Even if you’re not aiming for such a high ATAR, doing well in any level of mathematics will get you closer to your goal in no small amount.

With every subject, there’s the content, then there’s technique. This fact is especially significant in maths extension 2 – things like knowing shortcuts / quick methods to verify the correctness of answers, how well you know your way around your Board-approved calculator, how well you understand the marking process and how to get partial marks, etc.

We all know the commonly told tips that your teachers at school have no doubt told you many times by now (e.g. writing partial answers for partial marks). Here’s a few lesser known tips our maths tuition classes give our students as part of our Extension 2 maths course:

1. Volumes – easy way to check answer for volumes

Here’s a useful tip for the Volumes topic in Maths Extension 2 – Pappus’ Centroid Theorem. This theorem states that the volume of a solid of revolution generated by rotating a plane figure about an external axis is equal to the product of the cross-sectional area and the distance travelled by the cross section’s centroid.

Using this theorem, we can calculate all questions involving rotating a shape around an axis (e.g. circle around an axis to produce a torus) simply by finding the area of the cross section, then multiplying this with the distance travelled by the centroid of the cross section. It’s much quicker and reliable (because it’s simpler – less can go wrong) than using any of the prescribed methods, e.g. cylindrical shells or adding slices.

For example, what’s the volume of a circle, centre origin, radius 4, rotated about the line x=6? The answer is simply: the area of the circle – 16π multiplied by the distance travelled by the centre of the circle, which is 12π. The volume is therefore 192π^2 or 1894.96 cubic units. You can verify this with the cylindrical shells method.

Of course, in answering an exam question, you need to use the cylindrical shells method. The Pappus’ Centroid Theorem is just a useful tool to quickly check the correctness of your final answer. If there’s disagreement, you know you need to check your answer, and this would have easily saved you some lost marks!

1. Volumes, integration and other applications – the area of an ellipse and the volume of an ellipsoid

Here’s another tip to make you quicker at checking your integration results. It’s also helpful for other topics (e.g. volumes).

It’s a good idea to memorise the area of an ellipse:

And the volume of an ellipsoid:

These are not too hard to remember because in the special case of the circle and sphere, r=a=b and r=a=b=c respectively, which causes both equations to reduce to the equations of the area and volume of a sphere respectively.

There are occasionally situations where you have a definite integral you need to evaluate, where the expression to be integrated is in the form of an ellipse. For volume questions, solids of rotation would often form ellipsoids and knowing the simple formula could give you a quick tool to check the correctness of your answer.

1. Don’t be afraid to include explanations as part of your solution

Here’s a tip our extension 2 maths tutors love to give our students. In many situations in mathematics, especially in maths extension 2, some explanation saves you a lot of calculation.

For example, if you’re doing an integration and the form is that of a circle, instead of going through the x=sinα substitution, it’s easier to just write “This represents the area of a semi circle with radius r” – that’s all you need, then you can write the answer. Or if you need to evaluate a definite integral of an odd function with symmetrical limits, then the answer is always 0.

Sometimes you need to take cases – explain why some cases are impossible and this will save you time because you don’t need to cover them. Sometimes a graphical solution will show why one graph will never intersect another graph (hence no real roots to a related equation). The point I’m making is explanation goes a long way. The entire Mathematical Induction topic in 3 unit is based on explaining rather than using purely numbers, symbols and algebra.

Another situation would be proof questions (and we get many of these in Extension 2) that require you to prove LHS = RHS. Instead of starting somewhere random, you can start with what you’re trying to prove and write “If the above is true (what we’re required to prove), then:” and you can proceed to manipulate what you’re trying to prove. Just write ‘Then’ at the beginning of each line to signify you haven’t proven it yet, but if the original LHS = RHS, THEN so far these lines of working must all be true. Eventually, when you reach a situation that is in fact true (e.g. 1=1) then you’re finished, just write “And since the final line is true, the original LHS = RHS).

1. Integration by parts – LIATE

We all should know how to do integration by parts by now – but sometimes choosing which one should be u and which should be v can get tricky. A reminder, here is the definition of integration by parts:

OR

Choosing which one should be u (or f(x) if you prefer the first line) is an important decision – choose the wrong one and you’ll waste precious exam time going down a path that may lead to an impossible integral.

The general rule of thumb is to remember LIATE. LIATE stands for:

• Logarithm
• Inverse trigonometric functions
• Algebra (general polynomials)
• Trigonometric functions
• Exponential

You should give preference to the left-most function to be set as u. The reason for this is as you move from the left to the right in LIATE, functions become easier to integrate, so you should prefer to integrate the easy ones (e.g. trigonometric functions and exponentials are easy to integrate) and differentiate the hard ones (e.g. logarithms can’t easily be integrated).

