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NCEA ANALYSIS for Teaching and Learning

Cheryl Harvey - Secondary Literacy and Te Kotahitanga Facilitator

Jennifer Glenn - Secondary Facilitator – Specialist Classroom Teachers

Cheryl Harvey and Jennifer Glenn, TEAM Solutions, 2007 1

MATHEMATICS

NCEA ANALYSIS for Teaching and Learning

INTRODUCTIONThis resource has been developed to support secondary schools and teachers as they consider NCEA evidence and implications for their classroom practice. It provides a model for looking at the data through a different lens. The material in the resource comes from an analysis of examiners’ reports, moderators’ reports and explanatory notes of the Achievement Standards for 2004 and 2005. This model can be updated and adapted according to the needs of an individual school or department.

Reports have been completed for the following subjects:English Science Technology Media Studies Biology Physical EducationVisual Arts Chemistry Classical Studies French Physics DramaMusic Accounting Geography History Economics MathematicsHome Economics Graphics

FRAMEWORKS FOR ANALYSISThe reports have been analysed according to:

1. Literacy and Language:comments from the reports are categorised according to : Surface features – such as neatness, layout, spelling Vocabulary Writing Reading Information skillsThis analysis is divided into NCEA Levels 1, 2 and 3.


2. Thinking:comments linked to thinking from the reports are allocated to one of the 6 levels of Bloom taxonomy. This was chosen as it links most directly to the language of NCEA. Comments from the reports are categorised according to: Remember Understand Apply Analyse Evaluate CreateThis analysis is divided into NCEA Levels 1, 2 and 3.

3. Extension features - towards Merit and Excellence:Specific comments are included which describe what students have done which lifts their responses into either Merit or Excellence at all three levels.

4. Main reasons for Failure:Specific comments are included which describe the reasons for student failure at all three levels.

5. Specific Directives to Teachers: Often in the reports, there are direct suggestions made to teachers about what is needed to improve the learning and the subsequent student outcome.

6. Vocabulary:The report includes key vocabulary , encountered either in the assessment reports, or in the explanatory notes which accompany the standard. The words are simply listed for teachers in that curriculum area to note and to use. Reference back to the Assessment Reports and the Achievement Standards would give the context and further detail. One result of this analysis has been an awareness of the amount of vocabulary a student will have to master across a range of subjects in a given year.


USING THE ANALYSIS

There are many ways using the resource to inform teaching and learning. It is possible to cut – and paste – the material in a range of ways. For example:

to link to a school goal or initiative – eg a target group such as Level 1 Literacy can be cut across all curriculum areas to look at both curriculum specific and generic learning needed for success

a particular aspect of Literacy can be cut through all levels to note the development and progression – eg Information Skills in the development of a cross-curricular research process where skills are built from Year 9

where a school seeks to raise the numbers of students gaining Literacy and Numeracy, or gaining NCEA Level 1, the Reasons for Failure category can help departments set goals for improvement

a department seeking to lift the numbers of students reaching Merit and Excellence across the board may look to the Extension Features for Merit and Excellence for guidance

where a particular standard has been identified as a target in a curriculum area, the combination of general comments and Specific Directives may be used to develop goals

If you are using this resource please acknowledge our work. We would like to acknowledge the literacy template that originally came from work done at Thames High School and which we further developed and used to produce this resource.

Cheryl HarveyJennifer GlennTeam SolutionsUniversity of Auckland.


SUBJECT: MATHEMATICS

TOPIC: Language and Literacy

Surface Features Vocabulary Reading Writing Information Skills

LEVEL ONE

Show working where requested

Write clearly Use pen if work is to be

eligible for reconsideration

Use ruler for straight lines

Draw neat graphs with relevant features

Round appropriately Presented clearly

showing steps taken

LEVEL ONE

Use appropriate mathematical terms

Understand and use key words in question

Recognise probability situations that produce outcomes that were not equally likely from key words such as as if

Understand the conventions for naming angles

Read questions carefully Interpret in context Understand what is being

asked in the question and diagrams

Communicate mathematical ideas clearly and concisely

Write clear statements to show understanding

Give well-structured answers

Use correct mathematical statements to communicate method of solution

Write probability in an acceptable form

Avoid assumptions from diagrams

Use a systematic approach

Interpret diagrams Present reasons in a

logical order, alongside/after each step

Select relevant information from a context


This material is summarised from the Assessment Reports, Moderation Reports and Explanatory Notes to be found on the NZQA website. There is far more detail included in that material.


