VCE Specialist Maths tutoring

The assumed knowledge is explicit. The study design says Specialist Units 3 and 4 assume the key knowledge of Mathematical Methods Units 1 and 2 and Specialist Units 1 and 2, plus "concurrent study or previous completion of Mathematical Methods Units 3 and 4". In practice nearly every Specialist student is taking Methods in the same year.

The vector proofs it lists are classic and worth practising early: that the diagonals of a rhombus are perpendicular, that the medians of a triangle are concurrent, and that the angle subtended by a diameter in a circle is a right angle.

Proof is not decoration in this study. The discrete mathematics area of study covers conjectures, connectives, quantifiers, examples and counter-examples, so students are expected to write an argument that a stranger could check line by line, not just reach an answer.

Technology still matters. Outcome 3 asks students to apply computational thinking and use technology to develop ideas, and Euler’s method is literally an algorithm. A student who has coded a few steps of it understands its error far better than one who has only read about it.

The sums are zero, as symmetry predicted; the tiny non-zero values are rounding error in floating-point arithmetic, which is itself a useful thing for a Specialist student to recognise. The products alternate exactly as predicted, and both facts can be proved in a few lines once the polygon picture suggests them.

The perimeter column is a bonus. Each side has length 2 sin(π/n), so the perimeter is 2n sin(π/n), which climbs towards 2π = 6.28319 as n grows: the polygon becomes the circle. At n = 96 the perimeter is within about 0.0011 of 2π, the same polygon Archimedes used.

The same picture solves exam-style problems quickly. To find the cube roots of 8i, write 8i in polar form as 8 with argument π/2, take the cube root of the modulus (2) and divide the argument by 3: the first root has argument π/6, and the other two are spaced 2π/3 apart, at 1.7321 + i, −1.7321 + i and −2i. A triangle on a circle of radius 2.

De Moivre’s theorem, also in the key knowledge, is what makes "divide the argument" legitimate. Proving it for integral powers by induction joins the complex numbers and the logic and proof areas of study in a single exercise, which is exactly how Examination 2 likes to combine topics.