NCEA Level 3 Mathematics Achievement Standards:
Note: Under the alignment process 1 credit is intended to be approximately equivalent to 10 hours of
student engagement time (classroom instruction and/or homework)
For all internal L3 mathematics standards:
 To be used as evidence for the award of Achieved, a ‘method’ must be relevant to the solution of the problem and at the appropriate curriculum level.
 At all levels there is a requirement relating to the communication of the solutions.
 Students need to be familiar with the context of any task. It is acceptable for them to know and research the context before the assessment.
In general, a selection of methods means more than half of those listed. The number of methods required for achieved depends on the difficulty of the task.
No formula sheet is provided for L3 internal standards. Teachers will provide support appropriate for the task.
Teachers should refer to the Conditions of Assessment and Clarifications and Moderators’ Newsletters when planning the teaching and assessment of each standard.
Exemplars are a useful resource for internals but are NOT to be used for assessment without modification, as they are publicly available on the web.
Assessment Specifications should be referred to when planning teaching and learning for externals.
AS91573 [3.1] Apply the Geometry of Conic Sections in solving problems
A.O.:
M8-1 Apply the geometry of conic sections
methods: - graphs and equations of the circle, ellipse, parabola, and hyperbola
- Cartesian and parametric forms
- properties of conic sections
- tangents and normals.
Internal
3 credits
AS91574 [3.2] Apply linear programming methods in solving problems
A.O.:
M8-4b Use linear programming techniques
Methods - linear inequalities
- feasible regions
- optimisation.
Internal
A.O.:
M8-6 Manipulate trigonometric expressions
M8-7 Form and use trigonometric equations
(M8-2 Display and interpret the graphs of functions with the graphs of their inverse and/or
reciprocal functions)
Methods: -trigonometric identities
-properties of trigonometric functions
-reciprocal trigonometric functions
-solving trigonometric equations
-general solutions
A.O.:
M8-5 Develop network diagrams to find optimal solutions, including critical paths
Methods: - precedence tables
-scheduling
- network diagrams
-float times
- critical events
2 credits
AS91587 [3.15] Apply systems of simultaneous equations in solving problems
A.O.:
M8-8 form and use systems of simultaneous equations, including three linear equations in three
variables, and interpret the solutions in context
Methods: - forming systems of simultaneous equations
- solving systems of simultaneous equations
- the nature of solutions to systems
Internal
A.O.:
M8-9 Manipulate complex numbers and present them graphically
M8-7b Form and use polynomial, and other non-linear equations
Methods: -quadratic and cubic
- manipulation of complex numbers
equations with complex roots
- loci
- Argand diagrams
- De Moivre’s theorem
- polar and rectangular forms
- equations of the form zn=rcisθ, or zn=a+bi where a, b
- manipulation of surds
are real and n is a positive integer.
A.O.:
M8-10 Identify discontinuities and limits of functions
M8-11b Choose and apply a variety of differentiation techniques to functions & relations using
analytical methods
Methods: -derivatives of power, exponential, and
-related rates of change
logarithmic (base e only) functions
-derivatives of parametric functions
-derivatives of trigonometric (including
-chain, product, and quotient rules
reciprocal) functions
-properties of graphs (limits, differentiability,
-optimisation
continuity, concavity).
-equations of normals
-maxima and minima and points of
inflection
Note the assessment specification states that complex numbers
include real numbers. Knowledge of logarithms from NZC L7 is
assumed.
Teachers should refer to the notes and resources at AO M8-9 and
AO M8-7b in the TKI senior secondary guides when planning their
teaching for this standard.
Differentiation is often seen as a topic for University pathways to study
mathematics, economics, finance, and engineering. AS91578 does not
include implicit differentiation. Differentiation from first principles is a
useful teaching approach but will not be examined.
Teachers should refer to the notes and resources at AO M8-10 and
AO M8-11b in the TKI senior secondary guides when planning their
teaching for this standard.
6 credits
AS91579 [3.7] Apply integration methods in solving problems
A.O.:
M8-12 Form differential equations and interpret the solutions
M8-11a Choose and apply a variety of integration and anti-differentiation techniques to functions &
relations using both analytical and numerical methods
Methods: -integrating power,
-areas under or between graphs of functions, by integration
polynomial, exponential
-finding areas using numerical methods, eg the rectangle or
(base e only), trigonometric, trapezium rule
and rational functions
-differential equations of the forms y' = f(x) or y" = f(x) for
-reverse chain rule,
the above functions or situations where the variables are
trigonometric formulae
separable (eg y' = ky) in applications such as growth and
-rates of change problems
decay, inflation, Newton's Law of Cooling and similar
situations.
