# Mathematical Proof

Scheme of work: GCSE Higher: Year 11: Term 1: Mathematical Proof

#### Prerequisite Knowledge

• 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

#### Success Criteria

• Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagorasâ€™ 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 to generalise 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.

## Mathematical Proof Resources

### Mr Mathematics Blog

#### Regression, Correlation and Hypothesis Testing

A-Level Scheme of work: Edexcel A-Level Mathematics Year 2: Statistics: Regression, Correlation and Hypothesis Testing

#### 3D Vectors – Year 2

A-Level Scheme of work: Edexcel A-Level Mathematics Year 2: Pure 2: 3D Vectors

#### Parametric Equations

Edexcel A-Level Mathematics Year 2: Pure 2: Parametric Equations