Level 3, 2005 Calculus Differentiate and use derivatives to solve problems (90635) Integrate functions and solve problems by integration, differential equations or numerical methods (90636) Manipulate real and complex numbers and solve equations (90638) Sketch graphs and find equations of conic sections (90639) New Zealand Qualifications Authority, 2005 All rights reserved. No part of this publication may be reproduced by any means without prior permission of the New Zealand Qualifications Authority.
NCEA Level 3 (Calculus) 2005 page 2 Calculus, Level 3, 2005 Commentary Candidates were well prepared for all four standards assessed and results showed an improvement from the previous year. The four assessment activities of the 2005 paper were shorter than those of the 2004 paper, and it appears candidates achieved better results in 90639 Conic Sections because they did not run out of time. The Excellence questions were more accessible to candidates. Numbers of candidates completed the Merit questions without error and left the Excellence questions blank. These candidates need encouragement to attempt all questions. Algebraic manipulation skills are still a cause for concern. Errors abounded in algebraic manipulation, substitution, expansion, rearrangement and dealing with negative terms. Calculus: Differentiate and use derivatives to solve problems (90635) 7,209 25.5% 49.0% 20.6% 4.9% ability to carry out differentiation of exponential, log, trig and functions of the form ax n, where n is a real number ability to recognise when the chain rule was required and to use the chain rule ability to find the equation of a tangent ability to solve a related rates of change question, involving two directly related rates.
NCEA Level 3 (Calculus) 2005 page 3 Candidates assessed as Not Achieved commonly lacked the following skills and / or ability to apply their differentiation skills in a problem-solving situation good algebra skills that allowed them to correctly simplify the equation of the tangent ability to interpret the information given in the application of differentiation problems, especially in the related rates of change question understanding of the need to provide a derivative for any question that could be solved using the graphic calculator. ability to display superior algebra skills ability to be accurate ability to proof-read manipulations ability to think logically and set out working logically good understanding of applications of differentiation ability to form a model for a given situation. Performance of the 2005 cohort The proportion of candidates achieving the standard increased from 2004 to 2005 as a result of the changed sufficiency, which was reduced to three of four code A in 2005, compared to four of five code A in 2004. There was some improvement in the ability of the candidates to solve related rates of change problems. Question 3 in the 2005 paper was mainly done well. Otherwise, the cohorts of 2004 and 2005 were pretty similar, displaying poor algebra skills and the inability to solve differentiation problems accurately. Candidates demonstrated a poor understanding of concavity in Question 6. The majority of candidates solved the first derivative equal to zero rather than the second derivative to find the boundaries for the part of the graph where the function was concave down. Calculus: Integrate functions and solve problems by integration, differential equations or numerical methods (90636) 7,126 32.1% 49.2% 14.0% 4.7%
NCEA Level 3 (Calculus) 2005 page 4 ability to integrate composite functions, applying and modifying information from tables and the formulae booklet ability to calculate h and use correct values when applying Simpson s Rule ability to separate variables and find the constant in Question 3 ability to simplify a rational expression prior to integration in Question 1c ability to understand and use notation correctly eg integral signs were often omitted or incorrectly placed. While this did not preclude achievement it did suggest limited understanding. Candidates assessed as Not Achieved commonly lacked the following skills and / or ability to utilise algebraic and numeric skills for simplification and manipulation in Questions 1c and 3 ability to separate variables. This skill was even lacking for some candidates who were able to integrate the Merit questions involving substitution and trig formulae. Other candidates could not find c ability to show understanding of indices in Question 1a, and could not deal with converting from ln to exp in Question 3 An ability to set out work carefully, for example brackets were frequently missing, integral signs were omitted. This often led to subsequent errors. ability to use suitable substitution and remember to change to du in Question 4a. Many candidates at Achievement level showed a weakness in substitution ability to correctly convert a trig product to a sum in Question 4b, keeping the correct coefficient throughout (often by correct use of brackets) understanding of the details of Question 5 and ability to pursue the solution to the end, including writing down and solving a DE; and remembering to add 20 if they started with t = 0 for 120 rads. These candidates were able to calculate the coefficient of integration recognition of the need to rotate around the y axis, between correct limits in Question 6 ability, in Question 7, to find the areas between three curves by dividing into appropriate sections and then integrating difficult functions, following a number of steps accurately. This demonstrated understanding of the negative value of the integral below the axis, but its positive area. These candidates could also do more than just programme their graphics calculators. Several candidates obtained correct numerical answers, but the incorrect or missing intermediate step meant N.
