Reflections on Higher School Certificate examinations
Learning from their mistakes, HSC 1998

Robert Yen, Hurlstone Agricultural High School

This article summarizes common errors made by students in the 1998 HSC Mathematics examinations. Every year, the HSC examiners (markers) publish a detailed report on the performances of students in the previous year’s exams. Their findings are also reported at MANSW’s annual Examiners’ Day in March at Macquarie University (open to all). The report and seminars review all exam questions, explaining their rationale and marking schemes, showing performance statistics and sample student solutions. This article extracts the common themes from the exam report with the aim of helping this year’s Maths students and teachers prepare for the HSC. Another valuable guide is ‘Ten common mistakes made by HSC Maths students,’ published in Reflections 20(4), 1995 and based on a decade of examiners’ reports.

General Errors

Top 10 errors made by HSC Maths students

2/3 Unit errors

Maths in Society/Maths in Practice errors

10 hot tips for tackling the exam

General errors

1. Not showing enough working

Insufficient working shown by students is a perennial problem for HSC markers. An explanation of the marking process should illustrate the rationale for showing working. Examiners are always trying to award marks to students, and each progressive mark in a question is granted when the student has passed a particular stage in the solution successfully. There are no ‘half-marks’ awarded, but once a mark has been scored it cannot be taken off.

For example, Question 1 (f) of the 3 Unit exam was a
4-mark question: Evaluate . The breakdown of the marks was: rewriting into a form that could be integrated, finding the integral, substituting into the integral, and finally evaluating the answer. However, students often skipped steps in their working and when their final answers were wrong, it was difficult to award the part marks because there was no evidence that the required steps had been carried out. A 4-mark question should have at least four lines of working.

For these reasons, it is especially important that enough working is presented in ‘show that’ questions, where the answer is actually given in the question. In Question 8(b)(i) of the 2 Unit exam, students had to ‘show that the volume of the solid is given by ’. This was a 3-mark question, but many students did not score the third mark because they went straight from or to without justification. Students should make sure that they don’t quote the required answer too early in their solutions.

And to students who prefer to write single, bald answers without any working, be warned:

As in previous years, examiners recommend that working be spread out and they advise against a popular practice of ruling a page into two columns. Handwriting and diagrams should be large and legible. All working to a question, including rough working, should be written in the same answer booklet, but students who forget to do this will not be penalized. Because examiners are always looking to award marks, students should not obliterate incorrect or rough working by erasing it, scribbling over it, or using correction fluid. Instead, a neat line crossed through incorrect work will sometimes help examiners understand the student’s reasoning and approach, so even crossed-out work can earn some marks. If they wish, students may write their rough working on the unruled left side of the page.

2. Not reading or answering the question

As in previous years, students often neglected to read the key words of a question and failed to answer the question, either answering only one part of the question, answering more than what was asked, or answering a different but often-practised type of question.

The very first question of the 2 Unit paper, ‘Express as a recurring decimal’, a simple one-mark problem, was deemed too easy by the many students who wasted time and went on to prove that as well. In the coordinate geometry question (Question 3), students often proved the same equation of a line twice by applying to two different points on the line, and when asked to find the value of tan q in the same question, some found the value of q to the nearest minute also.

With double-barrelled questions, students often answered the first part only, forgetting the second, for example, ‘State why triangle BCD is isosceles, and hence find CBD’ (Question 5); ‘Find the value of x for which has a point of inflection and determine where the graph is concave upwards’ (Question 6). Also, in maximization or curve sketching problems, students often found the correct value of x, but forgot to substitute to find y. In Question 8, many students found t, when the maximum rate of flow of sand occurred, but did not substitute to evaluate R, what this maximum flow actually was.

3. Reading or drawing scales on a graph

The MIS and 2 Unit papers both had questions involving graphs in which both axes had different scales, and this caused problems. In Question 23 of the MIS paper, students plotted a curve from a table of values on 2 mm grid paper and then read off values and calculated the gradient of a tangent. This was difficult for them as 1 cm represented 0.25 on the horizontal axis but 0.1 on the vertical axis. More learning time needs to be spent on reading the scales of a graph. In Question 6 (b) of the 2 Unit paper, students graphed but because its range was y ¾ 1.2, many had trouble expanding the scale on the y-axis, and as the y-values did not vary much, there were many compressed-looking graphs.

