Specialist Mathematics ​Units 1&2

• Study Design
• Unit Materials
• VCAA Resources
• Changelog
• Other

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VCAA Study Design Unit 1
Unit 1 - Introduction
Unit 1 - ​Areas of Study
Unit 1 - Mathematical Investigation
Unit 1 - Outcome 1
​Unit 1 - Outcome 2
​Unit 1 - Outcome 3
​Unit 1 - Assessment

VCAA Study Design Unit 2
Unit 2 - Introduction
Unit 2 - ​Areas of Study
Unit 2 - Mathematical Investigation
Unit 2 - Outcome 1
​Unit 2 - Outcome 2
​Unit 2 - Outcome 3
​Unit 2 - Assessment
​
Specialist Mathematics Changes 2023​

VCE Information and Study Tips
Specialist Mathematics 1&2 Course Map (pdf) 2016-2022
Suggested Course Outline for Specialist Mathematics 1&2 2016-2022
​​
Greek Alphabet

​General CAS Information [TI-nSpire]
​General CAS Information [Casio]
TI Student Webinars

Specialist File Explorer

VCAA Specialist Mathematics
​
Bound Reference Specifications
​VCAA Formula Sheet Modified for Units 1&2 (2016-2022)

​Specialist Mathematics Implementation Videos
​Advice for Teachers (2016-2022)
Planning: Sample Course, Employability Skills, Resources
​
​Sample SACs (2016-2022)
Assessment: Scope of Tasks, Sample SACs, Performance Descriptiors

​Australian Mathematics Competition (Australian Maths Trust AMT)
Past Maths Competition and Olympiad Papers Archives (Australian Maths Trust AMT)
AMC Past papers: PDF (Australian Maths Trust AMT)

• Unit 1
• Unit 2

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Quick Links:
​Area of Study 1 - Algebra, Number, and Structure

• Proof and Number
• Graph Theory
• Logic and Algorithms

Area of Study 2 - Arithmetic and Number

• Sequences
• Combinatorics
• Matrices

 

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Area of Study 1 - Algebra and Structure

Proof and Number
​Study Design
Proof and Number Summary ​

• Notes
• Videos & Interactives
• Questions

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Number Systems
Sets and Set Notation
Prime Numbers
Conversion between Fraction and Decimal Forms
Principles of Proof 
​Simple Proofs

Mathematician Explains Infinity in 5 Levels of Difficulty (Emily Riehl on WIRED Video)
​
Mathematical Notation: ∀ "For all" and ∃ "There Exists" (Joel Speranza Math Video)
An introduction to mathematics theorems (TED-Ed Video)

Direct Proof (Joel Speranza Math Video)
Proof by Contradiction (Joel Speranza Math Video)
Proof by Contrapositive (Joel Speranza Math Video)
Disproving Statements (Joel Speranza Math Video)
Equivalent Statements (Joel Speranza Math Video)

​Proof by Contradiction

• ​How does it work? (Eddie Woo Video)
• ​Infinite Primes Example (Eddie Woo Video)

Proving a Converse Statement 

• Investigating original proposition (Eddie Woo Video)
• Considering the contrapositive (Eddie Woo Video)
• Using the inverse (Eddie Woo Video)

Four Basic Proof Techniques Used in Mathematics (patrickJMT Video)

Introduction to the Nature of Proof

• Prologue (Eddie Woo Video)
• Building blocks (Eddie Woo Video)
• Liars & truth-tellers (Eddie Woo Video)

​
Foundations of Proof

• Statements, implications, negations (Eddie Woo Video)
• Equivalence & quantifiers (Eddie Woo Video)

Negating Compound Statements

• Reviewing the basics (Eddie Woo Video)
• And, Or, Quantifiers (Eddie Woo Video)

Implications & Contrapositives

• How do they relate? (Eddie Woo Video)
• Further examples (Eddie Woo Video)

Proof: a³ - a is always divisible by 6 

• Two different approaches (Eddie Woo Video)
• Proof by exhaustion (Eddie Woo Video)
• By contradiction (Eddie Woo Video)

​
Proof by Contraposition (Eddie Woo Video)
Proof by Contradiction: log₂5 is irrational (Eddie Woo Video)

Proof - Odd composite numbers

• Setup (Eddie Woo Video)
• Understanding the negation (Eddie Woo Video)

Proving Algebraic Inequalities

• Introductory principles (Eddie Woo Video)
• Using the sign (Eddie Woo Video)

​
Proof: √3 + √2 is irrational (Eddie Woo Video)
Proof: Mersenne primes (Eddie Woo Video)

Irrational Logarithm Proof

• By cases (Eddie Woo Video)
• By graphing (Eddie Woo Video)

Mathematical Induction - Base case above 1

• Introduction (Eddie Woo Video)
• Completing the proof (Eddie Woo Video)
• Additional method (Eddie Woo Video)

Mathematical Induction (increments greater than 1) (Eddie Woo Video)

Induction with Sigma Notation 

• Basic example (Eddie Woo Video)
• Binomial Theorem - setting up proof (Eddie Woo Video)
• Binomial Theorem - completing proof (Eddie Woo Video)
• Binomial Theorem - Pascal's Identity (Eddie Woo Video)

Induction - Recursive Formulas 

• Basic example (Eddie Woo Video)
• Sum of an arithmetic progression (Eddie Woo Video)

Divisibility Proof 

• Sum of 7 consecutive integers (Eddie Woo Video)
• Using algebraic expansion (Eddie Woo Video)

How many subsets in a set? 

