Mathematical Proof

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 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 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.

 

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes:

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Mr Mathematics Blog

Getting Ready for a New School Year

When getting ready for a new school year I have a list of priorities to work through. Knowing my team have all the information and resources they need to teach their students gives me confidence we will start the term in the best possible way.  Mathematics Teaching and Learning Folder All teachers receive a folder […]

Mathematics OFSTED Inspection – The Deep Dive

Earlier this week, my school took part in a trial OFSTED inspection as part of getting ready for the new inspection framework in September 2019. This involved three Lead Inspectors visiting our school over the course of two days. The first day involved a ‘deep dive’ by each of the Lead Inspectors into Mathematics, English […]

How to Solve Quadratics by Factorising

The method of how to solve quadratics by factorising is now part of the foundational knowledge students aiming for higher exam grades are expected to have.   Here is an example of such a question. Solve x2 + 7x – 18 = 0 In my experience of teaching and marking exam papers students often struggle with […]