Secondary 3/4 E-Math

E-Math is the broader O-Level syllabus, covered steadily across Secondary 3 and 4 so every topic area — number, geometry, and statistics and probability — gets real practice, not a rush in the final term.

Number, algebra and functions

Indices, standard form, and a more formal treatment of the real number system.
Expanding, factorising and manipulating algebraic expressions and formulae, including algebraic fractions and quadratic identities.
Linear, quadratic and exponential functions and their graphs, including gradient and the tangent to a curve.

Equations, sets, vectors and matrices

Linear, quadratic and simultaneous equations, and linear inequalities.
Sets and Venn diagrams, and matrices.
Coordinate geometry: gradient, the equation of a straight line, and distance between two points.
Vectors in two dimensions: notation, magnitude, and vector addition and subtraction.

Ratio, rate and percentage, revisited

Ratio and proportion extended to direct and inverse proportion.
Rate and speed problems with more moving parts, including average speed over a journey in stages.
Percentage problems that run in reverse — finding an original value from a value after a percentage change.

The rest of shapes, statistics and probability

Angle properties of polygons and geometric constructions, continued from Secondary 1 and 2.
Composite area, volume and surface area problems across several solids at once.
Grouped data: estimating the mean from a frequency table, and reading quartiles and percentiles.
Probability of combined events, using possibility diagrams and tree diagrams.

Similar figures: area and volume don't scale like length

When two figures are similar, it's tempting to assume doubling the lengths doubles the area and volume too. It doesn't — area scales with the square of the length ratio, and volume with the cube. This one idea accounts for a large share of avoidable marks lost in this topic.

Qn 1

Two similar cylinders have radii in the ratio 2:5. The smaller cylinder has a volume of 40 cm³. Find the volume of the larger cylinder.

Volume ratio = 2³ : 5³ = 8 : 125

Larger volume = 40 × (125/8)

The larger cylinder's volume is 625 cm³ ✓

Beyond right-angled triangles

Once a triangle isn't right-angled, Pythagoras and the basic trigonometric ratios stop working — this is where the sine rule and cosine rule take over, often inside bearings questions that also test reading a compass direction correctly.

Qn 2

In triangle ABC, angle A = 40°, angle B = 65°, and side a (opposite A) = 8 cm. Find side b, to 1 decimal place.

b / sin B = a / sin A

b = (8 × sin 65°) ÷ sin 40°

b ≈ 11.3 cm ✓

Circle theorems

Circle theorems give a small set of angle relationships — the angle at the centre being twice the angle at the circumference, angles in the same segment being equal, and others — that only hold under the exact conditions each theorem states. The arithmetic is usually just halving or doubling; the real skill is identifying which theorem the diagram is actually showing.

Qn 3

O is the centre of a circle, and A, B, C lie on its circumference. Angle AOB = 100°. Find angle ACB, the angle subtended by AB at the circumference.

Angle at centre = 2 × angle at circumference

Angle ACB = 100° ÷ 2

Angle ACB = 50° ✓

Arc length, sector area and radian measure

Radians are a second way to measure an angle, alongside degrees, introduced specifically because the arc length and sector area formulas become much simpler in radians. The main adjustment is remembering which unit a formula expects — mixing degrees and radians in the same calculation is the most common slip.

Qn 4

A sector of a circle has radius 6 cm and an angle of 1.5 radians at the centre. Find the arc length and the area of the sector.

Arc length = rθ = 6 × 1.5

Area = ½r²θ = ½ × 36 × 1.5

Arc length = 9 cm, area = 27 cm² ✓

Standard deviation: spread, not just average

The mean, median and mode describe where data is centred; standard deviation describes how spread out it is around that centre — two classes can have the same average score with very different consistency. E-Math doesn't require deriving the formula from scratch, but does expect it to be used correctly, including recognising when a smaller standard deviation is the better answer, and when it isn't.

Qn 5

Find the standard deviation of the data set: 2, 4, 4, 4, 5, 5, 7, 9.

Mean = (2+4+4+4+5+5+7+9) ÷ 8 = 5

Sum of (deviation)² = 9+1+1+1+0+0+4+16 = 32; variance = 32 ÷ 8 = 4

Standard deviation = √4 = 2 ✓

Also see A-Math

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