# Find the Gradient Between Two Points

In this worksheet, students will find the gradients of straight lines when given the coordinates of two points on the line.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra, Graphs of Equations and Functions

Curriculum subtopic:   Graphs, Straight Line Graphs

Difficulty level:

### QUESTION 1 of 10

The gradient of a line is a number defining its steepness.

We can find this by using the coordinates of two points.

If the first point has coordinates (xa, ya) and the second has coordinates (xb, yb), we can use the formula below to find the value of the gradient:

Let's have a look at an example to see this in action.

e.g. Find the gradient of a line passing through (1, 3) and (4, 9).

The formula above means that we need to subtract the y-coordinates of both points, then subtract the x-coordinates and divide the former by the latter.

We can do this in stages or in one go using a calculator.

In stages:

9 - 3 = 6

4 - 1 = 3

6 ÷ 3 = 2

Using a calculator:

This is what we need to put into our calculator:

In this activity, we will find the gradients of straight lines when given the coordinates of two points on the line.

If you prefer the second method in the options above, you may want to have a calculator handy to support you.

Two points have coordinates (xa, ya) and (xb, yb).

What is the correct formula for finding the gradient of a line passing through these two points?

Find the gradient of a line passing through the points (3, 6) and (5, 8).

Find the gradient of a line passing through the points (5, 6) and (3, 10).

Find a gradient of a line passing between the points (10, -10) and (-2, -4).

-1/2

1/2

-2

2

Match each pair of point coordinates below with the gradients of lines between them.

## Column B

(-9, 2) and (9, 12)
12/7
(5, 3) and (-2, -9)
2/5
(-3, 1) and (5, -12)
-13/8
(-15, -3) and (0, 3)
5/9

A line passes through two points: (-4, -1) and (-1, 4).

What is the gradient of this line?

5/3

3/5

-3/5

-5/3

True or false?

The gradient of the line passing between the points (-3, -3) and (-4, 0) is -3.

True

False

True or false?

The gradient of the line passing through the points (4, 8) and (3, 14) is 6.

True

False

Match each pair of points below so that each has a line passing through them with a gradient of 3.

## Column B

(1, 8)
(4, 8)
(-1, -1)
(1, 5)
(2, 2)
(0, -3)
(-3, -12)
(2, 11)

Find the gradient of a line which passes between the points (-5, -1) and (-3, -7).

• Question 1

Two points have coordinates (xa, ya) and (xb, yb).

What is the correct formula for finding the gradient of a line passing through these two points?

EDDIE SAYS
We need to subtract the y-coordinates of both points, then subtract the x-coordinates and divide the former by the latter. The correct formula is this one:
• Question 2

Find the gradient of a line passing through the points (3, 6) and (5, 8).

1
EDDIE SAYS
We need to apply our formula here. Let's start with yb - ya: 8 - 6 = 2 Then let's work out xb - xa: 5 - 3 = 2 Finally, let's divide our first answer by our second: 2 ÷ 2 = 1 So the gradient of this line is 1.
• Question 3

Find the gradient of a line passing through the points (5, 6) and (3, 10).

-2
- 2
EDDIE SAYS
Remember to subtract x and y coordinate values first and then divide the result of y-coordinates by this of x-coordinates. 10 - 6 = 4 3 - 5 = -2 (watch out for this minus here!) 4 ÷ -2 = -2
• Question 4

Find a gradient of a line passing between the points (10, -10) and (-2, -4).

-1/2
EDDIE SAYS
There are quite a few negative numbers here, so be careful, even if you are using a calculator! -4 - (-10) = -4 + 10 = 6 -2 - 10 = - 12 6 ÷ -12 = -6/12 = - 1/2 Did you manage to keep those positive and negative values straight?
• Question 5

Match each pair of point coordinates below with the gradients of lines between them.

## Column B

(-9, 2) and (9, 12)
5/9
(5, 3) and (-2, -9)
12/7
(-3, 1) and (5, -12)
-13/8
(-15, -3) and (0, 3)
2/5
EDDIE SAYS
All these answers are fractions which is quite common at GCSE level. If your answer is not an integer (whole number), then it's usually best to leave it as a fraction. Let's have a look at (-3, 1) and (5, -12) as an example. -12 - 1 = -13 5 - (-3) = 8 -13 ÷ 8 = -13/8 Can you use this example and what you have learnt so far to find the remaining three matches independently?
• Question 6

A line passes through two points: (-4, -1) and (-1, 4).

What is the gradient of this line?

5/3
EDDIE SAYS
How are you getting on with those negative numbers? Here's the method we need to follow to find our gradient: 4 - (-1) = 5 -1 - (-4) = 3 5 ÷ 3 = 5/3 So the gradient of the line is 5/3.
• Question 7

True or false?

The gradient of the line passing between the points (-3, -3) and (-4, 0) is -3.

True
EDDIE SAYS
This statement is true! Let's follow our process to find the gradient of the line: 0 - (-3) = 3 -4 - (-3) = -1 (not 7!) 3 ÷ (-1) = -3
• Question 8

True or false?

The gradient of the line passing through the points (4, 8) and (3, 14) is 6.

False
EDDIE SAYS
Watch out! This statement is false. Let's find the gradient of the line: 14 - 8 = 6 3 - 4 = -1 6 ÷ -1 = -6 So the gradient of the line passing between (4, 8) and (3, 14) is - 6 not 6. Always check the signs in your calculations carefully to make sure you have the correct answer.
• Question 9

Match each pair of points below so that each has a line passing through them with a gradient of 3.

## Column B

(1, 8)
(2, 11)
(-1, -1)
(1, 5)
(2, 2)
(4, 8)
(-3, -12)
(0, -3)
EDDIE SAYS
This is a slightly harder task as we need to check which coordinates go together to make a line with a gradient of 3. The best approach is to try some pairs and see if they give you an answer of 3 when you use the formula:
• Question 10

Find the gradient of a line which passes between the points (-5, -1) and (-3, -7).

-3
- 3
EDDIE SAYS
Watch out for these negative signs! It's always worth writing your calculations out, as this helps you with checking for mistakes later. -7 - (-1) = -7 + 1 = -6 -3 - (-5) = -3 + 5 = 2 -6 ÷ 2 = -3 So the gradient of a line passing through these points is -3. Congratulations on completing this activity! Now that you have learnt how to find the gradient of a line between two points, why not move on to explore the midpoint?
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