Algebraic Methods

Scheme of work: Year 12 A-Level: Pure 1:

Simplify expressions using the rules of indices
Solve quadratic equations by
- Factorisation
- Completing the square
Apply angle facts, triangle congruence, similarity and properties of quadrilaterals and coordinate geometry to obtain simple proofs
Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

Understand and use the structure of mathematical proof,
Construct a mathematical proof from given assumptions through a series of logical steps to a conclusion.
Use methods of proof, including proof by deduction, proof by exhaustion and construct a disproof by counter exampleÂ
Proof by contradiction (including proof of the irrationality of âˆš2 and the infinity of primes, and application to unfamiliar proofs)Â
Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation, and simple algebraic division,
Be able to use and apply the factor theoremÂ
Simplify rational expressions including by factorising and cancelling, and algebraic division by linear expressions only

Division to find the roots of a cubic graph.
Applying the factor theorem often leads to simultaneous equations and long division.
When disproving mathematical statements, students should show all trials as part of their work.
Students need to consider the variable as odd and even when proving that an expression is or is not divisible by a constant.
When demonstrating that a conjecture is ‘sometimes true,’ students need to show a case for each.

Students often forget to give a written conclusion as the final part of their proof.
When solving cubic equations, mistakes are sometimes made when substituting in negative values of x, particularly with the cubic term.
Some students try to use the long division method to factorise quadratics which can be more easily solved by factorisation.
Students often lose marks when asked to prove whether an expression is divisible by or a multiple of a constant. Encourage them to consider the variable as an odd and an even value.
Students often lose marks by not concluding that a squared term can never be negative.