External
This is often seen as a topic for University pathways into Maths, Finance,
Economics.
Teachers should refer to the notes and resources at AO M8-8 in the TKI
senior secondary guides when planning their teaching for this standard.
5 credits
AS91578 [3.6] Apply differentiation methods in solving problems
External
This is a newly developed area for assessment in at NCEA L3 and
provides a pathway for learning from L2 AS91260 Apply network
methods in solving problems. Critical path analysis is commonly used in
business applications, so this is a useful standard linking mathematics to
the real world.
Teachers should refer to the notes and resources at AO M8-5 in the TKI
senior secondary guides when planning their teaching for this standard.
3 credits
AS91577 [3.5] Apply the algebra of complex numbers in solving problems
External
Pulls together 3 AO’s. There is the possibility of parts of AO’s not being
covered in assessment or teaching program (e.g. assessment could assess
modelling only or trig proofs only or finding general solutions of trig
equations in a maths context)
Teachers should refer to the notes and resources at AO M8-6, M8-7 and part
of M8-2 in the TKI senior secondary guides when planning their teaching for
this standard.
4 credits
AS91576 [3.4] Use critical path analysis in solving problems
Internal
No non-linear functions included here.
Teachers should refer to the notes and resources at AO M8-4 in the TKI senior
secondary guides when planning their teaching for this standard.
3 credits
AS91575 [3.3] Apply trigonometry methods in solving problems
Internal
Problems in exemplar tasks are set in a ‘real-life’ context.
Properties include features, transformations, loci of conic sections. Cartesian and
parametric forms include equations linking the different forms.
Teachers should refer to the notes and resources at AO M8-1 in the TKI senior
secondary guides when planning their teaching for this standard.
Volumes of revolution are not assessed as they are no longer in NZC.
Although skills such as ‘integration by parts’ or ‘integration by
substitution’ are not mentioned explicitly in the standard or SSTLGs
they are tools that students will likely find useful when integrating
functions mentioned in the standard – so do teach them as useful tools
for solving problems.
Teachers should refer to the notes and resources at AO M8-12 and
AO M8-11a in the TKI senior secondary guides when planning their
teaching for this standard.
6 credits
Sequences and series are assessed at NCEA L2.
AO M8-2 Display and interpret the graphs of functions with the graphs of their inverse and reciprocal functions is only partially assessed in the algebra and calculus standards (e.g. log is the
inverse of exponential and reciprocal functions are covered in the trig standard). AO M8-3 is not assessed in the Mathematics standards
revised September 2013
NCEA Level 3 Statistics Achievement Standards:
For all internal L3 statistics standards:
revised September 2013
 The assessment requires multiple sessions to allow time for students to research the context and develop a purpose prior to completing the analysis.
 Students need to have knowledge about the context and statements that are made need to be related to the context.
 Use of a statistical graphing package is expected.
 Students need to provide evidence of each component of the statistical enquiry cycle detailed in Explanatory Note 3 of the standard.
No formula sheet is provided for L3 internal standards. Teachers will provide support appropriate for the task.
Teachers should refer to the Conditions of Assessment and Clarifications and Moderators’ Newsletters when planning the teaching and assessment of each standard.
Exemplars are a useful resource for internals but are NOT to be used for assessment without modification, as they are publicly available on the web.
Assessment Specifications should be referred to when planning teaching and learning for externals.
Census at School has many useful resources for teaching Statistics at all levels.
AS91580 [3.8] Investigate time series data
AO S8-1 Carry out investigations of phenomena, using statistical enquiry cycle: using existing data sets;
finding, using, and assessing appropriate models, seeking explanations, and making predictions; using
informed contextual knowledge; communicating findings and evaluating all stages of cycle.
Methods: - using existing data sets
- identifying features in the data and relating this
- selecting a variable to investigate
to the context
- selecting and using appropriate
- finding an appropriate model
display(s)
- using the model to make a forecast
Internal
4 credits
AS91581 [3.9] Investigate bivariate measurement data
The scatter plot needs to be inspected visually before any model is fitted. It is possible
a linear model may not be appropriate for the data that is being investigated and this
should be determined by looking at a scatter plot of the raw data.
