2017 Specialist Mathematics - Subject Assessment Advice PDF

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fSpesh maths work realted course work detailing the study design and general expectations of the entirety of the course....

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Specialist Mathematics Subject Assessment Advice Overview Subject assessment advice, based on the previous year’s assessment cycle, gives an overview of how students performed in their school and external assessments in relation to the learning requirements, assessment design criteria, and performance standards set out in the relevant subject outline. They provide information and advice regarding the assessment types, the application of the performance standards in school and external assessments, and the quality of student performance. Teachers should refer to the subject outline for specifications on content and learning requirements, and to the subject operational information for operational matters and key dates.

School Assessment Assessment Type 1: Skills and Applications Tasks (50%) Students complete six skills and applications tasks, completed under direct supervision of the teacher. The equivalent of one skills and applications task must be undertaken without the use of either a calculator or notes. Students provide evidence of their learning in relation to the following assessment design criteria: •

concepts and techniques

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reasoning and communication.

Teachers can assist more successful responses by: •

ensuring there are both routine questions, and questions that include enough complexity to allow achievement at higher grade bands

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not using examinations (e.g. a mid-year examination) as one of the six skills and applications tasks

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ensuring the subject outline is used to guide the content taught and assessed. Content from texts that is not specified in the subject outline should not be assessed in skills and applications tasks

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including marks and teacher comments to provide students with appropriate feedback for improvement

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ensuring conjecture and proof is covered through this assessment type. It can be difficult to independently assess conjecture and proof through a mathematical investigation, depending on the choice of investigation task.

The more successful responses commonly: •

showed all algebraic working giving all relevant steps required, particularly for “show” type questions

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stated any theorems, properties etc. that were being applied

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used mathematically correct notation e.g. vector notation and integration

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labelled axes of graphs correctly and used a graphics calculator efficiently to draw both Cartesian and parametric functions, paying attention to properties of the functions such as asymptotes and shape

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paid close attention to all details given in questions and provided the required detail in the solution/response.

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Page of Stage 2 Specialist Mathematics – 2017 Subject Assessment Advice Ref: A699629 © SACE Board of South Australia 2018

The less successful responses: •

often didn’t attempt questions

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made many arithmetic and algebraic mistakes that prevented the student from generally “nice” working and through errors made the problems more complicated e.g. polynomials that didn’t factorise as planned

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lacked the appropriate detail and gave one statement for questions in which several marks were awarded and clearly more working (or a more thorough response) was required

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used incorrect notation and didn’t know algorithms required for solving problems

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seemed unfamiliar with the capability of graphics calculators and therefore didn’t always use efficient methods to solve problems using electronic technology.

Assessment Type 2: Mathematical Investigation (20%) The subject of the investigation may be derived from one or more subtopics. Teachers should provide minimal direction and tasks must allow the opportunity for students to extend the investigation in an openended context. Students are encouraged to use a variety of mathematical and other software to enhance their investigation. It must be completed in a report format and must be no longer than 15 A4 pages with minimum font size 10. Appendices may be used to support the report but are not part of the assessment decision unless they are part of the 15 pages. Teachers should provide feedback where appropriate on the suitability of the direction a student may take with their investigation, especially where the investigation topic was chosen by the student. Students provide evidence of their learning in relation to the following assessment design criteria: •

concepts and techniques

•

reasoning and communication.

Teachers can assist more successful responses by: •

ensuring the format of the investigation allows for an open-ended context (tasks that are question and answer style disadvantage students)

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encouraging correct use of notation and labelling of graphs etc.

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assisting students in the use of unfamiliar software so students can represent graphs etc. with appropriate information attached

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providing feedback through drafting/discussing the direction taken to ensure what they do will provide them with the opportunity to reach the highest standards

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explaining clearly the 15-page limit and the appropriate use of appendices. Appendices should not include one-off mathematical calculations, but rather repetitious calculations with an example of the calculation in the main body of work. Repetitive calculation solutions should be recorded in tables within the main body of the task.

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not using investigations that have published solutions to ensure authenticity of student work

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providing one opportunity for students to submit their response for drafting, and providing feedback to guide the students about areas for improvement and/or further development in their response.

The more successful responses commonly: •

provided detailed information about the investigation and the context in the real world

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included detailed explanations of all algebra and graphical work produced

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included graphical representations appropriately labelled to enhance the discussion within their investigation

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successfully developed a modelling situation with clear explanations provided for all decisions made

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demonstrated understanding of the reasonableness of their mathematical results and the limitations of their modelling process

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included use of appropriate mathematical software to enhance the quality of the investigation

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provided evidence of appropriate mathematical notation, representations and terminology.