LEVEL TWO

Cross out abandoned work

Set work out logically

LEVEL TWO

Know what is required for standard mathematical terms – eg solve, differentiate, expand

Know the meaning of terms – e.g. Parallel, medians, altitude

Use correct mathematical language

Apply terms to show understanding rather than rote-learned formulae/terms

Know the word ‘justify’ means statements need to be supported by, and linked to, valid conclusions or data

Read question carefully to determine what is required

Answer question asked Read scales on axes Interpret the question in

context

Use correct math. notation

Set work out logically



LEVEL THREE

Set work out logically and carefully – e.g. brackets, integral signs

Present clear working Round correctly

LEVEL THREE

Understand key words – e.g. iterate

Read and interpret instructions carefully – e.g. find an answer in the required form

Read questions accurately

Check final answers are correct

Interpret questions in context

Read correctly off a graph

Read and interpret given information

Proof-read manipulations Write confident answers

in a suitable form Communicate statistical

understanding with precision

Use accepted notation – e.g. probability

Interpret and use information given in a problem-solving situation

Think logically Sketch graphs for

contextual problems to help clarify

Speculate from information


DEEPER FEATURES – THINKING SKILLS

Remember Understand Apply Analyse Evaluate Create

LEVEL ONE

Recognise type of graph – linear or quadratic from the form of the equation

Round appropriately

LEVEL ONE

Do what questions ask

Recognise and select correct feature

Show understanding – e.g. sequence of steps to reach an answer

Be familiar with grid references

Give reasons Recognise

probability situations that produce outcomes that were not equally likely from key words such as ‘as if’

Understand conditional probability

Understand the meaning of key words

Use a calculator accurately – e.g. in degree mode

Use straight-forward algebraic methods correctly

Solve equations in context Manipulate/find equations–

linear graphs Use appropriate methods to

solve problems, equations Accurately calculate – e.g.

coordinates, angle, probability Accurately draw graphs- e.g.

smooth parabolas, correct intercepts, symmetry

Use working as evidence – e.g. to identify method, invoke ‘minor error ignored’ or ‘count-back’

Use/choose between Pythagoras’ theorem and trig. ratios to solve

Use a logical chain of reasoning

Interpret features – e.g. parabola

Interpret diagrams Determine the equation

for the situation modelled and apply it

Correctly select skills/methods to apply to a problem

Give correct mathematical reasons

Match intermediate calculations with corresponding geometric reasons

Prove assumptions Make links Choose appropriate

strategy – e.g. Pythagoras’ /trig. to best solve problem

Interpret features of graph and relate them to context



LEVEL ONE

LEVEL ONE

Understand key features

Understand and make use of implications inherent in given context

Understand conventions for naming angles

Convert a fraction into a decimal or percentage accurately

Use tree diagrams Devise appropriate method to

solve a theoretical probability problem

Construct a list of possible outcomes to help solve probability problems

Complete the simulation correctly

Use signs carefully when re-arranging, expanding, factorising

Determine kind of graph from its equation

Work with numbers in standard form

Select relevant information from context

Measure real objects Produce object/model with 2+

transformations, 2+ constructions

Set specific question for investigation/basis for comparison

Interpret questions in context

Solve problem with own measurements



LEVEL TWO

Know basic formulae

Identify type of sequence

Round appropriately

Recall appropriate terms

Recognise e.g. periodic nature of trigonometric equations

LEVEL TWO

Select correct approach

Understand nature of features/what is required

Use correct math. language

Communicate math. idea about how a graph was effected – e.g. with a sketch

Use logical arguments

Understand relationship – e.g. integral and area under a curve; derivative and gradient; between velocity, gradient and acceleration; distance formula and Pythagoras

Understand different calculus notations

Work from understanding rather than formulae

Identify type of sequence

Understand basic concepts

Write equations Factorise, expand, reduce,

simplify, solve equations/expressions

Demonstrate basic numeric/algebraic skills

Use key skills – e.g. logs, manipulation of indices, writing, factorising, expanding, simplifying, solving equations

Sketch and draw graphs with key features clearly and correctly placed – e.g. intercepts, asymptotes

