Units 1 & 2

The nature of proof — implication, converse, contrapositive, contradiction, counter-examples and quantifiers; circle properties and proofs.

• Wk 1 The nature of proof · implication · converse · equivalence · negation · inverse · contrapositive · contradiction · counter-examples · quantifiers (live)
• Wk 2–4 Circle properties · prove unknown angles and lengths · prove further results using circle properties and converses (live) Task 1 · Wk 4

Permutations and combinations, the multiplication and addition principles, inclusion-exclusion, the pigeon-hole principle, and identities from Pascal's triangle.

• Wk 5 Permutations · ordered arrangements · factorial notation · multiplication and addition principles (live)
• Wk 6 Inclusion-exclusion principle · pigeon-hole principle · proofs and applications (live)
• Wk 7 Combinations · unordered selections · Pascal's triangle identities (live) Task 2 · Wk 7

Geometric vectors, magnitude and direction, the triangle and parallelogram rules; algebra of vectors in component form, scalar product, projection; vector proofs of properties of parallelograms.

• Wk 8–9 Vectors as directed line segments · magnitude · direction · scalar multiple · triangle and parallelogram rules (live)
• Wk 10–12 Algebra of vectors · component form · unit vectors · scalar product · parallel and perpendicular · projection · displacement, force, velocity (live) Task 3 · Wk 12
• Wk 13–14 Geometric vectors and proof · properties of parallelograms via vectors (live)
• Wk 15 Semester 1 examination Task 4

Graphs of f(a(x − b)) + c for sin, cos, tan; compound angles, sum/difference, double angle, Pythagorean identities, products to sums, R cos(x ± α) form; reciprocal trig functions; modelling periodic phenomena.

• Wk 1 Basic trig functions · solve f(a(x − b)) = c · graph y = f(a(x − b)) + c for sin, cos, tan (live)
• Wk 2–3 Compound angles · sum/difference · double angle · Pythagorean · products as sums and differences · convert a cos x + b sin x to R cos(x ± α) (live) Task 5 · Wk 3
• Wk 4 Reciprocal trig functions · transformations · applications to periodic phenomena (live)

Matrix arithmetic, the 2 × 2 determinant and inverse; transformations of the plane (translations, dilations, rotations, reflections), composition and inverses; systems of linear equations in two variables.

• Wk 5–6 Matrix arithmetic · addition, scalar and matrix multiplication · 2 × 2 determinant and inverse · solving AX = B (live)
• Wk 7–8 Transformations in the plane · translations, dilations, rotations, reflections · composition and inverses · determinant and area (live) Task 6 · Wk 8
• Wk 9 Systems of linear equations in two variables · matrix algebra solution (live)

Proofs about numbers, rational and irrational numbers; mathematical induction for sums and divisibility; introduction to complex numbers, the complex plane, conjugates and arithmetic, complex roots of real quadratics.

• Wk 10–11 Proofs involving numbers · rational and irrational · terminating and recurring decimals · prove √2 irrational by contradiction (live)
• Wk 12 Mathematical induction · sums and divisibility results (live) Task 7 · Wk 12
• Wk 13 Complex numbers · i as a root of x² = −1 · rectangular form · the complex plane · conjugates and arithmetic (live)
• Wk 14 Roots of equations · quadratics with complex roots · conjugate pairs · linear factors of real quadratics (live)
• Wk 15 Semester 2 examination Task 8