• Induction proof (Eddie Woo Video)
• Combinatorial proof (Eddie Woo Video)

Divisibility Tricks (Numberphile Video)
Key to the Tower of Hanoi (Numberphile Video)
Why we need proof by mathematical induction (Joel Speranza Math Video)
Proof by Mathematical Induction: Mathematical Dominoes (Joel Speranza Math Video)
Proof by induction partial sums worked examples (Joel Speranza Math Video)

​Mathematical Induction Divisibility Proof

• What is Inductive Logic? (Eddie Woo Video)
• Test and Assumption (Eddie Woo Video)
• ​Proving the k+1 Case (Eddie Woo Video)

Mathematical Induction - Inequalities

• General Pointers (Eddie Woo Video)
• Rearranging the k+1 case (Eddie Woo Video)
• Starting with the Assumption (Eddie Woo Video)
• Manipulating the Inductive Hypothesis (Eddie Woo Video)

Induction Inequality Proofs

• Unusual Properties of Inequalities (Eddie Woo Video)
• Considering One Side of the Inequality (Eddie Woo Video)
• Introducing & Transforming the Inequality (Eddie Woo Video)
• Beginning with the Assumption (Eddie Woo Video)

Mathematical Induction: Polynomial Division Proof (Eddie Woo Video)
What does mathematical induction really look like? (Zach Star Video)

Making sense of irrational numbers  (TED-Ed Video)

Proof with shapes (MathsHKO)

 

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Graph Theory
​Study Design
Graph Theory Summary ​

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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Graphs
Degree of a Vertex
Simple Graphs, Subgraphs, and Isomorphism
Connectedness, Complete and Complement Graphs
Bipartite Graphs, Trees, and Regular Graphs
Planar Graphs
Types of Walks on Graphs

Rediscovering Euler's Formula with a Mug (not that Euler's Formula) (3Brown1Blue Video)
​Euler's Formula and Graph Duality (3Brown1Blue Video)

The Bridges of Konigsberg

• Introduction (Eddie Woo Video)
• ​Digging Deeper into the Problem (Eddie Woo Video)
• ​Understanding Basic Network Theory (Eddie Woo Video)

Planar Graphs (Numberphile Video)

​​Graph Creator (yED Live Interactive)
​
Vertices and Edges (D3 Graph Theory Interactive)
Degree of a Vertex (D3 Graph Theory Interactive)
Regular Graph (D3 Graph Theory Interactive)
Complete Graph (D3 Graph Theory Interactive)
Bipartite Graph (D3 Graph Theory Interactive)
Complete Bipartite Graph (D3 Graph Theory ​Interactive)
Walk (D3 Graph Theory Interactive)
Open vs Closed Walks (D3 Graph Theory Interactive)
Connectivity (D3 Graph Theory Interactive)
Eulerian Circuit (D3 Graph Theory Interactive)
Eulerian Trail (D3 Graph Theory Interactive)
Graph Colouring (D3 Graph Theory Interactive)
k-Colourable Graph (D3 Graph Theory Interactive)
Chromatic Number (D3 Graph Theory ​Interactive)
Trees (D3 Graph Theory Interactive)
Spanning Tree of a Graph (D3 Graph Theory ​Interactive)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the General Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. ​

​Constructing Graphs and Networks
Features of Graphs and the Degree of a Vertex
Matrix Representation of Graphs
Euler's Formula and Planar Graphs
Types of Graphs and Walks
Exploring and Travelling Problems

 

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Logic and Algorithms
Study Design
​Logic and Algorithms Summary ​

• Notes
• Videos & Interactives
• Questions

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Propositional Logic and Truth Tables
Karnaugh Maps
Boolean Algebra
Binary Number System
Place Value and Bases Other than Ten (Optional)
Arguments and Validity
Logic Gates and Circuits
Algorithms
Pseudocode and Algorithms

Introduction to Fundamental Math Proof Techniques (PatrickJMT Video)
Truth Table Generator (Stanford Interactive)
Truth Table to Karnaugh Map (Charlie Coleman Interactive)
Boolean Algebra Solver (Boolean Algebra Interactive)
​​​​Representing Numbers and Letters with Binary (Crash Course Video)
Boolean Logic & Logic Gates (Crash Course Video)
Logic Gate Truth Table Check (GeoGebra Interactive)
Logic Circuit Simulator (Logic.ly Interactive)
Computer Science Basics: Sequences, Selections, and Loops (GCF Learn Free Video)
​Python - VCAA Pseudocode (Simon Tyler Notes)

 

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Area of Study 2 - Arithmetic and Number

Sequences and Series
​Study Design​
Sequences and Series Summary ​

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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​Sequences and Series
The Arithmetic Sequence and Series
The Geometric Sequence and Series
Converting Implicit Equations to Explicit Equations

Sequences and Series Technology
Casio
TI-nSpire

Partial Sums / Series

• ​Adding up Terms in an Arithmetic Progression (Eddie Woo Video)
• ​Devising a Formula (Eddie Woo Video)

Sum of an Arithmetic Progression

• Visual Derivation of Formula (Eddie Woo Video)
• Developing & Using Alternate Formula (Eddie Woo Video)
• Determining Series Length (Eddie Woo Video)

Arithmetic Series Formula Where it comes from and how to use it (Joel Speranza Math Video)
Arithmetic Series resulting in a quadratic (Joel Speranza Math Video)

Intro to Geometric Progressions

• Definitions (Eddie Woo Video)
• Algebraic Derivation of Sum Formula (Eddie Woo Video)
• Different Versions of Sum Formula (Eddie Woo Video)

 
Series & Sequences: Finding a General Term from the Sum Formula (Eddie Woo Video)

Limiting Sum

• ​Introduction (Eddie Woo Video)
• Application to Recurring Decimals (Eddie Woo Video)
• ​Rigorous Definition (Eddie Woo Video)

Limiting Sum Applications

• Derivation of Formula (Eddie Woo Video)
• Visual Interpretation & Recurring Decimals (Eddie Woo Video)
• Probability Problem (Eddie Woo Video)

Annuities

• Worked example introduction (Eddie Woo Video)
• Generating the nth term (Eddie Woo Video)
• Determining value after 10 years (Eddie Woo Video)
• When does the money run out? (Eddie Woo Video)

Present & Future Value

• Establishing the geometric series (Eddie Woo Video)
• Evaluating the series (Eddie Woo Video)
• Visualising the relationship (Eddie Woo Video)

Generating Sequences from the nth Term (Minimally Different)
Linear Sequences - Structured Variation Grid (Maths Bot)
nth Term of a Geometric Sequence (Variation Theory)
Is a Term in a Given Sequence (Minimally Different)
Sequences (Corbettmaths Exam Style)
Sequences with Fractions (Interwoven Maths)
Sequences with Surds (Interwoven Maths)
Sequences with Standard Form (Interwoven Maths)
Limiting value of sequences (pattern) (Variation Theory)
​Limiting value of a sequence (Variation Theory)
Sequences Completion Table (Interwoven Maths)

Algebraic Proof (Corbettmaths)
Algebraic Proof (Corbettmaths Exam Style)
Thinking about Proof (University of Southhampton Booklet)

Reciprocals (MathsHKO)

Absolute Value of Integers (mathsworksheets4kids)
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​Absolute Value Equations

• ​Absolute Value Equations (Interactive Practice)
• ​Absolute Value Equations 2 (Interactive Practice)

​Absolute Value Inequalities​

• ​Absolute Value Inequalities (Interactive Practice)
• Absolute Value Inequalities 2 (Interactive Practice)
• Absolute Value Inequalities 3 (Interactive Practice)

​Graphing Absolute Value Equations (Interactive Practice)

Real Part and Imaginary Part (mathworksheets4kids)
Plotting Complex Numbers (mathworksheets4kids)
Complex Numbers: Complex Conjugate (Variation Theory)
​Complex Numbers: Adding and Subtracting (Variation Theory)
Adding and Subtracting Complex Numbers (mathworksheets4kids)
Complex Numbers: Multiplication (Variation Theory)
Multiplying Complex Numbers (mathworksheets4kids)
Complex Numbers: Division (Variation Theory)
Rationalise the Denominator (mathworksheets4kids)
Simplifying Complex Numbers (mathworksheets4kids)
Absolute Value of Complex Numbers (mathworksheets4kids)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the General Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 

​First Order Recursion Relations
Recursion (Applications)
Recursion (Finance)
Using Rules (Applications)
Using Rules (Interest)
Using Rules (Depreciation)
Graphs of Sequences
Type of Sequence and Common Differences and Ratios
Series

 

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Combinatorics
​Study Design
Combinatorics Summary ​

• Notes
• Videos & Interactives
• Questions

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Countability
Pigeon-Hole Principle
Inclusion-Exclusion Principle
Permutations 
Combinations 
Combinations and Pascal's Triangle 
​
Combinatorics Technology
Casio
TI-nSpire​

Things get weird at infinity (Zach Star Video)
This set has infinite size, zero length, and a fractional dimension (Zach Star Video)

Addition and Multiplication Principles (Joel Speranza Math Video)
Factorials! (Joel Speranza Math Video)

Pigeonhole Principle (Joel Speranza Math Video)
​The Pigeon Hole Principle: 7 gorgeous proofs (Mathologer Video)
Basic Set Theory (Joel Speranza Math Video)
The Inclusion Exclusion Principle for 2 and 3 Sets (Joel Speranza Math Video)

Pigeonhole Principle

• Establishing a pattern (Eddie Woo Video)
• Generalising the cases (Eddie Woo Video)

Permutations (Joel Speranza Math Video)
Permutations of Like Objects (Joel Speranza Math Video)
Permutations with Restrictions (Joel Speranza Math Video)
Arrangements in a Circle (Joel Speranza Math Video)

Combinations (Joel Speranza Math Video)
Pascal's Triangle and Subsets of a Set (Joel Speranza Math Video)
Probability with permutations & combinations example: taste testing (Khan Academy Video)
Combinations with Restrictions (Joel Speranza Math Video)
Compound Probability: Counting (Seeing Theory Interactive)
​The mathematical secrets of Pascal's Triangle (TED-Ed Video)

Product Rule for Counting (Corbettmaths)
​Product Rule for Counting (Corbettmaths Exam Style)

 

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Matrices
Study Design
Matrices Summary

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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​Matrices and the Dimension of a Matrix
Elementary Matrix Operations
Identity Matrices and the Inverse of a Matrix
Determinant and Inverse of a Matrix Manually
Systems of Linear Equations

Matrices Technology
TI-nSpire
​Casio

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the General Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. ​

​Numerical Information in Matrices
Order and Elements of Matrices
Matrix Arithmetic
Matrix Arithmetic Applications
Inverse Matrix
Systems of Linear Equations

Quick Links:
Area of Study 1 - Statistics: ​Simulation, Sampling, and Sampling Distributions
Area of Study 2 - Space and Measurement

• Trigonometry
• Transformations
• Vectors in the Plane​

Area of Study 3 - Algebra, Number, and Structure: Complex Numbers​
Area of Study 4 - Functions, Relations, and Graphs

 

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Area of Study 1 - Statistics

Simulation, Sampling, and Sampling Distributions 
Study Design
Simulation, Sampling, and Sampling Distributions Summary

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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Statistics of Discrete Random Variables
Proof of Linear Combinations of Discrete Random Variables
Sum and Double of Discrete Random Variables
Probabilities of Sums of Discrete Random Variables 
Random Experiments and Simulation
Sampling Technology [TI-nSpire]
Sampling Technology [Casio]
Parameters, Statistics, and Sampling Distributions
Simulation of Sampling [TI-nSpire]
Simulation of Sampling [Casio]

Binomial Variables (Khan Academy Video)
​Recognising Binomial Variables (Khan Academy Video)​
​The Hypergeometric Distribution - A Basic Example (PatrickJMT Video)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the Mathematical Methods Past Examinations and Examination Reports page or the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 


Sampling without Replacement (Methods 2016-2022 Study Design Questions)
Sampling without Replacement (Methods 2000-2015 Study Design Questions)
​Discrete Random Variables: Statistics (Methods 2016-2022 Study Design Questions)
​Discrete Random Variables: Statistics (Methods 2000-2015 Study Design Questions)
Binomial Random Variables (Methods 2016-2022 Study Design Questions)
​Binomial Random Variables (Methods 2000-2015 Study Design Questions)
​Sample Distribution of Proportions (Methods 2016-2022 Study Design Questions)
​​Sample Distribution of Proportions: Statistics (Methods 2016-2022 Study Design Questions)
Sample Distribution of Means (Specialist 2016-2022 Study Design Questions)

 

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Area of Study 2 - Geometry, Measurement, Trigonometry

Trigonometry
Study Design
Trigonometry Summary

• ​Notes
• Videos & Interactives
• Questions
• Exam Questions

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Radians and Geometry
Sine Rule and Cosine Rule
Applications of the Sine and Cosine Rules
Pythagorean Identities 
Compound, Double Angle, and Factorisation Formulas 
​​Linear Combination of Trigonometric Ratios

New Trig ID's from Similar Right Triangles (GeoGebra Interactive)
Exact Values of Reciprocal Trigonometric Functions (Joel Speranza Math Video)
Solving Reciprocal Trig Functions (Joel Speranza Math Video)
The Pythagorean Identity and Reciprocal Trigonometric Functions (Joel Speranza Math Video)
2 More Pythagorean Identities (Joel Speranza Math Video)
Angle Sum and Difference Identities (Joel Speranza Math Video)
Angle Sum and Difference Identities Part 2 (Joel Speranza Math Video)
Using the tangent angle addition identity (Khan Academy Video)
​Sine & Cosine of a Sum: Discovery (GeoGebra Interactive)
​Tangent of a Sum: Discovery (GeoGebra Interactive)
​Tangent of a Difference: Discovery (GeoGebra Interactive)
​Cotangent of a Sum: Discovery (GeoGebra Interactive)

Trig Identities

• A New Perspective for Trigonometric Equations (Rosalind Carson Video)
• Trigonometry Identities Revealed from a Circle (Rosalind Carson Video)
• Identities from Circle Geometry (Rosalind Carson Video)
• Trig Identities with just a Rectangle and a Circle (Rosalind Carson Video)

Proving Trigonometric Identities (Joel Speranza Math Video)
Double Angle Identities and Proofs and an Example (Joel Speranza Math Video)
More Proofs of Trig Identities (Joel Speranza Math Video)
Express the Sum of Trigonometric Functions as a Single Function (Joel Speranza Math Video)
Express the Sum of Trigonometric Functions as a Single Function Part 2 (Joel Speranza Math Video)
Product Sum Trigonometric Identities (Joel Speranza Math Video)
Sum to Product Trigonometric Identities (Joel Speranza Math Video)
Trigonometry Fundamentals (3Blue1Brown Video)

​Sine and Cosine Rules
​The Sine Rule

• What does it actually mean? (Eddie Woo Video)
• ​Proving & Using the Formula (Eddie Woo Video)
• ​Proof by Area of a Triangle (Eddie Woo Video)
• ​Ambiguous Case (Eddie Woo Video)

Ambiguous Case (SSA) Illustrator: All Cases (GeoGebra Interactive)
​
​The Cosine Rule

• Proof of the Formula (Eddie Woo Video) 
• Basic Example of Finding Sides (Eddie Woo Video)
• ​How to Tell Which Rule to Use (Eddie Woo Video)

Cosine Rule (GeoGebra ​Interactive)
Law of Cosines, Area, Law of Sines (James Tanton Video)
Identities from Circle Geometry (Rosalind Carson Video)
​Area of a Triangle - Deriving the Trigonometric Formula (Eddie Woo Video)
​
​Bearings

• Compass Bearings (Eddie Woo Video)
• ​True Bearings (Eddie Woo Video)
• ​Converting Between Forms ​(Eddie Woo Video)

​Trigonometric Functions of Special Angles (Interactive Practice)
Trigonometric Identities (Corbettmaths Exam Style)
Addition and Double Angle Formulae (Starting Points Maths)
Wave Function Representations (Starting Points Maths)

​​Sine Rule: Finding Missing Sides (Minimally Different)
Sine Rule (Corbettmaths)
Sine Rule? (Mr Day Maths)
Cosine Rule (Corbettmaths)
Cosine Rule: Missing Angles (Variation Theory)
Cosine Rule (Variation Theory)
Area of any Triangle (Corbettmaths)
​​Sine Rule, Cosine Rule, and Area of any Triangle (Corbettmaths Exam Style)
​Can this be a Bearing? (Variation Theory)
Bearings (Don Steward)
Not Right Triangles with Exact Values (Udall Worksheet)
​Enough Information for Non-right Angled Triangles (Miss E Resources)
​Advanced Trigonometry (MathsHKO)
​Trigonometry (Maths Bot Interactive Practice)
Trigonometric Identities Unit Test (Andrew's Guide to IB Mathematics Universe)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the General Mathematics Past Examinations and Examination Reports page or the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 

Area of a Triangle using Included Angle (General)
Circle Mensuration (General 2016-2022)
Circle Mensuration (General 2000-2015)
​Trigonometric Formulas and Identities (Specialist 2016-2022)
​Trigonometric Formulas and Identities (Specialist 2000-2015)
Non-Right Angled Triangles (General)
Bearings using Non-Right Angled Triangles (General)

 

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Transformations
Study Design​
Transformations Summary ​

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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Matrix Transformation
Linear Transformations of R²
More Transformations of R² (Optional)
Inverse Transformations
Transformation on Subsets of the Plane
​Invariant Points and Lines
Eigenvalues and Eigenvectors (Optional)
Determinant and Area
Matrix Multiplication Proofs

Linear Transformations Introduction (Joel Speranza Math Video)
Transforming a Figure using Matrix Transformations (Joel Speranza Math Video)
What is actually happening in a linear transformation? (Joel Speranza Math Video)
Transformations in One Dimensional Space (Joel Speranza Math Video)

Matrices as transformations of the plane (Khan Academy Video)
Working with matrices as transformations of the plane (Khan Academy Video)
Using matrices to transform the plane: Mapping a vector (Khan Academy Video)
Using matrices to transform the plane: Composing matrices (Khan Academy Video)

Matrix Transformations (GeoGebra Interactive)
Transformations using Matrices (GeoGebra Interactive)
Visualising Matrix Transformations (GeoGebra Interactive)
Translation (GeoGebra Interactive)
2×2 Matrices and Linear Transformations (Boss Maths Interactive)
The Matrix Arcade (Yizhe-Ang Interactive)
​
Dilation Transformations using Matrices (Joel Speranza Math Video)
Rotating about the Origin using Matrices (Joel Speranza Math Video)
Reflecting in the x and y Axis using Matrix Transformations (Joel Speranza Math Video)
Reflecting in the Line y=mx with Matrix Transformations (Joel Speranza Math Video)
Projections of an Image onto the x Axis or y Axis using Matrix Transformations (Joel Speranza Math Video) [Optional]​
Shear Transformation using Matrices (Joel Speranza Math Video) [Optional]
Composition of Matrix Transformations (Joel Speranza Math Video)
Representing systems of equations with matrices (Khan Academy Video)
Inverse matrix introduction (Khan Academy Video)
Inverse matrices and matrix equations (Khan Academy Video)
Invertible matrices and transformations (Khan Academy Video)
Inverse Matrix Transformation (Joel Speranza Math Video)
Transforming Multiple Points using a Single Matrix (Joel Speranza Math Video)
Finding an Unknown Matrix of Transformation (Joel Speranza Math Video)
Transforming Relations using Matrices (Part 1) (Joel Speranza Math Video)
Transforming Relations using Matrices (Part 2) (Joel Speranza Math Video)
Transforming Relations using Matrices (Part 3) (Joel Speranza Math Video)
Transforming Relations No Shortcuts (Joel Speranza Math Video)
Finding Area of a Transformation using the Determinant (Joel Speranza Math Video)
General Transformations (Part 1) (Joel Speranza Math Video)​ [Optional]
General Transformations (Part 2) (Joel Speranza Math Video)​ [Optional]
Invariant Lines of Matrix Transformations (GeoGebra Interactive)
Intro to determinant notation and computation (Khan Academy Video)
Interpreting determinants in terms of area (Khan Academy Video)
Finding area of figure after transformation using determinant (Khan Academy Video)
Proof: Matrix determinant gives area of image of unit square under mapping (Khan Academy Video)
Invertible matrices and determinants (Khan Academy Video)
Transformation Toy (Berkley Interactive)

Matrix Transformations (Corbettmaths Exam Style)
Matrix Transformations (Dr Austin Maths, scroll down)
Tricky Transformations (Interwoven Maths)
Self-inverse Matrices (Mr Southern Maths)
Invariant Points (Corbettmaths)
Invariant lines exercise (Susan Whitehouse)
Invariant lines exam style questions (Susan Whitehouse)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the Mathematical Methods Past Examinations and Examination Reports page or the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 
​
Matrix Transformations (Methods)

 

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Vectors in the Plane
Study Design
Vectors in the Plane Summary ​

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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Vectors 
Orthogonal Unit Vectors
Resolution of a Vector into Rectangular Components 
Scalar Product / Dot Product 
Scalar and Vector Resolutes 
Vector Proofs and Lines and Ratios 
Vector Proofs and Triangles 
Vector Proofs and Quadrilaterals 
Vector Proofs and Circles 
Navigation and Orienteering using Vectors 
Motion of a Body in Equilibrium - Statics 
Vectors Technology [TI-nSpire]
Vectors Technology [Casio]

Vectors Introduction (Joel Speranza Math Video)
Adding Vectors Graphically (Joel Speranza Math Video)
Vector Addition (PhET Interactive)
Vectors Multiplying by a Scalar (Joel Speranza Math Video)
Vectors Make Sense (Looking Glass Universe Video)
What is a Vector? (TED-Ed Video)
Vectors in component form and operations (Joel Speranza Math Video)
Adding Vectors in Component Form (Joel Speranza Math Video)
Adding Vectors Geometrically (GeoGebra Interactive)
Determining a vector between two points (Joel Speranza Math Video)
Vectors Multiplying by a Scalar in Component Form (Joel Speranza Math Video)
Scaling Vectors (GeoGebra Interactive)
​Defining a Vector between 2 Points (Joel Speranza Math Video)
Vectors Finding the Magnitude (Joel Speranza Math Video)
Vectors Finding the Unit Vector (Joel Speranza Math Video)
Unit Vectors (Eddie Woo Video)
Finding Direction of a Vector (Joel Speranza Math Video)
​Defining a Vector between 2 Points (Joel Speranza Math Video)
​​Dot Product Insight (GeoGebra Interactive)
The Dot Product (Part 1) (Joel Speranza Math Video)
The Dot Product (Part 2) (Joel Speranza Math Video)
The Dot Product (Part 3) (Joel Speranza Math Video)
Dot Product Basics (Eddie Woo Video)

Dot Product, Continued

• Cosine rule (Eddie Woo Video)
• Multi-part worked example (Eddie Woo Video)

Projections

• Scalar projection (Eddie Woo Video)
• Vector projection (Eddie Woo Video)
• Introductory example (Eddie Woo Video)
• Further examples (Eddie Woo Video)

What's a Vector Projection? (Joel Speranza Math Video)
Calculating Vector Projections (Joel Speranza Math Video)
Resolving a Vector into Components Parallel and Perpendicular to a Second Vector (Joel Speranza Math Video)
​Vector Projections (GeoGebra Interactive)
​2D Projection of Vectors (GeoGebra Interactive)

Vector Midpoints (Joel Speranza Math Video)
​Vectors in 2D shapes (Joel Speranza Math Video)
Vectors in Linear Combinations (Joel Speranza Math Video)
Geometric Proofs using Vectors (Part 1) (Joel Speranza Math Video)
Geometric Proofs using Vectors (Part 2) (Joel Speranza Math Video)
15 Vector Questions You Should Know How to do by the End of the Year (Joel Speranza Math Video)
Vectors Proof: A fun parallelogram proof style question (Joel Speranza Math Video)
Vector Proof: "Unspecial" Quadrilaterals & Parallelograms (Eddie Woo Video)

Vectors: Adding and Subtracting (Variation Theory)
Vectors: Multiplying by a Scalar (Variation Theory)
Vectors (Corbettmaths Exam Style)
Determining if Vectors are Parallel (Minimally Different)
Dot Product: Basic Synthesis Questions (Interactive Practice)
Properties of Scalar Product (Starting Points Maths)
Vectors in 2 Dimensions (Starting Points Maths)
Particles in Equilibrium (Sam Blatherwick)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 

Vector Operations
Scalar and Vector Resolutes
Vector Proofs
​Motion of a Body in Equilibrium - Statics

 

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Area of Study 3 - Algebra, Number, and Structure

Complex Numbers
Study Design
Complex Numbers Summary

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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Complex Numbers
Arithmetic of Complex Numbers
Arithmetic of Complex Numbers with Dienes Blocks
​​Quadratic Functions over C

​Relations of the Complex Plane
Loci Definitions using Arguments on the Complex Plane
Conjugates and Relations on the Complex Plane
Distance Definitions of Lines, Circles, and Ellipses
Distance Definitions of Hyperbolas and Conic Sections (Optional)
Regions of the Complex Plane

Arguments and Polar Form of Complex Numbers
Multiplication and Division in Polar Form
Proof of Basic Identities involving Modulus and Argument
Powers of Complex Numbers

Complex Number Technology
TI-nSpire
Casio

​A Visual, Intuitive Guide to Imaginary Numbers (Better Explained Article)
​This Problem Broke Math (and led to quantum physics) (Veritasium Video)
Dienes Blocks (Maths Bot Interactive)​

​Introduction to Complex Numbers

• Backstory (Eddie Woo Video)
• Why Algebra Requires Complex Numbers (Eddie Woo Video)

​Complex Arithmetic

• ​Addition & Multiplication (Eddie Woo Video)
• ​Conjugates & Division (Eddie Woo Video)

Complex Arithmetic

• Basic operations (Eddie Woo Video)
• Trigonometric identity (Eddie Woo Video)
• Identity from ℝ component (Eddie Woo Video)

​Complex Numbers as Points

• Geometric Meaning of Addition (Eddie Woo Video)
• Geometric Meaning of Subtraction (Eddie Woo Video)
• ​Geometric Meaning of Multiplication (Eddie Woo Video)
• ​Second Multiplication Example (Eddie Woo Video)

