Unit 1
Specialist Mathematics (Queensland)
Topic 1: Combinatorics

Addition and Multiplication principles

Factorials!

Introduction to Permutations

Permutations

Combinatorics on the Casio FX CG50 Calculator (Factorials, Permutations, Combinations)

Arrangements in a circle

Arrangements from like objects

Addition and multiplication principles revision

Factorials and permutations revision

Permutations of like objects

Permutations with restrictions

Combinations

Combinations with restrictions

Pascals triangle and subsets of a set

Pigeonhole principle

Basic Set Theory

The inclusion exclusion principle for 2 and 3 sets
Topic 2: Vectors in the Plane

Vectors Introduction

Adding Vectors Graphically

Vectors Multiplying by a scalar

Vectors in 2D and 3D Shapes

Vectors in component form

Adding Vectors in component form

Vectors multiplying by a scalar in component form

Vectors finding the magnitude

Vectors finding the unit vector

Finding direction of a vector

Defining a vector between 2 points

Vectors: Polar form (converting from component to polar and polar to component)

Vector Revision part 1

The dot product part 1

The dot product part 2

The dot product part 3

The scalar product or dot product of a vector revisited

Force Vector Diagrams

Resolving Force Vectors

Resolving forces into their components

Resolving Forces into components part 2

Vector midpoints

Vectors in linear combinations

Vectors in component form recap

Vectors in polar form

What’s a vector projection?

Calculating vector projections

Resolving a vector into components parallel and perpendicular to a second vector

Applications of vectors displacement and velocity

Applications of vectors displacement and velocity collisions

Applications of vectors relative velocity

Relative Velocity: EVERY QUESTION: fully worked solutions

Triangle of Forces: Constructing the triangle from a force diagram

Triangle of forces: Worked Example

Triangle of Forces: Inclined Plane

Triangle of forces

Inclined plane recap part 1

Solving equilibrium questions using resolution of forces

Vectors in component form and operations

Determining a vector between two points

Vectors in 2D shapes
Topic 3: Introduction to Proof

The Real Number System

Expressing a recurring decimal as a fraction

Mathematical Notation: ∀ “For all” and ∃ “There Exists”

Direct proof

Proof by contrapositive

Proof by contradiction

Equivalent statements

Disproving statements

Geometric proofs from the ground up

Proving the angle in a semicircle is a right angle

Prove the angle at the centre theorem

The angle on the same arc theorem

Proving Chord of Circle Theorem

Proving Tangent to a circle and the radius are perpendicular Using Proof by Contradiction

Proving the tangents from an external point theorem

Proving the cyclic quadrilateral theorem

Alternate Segment Theorem Proof

Intersecting Chord Theorem Proof

Proving the tangent secant theorem

Equidistant chord Theorem Proof

Intersecting Secants Theorem Proof: Teacher vs Student

Geometric Proofs using Vectors part 1

Geometric proofs using vectors Part 2