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Page of Stage 2 Specialist Mathematics – 2017 Subject Assessment Advice Ref: A699629 © SACE Board of South Australia 2018

•

communicated mathematical ideas effectively and provided reasoning to develop logical arguments

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made good use of appendices for repeated algebraic calculations to arrive at results.

The less successful responses commonly: •

gave little insight into what the investigation was about with a limited introduction

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failed to explain their reasoning for decisions made

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made poor use of notation and often didn’t explain what graphs represented

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provided insufficient or non-existent labelling

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completed the mathematical investigations guided by the early direction given in the task, however did not attempt the open-ended part of the investigation.

Assessment Type 3: Examination (30%) The new content in this course was approached well and indeed some of the new styles of questions were very well done. Candidates did not appear to run out of time and the new style of exam, presented in two booklets, did not seem to cause any problems for students. A reminder to advise students that if extra space is required to respond to a question, and the spare page is used in the booklet, they need to label their answers clearly and only use the page in a booklet that is relevant to that section of questions. A few general comments follow regarding styles of answers and areas that require improvement: •

‘Show that’ style questions require working to be shown no matter how obvious some steps may appear.

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An ‘exact answer’ requires the answer to be in rational or irrational form without approximations to decimal values.

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Knowledge of, and the use of, a graphics calculator is assumed. Parametric graphs or graphs of functions using the graphics calculator were not always handled well by students.

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Poor notation often appeared in student responses. The most common examples of poor notation were seen in vector and integration problems.

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Students should recognise that earlier parts of questions are relevant to the following parts within a question.

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Students should always ensure that they check all pages in each examination booklet to ensure they have not missed any questions, or parts of questions. Instructions are provided on each page such as ‘Turn over’ and “Question 9 begins on page 22” to support students to find and respond to all questions.

Specific comments pertaining to the questions in both booklets follow: Booklet One Question 1 Most students achieved above half marks for this question. Part (a) caused most difficulty when students attempted using

1 1 sin2 t= − cos 2 t rather than recognising the simpler method. Part (b) was 2 2

attempted well and even if the part (a) answer was incorrect students followed through. Only some students did not attempt the second part of this question. Question 2 Throughout this question the responses were fair with most students scoring around half marks. Although the form required for a response was given in part (a) some students chose to write their responses in the form

R(x ) S( x ) =Q ( x ) + D (x ) D (x )

and some also guessed a Q(x). The second part of (a) saw varied

responses with some students using the answer S(-2)=-1 from the remainder theorem in their work for part (b). Part (b) was attempted well by some students but some failed to link the earlier work. Question 3 This question saw a lot of students not using vector notation, especially in part (b)(i). This poor notation lead to some poor marks. Those who wrote ( p+r ) (r− p ) =r 2− p2 have not used vector notation nor

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Page of Stage 2 Specialist Mathematics – 2017 Subject Assessment Advice Ref: A699629 © SACE Board of South Australia 2018

have they understood vector algebra. Students were expected to show that for instance

and . The last part of the question was completed quite well with students’ recognising the sides of a rhombus are equal in length. Question 4 Responses to the first part of this question provided examples of poor use of the graphics calculator as well as poor attention to detail when drawing the curve. Part (b) was well done, but part (c) saw some students attempting the problem without using implicit differentiation as directed. Given that implicit differentiation is now part of this course it is important students recognise the times to use it. The last part was well done and overall most students achieved well in this question. Question 5 The formal proof by the Principle of Mathematical Induction is back in the course. It is important to note that the proposition, P(n), must be defined before a solution can venture into considering P(1) or P(k) and P(k+1). Together with the definition of P(n) the statement that “since P(1) is true and P(k) implies P(k+1) is true then by PMI P(n) is true for n > 0” was awarded a mark (provided there was relevant working within the proof). Students should acknowledge the use of the assumed P(k) in their algebra and be aware that in this particular problem P(k) is not just the last term of the sum. This incorrect step often halted progress in finding P(k+1) correctly. Many students approached this first part of the question very well. In part (b) many students did not realise that the first two terms had to be subtracted from P(10). Question 6 This question scored very highly with the cohort of students. The first part of this question is an example where students need to show all working to gain the marks allocated given that it is a ‘show that’ style of question. Students responded very successfully to the second part of this question, and the final part of the question was done reasonably well however some candidates did not adhere to the requirement of four significant figures in their answer. Question 7 This question was challenging for most students. In part (a) many recognised that r = 15 but not many answered the requirement for the equation to describe the set of points on the circumference of the circle in terms of z. In part (b)(i) many saw that the length of w was 20 but not many explained their reasoning well. The rest of part (b) was poorly attempted with only a few able to connect previous work in their responses. Question 8 Many students achieved higher than 50% for this question. Given that this topic is a new inclusion in the course students did attempt it well. The graphing of the functions needed more care at times. The major error being in (b)(iii) - not recognising the vertical asymptotes and also showing the regions where y = 0 when x > 2 and x < -3. Most students did not recognise that the minimum of the region between -3 and 2 for x was higher than in the previous graphs. Part (c) was not well done by the majority, with many trying to integrate a modulus function instead of recognising the doubled area. Also, many students did not find the exact value as required. This last part of the question is where most students lost marks. Question 9 Many students scored full or high marks for this question. In this question students used techniques new to the course. In part (a) some chose u and v’ wisely and produced appropriate responses. In part (b) some students did not square the y and some omitted pi, but on the whole made a good attempt. Again, a reminder that it is not correct to approximate an answer if an exact answer is required. Throughout this question integral notation was poor; often the “dx” was left out of the integral expressions. Question 10 Most students scored around half marks for this question with only a low number scoring full marks. Attention to detail is needed with many students not answering part (a) (i) - an easy mark to gain. Part (a)(ii) was well done but the trigonometric algebra within part (b)(i) caused some issues. Students should realise it is a good habit to state when an identity is used in a solution. Recognising the triangle inequality for part (b)(ii) was well done but in part (c)(i) students not only needed to state the vectors are parallel but that they are also in the same direction. Different methods of solution appeared for part (c)(ii) resulting in the correct value for theta. Some verified the final result in part (c)(iii).

