Gradient as rate of change
OCR J560 expects students to read a gradient as a real-world rate, complete with units. The skill spans every applied context — finance, physics, biology.
Gradient with units
If a graph plots y (units U_y) against x (units U_x), the gradient has units U_y / U_x.
| Plot | Gradient is... | Units |
|---|---|---|
| Distance (m) vs time (s) | Speed | m/s |
| Volume (litres) vs time (min) | Flow rate | litres/min |
| Cost (£) vs items (n) | Cost per item | £/item |
| Population vs years | Growth rate | people/year |
| Mass (g) vs volume (cm³) | Density | g/cm³ |
Reading a gradient from two points
Pick (x₁, y₁) and (x₂, y₂) on the line. Gradient = (y₂ − y₁) / (x₂ − x₁). Always show this calculation in OCR working.
Proportion graphs
A graph of y = kx (direct proportion) is a straight line through the origin with gradient k.
Identifying:
- If the line passes through (0, 0) and is straight → direct proportion.
- The gradient gives the constant of proportionality k.
A graph of y = k/x (inverse) is a hyperbola — never crosses an axis.
✦Worked example
A petrol tank is filled at a constant rate. After 0 seconds the tank holds 0 litres; after 30 seconds it holds 24 litres. The graph is a straight line through the origin.
Gradient = 24 / 30 = 0.8 litres/s. So petrol flows at 0.8 litres per second.
This is direct proportion: V = 0.8t, with k = 0.8 litres/s.
Variable gradient (curves)
For a curve, the gradient at a point is the gradient of the tangent at that point. This is the instantaneous rate of change. Drawn as a tangent line on the curve and computed from two points on the tangent.
OCR mark scheme conventions
- M1 for picking two points and computing (Δy)/(Δx).
- A1 for the numerical value.
- B1 for stating the rate "with units and meaning" (e.g. "the volume rises at 0.8 litres per second").
⚠Common mistakes
- Quoting a gradient without units when units are required.
- Reading off a curve as if it were a straight line.
- Misreading the scale on either axis.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves