Year 12 Mathematics Extension 1
Proof by mathematical induction and vectors in two and three dimensions.
- Each week
- One 3-hour lesson · 30-minute break
- In person · St Leonards
- Weekly from Saturday 17 October, decide by Saturday 24 October · 10:00am or 2:00pm
- Online · live from home
- Weekly from Sunday 18 October, decide by Sunday 25 October · 10:00am or 2:00pm
- Price
- $1,000 instead of $1,500 for all 10 lessons, for the first 5 students to sign up. Try 2 lessons before you pay
The term price covers all 10 lessons, the first 2 included. Pay up front, after the first lesson or after the second; if the class is not right by then, there is nothing to pay.
The term, lesson by lesson
In person Saturday 17 October to Saturday 19 December, online Sunday 18 October to Sunday 20 December. Follows the new NESA Mathematics Extension 1 11–12 Syllabus (2024), which this cohort is the first to sit in the 2027 HSC, in the order most schools teach it.
The term at a glance
| Lesson | Topic | In person | Online |
|---|---|---|---|
| Lesson 1 | Proof by mathematical inductionThe idea of proof by induction | Sat 17 Oct | Sun 18 Oct |
| Lesson 2 | Proof by mathematical inductionInduction for sums | Sat 24 Oct | Sun 25 Oct |
| Lesson 3 | Proof by mathematical inductionInduction for divisibility, and faulty proofs | Sat 31 Oct | Sun 1 Nov |
| Lesson 4 | Introduction to vectorsWhat a vector is | Sat 7 Nov | Sun 8 Nov |
| Lesson 5 | Introduction to vectorsVectors in component form | Sat 14 Nov | Sun 15 Nov |
| Lesson 6 | Introduction to vectorsVectors in three dimensions | Sat 21 Nov | Sun 22 Nov |
| Lesson 7 | Introduction to vectorsThe scalar product | Sat 28 Nov | Sun 29 Nov |
| Lesson 8 | Introduction to vectorsProjections and geometric proof | Sat 5 Dec | Sun 6 Dec |
| Lesson 9 | Introduction to vectorsVectors and motion | Sat 12 Dec | Sun 13 Dec |
| Lesson 10 | Term reviewConsolidation and term test | Sat 19 Dec | Sun 20 Dec |
Proof by mathematical induction
The idea of proof by induction
- Year 11 review
- ME1-12-01
- A quick review of polynomials, inverse functions and the binomial theorem
- What a proof by induction is: the base case, the assumption and the inductive step
- Why the steps together prove a statement for every positive integer
Key skill: Setting out an induction proof the way markers expect to see it.
Induction for sums
- Proof by induction
- ME1-12-01
- Proving formulas for the sum of a series
- Working with sigma notation inside a proof
- Managing the algebra in the k + 1 step, including fractions and powers
Key skill: Using the assumption at exactly the right point in the inductive step.
Induction for divisibility, and faulty proofs
- Proof by induction
- ME1-12-01
- Proving that an expression is always divisible by a given number
- Harder algebraic set-ups
- Spotting flawed proofs where one of the steps does not hold
Key skill: Rearranging an expression so the divisor appears.
Introduction to vectors
What a vector is
- Vectors
- ME1-12-02
- Magnitude and direction, and how vectors differ from scalars
- Vector notation, position vectors and when two vectors are equal
- Representing vectors as directed line segments
Key skill: Moving confidently between a diagram and vector notation.
Vectors in component form
- Vectors
- ME1-12-02
- Writing vectors with i and j, as column vectors and as ordered pairs
- Adding and subtracting vectors with the triangle and parallelogram rules
- Scalar multiples, parallel vectors, magnitude and unit vectors
Key skill: Calculating with components quickly and accurately.
Vectors in three dimensions
- Vectors
- ME1-12-02
- Coordinates in three dimensions and the coordinate planes
- Distance and midpoint in three dimensions
- Vectors written with i, j and k, their magnitude and unit vectors
Key skill: Picturing a three-dimensional situation and calculating in it.
The scalar product
- Vectors
- ME1-12-02
- The scalar (dot) product in component form and in terms of the angle between vectors
- Finding the angle between two vectors
- Tests for perpendicular and parallel vectors
Key skill: Using the scalar product to find angles and test for perpendicularity.
