Assessment Report. Level 2, Mathematics

Transcription

1 Assessment Report Level 2, 2006 Mathematics Manipulate algebraic expressions and solve equations (90284) Draw straightforward non-linear graphs (90285) Find and use straightforward derivatives and integrals (90286) Use coordinate geometry methods (90287) Solve straightforward problems involving arithmetic and geometric sequences (90290) Solve straightforward trigonometric equations (90292) New Zealand Qualifications Authority, All rights reserved. No part of this publication may be reproduced by any means without prior permission of the New Zealand Qualifications Authority.

2 NCEA Level 2 (Mathematics) 2006 page 2 Mathematics, Level 2, 2006 Commentary Candidates should attempt all questions and show working. All working can be evidence that verifies achievement. Minor errors may be ignored if the working demonstrates knowledge required by the standard being assessed. Care must be taken when reading questions and candidates should consider how sensible their answer is in relation to the question. Practise at solving and interpreting contextual questions is useful. Although rounding is not assessed, rounding every step to 1 decimal place, or rounding to inconsistent numbers of significant figures, or truncating can cause considerable inaccuracy. With scientific or graphic calculators, intermediate answers need not be rounded. Graphing calculators advantage candidates who use them correctly. When candidates do not know how to use these calculators correctly they often give solutions over the wrong domain and cannot relate the solutions to the questions, especially if the question is in context. Care needs to be taken when using integers and fractions, and when manipulating equations. Candidates who attempt questions more than once should clearly indicate what working they want marked. Crossed out working indicates candidates do not want that work to be marked. Manipulate algebraic expressions and solve equations (90284) expanded brackets correctly accurately simplified terms in an algebraic expression understood fractional exponents solved quadratic equations by factorising and so provided further evidence for algebraic manipulation accurately used the quadratic formula. manipulated logarithmic expressions to solve an equation solved quadratic equations giving only one solution, or made errors using the quadratic formula when this was selected as the method of solution simplified rational expressions inaccurately eg a correct numerator and a missing denominator or a correct addition that had further working causing a wrong answer to be given demonstrated poor calculation skills eg 16 1/2 3 = 8 and 64 = 24 incorrectly used graphics calculators found the numerical quantity 2 2 but did not have the correct sign when solving the 3 inequation did not understand the meaning of = did not take all opportunities to demonstrate manipulation or solving skills.

3 NCEA Level 2 (Mathematics) 2006 page 3 were able to form equations and correctly solve them could use a graphing calculator well solved simultaneous equations with one non linear equation understood that coordinates of both points meant both x and y were expected solved an exponential equation by using logarithms or a graphics calculator expanded a perfect square correctly. Draw straightforward non-linear graphs (90285) identified both the x and y intercepts of curves where required sketched well-shaped curves and recognised the correct orientation showed relevant working alongside their graphs. showed carelessness in identifying key features such as a turning point used straight lines in joining points for the parabola and cubic had difficulty using their graphic calculator and presented a parabola instead of sketching a circle discontinued their graphs at the x-axis. understood and were able to draw translations showed asymptotes clearly as dotted lines where required and showed that the curve approached the asymptote correctly inserted the "y =" in front of required functions followed bold instructions set out working clearly and logically. Find and use straightforward derivatives and integrals (90286) correctly differentiated or integrated expressions and used these to find a gradient, form an equation from a derived function, the area under a curve over an interval, and find the coordinates of a point at which the derived function has a known slope showed the derivative or integral used supported answers with clear and appropriate working so that mathematical error ignored (MEI) could be applied even if an incorrect final answer was given attempted merit questions providing sufficient evidence for achievement when mistakes had been made in earlier questions formed and solved a linear equation substituted values into correct expressions. did not know whether to differentiate or integrate for the given problem

