Functions, relations and graphs
What you must understand
A relation is any set of ordered pairs; a function is a relation where each input has exactly one output. You must be able to read and plot points on the Cartesian plane, identify the independent variable (horizontal axis) and dependent variable (vertical axis), and interpret the meaning of intercepts and gradients in context. Know the shapes of the key graphs: straight lines, parabolas, and simple exponential growth and decay curves.
For linear relationships, understand y = mx + c: m is the gradient (change in y divided by change in x) and c is the y-intercept. Be able to find a gradient from two points, recognise a positive, negative, zero or undefined gradient, and use a line of best fit drawn through scatterplot data to make predictions. Understand that interpolation (predicting inside the data range) is more reliable than extrapolation (predicting beyond it).
Traps to watch
- Reversing the gradient formula - it is (y₂ − y₁) ÷ (x₂ − x₁), keeping the points in the same order.
- Confusing the gradient with the intercept when reading a rule from a graph.
- Reading a value off the wrong axis, or forgetting the units of the gradient (for example dollars per hour).
- Treating every curve as linear between widely spaced points.
- Trusting an extrapolated prediction far beyond the plotted data.
Worked example
A line passes through (2, 5) and (6, 13). Gradient = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2. Using y = mx + c with (2, 5): 5 = 2(2) + c, so c = 1 and the rule is y = 2x + 1. Check with the other point: 2(6) + 1 = 13. Rule of thumb: always verify a derived rule against both original points.
Space, measurement and geometry
What you must understand
This domain is about measuring and reasoning with shapes in two and three dimensions. You must know and apply perimeter and area formulas for rectangles, triangles, parallelograms, trapeziums and circles, and volume formulas for prisms and cylinders. Understand that volume of a prism is the area of its cross-section multiplied by its length, and that capacity in litres relates to cubic centimetres (1 cm³ = 1 mL, 1000 cm³ = 1 L).
Know angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, vertically opposite angles are equal, and angles in a triangle sum to 180°. Recognise parallel lines and the corresponding, alternate and co-interior angle relationships. Be able to use Pythagoras' theorem to find a missing side of a right-angled triangle, and work with similarity and scale factors, remembering that areas scale by the square of the factor and volumes by its cube.
Traps to watch
- Mixing units - centimetres with metres in the same calculation.
- Using the diameter instead of the radius in circle formulas.
- Forgetting the half in the triangle area formula, or using slant height instead of perpendicular height.
- Applying Pythagoras to a triangle that is not right-angled.
- Scaling a length by a factor and assuming area and volume scale by the same factor.
Worked example
A cylindrical tank has radius 50 cm and height 120 cm. Volume = πr²h = π × 50² × 120 ≈ 942 478 cm³, which is about 942 litres. If a similar tank has all lengths doubled, its volume becomes 2³ = 8 times as large, about 7540 litres. Rule of thumb: write the units beside every step so mismatches are caught early.