# Algebraic Methods

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

#### Prerequisite Knowledge

• Simplify expressions using the rules of indices
• 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

#### Success Criteria

• 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

#### Key Concepts

• 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.

#### Common Misconceptions

• 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.

## Algebraic Methods Resources

### Mr Mathematics Blog

#### Planes of Symmetry in 3D Shapes

Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

Use isometric paper for hands-on learning and enhanced understanding.

#### GCSE Trigonometry Skills & SOH CAH TOA Techniques

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.

#### Regions in the Complex Plane

Explore Regions in the Complex Plane with A-Level Further Maths: inequalities, Argand diagrams, and geometric interpretations.