4. Not looking for the simplest approach

Often when solving a mathematical problem, HSC students don’t stop to think and consider the range of strategies and tools available to them, instead launching into some routine or formula which they have identified to ‘match’ with the problem. A lot of wasted working-out time and effort can be eliminated in an exam if the student spends a moment identifying the most effective method of answering a question. Students should use the number of marks allocated to a question as a guide to how much working is required.

One recurring theme to emerge from the MIS paper was that some students had now become so accustomed to using the sine and cosine rules that they often used them unnecessarily on right-angled triangles where the elementary sine, cosine and tangent ratios would suffice. Question 22 (a) required students to find the angle of inclination of a kite, and then its height above the ground. Although the sine and cosine rules still worked here, it would have been more time consuming and prone to error. Similarly, in calculating the pitch angle of a roof in the Maths in Construction option (Question 28), many students did not use the tangent ratio, instead complicating things by using Pythagoras’ theorem to find the hypotenuse and then using the sine or cosine (or even worse, the sine or cosine rules) to find the missing angle.

In dealing with probability problems, there are still many students who don’t realize that the probability of ‘at least one’ occurring is the complement of ‘none’ occurring. It is usually easier to calculate the probability of ‘none’ and subtract it from 1, than to add the individual probabilities of 1, 2, 3, etc., occurring. This happened in Question 21 (a) of the MIS paper and Question 5 (c) of the 3 Unit paper.

4 Unit students were not exempt from using long-winded approaches. Question 1 (c) of their paper was the integral . This could be solved easily using substitution or trigonometric identities, but there were those who persisted with the cumbersome t-formulas and failed.

5. Inability to answer ‘Explain why’ questions

An ‘explain why’ question is usually worth one or two marks and requires a simply worded answer. It is usually found in a geometry or trigonometry question. The answer is usually in the form of one or two short sentences.

Question 4 (c) of the 3 Unit paper involved circle geometry:

‘(i) State a reason for (Sonya’s) correct statement that EDCB is a cyclic quadrilateral.

(ii) State a reason why she should then correctly conclude that AED = ACB.’

This type of question still confounded many students, who:

In the Land and Time Measurement question of the MIS paper (Question 26) students used traverse and radial surveys to calculate the same length of a playground, and then were asked:

‘(v) Explain why the distances calculated in parts (ii) and (iv) are slightly different.’

However,

Students need more practice in interpreting results and writing about their mathematics, especially in keeping their explanations short. Perhaps there are not enough ‘Explain why’ questions in textbooks, and students should study past HSC papers for further examples.

Top 10 errors made by HSC Maths students

A more general list from a decade of HSC exams

1. Errors in basic algebra
2. Not showing working
3. Using liquid paper
4. Not drawing a diagram
5. Rounding off too early
6. Not answering the question asked
7. Poor reasoning and proofs
8. Errors in differentiating and integrating
9. Not noticing key words
10. Errors in calculator use.

2/3 Unit errors

1. Careless algebraic errors

Students continued to make silly mistakes in their algebra work, with the most common being of the form In Question 3(b) of the 3 Unit paper, many wrote the answer to as . Students had difficulty making r the subject of the equation in Question 7(c) of the 2 Unit paper, and possibly too skilled in using the quadratic formula, many had trouble solving the simple quadratic equation in Question 5(b) of the 3 Unit paper.

Students made errors in solving inequalities in Question 7(a) of the 2 Unit paper, giving the answer to as or k ¾ 3, and when simplifying a trigonometric expression in Question 5(a), there were these sloppy algebraic manipulations:

Problems involving incorrect formulas continued to prevail. In the 3 Unit paper, there were too many alternatives for in Question 2(b), while in Question 1(b), often had its + and - signs interchanged.

2. Differentiation and integration

Common errors in the calculus work were differentiating instead of integrating (Questions 1(d), 2(b)), not using the table of standard integrals to integrate (Question 2(b)), and forgetting the constant of integration (Question 8(a)(iv)). Some students were unaware that ‘primitive’ meant integral in Question 1(d). 3 Unit students had trouble evaluating the limit in Question 1(c), not knowing how to create the appropriate ‘factor’ with the 3 and 5.

When differentiating and integrating, students who set out their working and listed separately the expressions for u and v in the product rule, quotient rule (Question 7(c)(ii)), and integration by parts (Question 1(b), 4U) generally did better than students who didn’t. In differentiating x sin(x + 1) using the product rule in Question 2 (a)(ii), many showed their poor understanding of trigonometric functions by writing u = x sin, v = x + 1.

Question 4(a) involved approximating a definite integral by using both the trapezoidal and Simpson’s rules. Mistakes included substituting x- instead of y-values, confusion between ‘odd’ and ‘even’ y-values, using instead of and vice versa, misnaming the rules, and an inability to find h (width of strip). Students who used the more complicated expression for h often used n = 3 instead of n = 2. As mentioned in the last two years, students who used a ‘table of values’ approach with the required weights listed underneath experienced more success in their calculations.

3. Curve sketching

Some students still need to practise their ‘applications of the derivative’ skills. This was evident in Question 6 (b) where students had to graph . While they had few problems in finding a stationary point and determining its nature, it was the work involving concepts outside the ‘curve-sketching recipe’ that caused problems, namely determining where the graph was increasing and where it was concave up.

Judging by the answers to Question 4 (d), many students had not learnt the equation of a parabola in the form so they expanded instead. Some didn’t know the meaning of vertex, focus and directrix and their relation to the parabola, and some used differentiation unnecessarily to find the vertex.

In Question 4(b) of the 3 Unit paper, many did not understand the meaning of the words horizontal, vertical or asymptote when discussing the graph of the function

4. Recipe learning

2/3 Unit students are often guilty of rote learning a mathematical recipe or formula without really understanding the theory behind it, becoming unhinged when an atypical question arises in the HSC exam. It is important that students know exactly which formulas can be used in an exam without proof and which formulas need not be memorized because they are to be derived.

Question 3(b) of the 3 Unit paper involved the solving of a trigonometric equation using the auxiliary angle method.

Despite the persistence of some misleading study guides, formulas for superannuation and loan repayments should not be memorized or used, since every financial application of series is different, and any such formula used in an exam should be proved. Only half of those students who quoted a superannuation formula in Question 2(b) of the 3 Unit paper got it correct.

Students who follow a ‘curve-sketching recipe’ often become inflexible in their approach, especially when determining the nature of stationary points. Sometimes the second derivative test is too complicated to perform and it is easier to test the first derivative on either side of the stationary point. This was the case last year and certainly was true in Question 7(c) this year where the function had to be maximized.

Question 3(b) of the 3 Unit exam also asked for the general solution to a trigonometric equation. Many students memorized a formula for the general solution but had trouble adapting this formula to this question. The exam report recommends that students list a number of consecutive solution values first, then try to establish the pattern and write the general formula. The correct answer was , for any integer n. A similar approach is prescribed for Question 5(a) in the same paper, where a series involving cubed numbers needed to be written in sigma notation.

5. Logarithms and exponents

Properties of logarithms and exponentials featured more often than usual in the 2 Unit paper, and students demonstrated that they still had a fairly superficial understanding of them.

In Question 6(b), students had to graph , so in determining stationary points and points of inflection, the equation came up often, causing a large number of students to conclude the incorrect answer x = 0. Students often didn’t realize that is never zero, or that it is always positive. Question 9(a) was the equation . Again, students became confused and claimed that or even ‘divided’ both sides of the equation by ln to get 7x &emdash; 12 = 2x. Question 1(d) of the 3 Unit paper was ‘Given that find ,’ and many claimed that

Question 8(b) of the 2 Unit exam was an area of integration involving the function and students needed to show that . They couldn’t.

Question 10 was a difficult application of series: a loan repayments question applied to the harvesting of fish. At one stage, the equation reduced to Many could not proceed past this point and use logarithms to solve for n. Even a trial-and-error approach would have been a viable alternative in these circumstances.

Like Question 7(c) of last year’s 2 Unit paper, Question 3(c) of this year’s 4 Unit paper was about exponential decay. Students don’t need to prove that is the solution to by integrating (it is beyond the scope of the course), only show that it satisfies the differential equation by differentiating.

