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Assessment Report
Level 2 Mathematics and Statistics 2016
Standards 91261 91262 91267
Part A: Commentary
Candidates were assessed on the application of knowledge of mathematics from level 7 of the New Zealand Curriculum, however candidates are reminded that knowledge from level 6 or lower is also expected, e.g. graphing knowledge within the algebra assessment and algebra knowledge within the calculus assessment.
Part B: Report on Standards
91261: Apply algebraic methods in solving problems
Candidates who were awarded Achievement commonly:
 solved simple logarithmic equations, showing an understanding of the properties of logarithms and logarithmic statements
 applied logarithmic rules to simplify an expression
 manipulated expressions involving a range of exponents and indices
 interpreted, in a mathematical context, the information presented in a question and could set up a mathematical model to solve that problem
 factorised quadratic equations with the coefficient of x^{2} greater than one and determined the solutions
 derived a quadratic equation from known solutions
 manipulated an equation with indices to create a common base
 used algebraic techniques to write parabolic equations (including the coefficient for x^{2}) which modelled practical situations represented in graphs
 rewrote an algebraic rational equation and gathered terms in preparation for making a new subject
 demonstrated they could complete the square for a quadratic expression
 understood the meaning of 'solutions of an equation' and their relationship to the discriminant
 applied the quadratic formula to obtain solutions to equations with nonnumeric coefficients
 used a graphics calculator to solve problems without algebraic evidence, so depriving themselves of the opportunity to gain a higher grade.
Candidates who were assessed as Not Achieved commonly:
 did not fully demonstrate an understanding of the knowledge required by the standard with respect to indices, discriminants, roots and solutions to equations
 appeared unfamiliar with the process of completing the square
 were unable to convert a rate of depreciation into a relevant decimal
 were unable to rewrite a logarithmic equation into index form
 manipulated fractions incorrectly using their calculator
 did not form a common base when solving an equation involving multiple indices terms
 failed to recognise that a graph cutting the x axis corresponds to y = 0
 omitted the use of brackets and incorrectly substituted values (especially negatives) into rules
 misunderstood that the coefficients for the discriminant relate to the quadratic equation being in the form “= 0”
 were unable to write an equation(s) to model a practical situation mathematically
 did not realise when an answer was correct and so went on to undo a correct response by applying incorrect simplification techniques
 did not use a graphics calculator to solve straight forward quadratic equations or check the validity of their answers.
Candidates who were awarded Achievement with Merit commonly:
 demonstrated the ability to manipulate a range of algebraic expressions including logarithms, indices, fractions, rational expressions and factorising
 found a model for a situation in context, and derived an appropriate and meaningful solution(s) in context
 solved a range of equations including logarithmic and exponential to provide an answer in context, when appropriate
 used the solutions of a pair of quadratic equations to demonstrate the stated relationship between the solutions
 formed a quadratic equation from solutions and identified the coefficients for each term
 rearranged a rational expression
 applied the discriminant to an equation and determined the solutions written as an inequality
 related the concepts and solution from one question to a subsequent question.
Candidates who were awarded Achievement with Excellence commonly:
 applied algebraic and logarithmic skills to solve a problem and determine all valid solutions
 used the quadratic formula to find solutions to a pair of equations, then determined the relationship between the solutions
 understood the properties of the discriminant and successfully used it to find all relevant solutions to a given problem, identifying all constraints
 developed an equation or equations to model a contextual situation and used algebraic techniques to answer the problem in context
 applied the laws of indices, logarithms and algebra to solve an equation with valid solutions
 used their knowledge of quadratics and applied the information provided on a graph to solve a problem after having made suitable substitutions, then determine a valid solution in context.
Standardspecific comments
To reach Achieved or higher grades, candidates needed to demonstrate algebraic techniques.
The vocabulary of algebra needs to be understood so that candidates fully understand the meaning of each question.
Any constraints which are relevant to either the context or algebraic answer need to be applied to final solutions.
Consideration should be given by candidates to checking answers to ensure their validity and that the final answer represents the context of the question being asked.
