GCSE Maths: Formulae, Functions and Proof

Many of the harder algebra marks on a GCSE Mathematics paper come from skills that sit around equations rather than inside them: using and rearranging formulae, recognising an identity, working with function notation and, on the Higher tier, algebraic fractions, composite and inverse functions, iteration and 'show that' or 'prove that' questions. This material covers that part of the algebra content in the GCSE subject content for England.

The quiz has fourteen questions. You substitute negative values into $v=u+at$, make $t$ the subject of the same formula and $r$ the subject of $A=\pi r^2$, decide which statements are identities rather than equations, and evaluate a function such as $f(x)=3x-4$. The Higher-tier questions make $x$ the subject when it appears twice, simplify an algebraic fraction by factorising, add two algebraic fractions, find a composite function in the right order and an inverse function, carry out one step of an iteration, and choose a valid algebraic proof about consecutive integers, odd and even numbers, or the difference of two squares of odd numbers. The wrong options are real errors: taking the square root of each term, cancelling single terms instead of whole brackets, applying $f$ before $g$ in $fg(x)$, undoing operations in the wrong order, and offering one numerical example as if it were a proof.

The flashcards cover the subject of a formula, the identity symbol, what a function is, the order in a composite function, the method for an inverse, iteration, the algebraic forms of even, odd and consecutive numbers, and the methods for simplifying and adding algebraic fractions.

The printable written work has eight problems with full working: rearranging two formulae and using one, changing the subject when it appears twice, simplifying an algebraic fraction, solving an equation with algebraic fractions, a set of function questions ending with $f(x)=f^{-1}(x)$, an iteration that homes in on the root of a cubic, two short algebraic proofs, and a comparison between an identity and an equation built from the same expression.

Formulae, substitution, identities and function notation are content for both tiers; algebraic fractions, composite and inverse functions, iteration and algebraic proof are Higher tier. The material suits Year 10 and Year 11 students, especially those aiming for the top grades on the Higher tier.

• Substitute values, including negative numbers, into formulae
• Rearrange formulae to change the subject, including when it appears twice (Higher)
• Distinguish identities, equations, formulae and expressions
• Use function notation and find composite and inverse functions (Higher)
• Simplify, add and solve with algebraic fractions (Higher)
• Use iteration to approximate a root and write simple algebraic proofs (Higher)

Practice material written by Zestly, based on the DfE GCSE mathematics subject content (DFE-00233-2013), Algebra A2, A3, A4, A5, A6, A7 and A20.