Performance of the 2005 cohort NCEA Level 3 (Calculus) 2005 page 5 Candidates performed better at all levels of achievement compared to the 2004 cohort, especially at Excellence level. Some candidates still do not provide evidence of the integrated function when they use their graphic calculator to solve integration problems. Calculus: Manipulate real and complex numbers and solve equations (90638) 7,032 30.7% 42.6% 22.8% 3.9% ability to manipulate complex numbers in rectangular and polar form ability to use De Moivre s theorem for writing the power of a complex number in polar form ability to use the quadratic formula to express solutions in a + ib form where a and b are rational numbers ability to use logarithms to solve an exponential equation ability to use basic algebraic skills such as rationalising a denominator, expanding brackets and simplifying square roots and fractions understanding of how to use a graphics calculator to their advantage when other working was required or when an answer needed to be given in a different form, particularly when solving a cubic equation ability to read and interpret instructions correctly ie find an answer in the required form. Candidates assessed as Not Achieved commonly lacked the following skills and / or knowledge of what a conjugate is or how to remove brackets when negative coefficients are present understanding of the mode the calculator should be in, with the result that they did not understand whether the argument was in degrees or radians understanding that all expressions should be shown in simplest form, whether asked for or not, or the ability to simplify an expression when there were two parts to the numerator the knowledge necessary to use the quadratic formula or completing the square method to find solutions and the skill to simplify expressions using square roots understanding of how to use the logarithm rules or that x was to be in terms of p
NCEA Level 3 (Calculus) 2005 page 6 ability to interpret the answers they found on their graphics calculator ie not just write down the answer they found on them knowledge of the difference between a factor and a solution. ability to manipulate surds, logarithms and change the subject of an expression correctly ability to read the question that asked for x in terms of p accurately ability to correctly expand a squared binomial that contained a surd The knowledge that r is always positive and the ability to correctly change from rectangular form to polar form. Many excellence students were able to work in terms of pi. They showed the ability to use De Moivre s theorem to correctly obtain multiple solutions. Performance of the 2005 cohort Candidates had to interpret the answers found on their graphics calculators in the equation-solving questions in order to achieve this year. Even though the Excellence question was more accessible this year, not as many candidates attempted it. Calculus: Sketch graphs and find equations of conic sections (90639) 6,542 29.4% 46.4% 18.8% 5.4% Graph sketching: ability to sketch graphs of straightforward conic sections (ellipse, circle and hyperbola) with some accuracy ability to complete the square understanding of the properties of a hyperbola as it approaches asymptotes understanding of the relation of transformations of a graph to changes in algebraic form. Writing equations:
NCEA Level 3 (Calculus) 2005 page 7 ability to write equations of the conic sections from the graph or from a description of the graph ability to substitute appropriately ability to correctly rearrange in order to evaluate parameters understanding of the relation of transformations of a graph to changes in algebraic form. Candidates assessed as Not Achieved commonly lacked the following skills and / or Graph sketching: ability to sketch graphs of straightforward conic sections with sufficient accuracy. Although candidates were not penalised for poor drawing skills, their sketch needed to show they understood what the graph should look like. A number of candidates sketched ellipses that had sharp corners and circles that looked more like squares, and many sketches lacked any symmetry ability to sketch graphs that covered enough of the grid to provide evidence that they understood the shape of the hyperbola or its behaviour near asymptotes understanding of asymptotes eg for the circle and ellipse, some candidates sketched axes of symmetry that they labelled as asymptotes ability to sketch hyperbolae with branches that were parabolic and moved away from the asymptotes. Candidates did not achieve if asymptotes were not straight or had an incorrect gradient. Writing equations: understanding of the importance of the signs in the general equations of conics and the effect of graph transformations on equations understanding of the importance of evaluating all parameters in a general equation understanding of the importance of writing equations in full, including signs, brackets and both sides of an equation ability to carry out algebraic manipulation with care. Some candidates carried out inaccurate algebraic rearrangement when they would have been better to leave the required equations in the original form. General: An ability to check that their final answer is correct. A number of candidates calculated parameters correctly but then substituted them incorrectly or into an incorrect equation. Merit: ability to interpret questions in context. This involved the ability to impose axes onto a situation, use a matching coordinate system to obtain an equation and use this to answer the question in context. Candidates who used the simplest equation tended to have more success with completing the question than students who chose to use equations that included translations sound understanding of coordinate geometry, differentiation and algebra. Candidates need to be able to use simple geometric knowledge accurately eg area of a circle and equation of a straight line. At Merit and Excellence level candidates need to show how they have obtained their answer. Candidates needed to set work