6. Poor setting out of proofs

Question 3(a) was ‘Show that A(1, 0) and C(&emdash;1, 6) lie on the line 3x + y = 3.’ This simply required the substitution of both points into the equation, but some students actually proved the equation using the two points. Students who did this still earned the 2 marks, but probably lost some time. However, even those students who substituted correctly did not set out their proofs clearly, writing vague statements such as 3 + 0 = 3 and &emdash;3 + 6 = 0 rather than taking the LHS =     , RHS =      approach. The examiners looked for evidence of correct substitution to award marks.

Question 5(a) (ii) was a congruent triangle proof. Some students still have difficulty in the format of this type of proof, neglecting to give reasons for each step or inventing new congruency tests like ASS or AAA.

These errors also occurred with the similar triangles proof in Question 9(c). Students need more practice of geometry proofs.

Question 4(c) of the 3 Unit paper has already been mentioned, involving circle geometry and two ‘explain why’ questions. Many students failed here because they actually assumed results to be proved in their proofs, thus presenting a circular argument.

7. The calculus of motion

Possibly because it’s the last topic of the 2 Unit course and thus less often revised, many students had trouble interpreting results from physical applications of calculus. Question 6(a) was the interpretation of a displacement&emdash;time graph, and required understanding of how velocity and acceleration were related to the stationary points, points of inflection, gradients and concavity of the graph.

Many 3 Unit students did not know the chain rule as it is applied to rates of change in Question 4(a) (ii) of their paper. Question 6 involved (a) projectile motion beginning at the point (0, 1) and (b) velocity as a function of x (displacement).

Students had difficulty handling Question (b) (ii) ‘Will the particle ever return to the origin?’ because they could not interpret the meaning of negative velocities and accelerations.

Maths in Society/Maths in Practice errors

1. Language and terminology

Just as in the proposed 2 Unit General Mathematics course, terms and definitions make up a significant portion of the MIS and MIP courses, especially in the option topics. Considerable class time should be spent on learning the vocabulary as well as the mathematics of each application or theme. As in previous years, many students lost marks from not knowing or understanding simple facts and terms, for example:

Students and teachers should refer to the syllabus for definitions and clarifications and see what is covered by the MIS and MIP courses and what is therefore examinable, because a textbook does not necessarily cover everything. A student-generated glossary would be a valuable study resource.

2. Conversions of units and rates

Mixed units in measurement again caused confusion for MIS and MIP students, especially with units of length (commonly found in area, volume and Maths in Construction problems) and time (found in Space Maths and Land and Time Measurement problems). Some MIP students claimed that 1 year = 48 weeks (12x 4?) when calculating loan repayments (Question 31), while MIS students made many errors in subtracting times of day on a timesheet for a payroll calculation in the Personal Finance question (Question 27(b)).

In the Land and Time Measurement question (Question 26(b) (iv)), a significant number of MIS students could not convert from decimal hours to hours and minutes and interpreted a time difference of 7.73 hours as 7 hours 73 min. They should be reminded that the degrees&emdash;minutes&emdash;seconds key on a calculator can be useful for this conversion. MIS students often have trouble converting from seconds to years, and again this was the case in the Space Maths question (Question 24(a)), when they had to calculate the distance of 2.3 light years, given the speed of light.

Rate conversions usually feature prominently in MIS exams, but this year even the MIP paper had both fuel consumption and petrol cost rates in the same problem (Question 32(b)), asking students to calculate the cost of petrol required to travel between Sydney and Brisbane, given their distances on a map. This was a hard task.

Students also need more practice working on speed problems, for many still forget the relationship between speed, distance and time, which was required to find the speed of a satellite in the Space Maths question (Question 24(b)). In part (c), many simply quoted the formula for the distance between Mars and Earth, not realizing that the signal from Mars to Earth only travelled one way.

As with last year, MIS students had problems with Maths in Construction (Question 28) in converting calculated lengths in scale plans from centimetres to metres. Those who left the measurements in centimetres made things harder for themselves later when they had to calculate areas in square metres. All measurements should be expressed in metres, otherwise the conversion from square centimetres to square metres is too prone to error. In the same question, students were asked to calculate the cost of carpeting two rooms. Many didn’t realize that the carpet was sold by the metre and not by the square metre.

