NCEA 91947: demonstrate mathematical reasoning (Level 1 Maths)

What NCEA 91947 assesses

NZQA's standard says you demonstrate mathematical reasoning by using mathematical methods and concepts, and communicating with suitable representations. The three grades are three kinds of thinking:

• Achieved: use mathematical methods, concepts and terms, and communicate using appropriate representations.
• Merit: relational thinking. You carry out an appropriate sequence of steps (two or more methods within one problem) and use appropriate mathematical statements.
• Excellence: extended abstract thinking. You develop a clear chain of logical reasoning, form a generalisation or give a proof, or extend a method to investigate or solve a problem.

The exam is drawn from number and algebra (expressions, inequations, linear and quadratic equations, simultaneous equations, graphs and rate of change) and from measurement and geometry (Pythagoras' theorem and trigonometric ratios in 2D and 3D, similar shapes, surface area and volume). NZQA says a formula sheet will be provided.

The NCEA 91947 exam in 2026 and 2027

NZQA's 2026 timetable puts Level 1 Mathematics and Statistics on Tuesday 10 November, in the morning session. The 2027 timetable is not out yet: NZQA says exams run from 9 November to 3 December 2027 and the subject timetable is due in December 2026. The 2026 and 2027 specifications are the same for 91947: a printed paper of three questions, an approved calculator and a ruler.

• Follow the method asked for. NZQA's 2025 report says "using an algebraic method" means a graphing calculator's solver earns nothing without algebraic working, and "factorise AND solve" needs both.
• Correct answer only (CAO). An answer without working and justification gets grade u at best.
• Attempt every part. Each question has one opportunity for Excellence, and the parts are not in order of difficulty.

How NCEA 91947 is marked: grade score marking

Each of three questions scores N0 to E8 for a total out of 24, and each question has exactly one Excellence opportunity, so do not spend all your time on the first.

NCEA 91947 cut scores

These cut scores are NZQA's grade boundaries, out of 24. They move a little each year because the papers differ. The standard replaced the old Level 1 standards in 2023 (see the NCEA Level 1 Maths guide), so only three schedules exist. The 2026 cut scores will appear in NZQA's 2026 schedule once the exam has been marked.

Worked examples at each grade

Achieved: simultaneous equations by elimination

Solve 2x + y = 11 and x − 3y = −5 using an algebraic method. Multiply the first equation by 3 so the y terms cancel when you add:

Check in the second equation: 4 − 9 = −5. The method is the evidence, so show the elimination step, not just (4, 3). Practise it in the Algebra course.

Merit: a sequence of steps in 3D

A box is 12 cm long, 5 cm wide and 4 cm high. Will a straw 13.5 cm long fit inside? The longest straight line is the space diagonal. First find the diagonal of the base, then use it with the height:

Two Pythagoras steps in sequence is the Merit evidence. Finish with the statement the question wants: 13.5 cm is less than 13.6 cm, so the straw fits, with 0.1 cm to spare.

Excellence: a proof

Show that adding the squares of any two consecutive whole numbers always gives an odd number. Call the numbers n and n + 1:

Because 2(n² + n) is twice a whole number, adding 1 always leaves an odd number. Excellence rewards the generalisation and the closing sentence that links the algebra back to the claim. Now try a simultaneous equations question, then press Mark to check it in the course:

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the daily paper as a practice exam

Uni-Sine publishes a free daily paper on the NCEA Level 1 Maths track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. It draws on simultaneous equations, quadratic equations, inequalities, Pythagoras and trigonometry, similarity, and straight line and quadratic graphs, which are the core of 91947.

Common mistakes in NCEA 91947

• Letting the calculator do the algebra. NZQA's 2025 report says graphing calculator answers without working are marked as correct answer only.
• Weak simultaneous equations and quadratics. The report names solving a pair of simultaneous equations, and factorising then solving a quadratic, as gaps in many scripts.
• Not finding line and parabola equations. The report says many candidates could not find the equation of a straight line or a parabola from a graph.

Read next

New Zealand · NCEA Level 1NCEA Level 1 Maths: standards, exam date and practiceNCEA Level 1 Maths explained: the four standards, what the exam covers, the 2026 date, 2027 dates, the numeracy co-requisite and worked examples.