​Complex Locus Plotter (GeoGebra Interactive)

Graphs in the Complex Plane

• Introductory Examples (Eddie Woo Video)
• Shifting the Point of Reference (Eddie Woo Video)
• Where is the argument measured from? (Eddie Woo Video)

Further Graphs of the Complex Plane

• Geometrical Representation of Moduli (Eddie Woo Video)
• Algebraically Verifying Graphs Concerning the Moduli (Eddie Woo Video)
• Geometrical Representation of Arguments (Eddie Woo Video)

Graphs in the Complex Plane

• Perpendicular bisector - visual method (Eddie Woo Video)
• Algebraic method (Eddie Woo Video)
• Opposing rays (Eddie Woo Video)

Semi Circles on Argand Diagrams

• Relating the angles (Eddie Woo Video)
• Graphing the locus (Eddie Woo Video)
• Oblique example (Eddie Woo Video)

Algebraic Proof for Opposing Rays 

• Foundational knowledge (Eddie Woo Video)
• Generating the equation (Eddie Woo Video)
• Testing cases (Eddie Woo Video)

​
Arcs on the Complex Plane

• Exploring circle properties (Eddie Woo Video)
• Identifying length and direction (Eddie Woo Video)
• Cartesian equation (Eddie Woo Video)

Sketching Sets Defined on the Complex Plane (Maths Stats UNSW Video) [7:39 - 13:48] ​

​Square Roots of Complex Numbers

• ​Establishing Their Nature (Eddie Woo Video)
• ​Introductory Example (Eddie Woo Video)

Introducing Polar Form

• An alternative coordinate system ​(Eddie Woo Video)
• Relationship to rectangular form (Eddie Woo Video)
• Example conversion (Eddie Woo Video)

Complex Numbers - Mod-Arg Form

• ​​Introduction (Eddie Woo Video)
• Visualising Modulus & Argument (Eddie Woo Video)
• Calculating the Modulus (Eddie Woo Video)
• Conversion Example 1 (Eddie Woo Video)
• Conversion Example 2 (Eddie Woo Video)

Multiplying Complex Numbers in Mod-Arg Form

• ​Reconsidering Powers of i (Eddie Woo Video)
• ​Generalising the Pattern (Eddie Woo Video)

Using Inverse Tan to Find Arguments?

• Why it Doesn't Work... Sometimes (Eddie Woo Video) 
• Why it Works... Sometimes (Eddie Woo Video)

Working in Polar Form 

• Conjugate & negative (Eddie Woo Video)
• Square & reciprocal (Eddie Woo Video)

Exact value of cos(π÷12) (Eddie Woo Video)
Complex Numbers: A Very Fast Recap of i, Operations, Conjugates, Argand Diagrams & Modulus (Joel Speranza Math Video)
Complex Numbers: Polar Form: Fast Recap including Operations on Polar Form (Joel Speranza Math Video)
​Relationships Between Moduli & Arguments in Products of Complex Numbers (Eddie Woo Video)
​Understanding Complex Quotients & Conjugates in Mod-Arg Form (Eddie Woo Video)

Quotient of Complex Numbers

• Evaluating modulus (Eddie Woo Video)
• Evaluating argument (Eddie Woo Video)

Argument of the Complex Conjugate (Eddie Woo Video)
Working with Moduli and Arguments (Proof Question) (Eddie Woo Video)

​The Triangle Inequalities

• Sum of Complex Numbers (Eddie Woo Video)
• Discussing Specific Cases (Eddie Woo Video)
• ​Difference of Complex Numbers (Eddie Woo Video)

​
​De Moivre's Theorem (Eddie Woo Video)
​Using De Moivre's Theorem and Binomial Expansions (Eddie Woo Video)

Proving de Moivre's Theorem

• Prologue (Eddie Woo Video)
• Derivation & example problem (Eddie Woo Video)

Using de Moivre's Theorem - Example Question

• Purely real (Eddie Woo Video)
• Purely imaginary (Eddie Woo Video)

Factorising & Solving quadratics over C (Joel Speranza Math Video)
Completing the Square with Complex Numbers (Joel Speranza Math Video)
Solving Equations with Complex Numbers (Joel Speranza Math Video)
The Simpler Quadratic Formula (3Blue1Brown Video)

​Who cares about complex numbers? (Eddie Woo Video)
Complex Number Fundamentals (3Blue1Brown Video)

What does an imaginary power mean? (Eddie Woo Video) [Optional]

Informal Proof of Euler's Formula (Optional)

• Exponential calculus (Eddie Woo Video)
• Trigonometric calculus (Eddie Woo Video)

​
Trigonometric Expansions from Complex Numbers (Optional)

• Concept map (Eddie Woo Video)
• Double angle results (Eddie Woo Video)
• General compound angles (Eddie Woo Video)

​Imaginary Numbers are Real

• Introduction (Welch Labs Video)
• A Little History (Welch Labs Video)
• Cardan's Problem (Welch Labs Video)
• Bombelli's Solution (Welch Labs Video)
• Numbers are Two Dimensional (Welch Labs Video)
• The Complex Plane (Welch Labs Video)
• Complex Multiplication (Welch Labs Video)
• Math Wizardry (Welch Labs Video)
• Closure (Welch Labs Video)

PHANTOM GRAPHS (Philip Lloyd Notes)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 

