2009 Assessment Report. Mathematics Level 2

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1 National Certificate of Educational Achievement 2009 Assessment Report Mathematics Level Manipulate algebraic expressions and solve equations Draw straightforward non linear graphs Find and use straightforward derivatives and integrals Use coordinate geometry methods Solve straightforward problems involving arithmetic and geometric sequences Solve straightforward trigonometric equations CAS Mathematics Level Demonstrate an understanding of mathematical relationships Demonstrate an understanding of calculus methods Demonstrate an understanding of processes involving trigonometry and coordinates New Zealand Qualifications Authority, 2010 All rights reserved. No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

2 COMMENTARY NCEA Mathematics Level 2 Assessment Report, 2009 page 2 of 9 There was a changed format of these examination papers for Candidates with a graphics calculator who used it effectively, were advantaged in some standards, both in terms of time and in terms of the accuracy of their answers. However, some candidates using these calculators appeared not to understand the mathematical processes that are involved, as they did not arrive at correct answers to all questions. Premature rounding led to inaccurate answers in some cases. Some candidates did not attempt all questions, or attempted only the first page. Many of the candidates who did not achieve showed poor basic numeracy skills and algebra skills. STANDARD REPORTS MATHEMATICS LEVEL Manipulate algebraic expressions and solve equations expanded and simplified expressions correctly factorised quadratic expressions solved quadratic equations (whether by traditional methods or by using a graphics calculator) solved a multi-step inequation correctly. did not correctly factorise a quadratic expression could not find the square root of an algebraic expression showed poor number skills failed to reverse the sign of an inequation when dividing by a negative number had little understanding of the process of cancelling fractions could not understand log-equivalent form, often resorting to a calculator. related one part of a question to another (e.g. (i) and (iii) of Q1b) were able to perform a series of skills (e.g. substitute, expand and solve a problem) used the graphics calculator to good effect recognised that there were two intersections between a straight line and a circle. 2

3 NCEA Mathematics Level 2 Assessment Report, 2009 page 3 of 9 related algebra to other areas of mathematics (e.g. Pythagoras Theorem and the tangent property) proved or showed using an extended mathematical argument. In the Excellence questions there were many elegant and innovative solutions provided by candidates with an impressive depth of understanding. However, some of the candidates who were successful at this level did not correctly answer some of the preceding Achievement level questions Draw straightforward non linear graphs displayed the appropriate key features of each graph used a graphics calculators correctly to assist with the drawing of graphs sketched polynomial graphs correctly, and accurately showed intercepts and the vertex of a parabola. drew a hyperbola correctly and showed they understood the nature of an asymptote. drew insufficient graphs did not show understanding of the general shapes of the function drew incomplete graphs which lacked the necessary detail for the situation drew graphs that crossed or touched asymptotes did not take note of the scale on the axes made several attempts on one set of axes at drawing the graph used or relied on a graphics calculator without understanding the features of the graph ruled straight segments for some non linear graphs did not extend a logarithmic graph below the x axis. correctly identified translations and wrote an equation to define a translated graph derived the equation of the graph from key features applied the mathematics in a practical situation 3

4 NCEA Mathematics Level 2 Assessment Report, 2009 page 4 of 9 drew more complex graphs (logarithmic and circle), accurately showing the features appropriate for that graph identified that maximum corresponded to a turning point and correctly calculated this value. wrote the equation of a translated parabola and calculated the y intercept defined a piece-wise function using an equation and the domain understood how to interpret a reduced growth rate, and formed an equation to solve a problem (or simply used an iterative approach to model a reduced growth rate situation). Some candidates found questions involving Profit and Loss confusing. Some candidates did not show equations with a subject. Some candidates redrew graphs but did not make clear which graph was to be marked. Multiple attempts or solutions did not achieve. Some candidates redrew on the wrong additional grid with a scale different from the original axes Find and use straightforward derivatives and integrals found a gradient of a curve at a point found f(x) given the gradient function integrated and linked it to finding area under a curve. did not differentiate and apply to calculate gradients did not graph a gradient function did not integrate correctly used only the graphics calculator to answer questions. used calculus to determine a maximum value for a function. correctly calculated composite areas. 4

