Read coordinates and calculate gradient
Locate ordered pairs, interpret axes and find the gradient from a grid or two coordinates.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In the point (−3, 5), which coordinate is the horizontal position?
What you will learn
- Use Cartesian coordinates in two dimensions
- Find gradient from a grid for Core
- Calculate gradient between points for Extended
Read ordered pairs in a fixed order
Coordinates are written (x, y): move horizontally from the origin, then vertically. A point in quadrant II has negative x and positive y; label each axis and scale before reading.
For a straight line, gradient is vertical change divided by horizontal change: m = rise/run. A negative gradient falls as x increases; a horizontal line has gradient 0.
Use two points for Extended
For points (x₁, y₁) and (x₂, y₂), m = (y₂ − y₁)/(x₂ − x₁), provided the line is not vertical. Keep the coordinate order consistent in numerator and denominator.
Core questions find gradient from a grid. Extended candidates also calculate it directly from coordinates. A vertical line has undefined gradient because its horizontal change is zero.
Pause and explain
What is the gradient of a horizontal line?
Worked example
Find the gradient through A(−2, 1) and B(4, 10).
Show the worked solution
- The vertical change is 10 − 1 = 9.
- The horizontal change is 4 − (−2) = 6.
- Gradient m = 9/6 = 3/2.
Answer Gradient 3/2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →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.
Plot P(−1, 2) and Q(3, 2) conceptually, then find the gradient of PQ.
Give me a hint
The y-coordinate does not change; calculate rise over run.
Compare my reasoning
- Both points have y = 2, so the line is horizontal.
- Rise = 2 − 2 = 0.
- Run = 3 − (−1) = 4; gradient = 0/4 = 0.
Gradient 0.
Look for these in your work
- Ordered pairs read correctly
- Horizontal line identified
- Rise-over-run used
Find the gradient through (2, −1) and (6, 7), then state whether the line rises or falls from left to right.
Give me a hint
Use the same endpoint order for the y and x differences.
Compare my reasoning
- Change in y = 7 − (−1) = 8.
- Change in x = 6 − 2 = 4.
- Gradient = 8/4 = 2, so the line rises from left to right.
Gradient 2; it rises.
Look for these in your work
- Signed differences correct
- Ratio simplified
- Direction interpreted
A map uses one square per 2 km east and 1 km north. A route joins A(−2, 1) to B(4, 4). Find its gradient in grid units and explain why the physical map gradient depends on the axis scales.
Give me a hint
Calculate grid rise/run, then convert each axis change using its different scale.
Compare my reasoning
- Grid rise = 4 − 1 = 3 and run = 4 − (−2) = 6, so grid gradient is 1/2.
- Physical north change is 3 × 1 = 3 km; east change is 6 × 2 = 12 km.
- The physical ratio is 3/12 = 1/4 because one horizontal grid unit represents twice the distance.
Grid gradient 1/2; physical north/east gradient 1/4.
Look for these in your work
- Grid differences found
- Unequal map scales applied
- Units and meaning distinguished
Common mistakes
- Reversing x and y in an ordered pair
- Dividing run by rise
- Changing point order in only one difference
- Assigning a finite gradient to a vertical line
What can you explain now?
A line rises 6 units for a run of 3 units. What is its gradient?
Compare with the explanation
2
Gradient is rise divided by run.
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Make it stick
How do ordered pairs and signed changes determine the gradient, and what makes a vertical gradient undefined?
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.