NCEA Level 3 (Calculus) 2005 page 8 out logically in a step-wise development, and correct answers so that they are not penalised ability to identify the appropriate differentiation technique ie implicit differentiation, and carry out the procedure to find the gradient of the tangent, to evaluate this correctly and substitute it into the equation of the straight line. Simple errors in rearranging or differentiating were common, and were penalised as this is a basic skill at this level. Many candidates completed Question 7 correctly up to substitution of the gradient and the point into the straight line equation, but then made a mistake in rearranging the equation, with the most common error the incorrect multiplication of two negatives. Although students were not penalised for this error this year, they do need to take care in expanding and rearranging. Careful checking of working would help students identify mistakes. A small proportion of candidates showed a lack of understanding of what they were doing and calculated the equation of the normal rather than the tangent Excellence: ability to choose and use an appropriate strategy to solve a problem involving a number of different ideas and to follow the process through to the end. Candidates needed good algebra skills understanding of the geometry of the situation required for Question 8. Although there were a variety of elegant solutions, many candidates only reached part way through the question. A number of candidates did not differentiate to find the gradient, instead mistakenly getting the gradient by using a point on the curve as if it was also on the tangent. Performance of the 2005 cohort The 2004 examiner s report commented on the importance of students understanding what was meant by the key term sketch the graph. This advice appears to have been heeded, with far fewer candidates simply drawing a picture and many more showing some understanding of the need for intercepts and asymptotes to be accurately placed on the grid. This year s candidates were better at using the conic described in the question. In previous years candidates tended to model a situation with a conic other than the one described in the question. Several candidates only attempted some or all of Questions 1 through 4 and so provided limited options for replacement evidence.
Level 3, 2005 Statistics and Modelling Calculate confidence intervals for population parameters (90642) Solve straightforward problems involving probability (90643) Solve equations (90644) Use probability distribution models to solve straightforward problems (90646) New Zealand Qualifications Authority, 2005 All rights reserved. No part of this publication may be reproduced by any means without prior permission of the New Zealand Qualifications Authority.
NCEA Level 3 (Statistics and Modelling) 2005 page 2 Statistics and Modelling, Level 3, 2005 Commentary Candidates achieving the standard showed good mathematical knowledge and skills. There was an increase in the number of candidates not attempting Merit or Excellence questions. Candidates need to be encouraged to attempt all questions as their answers can be used as replacement evidence and can provide multiple opportunities to reach Achievement. With the increased use of graphics calculators, some candidates are showing little working. While working is not always required, candidates should show it where they can, or risk losing the opportunity to have minor errors ignored or to provide replacement evidence. Candidates need to be aware that graphics calculators can be an advantage in some Achievement standards, both with respect to demonstrations of the skills required and the time available to them to complete the assessment. While there appeared to be an improvement on the previous year, a large number of candidates are still rounding values prematurely, or not rounding a final answer to a degree of accuracy appropriate to the problem. While schedules allowed for a reasonable variation in rounding, particularly at Achievement level, candidates should be encouraged to provide answers to a sensible degree of accuracy. Where the term justify is used in a question, statements need to be supported by, and linked to, valid calculations or data. Statistics and Modelling: Calculate confidence intervals for population parameters (90642) 11,390 8.7% 37.6% 47.3% 6.4% ability to calculate a confidence interval for a population mean and a population proportion. This required the ability to determine the appropriate z-values for the specified precision. Candidates assessed as Not Achieved commonly lacked the following skills and / or ability to correctly use the information given in the question
NCEA Level 3 (Statistics and Modelling) 2005 page 3 ability to correctly determine the appropriate z-values for the specified precision and display the knowledge of the form of a confidence interval, in particular the form of the standard error ability to round figures in their working correctly. ability to calculate a confidence interval for the difference between the means of two populations and explain why there was a statistically significant difference between the two means ability to use the standard error term at a predetermined level of accuracy to calculate a minimum sample size. This allowed some candidates to achieve with Merit ability to obtain a solution to a problem and convey their reasoning clearly. This allowed candidates to achieve with Excellence. These candidates demonstrated a good understanding of what is required to justify an answer. Statistics and Modelling: Solve straightforward problems involving probability (90643) 11,185 49.7% 34.7% 12.0% 3.6% ability to use a range of techniques to solve probability problems. In particular they used Venn diagrams and tree diagrams appropriately to solve problems, as well as showing an understanding of independence, mutually exclusive events and the complement of an event ability to present clear working. Candidates assessed as Not Achieved commonly lacked the following skills and/or ability to read and interpret the given information ability to display a knowledge of basic probability terms and to recognise that a probability cannot be greater than 1.