Projections and geometric proof
- Vectors
- ME1-12-02
- The projection of one vector onto another
- Splitting a vector into parallel and perpendicular parts
- Proving geometric results, such as properties of triangles and quadrilaterals, with vectors
Key skill: Writing a clear, complete vector proof.
Vectors and motion
- Vectors
- ME1-12-02
- Position as a vector function of time
- Parametric equations and finding the Cartesian path
- Constant velocity problems, and boats and planes in a current or crosswind
Key skill: Splitting motion into horizontal and vertical components.
Term review
Consolidation and term test
- ME1-12-01 to ME1-12-03
- Mixed HSC-style questions on proof by induction and vectors, including the mistakes that cost marks most often
- A short test under exam conditions, marked with written feedback
- A plan for the summer holidays, built around the further calculus that opens Term 1
Key skill: Writing rigorous proofs under time pressure.
How every lesson runs
3 hours of teaching with a 30-minute break, following the same pattern every week.
Check in
A short quiz on last week’s content, and the homework questions that caused trouble worked through on the board before anything new is taught.
Teach
The new content taught at the whiteboard, with worked examples that build from the first idea up to the standard of an HSC question.
Practise
Questions at four levels: Assimilate, Consolidate, Integrate and Extend. Everyone starts together and moves up as they are ready, with written work checked as they go.
Exam practice
Past HSC questions on the week’s topic, answered under time and marked against the NESA marking guidelines, so students learn what earns the marks.
Take home
Session notes and homework set at each student’s level, so the week’s work is secure before the next lesson builds on it.
What your child receives
Every student in the class gets all of this, week by week and across the term.
- Every week
A check-in quiz
A short quiz on last week’s content opens every lesson, so a gap is found and fixed before new work is built on top of it.
- Every week
Session notes
Notes on what the lesson covered, to revise from at home. They are sent even when a lesson is missed.
- Every week
Homework at their level
Questions set at one of four levels, Assimilate, Consolidate, Integrate or Extend, so the work stretches each student without losing them.
- Every week
Marked HSC questions
Past HSC questions on the week’s topic, answered under time and marked against the NESA marking guidelines.
- End of term
A term test with written feedback
A short test under exam conditions in the final lesson, marked and returned with written feedback on where the marks were won and lost.
- End of term
A holiday plan
A plan for the holidays that follow, built around the work that opens the next term.
The rest of Year 12
Where the course goes next, in the order schools teach it.
- Term 1, 2027
Inverse trigonometric functions in full, and further calculus skills including integration by substitution. Vector motion and projectiles once integration is in place.
- Term 2, 2027
Further applications of calculus: related rates, volumes of solids of revolution and differential equations. The binomial distribution begins.
- Term 3, 2027
The sampling distribution of the mean and the central limit theorem. Then trial exams and revision for the HSC.
Questions about this class
Which syllabus does the Mathematics Extension 1 class follow?
The class follows the new NESA Mathematics Extension 1 11–12 Syllabus (2024), which this cohort is the first to sit in the 2027 HSC, in the order most schools teach it.
What if my school teaches Mathematics Extension 1 in a different order?
The class follows the plan on this page, which uses the order most schools teach Mathematics Extension 1 in. Where your school takes a topic earlier or later, that week’s lesson is either revision of work already done or a head start on what is coming. You keep the session notes from every lesson, so they are on hand when school reaches the topic.
Can I join partway through the term?
Yes. Classes run every week through the term, so you can start whenever suits and still have two lessons to decide before you pay. Tell us which course you are in and we will let you know what the class has covered so far.
What if I miss a week?
Tell us as early as you can. Where there is room, you can sit that week with the other day’s group. The Saturday and Sunday classes cover the same material and we will send the session notes either way.
Do I pay for the first two lessons?
Only if you stay. The term price covers every lesson, the first two included. Pay up front, after the first lesson or after the second. If the class is not right after two lessons, you pay nothing.
Sitting more than one?
Every course runs as its own class, each with its own plan.
Join the Mathematics Extension 1 class
Try 2 lessons before you pay. Tell us whether Saturdays in person or Sundays online suit, and we will confirm your session.