Year 12 Extension 1 Term 1 Topics

Optimisation Problems
Arguably the most difficult part of the 2 unit course. There has been many years where the final challenge problem in HSC papers is on this topic. And deserving so, as it combines/examines many core ideas from the course concurrently, and has many applications.

Integration
We will prove the Fundamental Theorem of Calculus, which reveals a surprising connection between two seemingly unrelated ideas. While this important theorem answers a curious mathematical question, it is also the foundation of many practical applications. I will show you why this is the case with an application in Physics, and continue with this once we have covered vectors.

Calculus with Exponential, Log and Trig Functions
We extend all the calculus tools we have learnt thus far to these other important functions.

Vectors
In this difficult Extension 1 topic, we will develop new objects that hold both magnitude and directional information. We will define operations on these objects that allows us to measure key elements in mechanics. While doing so, we will highlight similarities with familiar objects, and realise the idea of looking at share overall structure, which underpins the study of abstract algebra.

Year 10 Term 1 Topics

Year 11 Extension 1 Term 1 Topics

Surds, Algebra and Lines Equations
Extending upon these previously covered topics by doing harder year 10 level questions. While the new topics in the second half of year 10 are substantially harder than the topics covered leading into the half yearly exam, without solid fundamentals here, students often begin to struggle with maths. Without sound algebra skills, students have trouble learning new concepts from here on out, as they are overloaded trying to handle the increasing complexity of the algebra at the same time. 

Quadratic Equations and Parabolas
We can find nice rational solutions to equations using factorisation techniques, but we can easily write down an equation that cannot be factorised. Does this mean the equation does not have any solutions? Or do we simply lack the technique to find it? We will explore these questions in depth, and find the exact characteristics of equations that control the nature of its solutions. We will then explore visual representations and applications of these ideas through the graph of parabolas.

Measurement
We will calculate surface area and volume of various shapes. We will take on the surprising difficulty of relating the dimensions of the net of a cone to its surface area and volume.

Polynomials
Asking what are the roots of a polynomial is a seemingly simple question, but very difficult in practice. The road block behind this simple question has motivated much of the study and exploration in Mathematics, and has lead to many surprising results. We will look at techniques that allow us to bypass the task of finding roots directly, and instead derive equations which reveals the pattern relating the polynomials' roots and its coefficients. We will also see how differentiation surprisingly has a role to play in this.

Trigonometry Part I
We start this topic by checking good fundamentals. We will look at how trigonometry is not simply about triangles, instead, it is about how things rotate. Consequently, trigonometry has the ability to measure and keep track of how things spin, repeat in cycles, oscillate and is what allows us to encode periodicity. Since many things rotate, or have cycles, and maths is about finding patterns, so a solid foundation in trigonometry is important. We will also derive and use new trig identities and explore three dimensional problems.

The Derivative
We introduce the idea of limits and use it while analysing motion to motivate and define the derivative, a vital component of Calculus. We will also illustrate its applications in measuring rates and optimisation. We will then ask a curious question about rates and differentiation to discover most important exponential with the special base e. 

Functions, Graphs and Transformations
We will build upon previously covered graphs and transformations by introducing new notations, language, as well as brand new graphs and transformations which we will be needed for the rest of the course. We will also examine whether the order in which we apply successive transformations matters. We then introduce the important notion of functions along with their operations.

Exponentials and Logarithms
We will tackle harder index law and exponential questions. We will also pay special attention to key behaviour of exponential equations and their graphs, an important perquisite knowledge for later work. We also Introduce the notion of logarithms and the log rules that underpins their applications. Many students don't understand the purpose of logs, so I will show you examples that illustrates their application and importance.