Gradient and Coordinate Lines
Calculate gradient consistently and connect coordinate evidence to a line equation and geometric interpretation.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
Gradient between two points is the change in y divided by the corresponding change in x.
Exam cue: Write coordinate pairs in matching columns before subtracting.
Concept 2
The ordering of points may be reversed only if numerator and denominator are reversed together.
Exam cue: Simplify the gradient and use the same value in the line equation.
Concept 3
Gradient can represent steepness or rate of change and feeds directly into the straight-line equation.
Exam cue: Interpret zero or undefined gradient geometrically.
Risk pitfalls and guardrails
Subtracting x-values in a different order from y-values.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Using horizontal change over vertical change.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Confusing a negative gradient with a negative intercept.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Gradient formula
m = (y₂ − y₁)/(x₂ − x₁).
Order rule
Use the same point order in numerator and denominator.
Positive gradient
The line rises as x increases.
Negative gradient
The line falls as x increases.
Zero gradient
A horizontal line.
Undefined gradient
A vertical line has zero horizontal change.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find the gradient of the line through (−3, 4) and (5, 20).
A straight road profile passes through coordinate points (2, 7) and (8, −5). Calculate its gradient.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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