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Students learn how to construct a mathematical proof using algebraic notation and geometrical reasoning. This higher GCSE topic take place in Term 3 Year 11. Before learning about proof students should be able to manipulate algebraic expressions.

Mathematical proof is often the hardest topic to learn at GCSE as it requires students to form fluent and reasoned mathematical arguments using concise algebraic notation.

**Mathematical Proof Lessons**

**Prerequisite Knowledge**

- Know the difference between an equation and an identity;
- Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax
^{2}+ bx + c - Understand and use standard mathematical formulae; rearrange formulae to change the subject
- Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

**Success Criteria**

- Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ Theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs
- Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs
- Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
- Use vectors to construct geometric arguments and proofs

**Key Concepts**

- Know the difference between an equation and an identity;
- Simplify and manipulate algebraic expressions by factorising quadratic expressions of the form ax2 + bx + c
- Understand and use standard mathematical formulae; rearrange formulae to change the subject
- Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
- Parallel lines have vectors that are multiples of each other.

**Common Misconceptions**

- A common incorrect approach is to attempt to prove an algebraic and geometrical property through numerical demonstrations.
- Students often struggle generalising the rules of arithmetic to produce a reasoned mathematical argument.
- Some students expand brackets incorrectly when proving a quadratic identity.
- Students often lose marks when attempting to prove geometrical properties due to not connecting the various angle properties.
- Incorrect application of ratio notation leads to difficulty when proving geometrical properties.
- Students often fail to label vector diagrams sufficiently to identify known paths.
- Providing a proof of geometrical facts tends to separate the most able from the majority.

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