Be specific rather than general with features

Use a graphics calculator correctly/in right order

Draw graphs carefully – e.g. non-linear

Use correct mathematical language

Integrate / differentiate – show working

State and use derived function

Form own equations from information given

Tackle/present a proof Describe sampling method Calculate using sample

Relate solution back into context to solve

Interpret - e.g. formulae, symmetry

Discuss sampling method discuss how representative a sample is

Analyse data Make an inference

about the population from the sample analysis

Use a sample to make inferences about the population

Relate evaluation to sampling method

Critically evaluate sampling process

Consider implications of any limitations on inferences made

Speculate from data



LEVEL TWO Be confident to deal with multi-step problems

Apply terms to show understanding rather than rote-learned formulae/terms

Recognise symmetry

Recognise type of sequence, list terms

Substitute – e.g. integers into appropriate formulae

Calculate correctly – e.g. distance, mid-point Work in both degrees and

radians Use derivatives and integrals

to solve problems Support answer with clear

working Use arithmetic and geometric

formulae



LEVEL THREECALC

Recognise where chain rule is needed

Understand applications of differentiation

Understand notation Show understanding

of indices Understand how to

use a graphics calculator to advantage

Read and interpret instructions carefully – e.g. required form

Understand difference – e.g. between factor and solution

Understand use of logarithm rules

Understand the properties of a hyperbola as it approaches asymptotes

Understand the relation of transformations of a graph to changes in algebraic form

Read correctly off a graph

Use chain rule Find the equation of a tangent Solve a related rates of

change question Form a model for a given

situation Integrate composite functions Apply and modify information

from tables and formulae Calculate accurately Separate variable and find

constant Differentiate Use correct values Simplify a rational expression Use notation correctly – e.g.

placement of integral signs Utilise algebraic and numeric

skills for manipulation and simplification, to rationalise a denominator, to expand brackets, to simplify square roots and fractions

Convert Manipulate complex numbers

in rect. and polar form Use De Moivre’s theorem Use quadratic formula to

express solutions

Use differentiation skills to problem solve

Interpret information given in problem-solving situation

Think logically Interpret answers from

graphic calculator

Choose and use an appropriate strategy to solve a problem involving a number of different ideas and follow through the process to the end



LEVEL THREECALC

Use logarithms to solve an exponential equation

Use a graphics calculator to advantage and know modes

Sketch a graph of straight-forward conic sections with some accuracy

Complete the square Write equations of conic

sections from graph or description of graph

Substitute correctly Rearrange correctly in

order to evaluate parameters

Show how an answer is reached

LEVEL THREESTATS

Know the word ‘justify’ means statements need to be supported by, and linked to, valid conclusions or data

Understand independence, mutually exclusive events and the complement of an event

Identify a feasible region from constraints and from a graph in a Linear Programming problem

Identify vertices

Calculate a confidence interval for a population mean/proportion

Use information given Round working correctly Find z-values for the

specified precision Substitute into appropriate

formulae Manipulate fractions

Identify trends Speculate from information



Level 3Stats

Know form – e.g. confidence interval, standard error

Know basic probability terms

Recognise that probability cannot be greater than one

Avoid premature rounding

Read from tables accurately

Demonstrate knowledge of the use of a graphics calculator to solve problems using probability distributions

Communicate statistical understanding with precision

Understand definitions – e.g. events not independent

Understand key words – e.g. iterate

Use a range of techniques to solve probability problems

Use tree and Venn diagrams

Present clear working Solve equations – e.g. a

system of 3x3 simultaneous

Use parallel line test Find an optimal solution Complete at least two

iterations of either the Newton-Raphson or bisection methods

Correctly select/use a distribution and its parameters to problem solve

Find a model, use the model to solve problems and use trig. manipulation

Determine, describe and interpret trend in context

Quantify relationship Collect and use own data


EXTENSION FEATURES – towards MERIT and EXCELLENCE

LEVEL 1 LEVEL 2 LEVEL 3 CALC Formed and solved equations, related the

solution to the problem Demonstrated high level of knowledge,

problem-solving/ interpretation skills Interpreted an answer in context Correctly applied techniques Maintained a high degree of accuracy – e.g.

calculations Used correct mathematical reasons and

statements to communicate method of solution Supported all answers with reasons Presented reasons in a logical order,

alongside/after each step – e.g. to link 2 angles

Presented ‘complete’ reasons for the given situation

Manipulated equations in order to draw more complex graphs

Interpreted features – eg of a parabola Provided relevant features on clearly drawn

graphs Found equations of linear graphs Determined the equation for the situation

being modelled and apply it to the question asked

Gave clear, logical mathematically sound arguments

Showed understanding - e.g. sum to infinity concept, trig. graphs, points

Had a better grasp of percentage changes Applied terms to show clear understanding

and recall rather than rote-learned formulae/terms

Gave clear solutions supported by logical arguments

Checked answers in light of question to ensure sensible and rounding correct

Could work out correct solutions Used guess-and-check method effectively Able to write solutions in words if they did not

use inequality signs (Ex). Could interpret question in context Understood graph translations Attempted all questions Set out answers well to show understanding Understood and applied calculus techniques of

a variety of situations rather than using rote-learned

Displayed superior algebra skills Accurate Proof-read manipulations Could think logically and set work out logically Understood applications of differentiation Could form a model for a given situation Used suitable substitution Able to correctly convert a trig product to a

sum keeping the correct coefficient throughout (often by use of correct brackets)

