Sine Waves Superimposed to create a Square Wave
"Go down deep enough into anything and you will find mathematics"
We start by deepening the knowledge of the algebraic concepts learned previously: learning to rationalise the denominator; proving that completing the square and the quadratic formula are one and the same; expanding our knowledge of polynomials to cubics and quartics; seeing Euclidean geometry and Cartesian graphs merge.
Proof becomes a topic in its own right as we look at different methods: by exhaustion; by deduction; by contradiction. We look at Pascal's triangle, and how we can generalise with factorial notation leading to the binomial expansion. We study arithmetic and geometric sequences and learn of Gauss' schoolboy trick for summing,
Our study of trigonometry introduces radian measure and calculation, as well as looking at trigonometric identities and proofs of double angle formulae. Work on exponential relationships is extended by the learning of Napier's logarithms,
The completely new topic of calculus is introduced as we learn of Newton and Leibniz's concurrent discoveries, allowing us to differentiate and integrate. We expand our repertoire of techniques to cover the differentiation of almost all functions.
We also look at numerical methods such as the trapezium rule and the Newton-Raphson method. We change from Cartesian to parametric form.
The applied area of mechanics puts context to the pure side of the course, allowing for calculations between displacement, velocity and acceleration, bringing in calculus, and applying vector notation. We study Newton's laws of forces and consider projectiles and moments.
In Statistics, we look at the whole data collection cycle: sampling, calculation, representation and analysis,
In our study of probability, we build on prior knowledge to start looking at particular distributions: binomial and Gauss' Normal distribution. This leads to hypothesis testing to aid in our analysis of the statistics by understanding the significance of results.