Course Description

This course provides teachers with a comprehensive guide to the knowledge and techniques required to teach the new NSW Mathematics Extension 2 syllabus. 

The course goes into greater detail than the textbooks, so that teachers can become more comfortable in their ability to deliver the content. All 179 examples have been recorded, so that you can watch an experienced maths teacher talk through the examples and any important points to note.

EXT2#1 covers Proof and Vectors MEX-P1 and MEX-P2 (V1.1, V1.2 and V1.3), two topics that are paired with topics in Extension 1. They are best taught early in the course, once the equivalent topics in Extension 1 have been taught.

New and existing content is explored in detail, including interesting and more efficient techniques that experienced teachers may not have seen before. Important points to be covered with students are noted, and the limits of the syllabus are discussed. As more detail on the syllabus is released by NESA any changes required will be made. Also included are related skills that are beyond the syllabus that you can use to extend more capable students.

The course includes:

  • 179 recorded examples

  • 306 page course handout 

  • 179 optional practice questions matching the examples from the course, with fully worked solutions

 A suggested scope and sequence for Year 12 Advanced, Extension 1 and Extension 2 for combined classes is provided, plus a lesson by lesson breakdown of the Advanced, Extension 1 and Extension 2 courses for Year 11 and 12 to help you create your own scope and sequences.


Mathematics teachers who are preparing to teach the new Extension 2 syllabus. This course is suitable for teachers with any level of previous experience.

Teaching Standards

2.1.2 Proficient Level - Know the content and how to teach it - Content and teaching strategies of the teaching area:  Apply knowledge of the content and teaching strategies of the teaching area to develop engaging teaching activities

Do you teach in NSW? If so, this is relevant to you:

Completing Mathematics Extension 2: Proof And Vectors - EXT2#1 will contribute 10 hours of NSW Education Standards Authority (NESA) Accredited PD in the priority area of Delivery and Assessment of NSW Curriculum/Early Years Learning Framework, addressing Standard Descriptor 2.1.2 from the Australian Professional Standards for Teachers towards maintaining Proficient Teacher in NSW.


Course curriculum

  • 1


    • Course Administration

    • Introduction to the Mathematics Extension 2 In Depth Year 12 Series

    • Introduction to Part I

  • 2


    • Introduction to Proof

  • 3

    Proof 1: Language of Proof & Simple Proofs

    • Lesson 1 Overview

    • How should we use the Language of Proof?

    • Terms and Symbols

    • Implication

    • Quantifiers

    • Other Terminology

    • Symbols and Set Notation

    • Simple Proofs Involving Numbers

    • Lesson 1 Quiz

  • 4

    Proof 2: Proof by Contradiction & Contrapositive

    • Lesson 2 Overview

    • Converse

    • Negation

    • Contrapositive

    • Proof by Contrapositive

    • Proof by Contradiction

    • Lesson 2 Quiz

  • 5

    Proof 3: Equivalence & Disproofs

    • Lesson 3 Overview

    • Equivalent Statements (If and Only If)

    • Proving Equivalence

    • Disproof, Examples and Counterexamples

    • Lesson 3 Quiz

  • 6

    Proof 4: Inequality Proofs

    • Lesson 4 Overview

    • Inequalities

    • Setting out the Proof

    • The Basics of Inequalities

    • Equality

    • Properties of Positive Numbers

    • Geometrical Analogies

    • Miscellaneous Tips

    • Triangle Inequality

    • The Square of a Real Number is Non-negative

    • Lesson 4 Quiz

  • 7

    Proof 5: Arithmetic Mean - Geometric Mean Inequality

    • Lesson 5 Overview

    • The Four Means

    • Arithmetic Mean - Geometric Mean Inequality

    • Proving the AM-GM Inequality for 3 or 4 variables

    • Lesson 5 Quiz

  • 8

    Proof 6: Induction I - Harder Algebraic Proofs

    • Lesson 6 Overview

    • Mathematical Induction in Extension 2

    • The Initial Value and Step not 1

    • Sigma Notation

    • Divisibility Proofs in Extension 2

    • Inequality Proofs by Induction

    • Lesson 6 Quiz

  • 9

    Proof 7: Induction II - Other Proofs

    • Lesson 7 Overview

    • Choosing a Method in Unusual Questions

    • Calculus

    • Probability

    • Geometry

    • Classic Induction Questions

    • First Order Recursive Formula

    • Lesson 7 Quiz

  • 10

    Proof 8: Beyond the Syllabus (Optional)

    • Lesson 8 Overview

    • Fibonacci Sequence and Induction

    • Finding the Conjecture

  • 11


    • Introduction to Vectors

  • 12

    Vectors 1: Three dimensional Vectors

    • Lesson 1 Overview

    • Typing Vectors in Microsoft Office (Word, PowerPoint etc)

    • Vectors in Three Dimensions

    • Three Dimensional Coordinate System

    • Standard Unit Vectors

    • Addition, Subtraction and Scalar Multiplication

    • Magnitude of Vectors

    • Unit Vectors

    • Scalar (Dot) Product of Vectors

    • Angle Between Two Vectors

    • Lesson 1 Quiz

  • 13

    Vectors 2: Geometric Proofs

    • Lesson 2 Overview

    • Geometric Proofs

    • Lesson 2 Quiz

  • 14

    Vectors 3: Straight Lines

    • Lesson 3 Overview

    • Vector Equation of a Line in Two Dimensions

    • Lines in Three Dimensions

    • Sketching a Vector Equation

    • Finding the Vector Equation of a Line Through Two Points

    • Parallel and Perpendicular Lines

    • Does a Point Lie on a Line?

    • Point of Intersection of Two Lines

    • Converting Between Cartesian and Vector Equations

    • Lesson 3 Quiz

  • 15

    Vectors 4: Spheres and Curves

    • Lesson 4 Overview

    • Spheres

    • Curves in Two Dimensions

    • Transformations

    • Restrictions

    • Spirals

    • Miscellaneous Curves

    • Curves in Three Dimensions

    • Helices

    • Unusual Curves

    • Lesson 4 Quiz

  • 16

    Vectors 5: Beyond the Syllabus (Optional)

    • Lesson 5 Overview

    • Curves Involving Sine and Cosine

    • Roulettes

    • Other Curves

  • 17


    • Highlights of Proof and Vectors

    • Conclusion to Part I

  • 18

    Course Feedback

    • Course Feedback


Steve Howard

Steve Howard

Steve Howard has inspired and equipped hundreds of Maths teachers with his In-Depth Senior Mathematics courses, providing all the tips, examples and resources to teach the HSC syllabus with confidence. He loves to discover the most efficient techniques for solving mathematical problems, by analysing other experts' solutions or by developing his own approaches when he finds a better way. As a student, Steve did Unit 4 Mathematics, gaining a mark of 198/200, before training as an actuary. After retraining as a teacher, Steve has taught Maths at Cowra High School, NSW, for 28 years. He has also taught gifted and talented students online through Xsel and Aurora College, and developed all student course material for TTA's Year 12 Extension Maths Portal.

Other Courses by Steve Howard

Features of TTA Online PD

  • Availability

    Online courses are available 24/7. Designed to be done in your own time at your own pace.

  • Team Online

    All online courses are available for team purchase. Unlimited teachers from the one Campus for any course for $1250 + GST

  • Money back Guarantee

    If you complete less than 25% of an online course and aren't completely satisfied, let us know, and we will cancel your enrolment and provide a full refund.