Complex Numbers (2016-2022)
Complex Numbers (2000-2015)
Relations of the Complex Plane (2016-2022)
Relations of the Complex Plane (2000-2015)
​Regions on the Complex Plane (2000-2015)
Quadratics over C

 

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Area of Study 4 - Functions, Relations, and Graphs

Study Design
Functions, Relations, and Graphs Summary ​

• Notes
• Videos & Interactives
• Questions
• Exam Questions

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Equating Coefficients of Polynomial Identities
Rational Functions and Partial Fraction Decomposition
Partial Fraction Decomposition using Limits (Optional)

Reciprocal Functions
Defining the Circular Functions on the Unit Circle
Reciprocal Circular Functions 
​
Inverse Circular Functions
Graphs of Inverse Circular Functions

Conic Sections 
Locus Definition and Construction in the Plane
Polar Equations
Parametric Equations 
​​
Absolute Value Function

Functions, Relations, and Graphs Technology
TI-nSpire
Casio

Graphing Reciprocal Functions

• Linear Example (Eddie Woo Video)
• Considering Sign (Eddie Woo Video)
• Considering Symmetry (Eddie Woo Video)
• Cubic + Semi-Circle Example (Eddie Woo Video)

Reciprocals of Functions (Joel Speranza Math Video)
Sketching Reciprocal Functions Worked Example (Joel Speranza Math Video)
Reciprocals of Trigonometric Functions​​ (Joel Speranza Math Video)
Transformations of the Reciprocal Trigonometric Functions (Joel Speranza Math Video)
Exact Values of Reciprocal Trigonometric Functions (Joel Speranza Math Video)
Solving Reciprocal Trig Functions (Joel Speranza Math Video)

Graphing Reciprocal Trigonometric Functions

• Cosecant (Eddie Woo Video)
• ​Cotangent (Eddie Woo Video)

Introduction to Locus

• Exploring Shapes through Geometric Rules (Eddie Woo Video)
• ​Using Coordinates to Assist (Eddie Woo Video)
• ​Determining Locus Equations (Eddie Woo Video)

​Locus: A Surprising Definition of a Familiar Shape (Eddie Woo Video)

Locus of a Parabola

• ​Defining Features (Eddie Woo Video)
• ​Deriving the Basic Equation (Eddie Woo Video)
• ​The Standard Forms (Eddie Woo Video)

​Translations of the Parabola

• ​Find the Equation from its Features (Eddie Woo Video)
• ​Updated Standard Forms (Eddie Woo Video)

​Parabola: Locus Definition (GeoGebra Interactive)
​Conic Sections: Double Cone (GeoGebra Interactive) [Optional]
Double Cone Model (GeoGebra Interactive) [Optional] [Broken?]

Introduction to Parametrics

• Clarity & Convenience (Eddie Woo Video)
• ​How to Eliminate a Parameter (Eddie Woo Video)

Parametric Equations → Cartesian Equation

• Manipulating with Trigonometric Identities (Eddie Woo Video)
• ​What's the Graph? (Eddie Woo Video)

​
Parametric Curves with x(t) and y(t) Curves (GeoGebra Interactive)
Parametric Equations Grapher (GeoGebra Interactive)
Create Parametric Curves Graphically and Kinesthetically (Sine of the Times Interactive)
Polar Coordinates ​(KirstaKingMath Video)

Graphing a Polar Curve

• Graphing a Polar Curve - Part 1 (PatrickJMT Video)
• Graphing a Polar Curve - Part 2 (PatrickJMT Video)

How do you sketch a polar curve? (KristaKingMath Video)

Lissajous Figures (Codepen Animation)
Lissajous Curves (GeoGebra Interactive)
​​Polar Graphs (GeoGebra Interactive)
​Swirlly: Online Spirograph (Swirlly Interactive)

Cool Curves (with Applications) (Zach Star Video)


​Solving Absolute Value Equations Example (Eddie Woo Video)

Absolute Value Equations & Inequalities

• Visualising an Equation (Eddie Woo Video)
• Visualising the inequality (Eddie Woo Video)
• Separate intervals (Eddie Woo Video)
• Graphing to Avoid Unnecessary Algebra (Eddie Woo Video)

​Absolute Value Equations (GeoGebra Interactive)
Absolute Value Inequalities (GeoGebra Interactive)
​Absolute Value Function (GeoGebra Interactive)

This Problem Broke Math (and led to quantum physics) (Veritasium Video)

Identities (Mr Day Maths)
Graphing Ellipses (EQ in Standard Form) (Interactive Practice)
Parametric Equations (Mr Day Maths)

Solutions to past VCAA exam questions can be looked up in the appropriate examiner's report on the Mathematical Methods Past Examinations and Examination Reports page or the Specialist Mathematics Past Examinations and Examination Reports page or the Northern Hemisphere Timetable (NHT) Past Examinations and Examination Reports page. 

​Partial Fractions (2016-2022)
Partial Fractions (2000-2015)
Reciprocal Functions
Inverse and Reciprocal Trigonometric Functions 
​Curves Defined Parametrically (2016-2022)
Curves Defined Parametrically (2000-2015)
Circles, Ellipses, and Hyperbolas in Cartesian Form (2016-2022)
Circles, Ellipses, and Hyperbolas in Cartesian Form (2000-2015)
​Absolute Values (2016-2022)
Absolute Values (Methods 2000-2015)

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