Competences

• • a)Confidently read and write proofs using the appropriate language, symbols and presentation;
  • b)Apply the four fundamental proof techniques to several areas of mathematics;
  • c)Create lessons that incorporate appropriate proof techniques in various area of mathematics;
  • d)Guide students and ensure that they use appropriate presentation, mathematical language, terminology and symbols when presenting mathematical proofs;
  • e)Ensure that students recognise the elegant and rigorous processes involved in establishing and proving mathematical results including the ability to construct proofs of negative statements or counterexamples;
  • f)Guide students to ensure that they are aware of the insolvability of the general quintic equation by radicals showing competence in linking group theory with classical algebraic problems;
  • g)Apply the method of contrapositive reasoning to prove mathematical statements particularly those involving quantified expressions.

Knowledge

• • • a)Explain the logic underlying each of the four fundamental proof techniques;
    • b)Define what constitutes a mathematical proof;
    • c)Identify, use and interpret logical symbols and quantifiers;
    • d)Identify different types of mathematical statements, including theorems, propositions, lemmas, and corollaries;
    • e)Identify the difference between mathematical statements, namely: theorems, propositions, lemmas and corollaries;
    • f)Explain the meanings and roles of definitions, axioms/postulates, and conjectures;
    • g)Identify the structure of conditional statements and their logically equivalent contrapositives.

Skills

• • • a)Create and interpret truth tables for simple mathematical statements involving AND, OR, IF…THEN…, and IF AND ONLY IF;
    • b)Use logical statements, connectives and quantifiers in constructing arguments and expressing statements;
    • c)Write proofs according to standard guidelines in mathematical writing;
    • d)Choose the appropriate and most efficient proof technique according to the context;
    • e)Apply the four fundamental proof techniques in a variety of mathematical contexts;
    • f)Write proofs using the correct approach and presentation for each of the four fundamental techniques;
    • g)Construct direct proofs using deductive reasoning to prove  statements in various contexts such as algebraic relationships, trigonometric relationships, geometric properties, and number theory;
    • h)Apply the method of proof by contradiction to prove statements in various contexts such as number theory and algebraic relationships;
    • i)Apply the method of proof by contrapositive to prove statements in various contexts such as number theory and algebraic expressions;
    • j)Use inductive reasoning, the principle of mathematical induction, and apply it to prove statements on number theory, matrices, sequences and series, and inequalities, where the statements involve natural numbers;
    • k)Apply appropriate proof techniques to prove some standard theorems and statements such as De Moivre’s Theorem;