"Mathematics is the music of reason"
James Joseph Sylvester
Students begin to excel in the different areas of mathematics: fluency in calculations; spotting patterns in number and being able to formalise generalised relationships; manipulation of complex algebra; geometric reasoning and analysis of statistical measures.
We concentrate on the foundations of formal proof, creating logical and rigorous arguments: essential rhetoric for a mathematician.
Students are taught to approach unseen problems with a range of strategies and use mathematical insight to identify the most efficient. The lines between areas of mathematics start to blur as we begin to appreciate the inter-relatedness of the different disciplines.
We start with the most basic of polygons, the triangle, and learn of Pythagoras and the theorem named after him. This leads naturally on to trigonometric calculations for right-angled triangles and then onto generalised rules for all triangles. We return to Euclid and his work, looking at geometry, building up from angles in triangles to generalised polygons and then onto proofs of circle theorems.
This area of mathematics is based firmly in Ancient Greece, where only straight edges and compasses were available to make constructions and loci, which leads us to the quest to find the area of a circle. Archimedes then leads us to the volume of prisms and density calculations.
Statistics remains firmly grounded in calculations, extending the range of diagrams, and adding in more advanced analysis. Students are introduced to probability calculations and Venn diagrams, leading to an understanding of basic conditional probability,
Work on the number line leads us to an introduction to surds and irrational numbers, which completes the set of real numbers.
We look at standard form linking in with calculations in Science which leads to compound measures such as speed and pressure. Building on our earlier work on proportional relationships, students generalise direct and inverse proportion.
Algebra takes prominence at this level of study. Starting with graphical representation of algebraic functions, we analyse the fundamental link between solutions and graphs of equations, and this develops further to generalised graph transformations. The graph work is completed with some pre-calculus calculations of rates of change.
Building on previous work, we solve quadratic and simultaneous equations using a host of different methods and understanding the advantages of each.
We look at inequalities and equations which cannot be solved exactly, and thus extending students' repertoire to include iteration: a powerful tool used by computers. We link back to concrete examples developing more and more sophisticated models for real life situations such as exponential growth and decay,
The final part of the journey is to develop the rhetoric of algebraic proof, making logical and rigorous arguments and paving the way for further study.