NCEA Level 3 (Statistics and Modelling) 2005 page 4 understanding of conditional probability and knowledge of how to use combinations to determine probabilities. This was needed for Merit ability to follow through multi-step problems and represent probabilities in a variety of ways. This needed for Merit understanding of expectation theory. This was required to achieve Excellence ability to follow through and clearly communicate working in a multi-step problem. This was needed for Excellence. Performance of the 2005 cohort While the 2005 examination was a similar format to the previous year there seemed to be fewer Achievement grades awarded and more candidates assessed as Not Achieved for this standard. There was an apparent increase in the number of candidates who did not attempt the harder questions and, therefore, did not give themselves the opportunity for replacement evidence. Statistics and Modelling: Solve equations (90644) 11,265 28.1% 53.0% 14.5% 4.4% ability to solve a system of 3 by 3 simultaneous equations. While some candidates did this by systematically setting out their working, many were able to rearrange equations where needed and then use their graphics calculators to obtain a correct solution ability to identify a feasible region from constraints and from a graph in a Linear Programming problem. They were then able to either identify and study the vertices, or use the parallel line test, in order to find an optimal solution, while showing clear working ability to complete at least two iterations of either the Newton-Raphson or the bisection methods. Candidates assessed as Not Achieved commonly lacked the following skills and / or ability to algebraically rearrange equations in order to solve a system of 3 by 3 equations, whether by graphics calculator or by hand ability to understand Linear Programming or Numerical Methods, attempting only the Simultaneous Equations questions ability to investigate all of the relevant vertices and correctly apply the parallel line test in order to optimise the objective function when attempting Linear Programming
NCEA Level 3 (Statistics and Modelling) 2005 page 5 ability to understand when one iteration ends and the next one begins when using the bisection method for calculating numerical solutions ability to accurately substitute values into, and solve, complex numerical calculations when using the Newton-Raphson method. ability to accurately and concisely set out and communicate working ability to understand and justify what constitutes valid starting value(s) for the bisection or Newton-Raphson methods and then to demonstrate understanding of when an iterative method converges to a specified degree of accuracy ability to form a system of equations from word problems and accurately form constraints and an objective function from an applied Linear Programming problem ability to understand and justify the effect of removing a constraint from a Linear Programming problem. This was shown by candidates who achieved with Excellence ability to interpret the geometrical meaning of a system of 3 by 3 equations and compare iterative methods with respect to their rate of convergence. This was also shown by those achieving with Excellence. Statistics and Modelling: Use probability distribution models to solve straightforward problems (90646) 10,811 19.7% 45.6% 25.1% 9.6% ability to correctly select a distribution and its parameter(s) and use them to solve problems ability to demonstrate knowledge of the use of a graphics calculator to solve problems using probability distributions. Candidates assessed as Not Achieved commonly lacked the following skills and / or knowledge.
NCEA Level 3 (Statistics and Modelling) 2005 page 6 ability to clearly interpret questions, particularly statements such as no more than two understanding of the need to add 0.5, or subtract from 0.5 at times, when using tables in normal distribution problems understanding of when it is appropriate to use a continuity correction. These candidates used it when it was not needed. ability to correctly apply the inverse normal distribution ability to solve problems involving the sum of random variable ability to calculate a Poisson parameter in context and apply it to solve a problem ability to understand the conditions for probability distributions and correctly use one distribution to approximate another. This allowed candidates to achieve with Excellence.