3. Not using a ruler or geometrical instruments

As in previous years, failure to bring necessary equipment to the exam disadvantaged MIP and MIS students the most. Students needed a ruler to complete a tessellation (MIP Q34(a)) and sketch a diagram of a playground from a traverse survey (MIS Q26(c)). Compasses were needed to complete a design in MIP Q34(c).

4. Algebra and graphs

As in the past two years, many MIS students could not solve equations with the variable in the denominator:

And again, the graphing question (Question 23(a)) assessed understanding of tangents and the gradient as a measure of rate of change, both often-neglected parts of the syllabus. Both were poorly done.

5. Trigonometry

There were two MIS questions involving the cosine rule, Questions 22(c) and 26(c). Both were generally well done, with common errors being miscalculations, incorrect order of operations (for example, ), rounding off too early and forgetting to take the square root at the end.

When applying trigonometry to find the diameter of Mars in the Space Maths question 24(c), one common error was to forget to halve the angle 0.0026° when constructing the right-angled triangle.

As in previous years, students had trouble with compass bearings (Question 22(d)). They could not draw the appropriate diagram, thus making it impossible for them to apply the sine rule in the second part of the question. More class time needs to be spent on students practising the construction of diagrams from worded descriptions in trigonometry problems.

6. Probability

As mentioned earlier, MIS students attempting the Chance and Gambling option did not demonstrate understanding of the language of chance. In the core section, Question 21 had two sections on probability. A significant number of students still have trouble with tree diagrams: in drawing them, forgetting to list probabilities on branches, listing wrong probabilities, not drawing enough branches, making silly calculation mistakes in adding and multiplying fractions, and incorrectly using the product and addition rules.

7. Statistics

Students in MIP and MIS still have trouble finding the median of a distribution, whether from a list of home prices (MIP Question 33(a)) or from a frequency table (MIS Question 23(b)). One common mistake was to forget to put the home prices in order first.

Many MIS students can calculate standard deviation easily, but don’t know what it means, and so had trouble answering Question 23(b) (vi) regarding how eight new scores would affect the standard deviation of the distribution.

References

10 hot tips for tackling the exam

1. Find out all you can about the format of the exam.

• Time allowed, number of questions, marks and average time spent per question.
• Number of parts, topics tested.
• Do past HSC papers to be familiar with the format and standard.

2. Be prepared.

• Revise, revise, revise!
• Bring pens, pencil, calculator (check calculator works!) your brain (ditto!).
• Eat and sleep well, be early, be confident, be a little nervous.

3. Use the reading time to plan your exam.

• Read all instructions carefully.
• Skim through all questions to see the work that is ahead of you.
• Mark the difficult questions (*) which will require more time - plan your time!

4. Spend the first minute of each question planning and thinking.

• You don’t need to be writing all of the time. (What you’re writing may be wrong and a waste of time!)
• Read each question carefully and decide what needs to be found.
• Make sure you use all the information given.

5. Pace yourself - keep an eye on the time.

• Work steadily - make sure you are not spending too much time on one question.
• Don’t rush or you’ll make silly mistakes, and your work will be messy.
• Don’t panic if you run out of time - it is better to get most questions right than to get all questions wrong. Complete the work you do know, rather than rushing.

6. Write clearly, draw big diagrams.

• Show working, spread out your work neatly, use as much paper as you like.
• Demonstrate to the marker that you know your Maths.
• Write down the page, not across. Use words and diagrams if appropriate.
• Don’t use liquid paper. Draw a line through mistakes. Use pencil only for diagrams.

7. Make sure you have answered the question.

• Does it sound reasonable? Correct units included? Correct no. of decimal places?
• Highlight the final answer in a box. Should you write it in a sentence?
• Easier questions are at the front. Do them first to boost your confidence.
• Feel confident about yourself when you have answered a question correctly.

8. Attempt every question.

• Aim to earn some marks for every question, even if it requires an educated guess.
• Try to finish each question before moving on, so that you don’t have to worry about coming back to it.
• If a question is too hard, skip it and leave time to come back to it later.

9. Move on if you’re getting nowhere.

• If your working-out of a hard question is taking too long, then it’s probably wrong!
• If you’re stuck, don’t waste valuable time getting bogged down. Stop, retrace your steps, think about a simpler method, or start again. Sometimes it’s even better to skip the question and return to it with a fresh mind.

10. At the end of the exam, check your work, and go back and attempt harder or uncertain questions.