Use and interpretation of R2 is not expected at this level.
Teachers should refer to the notes and resources at AO S8-1 in the TKI senior
secondary guides when planning their teaching for this standard.
AO S8-1 Carry out investigations of phenomena, using statistical enquiry cycle: using existing data sets;
finding, using, and assessing appropriate models, seeking explanations, and making predictions; using
informed contextual knowledge; communicating findings and evaluating all stages of cycle.
Methods: - posing an appropriate relationship
- describing the nature and strength of the
question using a given multivariate data
relationship, and relating this to the context
set
- using the model to make a prediction
- selecting and using appropriate displays
- communicating findings in a conclusion
- identifying features in data
- finding an appropriate model
Internal
4 credits
AS91582 [3.10] Use statistical methods to make a formal inference
AO S8-1 Carry out investigations of phenomena, using statistical enquiry cycle: using existing data sets;
finding, using, and assessing appropriate models, seeking explanations, and making predictions; using
informed contextual knowledge; communicating findings and evaluating all stages of cycle.
AO S8-2b Making inferences from surveys or experiments; determining estimates and confidence intervals
for differences; using methods such as resampling to assess the strength of evidence.
Methods:
- posing -a comparison investigative question using a
- discussing sampling variability, including the
given multivariate data set
variability of estimates
- selecting and using appropriate displays
- making an appropriate formal statistical
and summary statistics
inference
- discussing sample distributions
- communicating findings in a conclusion
Internal
4 credits
AS91583 [3.11] Conduct an experiment to investigate a situation using
experimental design principles
AO S8-1 Carry out investigations of phenomena, using statistical enquiry cycle: using existing data sets;
finding, using, and assessing appropriate models, seeking explanations, and making predictions; using
informed contextual knowledge; communicating findings and evaluating all stages of cycle.
AO S8-2b Making inferences from surveys or experiments; using methods such as randomisation to assess the
strength of evidence.
Methods:
- posing –posing an investigative question about a
- selecting and using appropriate displays and
given experimental situation
summary statistics
- planning the experiment using
- making appropriate formal statistical inferences
experimental design principles
- communicating findings in a conclusion
- conducting the experiment
Internal
4 credits
AS91584 [3.12 ] Evaluate statistically based reports
AO S8-3 Evaluate a wide range of statistically based reports, including surveys and polls, experiments, and
observational studies (critiquing causal relationship claims, interpreting margins of error). [AO S8-1 Carry
out investigations of phenomena, using statistical enquiry cycle: using existing data sets; finding, using, and
assessing appropriate models, seeking explanations, and making predictions; using informed contextual
knowledge; communicating findings and evaluating all stages of cycle. AO S8-2b Making inferences from
surveys or experiments; determining estimates and confidence intervals for means, proportions, differences
recognising the relevance of the central limit theorem; using methods such as resampling or randomisation]
Concepts :
-the statistical enquiry cycle
-interpreting a wide variety of statistical
-principles of experimental design
tables and graphs
-surveys and polls, including potential
-analysing a wide variety of statistical situations
sources of bias
-critiquing causal-relationship claims
-interpreting statistical inferences
-interpreting margins of error.
External
4 credits
AS91585 [3.13] Apply probability concepts in solving problems
A.O.:
S8-4 Investigate situations that involve elements of chance
- calculating probabilities of independent, combined, and conditional events
Covers: - true probability vs model estimates vs experimental estimates
- randomness, independence, mutually exclusive events, conditional probability
- probability distribution tables & graphs
- two way tables, probability trees, venn diagrams
External
4 credits
AS91586 [3.14] Apply probability distributions in solving problems
A.O.:
S8-4 Investigate situations that involve elements of chance
- calculating and interpreting expected values and standard deviations of discrete random
variables
- applying distributions such as the Poisson, binomial, and normal
Covers: - discrete and continuous probability distributions
- mean and standard deviation of random variables
- distribution of true probabilities vs distribution of model estimates of probabilities vs
distribution of experimental estimates of probabilities
External
4 credits
There has been significant change from the way time series has been assessed in the
past.
The focus is on real data and ‘getting the story out of the data’.