Understood details of question Could pursue solution to the end Could calculate coefficient of integration Recognised need to rotate around the y axis

between correct limits Could find the areas between three curves by

dividing into appropriate sections and then integrating difficult functions, following a number of steps accurately

Understood negative and positive values



LEVEL 1 LEVEL 2 LEVEL 3 CALC Clearly understood connections between both

algebraic and graphical representations with the context

Measured real object Selected correct method for more complex

number problems Used reverse/inverse processes – e.g. to find

original quantity before % incr. Demonstrated confidence in working with

numbers in standard form Solved multi-step problems where a

systematic approach was required Could itemise a series of calculations and

perform them accurately Developed a strategy for solving a problem

requiring several calculations and carry it out successfully

Understood and communicated the Sequence of steps to reach the answer Gave well-structured answers with consistent

calculations Maintained accuracy of answers Gave evidence of checking answers

Could manipulate algorithms Found slope of a tangent Formed equations from derived functions and

information given Understood quadratic graphs Could separate an integral into 2 parts and

optimise functions Confidently used algebra to rearrange

equations, substitute values and solve simultaneous equations; surds

Higher level of algebraic skills – incl. ability to write and solve equations; use logs and indices

Understood compound interest Could deal with logs/use calculators Could justify Could use radian measure Could manipulate complex expressions Could select all solutions in a given domain Relate evaluation to sampling method (Exc.)

Could do more than just programme graphics calculator

Could manipulate surds, logarithms Could change the subject of an expression

correctly Read questions accurately Expanded correctly – e.g. a squared binomial

that contained a surd Knew that ‘r’ is always positive and could

correctly change from rectangular to polar form

Could work in terms of ‘pi’ (Exc.) Used De Moivre’s theorem correctly to obtain

multiple solutions Could interpret questions in context Imposed axes onto a situation , used a

matching coordinate system to obtain an equation and used this to answer in context

Showed sound understanding of coordinate geometry, differentiation and algebra

Used knowledge accurately – e.g. area of a circle, equation of a straight line



LEVEL 1 LEVEL 2 LEVEL 3 CALC Was familiar with grid references Proved before assumed – right-angled triangle Drew clear accurate diagrams Could interpret and visualise right-angled

triangles in more complex geometric situations Calculated angle correctly matches

intermediate calculations with corresponding geometric reasons

Proved any assumptions made from the diagram

Showed how they reached an answer Identified correct technique – e.g.

Differentiation and carry out procedure Could choose and use an appropriate strategy

to solve a problem involving a number of different ideas and follow through the process to the end (Exc.)

Calculated a confidence interval Explained statistically significant difference Used standard error term at a predetermined

level of accuracy to calculate a minimum sample size (M)

Could obtain solution to a problem and conveyed reasoning clearly – had a good understanding of ‘justify’ (Exc)

Understood conditional probability Knew how to use combinations to determine

probabilities (M) Followed through multi-step problems and

represent probabilities in a variety of ways (M) Understood expectation theory (Exc) Followed through and clearly communicated

working in a multi-step problem (Exc) Accurately and concisely set out and

communicated working Understood and justified – valid starting

value(s) for bisection or Newton—Raphson methods



LEVEL 1 LEVEL 2 LEVEL 3 STATS

Calculated probabilities involving combinations of events

Understood the concept of conditional probability from keywords such as i

Either avoided rounding intermediate results or rounded them appropriately/sensibly

Devised an appropriate method to solve a theoretical probability problem

Correctly used fractions to describe and calculated further probabilities

Could use tree diagrams and extend a diagram to include new information and solve problems

Could give a particularly clear description of a simulation, often giving a helpful example to further clarify their response

Used a diagram to include new information and solve problems

Formed a system of equations from word problems

Accurately formed constraints and an objective function from a Linear Programming problem

Understood and justified the effect of removing a constraint (Exc)

Interpreted the geometrical meaning of a system of 3X3 equations and compare iterative methods with respect to their rate of convergence (Exc)

Correctly applied inverse normal distribution Solved problems involving the sum of random

variable Calculated and applied a Poisson parameter in

context Understood the conditions for probability

distributions and correctly used on distribution to approximate another (Exc)

Commented in relation to context


MAIN REASONS FOR FAILURE

LEVEL 1 LEVEL 2 LEVEL 3 Did not understand key words and processes

implied by them – e.g. algebraic Did not recognise when the solution to a

problem has been reached Lacked care with graphs Did not understand key features of graphs Could not plot points correctly using features

implied from equation Could not interpret features because

incorrectly read graph scales Unable to understand or make use of

implications inherent in the context of the question

Lacked basic number skills, especially with fractions

Could not correctly select values from the context and/or perform appropriate operations with them

