Learn from your mistakes to succeed in HSC maths

Jack Coleman, who topped 2 unit advanced in the 2020 HSC.

For Jack Coleman, who came first in Mathematics Advanced in 2020, the best way to study is to revisit and learn from past mistakes.

When Jack made errors in his maths class, he saw it as an opportunity.

“If you’re getting something wrong, you need to address it by further studying – especially if it happens more than once,” he says.

So Jack recorded mistakes in a book. This way he could see common errors at a glance. These included forgetting the constant of integration or not writing the limit when using first principles.

‘If you’re getting something wrong, you need to address it by further studying – especially if it happens more than once.’

Thinking back, perhaps his biggest mistake in the lead-up to the HSC exams was missing a lesson on calculating loans and annuities using tables. Jack knew how to make these calculations longhand and thought that would be enough.

“As the exam approached, I had no clue how to attempt these problems and struggled to make sense of them,” he says.

It took a lot of extra study to get up to speed, but Jack still didn’t feel confident in tackling these questions in the exam. He checked his responses using the longhand method.

How to study with past papers

Here’s how Andrew Malcolm, who was first in Mathematics Standard 2, used past papers to study for his HSC exam.

Andrew likens exam preparation to physical training: “You can’t develop muscles from watching gym tutorials online,” he says. “You’ve got to actually put into practice the methods you’re taught.”

That’s why he downloaded past HSC exam papers from the NESA website and completed them under timed conditions.

But after repeatedly making careless mistakes, Andrew developed a checklist to review his answers:

Top tips from maths teachers

Maths teacher Michael Murton.

Michael Murton, Bowral High School
Member of NSW Government Best in Class unit

Getting to the right answer – a guide

To solve an HSC exam problem, look carefully at the information in front of you. Everything you need is there.

Check how many marks the question is worth. Use the marks allocated to the question to guide the number of steps in your solution.

Look for key words and phrases to help direct you. In this case, the decagon is “regular” and its perimeter is 80 centimetres. This allows you to find the length of AB.
Use familiar mathematics to guide you. The area of a decagon is an unfamiliar problem, but the area of a triangle is not. To find the area of the decagon, you need to find the area of the triangle AOB.
Draw on earlier learning. Knowledge and use of Stage 5 maths concepts are important. Geometry – knowledge of angles in a revolution and angle properties of an isosceles triangle – needs to be used to find the size of angles in triangle AOB.
Don’t forget the reference sheet. Use the triangle area formula that is on your reference sheet to solve this problem. The area of a triangle formula requires the lengths of two sides (OA and OB) and the size of the included angle that you should have already worked out.
Use the diagram to your advantage. Write information you find on your diagram. This will help you see connections and to understand what is required to solve the problem. By writing the size of the angles and length of side AB on the diagram, it should become clear that the sine rule is needed to find OA and OB. Therefore, the area of triangle AOB can be worked out and the area of the decagon revealed.

Students practise maths to prepare for exams. Credit:Louise Kennerley

Christopher Guy, Blaxland High School

Deputy Principal

Question checklist

• Get to know your reference sheet

The reference sheet won’t tell you what the formulas are used to solve or what the symbols refer to. Find out as much as you can about what the formulas can be for. Make sure you know what they mean and when to use them.

• Show all working out

If a question is worth four or five marks and you have a page of working-out space, then there are probably a few steps in getting to the answer. Show how you get from one step to the next as you are unlikely to get full marks by writing down one number. Provide all your working-out to support your answer.

• Check the validity of your answer

Answers need to make sense in the context of the question. For example, when calculating energy costs in running a household, the cost for one light globe should make some sense. You would have made a mistake if your cost was large (eg $500) as this is about how much a family would pay per quarter for all their electricity.

• Read the entire question … twice

Check what the question is asking and in what form you need to present the answer. For example, you might need to round the final answer (decimal places, significant figures, scientific notation) or convert to an annual amount (from weekly, fortnightly, quarterly).

• Convert units

Most students remember to convert common units such as litres to millilitres and kilometres to metres. Don’t forget to convert area units, volumes and energy.

• Double-check your calculation

Know how to calculate Pearson’s correlation coefficient, find mean and standard deviation and correctly calculate questions involving negative numbers.

