OT1: Mathematical argument, language and proof: Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable.

The idea

This outcome is about communicating mathematics as a reasoned argument, not just producing an answer. A diagram or sketch should be chosen to reveal structure: for example, sketching y = x² − 4x + 3 shows roots at x = 1 and x = 3, the line of symmetry x = 2 and minimum (2, −1), supporting a deduction about the range. Logical deduction means each line follows from the previous one, using 'since', 'therefore', 'if … then' and 'if and only if' accurately. Precise statements require correct vocabulary: a constant is fixed, a coefficient multiplies a variable, an expression is a collection of terms, an equation asserts equality, a function maps each input to one output, an identity holds for all values, an index shows repeated multiplication, a term is a component of an expression, and a variable represents a changing or unknown value.

What earns the marks

• Selects a diagram or sketch that exposes the relevant feature, such as intercepts, turning points, asymptotes or regions, rather than a decorative picture.
• Links successive steps with valid logical connectives, distinguishing implication from equivalence where the converse is not true.
• Uses the listed vocabulary precisely: for example, calling 3x² + 2x − 1 an expression, 3x² + 2x − 1 = 0 an equation, and sin²θ + cos²θ = 1 an identity.
• States the domain and range, or the variable being varied, when these affect the truth of a claim.
• Presents a complete argument in which the conclusion is explicitly justified by earlier statements, not merely asserted.
• Uses correct notation for indices, functions and coefficients, such as f(x) = 2x³ meaning coefficient 2 and index 3.

Common mistakes and how to put them right

• Treating an identity as an equation to be solved.

  Put it right: An identity is true for all permitted values, so it is proved by manipulation, not solved.

• Writing 'therefore' when only 'if' has been established, reversing the direction of an argument.

  Put it right: Check whether the converse is true; use 'if and only if' only when both directions hold.

• Using 'equation' for any algebraic string, so that 5x + 3 is called an equation.

  Put it right: An equation must contain an equals sign; 5x + 3 is an expression with terms 5x and 3, coefficient 5 and variable x.

• Sketching a graph without labelling axes, intercepts or key points, so the diagram cannot support the deduction.

  Put it right: Annotate the sketch with the features the argument uses.

Exam tips

• Before writing, decide whether a sketch or diagram will make the argument clearer; a labelled sketch often earns credit for structure as well as answer.
• Use a short glossary check: name each symbol in your working as constant, coefficient, term, variable or index, and correct yourself if the label is wrong.
• When a question says 'show that' or 'prove', write connected prose with reasons; when it says 'solve', an equation is expected and an identity would be inappropriate.
• Leave a clear final statement that answers the question, since an unexplained answer can lose communication credit even when the arithmetic is right.