Could not round sensibly Lacked awareness of the need to solve

problems in context Unable to use BOTH Pythagoras/Trig or

choose between them

Lacked care with basic numeric skills Did not attempt all questions Gave solutions “out of the air” from incorrect

working Presented two answers – didn’t cross out

abandoned work Provided only one answer – e.g. to quadratic

equations Did not recognise – e.g. need to factorise the

rational expression before simplifying and frequently cancelled incorrectly; pythagoras problem

Incorrect notation used within problem Failed to simplify indices correctly – e.g.

confused negative and fractional Used quadratic formula incorrectly Could not factorise Could not identify slope from equation Could not use a Graphics Calculator

effectively – failed to transfer key features from calculator to the sketch of their graphs

Didn’t know non-linear graphs must be drawn without a ruler

Rounded inappropriately

Made errors in algebraic manipulation, substitution, expansion, dealing with negative terms, rearrangement

Could not apply differentiation skills in a problem solving situation

Unable to interpret information given in the application of differentiation problems, especially rates of change

Did not provide a derivative for solutions with a graphic calculator

Understood concavity poorly Unable to utilise algebraic and numeric skills

for simplification/manipulation Could not separate variables Did not understand indices Could not deal with conversion Did not set work out carefully – e.g.

brackets/integral signs missing Lacked knowledge – conjugate; bracket

removal when negative coefficients present; quadratic formula; multiply - e.g. complex numbers in rect. form; simplify, expand a perfect square; deal with expression with linear factor as numerator/negative imaginary part



LEVEL 1 LEVEL 2 LEVEL 3 Had calculator in wrong mode Chose incorrect trig ratio Gave incomplete calculation – inverse trig /

square root processes Found area when length was asked for Guessed Unable to use appropriate methods Used wrong measurements Could not identify angle referred to Could not use geometric reasoning Didn’t attempt more than two questions Lacked knowledge of probability values Unable to select relevant information form the

context Unable to write probability in an acceptable

form

Used wrong/couldn’t decide on method – e.g. intercept rather than translation to draw parabola; differentiate/integrate

Incorrect placement - e.g. vertex, curve Did not give evidence of understanding the

nature of the features and what was required Unable to give/use correct math. language –

even basics - e.g. radius Read/interpreted question incorrectly Found communicating the mathematical idea

about how a graph was affected a challenge Omitted the constant of integration Had a problem selecting correct limits for

integration Worked from formulae/rote-learning rather

than understanding Unable to recollect even the most basic

formulae – e.g. midpoint Confused formulae – e.g. midpoint/gradient Unable to both differentiate and integrate

correctly Did not recognise type of sequence/did not

choose appropriate formula/did not use/made mistakes with correct order of operations

Unable to use both arithmetic and algebraic formulae, order operations

Tried to use general solution formula without the knowledge to correctly interpret them

Read scales/values of axes incorrectly

Did not understand mode calculator in so that did not understand whether argument was in degrees or radians

Did not understand/use simplest form Did not understand how to use logarithm

rules/symbols Could not interpret answers found on graphic

calculator – just wrote them down Did not know difference between a factor and

a solution Graphs lacked sufficient accuracy or evidence

of understanding – e.g. Ellipses with sharp corners, circles more like squares, lack of symmetry; enough of grid to show shape of hyperbola or its behaviour near asymptotes; hyperbolae with branches that were parabolic and moved away from the asymptotes

Lacked understanding of asymptotes – e.g. not straight or incorrect gradient

Lacked understanding – importance of signs; effect of graph transformations

Lacked understanding – e.g. evaluating all parameters in a general equation;

Did not understand the importance of writing equations in full, including signs, brackets and both sides of an equation

Did not check final answer was correct/valid Ignored instructions/reminders – e.g. that they

need not spend time to do the second derivative test



LEVEL 1 LEVEL 2 LEVEL 3 Could not meet sufficiency in both

manipulation of expressions and the solving of equations

Unable to truncate graphs Did not recognise symmetry Lacked care/understanding in drawing graphs

– e.g. cubic through y intercept Did not give enough of a graph to show

understanding – e.g. presence of asymptotes/hyperbola graph

Failed to describe the translation as an equation

Omitted the negative sign for the cubic Gave the answer as an expression rather than

an equation

Did only partial answers Did not read questions carefully Did not show integral used Did not take care over writing the order of

limits of an integral Had difficulty – e.g. integrate composite

functions successfully; find correct coefficients; set up appropriate integral

Did not read correctly off a graph Could not find correct values Did not choose an appropriate substitution Could not use methods – e.g. De Moivre’s

theorem, Simpson’s rule Confused as to degree of accuracy to sketch a

graph Did not understand – e.g. importance of

symmetry and shape; need for intercepts and asymptotes to be placed accurately on grid