5 NCEA Mathematics Level 2 Assessment Report, 2009 page 5 of 9 correctly applied principles of kinematic for the links between acceleration, velocity and distance derived an appropriate model for a parabola and linked integrals and area. Some candidates demonstrated an over-reliance on the graphics calculator as a tool without understanding the technique. These candidates were unable to give the derived or integrated function. Some candidates were poor at communicating mathematical ideas in a way that showed clear, logical working. Many candidates made frequent numerical and simple algebraic errors Use coordinate geometry methods knew and used the correct formulas for co-ordinate geometry, i.e. midpoint, distance, gradient correctly substituted values into formula located the gradient from the equation of a straight line formed an equation using a point and a gradient showed they understood parallel and perpendicular gradients. used incorrect formulas for co-ordinate geometry (e.g. they added x and y co-ordinates for distance) did not identify the x and y co-ordinates correctly, and confused formulae did not recognise that when they squared a negative number they wrongly gave a negative answer which was not accurate for a length question did not form equations did not show understanding of parallel and perpendicular gradients did not answer any part of the second question. worked with co-linear points and equated gradients containing one variable correctly applied the properties of a parallelogram 5

6 NCEA Mathematics Level 2 Assessment Report, 2009 page 6 of 9 selected appropriate formula to answer the questions. showed an extended, logical set of steps to solve a problem used variables correctly in a distance formula and then worked correctly with a formula knew and correctly applied a formula for calculating the shortest distance from a point to a line demonstrated good algebraic skills including solving simultaneous equations and applying their answer to solve the question. Many candidates did not correctly substituted negative numbers into formula. Candidates commonly made both arithmetic and algebraic errors after a correct substitution. Many candidates demonstrated very poor geometrical knowledge of parallelograms Solve straightforward problems involving arithmetic and geometric sequences identified an arithmetic progression or a geometric progression appropriate to the question identified key features of the progression substituted these key features correctly into an appropriate formula to come up with a correct solution. did not select the correct progression appropriate to the question did not show any evidence of checking their answer by performing some simple calculations substituted values incorrectly into known formulae. interpreted a wide range of questions correctly, discerning between summation questions and questions involving finding particular terms solved problems and checked for appropriateness correctly worked from the starting term in the geometric progression questions. 6

7 NCEA Mathematics Level 2 Assessment Report, 2009 page 7 of 9 solved more difficult problems with a confident algebraic approach displayed their working in a logical format Solve straightforward trigonometry equations used the graphics calculator solver to find x values found other solutions within the specified domain used the calculator in both degree and radian modes drew sketches of the trig functions. provided only one correct solution did not recognise the requested domain as being relevant to their answer left their calculator on one mode (degrees, radians or gradients) did not correctly rearrange the equation prior to finding the first solution. used the graphics calculator to form multiple intersections between functions correctly interpreted values as being above or below a particular stated value calculated time intervals between two values. used the graphics calculator to find relevant roots of functions recognised and listed the appropriate amplitude, vertical translation and period values to state the correct model. Most candidates who used the graphics calculator answered questions successfully. Among those relying on algebraic methods, there were more errors, but also some very precise and correct solutions. Some candidates did not answer solutions in the appropriate units given in the domain, or use appropriate rounding for all questions. 7

8 NCEA Mathematics Level 2 Assessment Report, 2009 page 8 of 9 STANDARD REPORTS CAS MATHEMATICS LEVEL Demonstrate an understanding of mathematical relationships worked between tables, graphs and equations appropriately used their calculators effectively to perform basic skills. did not correctly use their calculators to perform basic skills. demonstrated the ability to solve more complex problems across the range of mathematical standards. interpreted and applied mathematical concepts into general situations and problem solving situations. It appears that some candidates used CAS technology merely to obtain answers, rather than to enhance understanding and allow exploration. For example, they apparently did not look at the number of solutions to a trig equation in order to find the coefficient of x in a trig equation Demonstrate an understanding of calculus methods demonstrated the ability to perform basic differentiation and integration in finding gradients, equations and areas. 8

9 NCEA Mathematics Level 2 Assessment Report, 2009 page 9 of 9 did not solve problems that involved both differentiation and integration. solved problems involving calculus in more complex situations. related their calculus to more generalised situations. It appears that some candidates used the CAS technology merely to obtain answers, rather than to enhance understanding and allow exploration Demonstrate an understanding of processes involving trigonometry and coordinates solved problems involving coordinates. did not demonstrate any understanding of coordinate geometry and trigonometry. solved problems in more complex situations. generalised situations and solutions. It was disappointing to see that many candidates did not use CAS technology in solving problems involving either trigonometric or geometric situations. 9