The use of a programme such as iNZight is an advantage as it allows the student to focus
on interpreting the time series data in context.
Features of the data include the trend and seasonal pattern.
While students could investigate more than one time series or combine series this is not
a requirement of the standard. Teachers should refer to the notes and resources at AO
S8-1 in the TKI senior secondary guides when planning their teaching for this standard.
AS91582 requires making a formal inference (an empirically based inference derived
from resampling - bootstrapping is the resampling method if using iNZight) and
plotting the resampled distribution..
Computer use enhances student achievement in all statistics standards but is essential
in this standard.
The analysis will involve students determining if there is a difference between the
population medians (or means) and quantifying the difference in the medians (or means)
by using the bootstrap confidence interval.
A suitable investigative question would be ‘I wonder what the difference in median
heights is between NZ year 13 boys and NZ year 13 girls?’
Teachers should refer to the notes and resources at AO S8-1and S8-2b in the TKI senior
secondary guides when planning their teaching for this standard.
Randomisation methods mentioned in AS91583 refer to using a computer to randomly
re-sort the data into two groups a large number of times to get an experimental estimate
of the probability that the difference between the two treatments is due to chance alone.
Randomisation (random allocation) is also required for the experimental design.
Computer use is essential (iNZight recommended). There needs to be a clear and
meaningful purpose for the experiment. Students need to identify experimental units, the
treatment and the response variable and how the treatment is to be allocated to the
experimental units. Students need to make a formal statistical inference which will be a
causal inference based on the strength of evidence. This will involve the use of
randomisation. The conclusion needs to be consistent with the analysis and answer the
investigative question.
Teachers should refer to the notes and resources at AO S8-1and S8-2b in the TKI senior
secondary guides when planning their teaching for this standard.
While AS91584 only mentions one AO, S8-3, this standard assesses students
understanding of the PPDAC cycle in multiple data contexts (so students need to be
familiar with AO’s S8-1 and S8-2).
The approach used in interpreting the margins of error is new to the statistics
curriculum. It is based on making informed approximations or rules of thumb to
interpret reported margins of error and is linked to confidence intervals.
Solutions for problems may also require knowledge up to and including Statistics
Curriculum Level 7. Questions may be set in a statistical context. Questions may require
candidates to interpret their solutions in context.
Candidates will need to answer questions about statistically based reports. The questions
will require candidates to evaluate claims or conclusions made in the report by:
identifying and discussing potential sources of error associated with surveys; calculating
and interpreting margins of error; identifying and assessing causal relationship
inferences. Candidates will not be provided with formulae for margin of error.
Candidates should recall and use the “rules of thumb” based on 1/√n but will not be
penalised for using exact norm-based formula formulae appropriately.
Teachers should refer to the notes and resources at AO S8-3 in the TKI senior secondary
guides when planning their teaching for this standard .
AS91585 includes distinguishing between deterministic and probabilistic models and
demonstrating understanding of the relationship between true probability, model estimates, and
experimental estimates. Students may need to use simple expected values in solving problems, but
expected value will not be assessed formally. Sums, difference and linear combinations of random
variables are included only for discrete random variables, not for continuous random variables.
Probabilities may be expected to be calculated from formulae, a probability distribution table or
graph, tables of counts or proportions, simulation results or from written information. Candidates
should clearly show the method they have used to calculate probabilities and state assumptions
made. Questions will be set in real-life or statistical contexts. Candidates may be required to
interpret solutions in context. Sensible rounding is expected. Early rounding may be penalised.
Teachers should refer to the notes and resources at AO S8-4 in the TKI senior secondary guides
when planning their teaching for this standard.
AS91586 includes calculating probabilities from distributions presented as formulae, tables or
graphs of data, simulation results or from written information. Candidates should clearly
identify the distribution applied in solving the problem and state assumptions made. Candidates
will need to be familiar with the Normal, Poisson, binomial, uniform, and triangular
distributions. Normal, Poisson and binomial distribution tables will be provided in the
Formulae and Tables Booklet; however, appropriate use of a calculator will be acceptable.
Candidates may be expected to calculate or estimate the mean and standard deviation of a
random variable. Continuity corrections should only be used when a continuous probability
distribution is used to model a discrete random variable or for a variable where measurements
have been rounded.
Teachers should refer to the notes and resources at AO S8-4 in the senior secondary guides
when planning their teaching for this standard.