Unfamiliar with material on formulae sheet Used inappropriate formula – e.g. distance Couldn’t use information given in the question

correctly Couldn’t find z-values for the specified

precision Lacked knowledge – e.g. form of a confidence

interval, form of standard error Didn’t round correctly



LEVEL 1 LEVEL 2 LEVEL 3 - STATS Could not algebraically rearrange equations to

solve a system of 3X3 equations Lacked understanding – e.g. Linear

Programming; Numerical Methods; bisection method (iterations)

Could not investigate all of relevant vertices Could not correctly apply parallel line test Attempted only Simultaneous Equation

questions Could not accurately substitute values into and

solve complex numerical calculations when using Newton-Raphson method

Unable to clearly interpret questions – e.g. statements such as ‘no more than two’

Did not understand when to add/subtract 0.5 when using tables in normal distribution problems

Did not understand when it was appropriate to use a continuity correction – used it when it was not needed

Lacked logic Did not show understanding of basic

probability theory Comments about trend too lightweight


SPECIFIC DIRECTIVES TO TEACHERS

LEVEL1 LEVEL 2 LEVEL 3 CALC Encourage students to attempt all questions

as evidence for Achieved can be found anywhere in the paper

Ensure students are aware that working which clarifies intent can lead to minor errors being ignored

90149: students must measure real objects and not diagrams and solve at least one problem using own measurements

90149: scale drawings and drawings of objects are not acceptable

90149: for Merit, conversions etc. from explanatory notes must be used within the context of solving a problem

90149: for Exc. models need to be more complex than packaging rectangles into rectangular boxes

90150: need to be clear on evidence for Merit and Excellence

90150: best designs are kept simple although must include at least 2 constructions and 2 transformations – check detail specifications

90193:measures of centre and spread for the data is not required

Ensure that candidates understand they need to attempt all questions - evidence for achievement can be gained from all

It is important for teachers to have read both standard and specifications and to ensure students have a good understanding of their contents

Teachers need to be aware of the evidence required from a candidate who uses a graphics calculator so that working can enable credit where a minor error has occurred

Students must expect to get two solutions for problems involving quadratics and write equations in correct form using x and y values

Lack of key skills- use of logs, writing equations, manipulation of indices, factorising, expanding, simplifying, solving equations disadvantages candidates across the standards

Spend more time on tackling and presenting a proof

Encourage students to attempt all questions Errors in algebraic manipulation are a cause

for concern Students need a better understanding of

concavity Students need to understand that they must

provide evidence of the integrated function when they use their graphic calculator to solve integration problems

Students need to use efficient checking systems to avoid unnecessary errors

Teachers need to do more work with their students on the related rates of change topic

Teachers need to emphasise that students are expected to show the integral used

Students need practice at designing an appropriate sequence of statements for a ‘’show’ or ‘prove’ question

More students need to sketch graphs for contextual problems to help clarify



LEVEL1 LEVEL 2 LEVEL 3 STATS 90193: detail is given about the nature of

comments on student work – e.g. aspects of data collection

90153: ensure questions are specific so that comparisons can be valid and that the basis for comparison is clear and justified

Students need to be aware of/heed time allocation on examination papers

Students need to be aware space in paper not necessary an indication of how much is needed

Encourage students to take careful note of bold words in questions

90288: ensure students show consideration as to whether the sample is representative of the data; use their sample in making inferences about the population in context

90288: exc./merit evaluation should relate to sampling method

90288 need to provide less scaffolding which trivialises assessment tasks

90288: be clear as to the intention of the standard – e.g. re sample size, acceptable measures of spread (detail given 2005) and re requirements for excellence – e.g. critically evaluate the sampling process/limitations on inferences

Encourage students to attempt all questions as replacement evidence can lead to multiple opportunities for achievement

Encourage students to show working to gain opportunity to have minor errors ignored or to provide replacement evidence; confidence intervals and the use of shaded diagrams in normal distribution calculations also

Make sure students are aware of the advantages of graphics calculators

Be sure students are clear as to rounding – to degree of accuracy appropriate to the problem, sensible

Ensure that students know the word ‘justify’ means statements need to be supported by, and linked to, valid conclusions or data

Ensure students know to cross out any work that is not to be marked



GENERAL LEVEL 2 LEVEL 3 Assessment schedules from web tasks must

be adapted to give specific examples of acceptable responses

Be aware of changes in requirements – e.g. ensure practical components stay

Older versions disadvantage students Encourage use of appropriate technology in

internal standards – e.g. Level 3 stats. assume use of computer/graphic calculator – limited technology – e.g. ordinary calculators, is unfair to learners

Where standards involve solving problems, students are expected to choose technique

Where students are required to generate random numbers, they do not have to show calculator process

Too many scanty and rote type responses being accepted, often not linked to context – need to be for Exc.