Mathematics teacher Cheryl Rix.

Cheryl Rix, Gosford High School

Head Teacher, Mathematics

Down to the detail

Here are some finer points that might help you in the Mathematics Advanced exam.

When you read a question, write down the values of any variables you may be using in the solution (eg P(A)=0.2).
Use diagrams or tables in your working for questions involving probability or statistics. This will clarify your understanding of the information in the question and support your working.
Show any relevant substitutions. This shows the marker that correct mathematical processes are being used (eg show the substitution of an x-value into a second derivative in order to determine the nature).
If using a table in a calculus question, clearly label whether you are using the first or second derivative and include values rather than + or – signs in your tables.
Make sure graphs have a labelled scale on the axes and important points are clearly labelled.
If completing a proof, separate the left-hand side and right-hand side working to arrive at the equivalent expressions.
Consult the reference sheet for derivative and anti-derivative. Check that you can manipulate expressions into the form given on the reference sheet.
Remember that a rate means a derivative. Make sure you understand the difference between average rate of change (gradient between two points) and instantaneous rate of change (derivative at a point).

Degrees or radians?

Remember to use radians and not degrees for questions involving calculus, sketching graphs and circle trigonometry. Get in the habit of checking whether a question requires radians and check your calculator setting before you start working it out.

Solving multiple-choice questions: Mathematics Advanced

Question:

Suppose the weight of melons is normally distributed with a mean of µ and a standard deviation of σ.

A melon has a weight below the lower quartile of the distribution but NOT in the bottom 10 per cent of the distribution.

Which of the following most accurately represents the region in which the weight of this melon lies?

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Answer: C

Answer explanation

Using the percentages given on the reference sheet:

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This can be visualised on a line indicating the percentage of scores below the mark:

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The lower quartile would be located at 25 per cent on this number line, between 16 and 50 per cent, and 10 per cent would lie between 2.5 and 16 per cent.

This region is indicated in option C.

Mathematics standard 2

Question:

Joan invests $200. She earns interest at 3 per cent per annum, compounded monthly.

What is the future value of Joan’s investment after 1.5 years?

Answer: B

Answer explanation

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Top-scoring exam answers in 2020: Mathematics Advanced

Exam question:

The circle ?²– 6? + y²+ 4y − 3 = 0 is reflected in the ?-axis. Sketch the reflected circle, showing the co-ordinates of the centre and the radius.

Student answer:

?² – 6? + 9 – 9 + 4y + 4 – 4 – 3 = 0

(? – 3)² + (y + 2)² – 16 = 0

(? – 3)² + (y + 2)² = 16

∴ original centre = (3,-2)

original radius = 4

Reflected in x-axis: new centre = (3,2)

<>Credit:NESA

Marker comments

In this response, the process of completing the square was applied correctly. The original centre and radius were stated and the location of the new centre after reflection was given.

The sketch of the reflected circle shows the correct co-ordinates of the end points of the vertical and horizontal diameters.

Mathematics standard 2

Exam question:

The preparation of a meal requires the completion of all 10 activities A to J. The network diagram shows the activities and their completion times in minutes.

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(a) What is the minimum time needed to prepare the meal?

(b) List the activities that make up the critical path for this network.

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Student answer:

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(a) 46 minutes

(b) The critical path: C, D, E, F, H, I

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Marker comments:

In this response, the diagram has been used to determine the correct answers for parts (a) and (b). The final answers have been stated correctly in the spaces provided. The working for part (c) accompanies the solution and is both logical and correct.

Exam workbooks, which include more examples from top-scoring students, are available from the NESA Shop.

Why different maths exams have common questions

The maths exams have some questions that are the same.

You’ll find common questions across these exams:

Mathematics Standard 1 and Mathematics Standard 2.
Mathematics Standard 2 and Mathematics Advanced.

This is because students in these courses learn some of the same content. Common areas of study include statistics and financial maths.

This doesn’t mean the Mathematics Standard 2 exam is harder than it should be or that the Mathematics Advanced is easier – all students are tested on what they have learnt throughout the course.

Learn more at educationstandards.nsw.edu.au under Mathematics Standard.

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