4 NCEA Level 2 (Mathematics) 2006 page 4 could successfully differentiate or integrate but not both used a graphics calculator to find the area under a curve without showing the integrated function displayed incorrect numerical calculation skills along with insufficient working to enable MEI to be applied failed to include or successfully apply the constant of integration when integrating the derived function. made multiple arithmetic errors. knew which additional processes were required to solve problems having found a derivative or integral i.e. understood the relationship between distance, velocity and acceleration in kinematics problems, used point/slope equation, could find the area bound between a curve and the x. axis over an interval for a problem requiring the separation of an integral into two parts could find a value which optimises a function interpreted what the question required, formulated an appropriate calculus process and logically and systematically presented pertinent answers. Additional Although Question 3 had a misprint with the labeling of the axes, very few students appear to have picked up this error and there was no apparent evidence of any candidates being disadvantaged by it. Strategies were put in place to ensure no student would be disadvantaged by this error, however they were rarely required. In fact many candidates answered Question 3 better than other questions in the examination. Use coordinate geometry methods (90287) used an appropriate formula to find the equation of a line understood parallel and perpendicular gradients used integers. did not understand what a midpoint was were not able to substitute correctly into a formula did not choose the appropriate formula for the given situation gave midpoints that were outside the range of the two starting points. were confident when using algebra to solve linear equations understood how to set up multiple equations using perpendicular gradients. Solve straightforward problems involving arithmetic and geometric sequences (90290) identified the sequence type (arithmetic or geometric) from the context could tell when a term, a sum of terms or the sum to infinity was required

5 NCEA Level 2 (Mathematics) 2006 page 5 identified the common ratio r from a contextual situation recursively listed terms up to the required one and summed these if required or identified the appropriate variables and substituted them into the correct formula correctly performed calculations involving several steps, brackets and order of operations. incorrectly identified the particular type of solution required calculated incorrectly eg 750/ (1-0.85) incorrectly calculated as 750/ = set up and solved appropriate equations and simultaneous equations from the context given and found appropriate variables to use for the solution to the problem found the final investment value in a compound interest situation applied a range of knowledge and skills to an original situation to find a solution to the problem. Solving Trigonometric Equations (90292) solved basic trigonometric equations producing two consecutive solutions handled multiple solutions in a given domain understood the symmetry of the trig graphs in obtaining a solution understood when to use the inverse trig function on the calculator, and rearranged the equation first if required dealt with the negative angle that the calculator gave, or could use the positive acute angle correctly to find the solutions when the trig function had a negative value. used the unit circle and graphical methods correctly but the effective use of the general formulae was less common worked to more than one decimal place if working in radians. Candidates who did not achieve this standard lacked some or all of the skills and could not handle multiple solutions in a given domain so only gave one solution mixed radians and degrees when solving equations eg found the first solution correctly in radians and then found the next solution by doing ANS attempted to give answers from four quadrants to cover bases put π (pi) at the end of radian answers eg.66π ignored the negative angle and so proceeded to solve a different problem gave answers such as sin -1 (0.8) = = x + 1 so x = used the general formulae incorrectly or confused which one to use gave wrong solutions or extra wrong solutions within one cycle were careless when copying answers from their calculators, or when making calculations, eg added instead of subtracted or gave 26.6 or as an answer when the correct answer required was 26.6 and and no further working was given could not relate trig graphs and trig equations. understood what more than 2m meant, and did not use or 2.1 could work in degrees and change their answer to radians so that the resulting answer was appropriate for the situation eg they did not have the girl on the swing at 2m for over 90 seconds in one motion

6 NCEA Level 2 (Mathematics) 2006 page 6 could solve the equation with 2x as the argument (rather than dividing 4 by 2), and knew how to find the 4 solutions could solve equations with something other than x as the argument understood that whole number answers only were not sufficient and showed working to support their answers used appropriate rounding in the intermediate steps and in their final answers gave all solutions to problems where there was more than one answer accounted for all signs throughout their working understood amplitude and other properties of trig graphs could relate answers to the context of questions making sure they were accurate eg after getting A = 28 and B = 128 could see it should be 28 after looking at the graph or did not give times greater than 1 hour which should have indicated that there were problems with A and B answers.