90291: scale drawings are not acceptable – students must complete a practical measurement and solve a trig. task

90637:ensure students find the model, use the model to solve problems and use trigonometric manipulation

90641: help students to avoid trend comments that are too lightweight – e.g. ‘the trend is increasing’ and need to interpret in context

90641: assessors need to provide less scaffolding for assessment tasks

90641: Exc. need to make comments in relation to context

90645: students tended to remove outliers to give a better fit

90645: students must quantify relationship 90645: Exc. ensure that speculations do not

go beyond the information provided 90647: students are required for merit to have

collected and used own data 90647: students often used raw data rather

than the trend line


KEY VOCABULARY

LEVEL 1 LEVEL 2 LEVEL 390147factorise, expand, solve, simplify, substitute, exponent, algebraic expression, value, linear pattern, diagram, table, equation, inequation, quadratic patterns, formulae, modelling, rearrange, square, simultaneous linear equations

GeneralSolve, differentiate, expand, parallel, median, altitude, quadratics, coordinates, x and y values, manipulation, indices, factorise, simplify, logs,

90635differentiation, chain rule, equation of the tangent, related rates of change, concavity,

90148sketch, interpret, graph, charge per kilometre, fixed fee, length of trip, charge the same, gradient of a straight line, intercept, intersection, vertex, shape, axes of symmetry, parabola, scale, linear equation, quadratics, factored form, rates of change, coefficient, transformation, maxima and minima, algebraic and graphic representation

90284manipulation, algebraic expression, equation, Guess and check method, rational expression, notation – set and co-ordinate, cubic equation, exponential equation, brackets sums and products of roots, simultaneous equations, quadratic formula, roots of a quadratic, relevance, manipulate, linear, non-linear and quadratic equations, factorise, negative and fractional indices, order of operations, sufficiency, point of intersection, inequality signs, elementary properties of logarithms, multi-step linear equations, inequations, algebraic proof

90634

90149geometric shapes and solids, circle, triangle, rectangle, cuboid, cylinder, triangle prism, perimeter, circumference area, surface area, volume, mass, capacity, time, conversion, scale, rate, estimation, precision, limits of accuracy, limitations, effectiveness, rationale, extended sequence of measurement, unit, formula

90285intercepts, asymptotes, non-linear graphs, scale, axes, radius, centre, parabola, vertex, negative cubic, points of inflection or intersection, translation, variable, curve, rectangular hyperbolae, symmetry, quadrant, truncation, quadratics, factorised polynomials, coefficient, exponential/logarithmic functions, maxima and minima, non-zero values

90635differentiating function, derivative, power, exponential, logarithmic, trigonometric, chain, product and quotient rules, expanded polynomials, optimisation, rates of change, kinematics, equations of normals and tangents, maxima, minima, degree, implicit/parametric differentiation, limits, differentiability, discontinuity, gradients, concavity, turning points, points of inflection, modelling, verifying


These are lists of key words taken from Assessment Reports and from the Achievement Standard’s explanatory notes. They provide a starting point

KEY VOCABULARY

LEVEL 1 LEVEL 2 LEVEL 390150pattern, object, isometric drawings, net, construction, loci, cuboid, constraint, replication

90286integration, differentiation, derived function, integral, area under a curve, derivative, gradient, areas, compound area, calculus notations, constant of integration, limits, distance, velocity, acceleration, variable, algorithm, slope of a tangent, optimise, expanded form, natural number exponents, turning points, tangent, polynomials, optimisation, kinematics, notation

90636integration, differential equation, numerical method, composite function, Simpson’s Rule, Trapezium Rule, variable, constant, simplify, rational expression, notation, integral signs, substitution, manipulation, simplification, trig formulae, trig product, coefficient, rotate, revolution, axis, correct limits; difficult, composite, exponential, trigonometric, rational functions, negative and positive value, rates of change, kinematics, Newton’s Law of Cooling, growth, decay, inflation

90151ratio, fraction, percentage, decimal, what fraction, how many, what percentage, number skills, denominator, percentage change calculation, sensible, estimate, integers, mark-up, discount, GST, inclusive price, standard form, multi-step problems, sequential number calculation, budgeting, fund-raising, hire-purchase

