give it to 1 decimal place.

4. A ramp has to rise 0.75 m and the code requires a gradient no steeper than 1 in 12. What is the shortest horizontal run that satisfies it, in metres, to 2 decimal places?

5. Someone says a 1 in 10 slope and a 10° slope are about the same. What is the best reply?

Question 5. Someone says a 1 in 10 slope and a 10 degree slope are about the same. What is the best reply? They are, for small angles. 1 in 10 is 5.7°, so the 10° slope is nearly twice as steep. 1 in 10 is 10° by definition, so they are identical. You cannot compare a gradient and an angle at all.

Where the marks go

1 markRearranging to y = mx + c, shown.

1 markThe gradient, with its sign.

1 markThe intercept, or the second gradient.

The sign is a mark on its own more often than anything else in this sub-topic, because a general form with a positive y coefficient and a positive x coefficient always has a negative gradient, and that is not where the eye goes.