91262: Apply calculus methods in solving problems
Candidates who were awarded Achievement commonly:
 differentiated basic polynomials
 found the gradient at a point
 drew a parabola (gradient function) with the correct x intercepts
 drew a positive cubic given a gradient function of a parabola
 antidifferentiated basic polynomials, including finding the constant of integration
 answered kinematic problems using physics formulae.
Candidates who were assessed as Not Achieved commonly:
 did not know when to substitute and when to solve
 only differentiated but did not know what to do to solve the required question
 completed Achievementlevel questions but often included minor errors which could not be ignored at this level
 were unable to relate relevant features of function / gradient sketches
 did not find the constant of integration.
Candidates who were awarded Achievement with Merit commonly:
 interpreted given information and successfully used multiple pieces of information
 used calculus in answering a question without communicating exactly what the answer to the question was
 found a local minimum without justification
 found the equation of a tangent given a function and coordinate
 found an unknown variable in an expression given the gradient
 drew a cubic with the correct turning points given a gradient function of a parabola
 used kinematics without communicating exactly what the answer to the question was.
Candidates who were awarded Achievement with Excellence commonly:
 found an equation of a function given a gradient function with an unknown variable and a coordinate on the function
 found a local minimum and justified it accurately
 dealt with unfamiliar situations, like multiple variables
 communicated the answer to the question
 applied calculus to a rate of change problem involving pi
 solved a quadratic with a variable and gave the positive and negative solutions
 used inequality signs correctly
 answer kinematics problems using calculus
 showed working to make it clear when the constant of integration was zero
 communicated effectively with correct calculus notation, good algebra skills and no incorrect mathematical statements.
Standardspecific comments
Some candidates showed weak basic algebraic manipulation and solving skills. If this resulted in a minor error that did not affect the level of difficulty of the problem, this may have been ignored.
91267: Apply probability methods in solving problems
Candidates who were awarded Achievement commonly:
 drew a probability tree diagram from the information provided
 found straightforward probabilities from a probability tree diagram
 understood that proportion and risk are calculated in the same way as a probability
 calculated straightforward probabilities from information presented in contingency tables
 found probabilities involving a normal distribution, either using a graphic calculator or probability tables
 made valid comments on either a frequency histogram or a probability distribution graph
 found proportions from a frequency histogram
 understood the link between two tables of information that presented probabilities in a different form, i.e. two tables rather than in a tree diagram or a contingency table.
Candidates who were assessed as Not Achieved commonly:
 could not draw probability tree diagrams
 did not recognise the similarity between proportions and probabilities
 added or divided probabilities along branches on a probability tree
 could not solve problems with contingency tables, particularly when the sample space was reduced
 did not understand absolute risk
 could not use their graphic calculator to solve normal distribution problems
 did not make the link between the calculation of a Z value in the standard normal distribution and the probability associated with it when using tables
 did not use statistical language when describing comparisons between a normal probability distribution and a frequency histogram.
Candidates who were awarded Achievement with Merit commonly:
 calculated a relative risk
 calculated probabilities for multiple events using a probability tree diagram
 solved an inverse normal distribution problem when given a probability
 used statistical language effectively to compare two graphs
 completed a contingency table when provided with more complex information, and then used it to calculate a probability.
Candidates who were awarded Achievement with Excellence commonly:
 applied conditional probability correctly when given a complex probability problem
 calculated, and then interpreted a relative risk correctly
 had a comprehensive knowledge of the three aspects of centre, shape and spread when comparing graphs and used supporting numerical evidence
 applied their understanding of probability to complex probability situations
 solved a complex probability problem where two tables of probabilities were given and demonstrated an understanding of the relationship and connection between the tables
 linked answers from previous questions to a current question and evaluated a claim, providing appropriate evidence.
Standardspecific comments
While latitude was given in the marking, many candidates were inclined to round or truncate their answers at an intermediate stage, rather than at the end of a problem.
The concept of proportion and its relationship to probability and risk is not well understood by some candidates.
There was an improvement in the presentation of answers where distributions were compared. However, there was weakness in the use of statistical language and in using a methodical approach to compare shape, centre and spread.
Of note were the increased number of candidates who described shape well, and especially the concept of ‘skewness’.
Candidates were required to construct a tree diagram and contingency table without being directed to do so. Some candidates lacked this skill