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Gradient

FoundationAQAEdexcelOCR

Practise gradient with this free Foundation GCSE Maths worksheet from Virtus Academy. You'll work through calculating the gradient of a line, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. Gradient = change in y ÷ change in x (rise over run).

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Topic overview

The gradient of a straight line measures its steepness. It is the change in \(y\) divided by the change in \(x\), often remembered as rise over run.

A positive gradient means the line slopes upwards from left to right, and a negative gradient means it slopes downwards. A larger number, ignoring sign, means a steeper line.

Two special cases are worth knowing. A horizontal line has gradient \(0\), because there is no rise. A vertical line has an undefined gradient, because the run is zero and you cannot divide by zero.

Revision notes

Calculating the gradient

Pick two clear points on the line, then divide the vertical change by the horizontal change.

For points \((1, 2)\) and \((4, 11)\): the rise is \(11 - 2 = 9\) and the run is \(4 - 1 = 3\), so the gradient is \(9 \div 3 = 3\).

Positive and negative gradients

Read the line from left to right. Upward means positive, downward means negative.

A line through \((0, 6)\) and \((3, 0)\) falls by \(6\) over a run of \(3\), giving a gradient of \(-2\). The minus sign is part of the answer.

Choosing good points

Use points where the line crosses grid intersections exactly, and as far apart as possible.

Close or estimated points make small reading errors much more significant. Drawing a right-angled triangle on the line makes the rise and run easy to read.

Key points

  • Gradient is the change in y divided by the change in x.
  • Remember it as rise over run.
  • A positive gradient slopes up from left to right.
  • A negative gradient slopes down.
  • A horizontal line has gradient 0.
  • A vertical line has an undefined gradient.

Worked examples

Example 1

Find the gradient of the line through \((2, 3)\) and \((6, 15)\).

Working

\[15 - 3 = 12\]find the change in y
\[6 - 2 = 4\]find the change in x
\[12 \div 4 = 3\]divide rise by run

Example 2

Find the gradient of the line through \((0, 8)\) and \((4, 0)\).

Working

\[0 - 8 = -8\]the line falls, so the change in y is negative
\[4 - 0 = 4\]find the change in x
\[-8 \div 4 = -2\]the gradient is negative

Example 3

A line passes through \((1, 5)\) and \((7, 5)\). Find its gradient.

Working

\[5 - 5 = 0\]there is no change in y
\[0 \div 6 = 0\]the line is horizontal, so the gradient is zero

Common mistakes

  • Dividing run by rise.

    The gradient is rise over run, so the change in y goes on top.

  • Losing the negative sign.

    A line sloping down has a negative gradient, and the sign is part of the answer.

  • Reading points inaccurately.

    Use points at exact grid intersections and as far apart as possible.

  • Saying a vertical line has gradient zero.

    A horizontal line has gradient 0; a vertical line's gradient is undefined.

Exam tips

  • Draw a right-angled triangle on the line to read the rise and run.
  • Choose points far apart at exact grid intersections.
  • Check the sign matches the direction of the slope.
  • Show the subtraction for both changes in your working.

Key terms

Gradient
A measure of the steepness of a line.
Rise
The vertical change between two points.
Run
The horizontal change between two points.
Undefined
Having no valid value, as with the gradient of a vertical line.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.