Aimed at the UK Mathematical Challenges (JMC, IMC, SMC), this edition went well beyond exam drill. It built genuine mathematical understanding — from the history of numbers and fundamental constants, through algebra and proof, to geometry — and paired it with real past papers and a hands-on spreadsheet project.
Explored what numbers are and the history of numeral systems (Sumerian, Babylonian base-60, Roman), scientific notation, and the constants that can't be written simply — π, the imaginary unit i, and Euler's number e — alongside Euler's identity and trigonometric ratios. Homework was the JMC 2024 paper as a level check.
Introduced factorials and series, then explored square numbers in depth: a digit rule for square roots, why squares only end in 0/1/4/5/6/9, that n² is the sum of the first n odd numbers, and the difference of squares for fast mental arithmetic.
Practised square techniques for rapid mental multiplication, then introduced algebra's history (al-Khwarizmi) and its structures — domains, operations and axioms such as commutativity, associativity, identity and inverses — asking which number sets form Abelian groups.
Introduced Boolean algebra over {True, False} with AND, OR and NOT, then formalised elementary algebra: variables vs constants, coefficients, terms and exponents, and operations on expressions — simplifying like terms, expanding via the distributive law, and factorising.
Worked a full algebraic proof that n² equals the sum of the first n odd numbers, contrasting visual and algebraic proof. Then began spreadsheets — cells as variables, data types and functions — with a hands-on times-table project.
Solved an equilateral hexagon inscribed in a unit square using the Pythagorean theorem (and its history and proof), setting up and solving the resulting quadratic and discussing which root is valid.
Taught the laws of exponents — products, quotients, nested powers and fractional exponents (deriving that 3^½ = √3) — and applied them to simplify surd expressions, with spreadsheet practice on absolute ($) references.
Taught breaking numbers into prime powers and using prime factorisation to simplify fractions reliably and with fewer steps, worked through several examples, and revisited the hexagon problem as a full worked solution.
Recapped the course so far and introduced geometry — distance, shape, size and position — deriving the distance formula between two points from the Pythagorean theorem, with coordinate exercises.
Covered parts of a circle, proportion and ratios, arc length and sector area, then Eratosthenes' measurement of the Earth using shadow angles — set as a homework calculation.
Used the circumference problem to teach a method: extract the data, eliminate the noise, and work within the brain's 3–5 item working memory. Emphasised "learning to learn" and connecting new ideas to existing knowledge, turning the problem into algebra.