90287coordinate geometry, formulae, midpoint, gradient, distance, equation of a line, Pythagoras, proof, isosceles triangle, right angle triangle, integers, parallel, perpendicular, intersection, surds, 2 and 3 dimensional, median, bisector, altitude, proof, collinear

90637trigonometric function and equation, amplitude, period, frequency, data, manipulation, trigonometric identity, equations, reciprocal relationships, Pythagorean identities, compound and double angle formulae, sum and product formulae, general solution, specified domain

90153angle, triangle, line, parallel line, intersecting line, circle theorem, polygon, two-step process, 3 step reasoning, circle geometry, angle geometry, conjecture, proof

90289simulation, probability, normal distribution, dice, random number generator, expected number, probability trees, tables and informal conditional probability, sample space, prediction, theoretical probability, experimental probability, z-value, multi-step problems, inverse normal problems, limitations, model, spreadsheet

90639conic section, ellipse, circle, hyperbola, square, asymptotes, transformation, translation, parameter, symmetry, grid, axes, parabolic, gradient, coordinate system, implicit and parametric differentiation, tangent, intercepts, replacement evidence, Cartesian or parametric form, proof, chain of reasoning, loci, eccentricity, directrix



KEY VOCABULARY

90152Pythagoras’ theorem, trigonometric ratio, distance, height, angle, hypotenuse, grid references, degree mode, inverse, right-angled triangle, symmetry, isosceles triangles, 3D, vectors, bearings, grid references,

90288sample, inference, population, dataset, hypothesis, sampling method, standard deviation, mean, median, quartile, proportion

90638real and complex numbers, rectangular and polar form, De Moivre’s theorem, power, quadratic formula, rational numbers, equations: logarithmic, exponential, cubic, quadratic, irrational ; remainder and factor theorem, rationalising a denominator, expanding brackets, simplifying square roots and fractions, cubic equation, conjugate, negative coefficient, mode, degree, radian, ‘completing the square’ method, square roots, integer and complex roots, factor and solution, products and quotients of surds, squared binomal, ‘r’, ‘pi’, multiple solutions, loci, binomial expansions, integer exponents, Argand diagrams,

90154probability, multivariate statistical data, theoretical models and methods, randomly chosen, tree diagram, estimate, conditional probability, simulation, reverse process

90290sequence, sum to infinity concept, percentage changes, order of operations, reasonableness in context, arithmetic and geometric formulae, brackets, indices, indefinitely, simultaneous equations, compound interest, justify, general terms, partial sum, notation, ‘a’, ‘d’ ‘r’, radio-active decay, compound interest, pendulum, log equations

90291trigonometry, right-angled and non right-angled triangles, area, sine, cosine, circular measure, arc length, area of sector, bearings, relative velocity, 2D, 3D

90641trend, time series data, smoothing, moving averages, gradient, seasonal effects, cyclic effects, forecast, limitations,



KEY VOCABULARY

90292algebraic skills, trigonometric equation, periodic nature, multiple solutions, degree, radian, complex expression, domain, maximum and minimum points, manipulation, cos or tan forms

90642confidence interval, mean, population, standard error, proportion, z-values, specified precision, statistical significance, sample size, justify, seasonally adjusted data, index number series, population parameters, mean, proportion, justifying, refuting, pre-specified precision, estimation, standard error, Central Limit Theorem

90643theoretical and experimental probability, Venn and tree diagrams, permutations and combinations, independent events, mutually exclusive events, complementary events, combined events, conditional probability, multi-step problems, expectation theory, expected values, variance of random variables, discrete probability distribution, conditional probability, non-numeric reasoning, linear functions of independent random variables, proofs, table of counts, relative frequencies

90644simultaneous equation, feasible region, constraint, Linear Programming, vertices, parallel line test, optimal solution, iteration, Newton-Raphson or bisection methods, rate of convergence, specified degree of accuracy, system of equations, objective function, applied problem, justify, geometrical meaning, starting interval, derivatives of polynomials, optimisation,



KEY VOCABULARY

90645 continuous bi-variate data, linear model, variable, purpose statement, regression, appropriateness, correlation coefficients, r, coefficients of determination, interpolation, extrapolation, residual, causality, correlation, assumption, limitation, outliers, piecewise and linear models, data source and data collection models, relevance and usefulness of evidence,

90646distribution, parameter, probability ‘no more than two’, (inverse) normal, binomial, distribution, continuity correction, random and independent variable, Poisson, parameter, approximate

90647mathematical curve, model, raw data, power, exponential or piecewise functions, justification, variable, limitation



subject: · web viewknow the word ‘justify’ means statements need to be supported...

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