Reading a gradient from the grid
Find a line's signed rate of change using a large triangle and the axis scales.
Core and Extended: this lesson reads a gradient from a grid. Calculating it directly from two stated coordinates without a grid is the next, Extended-only lesson.
Before you begin
- Read coordinate scales on both axes.
- Simplify signed fractions.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Simplify 4/6.
What you will learn
- Find rise and run from a plotted straight line.
- Use the signed ratio vertical change ÷ horizontal change.
- Explain positive, negative, zero and undefined gradients.
Use a triangle on the line
Choose two clear grid intersections on the plotted line, preferably far apart. Draw or imagine a horizontal side from the first point and a vertical side to the second. Read the horizontal change, called the run, and vertical change, called the rise, from the axis scales. The gradient is rise divided by run, not the reverse.
On the diagram, moving six units to the right takes you four units up, so the gradient is 4/6 = 2/3. A larger triangle on the same line gives the same ratio because straight lines have a constant gradient. Using points far apart reduces the effect of a small plotting or reading error.
The sign belongs to the vertical change
Read from left to right so the horizontal change is positive. A rising line then has a positive vertical change and gradient; a falling line has a negative change and gradient. A fall of six units over a run of four means m = −6/4 = −1.5. Its steepness is 1.5, but the gradient includes the negative sign.
A horizontal line has no vertical change and gradient zero. A vertical line has no horizontal change; dividing by zero is undefined. Its gradient is undefined, not zero and not a very large exact number. These cases matter when choosing equations and comparing lines.
Count values rather than printed squares
Suppose a graph's triangle runs three squares right and rises four squares. If each horizontal square represents two units and each vertical square represents 0.5 units, the run is six and the rise is two, giving m = 1/3. The square-count ratio 4/3 would be wrong because the axis scales differ.
For a graph in context, attach units to the ratio. On a distance–time graph, kilometres divided by hours gives kilometres per hour. A negative gradient could mean decreasing distance from a reference point rather than negative speed. Use the quantities named by the axes to interpret the sign.
Pause and explain
A grid triangle runs 5 units right and falls 2 units. What is m?
Worked example
Use the diagram's gradient triangle from (−2, −1) to (4, 3) to find the line's gradient.
Show the worked solution
- From left to right, the horizontal side runs six grid units.
- The vertical side rises four grid units.
- Divide rise by run: m = 4/6 = 2/3; the line rises, so the sign is positive.
Answer 2/3
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A line's grid triangle runs 8 units right and rises 6 units. Find and interpret its gradient.
Give me a hint
Divide the vertical change by the horizontal change.
Compare my reasoning
- The rise is 6 and the run is 8.
- m = 6/8 = 3/4.
- For each extra horizontal unit the line gains 0.75 vertical units.
3/4
Look for these in your work
- Put rise in the numerator.
- Simplified and explained the positive rate.
A triangle on a graph runs 4 units right and falls 10 units. Calculate the gradient.
Give me a hint
A fall is a negative vertical change.
Compare my reasoning
- Record vertical change −10 and horizontal change 4.
- m = −10/4 = −5/2.
- The negative sign matches the line's fall from left to right.
−5/2
Look for these in your work
- Retained the negative sign.
- Used coordinate units from the grid.
A plotted distance–time line rises 3 vertical squares over 2 horizontal squares. The scales are 4 km per vertical square and 0.5 h per horizontal square. Find the rate.
Give me a hint
Convert both square counts into the quantities on the axes.
Compare my reasoning
- Distance change is 3 × 4 = 12 km.
- Time change is 2 × 0.5 = 1 hour.
- Rate is 12/1 = 12 km/h.
12 km/h
Look for these in your work
- Converted both axes before dividing.
- Included the rate's units.
Common mistakes
- Using run divided by rise.
- Ignoring unequal axis scales.
- Calling a vertical gradient zero.
What can you explain now?
What is the gradient of a horizontal line?
Compare with the explanation
0
The vertical change is zero for a non-zero horizontal change, so the ratio is zero.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, sketch a rising and a falling line. Mark a large triangle on each and explain its sign before calculating.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons; Geometry contains 15. Coordinate geometry and Geometry each have a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other five syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice and paper constructions are self-checked, not automatically graded. The checkpoints sample skills and do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources