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**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.

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