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Page of Stage 2 Specialist Mathematics – 2017 Subject Assessment Advice Ref: A699629 © SACE Board of South Australia 2018

Booklet Two Question 11 The majority of students achieved half marks for this question. Some of the major errors were in graphing, simple algebra and correct use of the calculator. In part (a)(i) many students do not have the entire graph drawn due to not setting the parameter on their calculators correctly. The three values of t for part (a)(ii) were -1, 0 and 1. Many students did not solve t2 =1 correctly to obtain the negative result. In part (a) (iii) students needed to be aware of the functionality of their calculator to evaluate this integral correctly. In part (b) again many students did not solve simple algebra for t correctly to find all solutions. The ‘show that’ requirement in part (b)(ii)(1) was not done well by students: many not substituting t=± √ x correctly into the equation for y. The integral calculation for part (b)(ii)(2) was well done but marred by some students changing their answer to be positive. Some candidates realised part(b)(ii)(3) is just double the positive answer from the previous part. Question 12 This vectors question was well attempted by the students. Many obtained full or close to full marks. The first part of the question finding the augmented matrix was well done, but unfortunately many arithmetic errors marred progress in the second part of part (a). Again, students are reminded to show steps of working. Some students chose to substitute the coordinates of the given points into all planes for part (b)(i) but some approached the problem more efficiently finding t = 0 or t = -1. Part (b)(ii) was very well done finding the dot product of the normals to the planes to be zero. Although most students used t as the parameter again in part (c)(i) the question was well done. Some found the coordinates of E well, whereas some used the incorrect plane. Finding the distance from E to the plane P1 was not treated well, with many students not using the formula correctly. The last part of the question had some good responses with sound reasoning, although careless use of formulae did let some students down. Question 13 Students struggled with this question in general; many who did attempt it did not use polar form correctly. It was evidence that De Moivre’s Theorem was not well known and drawing solutions on the Argand diagram was lacking appropriate detail. Part (a)(i) was poorly done with many candidates allowing r = -1 as part of their polar form for solutions. Drawing the solutions on the Argand diagram requires position as well as length detail. In this case a scale was required on the diagram to indicate length. The next three parts required students to answer ‘Show that’ style questions. All working and logic must be shown to be able to gain marks for these questions. Many students did not indicate the reasoning for 2 π /5 to be argument of sine in part (a)(v). This occurred later in part ((c)(i) as well. Students answered the sections of part (b) reasonably well. Question 14 This question was popular and approached well by students. Students drew the solution on the slope field quite well, but attention to the initial condition, not crossing over the limiting value of 50 and keeping the solution line from crossing slope field lines is required. The ‘Show that’ in part (a)(ii) was well done but the solving the differential equation in part (a)(iii) needed some care with moduli signs and clearly showing that c=49. Some students used a different variable symbol rather than using the symbol (A) in the question. Part (b)(i) was well done with most students seeing B=25 when d=0.874. In part(b)(ii) students had to realise the time t is now t = 1+0.84 and hence A = 2.48. The last part of the question required some statement indicating that bacteria B would dominate. Question 15 This question was well done even though it was the final question of the paper, as well as being based on new content in the course. The majority of students gained more than half marks, although the last part of the question was quite challenging. The first part of the question was well done although in part (a)(ii) it is advisable to show reasoning for no other roots. The best responses came from those who chose to factorise the function and show that the only roots other than x = 1 were complex, whereas some showed the discriminant of the quadratic factor is negative and hence has no real roots. Finding the composite function in part ...