1. L’Hopital’s rule

In the Graphs topic, there are many compound graphs that give rise to situations where you have infinite multiplied by 0. When this limit occurs, 3 possibilities arise:

• Graph turns to 0
• Graph turns to infinite
• Graph turns to a constant

To find out how, we suggest to students to learn how to use L’Hopital’s rule to discover the relative speed of curves. This rule only applies when the limit below turns to infinite or negative infinite, or 0.

For example, lets see what happens to the graph at y=xlnx close to x=0. We set f(x) = lnx and g(x) = 1/x so that f(x)/g(x) = xlnx. When we evaluate the limit for f’(x)/g’(x) we find it equals (1/x)/(-1/x^2) which equals –x. This would turn to 0 as x approaches 0 so this shows the graph y=xlnx turns to 0 instead of negative infinite at x near 0. Incidentally, this is also one of our HR manager’s favourite questions to ask potential candidates for maths tutors, even if they tutor lower levels of maths (you’d be surprised how many people apply to be a mathematics tutor but can’t even graph y=xlnx).

1. Use your calculator memory effectively

Become proficient in the use of your Board-approved calculator that you can take with you to the exam room. If you’re swift with using your calculator’s memory slots, this makes rechecking over your answers so much faster and accurate (if you happen to finish your exam with 20 minutes to spare, you can literally go through the entire exam once or even twice! If you’re good with your calculator, that is). And this helps even more in subjects like HSC Physics and HSC Chemistry where there’s lots of calculator work that requires a final definite numerical answer.

Another reason is exact values. Your calculator’s memory actually stores something to the order of 100 digits (much more than what’s only shown on screen when you press ‘=’). Sometimes when you’re supposed to get an exact value, if you put in your written down answer (that’s only rounded to 3-4 digits at most) you will get some answer like 1.99837734 but if you subbed in your memory-stored calculator, you will get an exact value of 2. In complex questions, even simple clues like knowing something is an exact value could be the difference between doing the question and skipping it altogether.

1. Multipart questions (attempt part 2)

We should all know to be able to attempt the next subpart of an exam question by now. E.g. if you can’t do part a, which required you to prove a result to be used for part b, you should use the result to do part b so that you can still score partial marks for the question.

Sometimes even within a single part worth multiple marks, there are two parts to the question (e.g. “Show that LHS = RHS and hence derive an expression for acceleration of the particle.”) In these examples, if you can’t do the question as they intended, just use the result you’re given and finish the second part of the question for partial marks. If the question was worth 3 marks, you should at least get a mark for your efforts.

1. Maths tutoring helps

Sometimes if you’re stuck with a teacher at school that’s not very knowledgeable, it would be wise to seek outside help. This situation is particularly common in disadvantaged schools – not all schools even offer extension 2, and for schools that do, not all their teachers assigned to the course are actually capable of teaching all parts of the course effectively. Getting outside help also gives you an important advantage over your peers when it comes to internal assessments (where you’re up against your peers for the top assessment ranks).

• Cross out incorrect answers with a single line

HSC Markers read everything that can be read, even if you’ve crossed out an answer. If you’ve written an answer but change your mind afterwards and write another answer, cross your old answer out with a single diagonal line using your pen. Do not use liquid paper. This ensures that even if your final answer is wrong, there’s more chance you’ll receive partial marks for the question (as long as the marker can see you did SOME things correctly).

• Show ALL working out

Some students prefer to write things out step by step – that’s generally the better / safer approach, as showing working out ensures you will get at least partial marks, even if your final answer is incorrect.

In the past, one of our top students (who later went on to achieve a state rank) preferred to do entire questions just by using his calculator’s memory, storing everything into the A, B, C … to M memory slots! We always had to remind him to remember to write out his ‘working out’ after he wrote his final answer – it was also a great way to check his answer.

• Look for clues from previous parts of a question

All HSC maths exams (from General maths, 2 unit to Extension 2) structure their questions in terms of part a, b, c, etc. Use the answer from the previous parts as a clue to your current part (even if it’s not a ‘hence’ or ‘hence or otherwise’ question).

• Use your calculator’s memory!

For questions / parts that require you to use a numerical result from a previous question / part, you’re better off using the stored number in your calculator rather than your rounded written answer. This applies especially true in subjects like HSC Physics and HSC Chemistry where you’ll be doing much more numerical calculations.

For Mathematics Extension 1 & 2 students

• Work a proof question from BOTH sides

For questions that require you to show LHS = RHS (e.g. typical induction questions like “Show that f(x) = g(x) is true for all x > 0″), realise that you don’t need to work strictly from LHS to RHS.

Instead, start with the LHS, see if you can simplify it / progress it as usual. Then when you’re stuck, check the RHS and try progressing with that. Usually you will find this approach makes equating LHS and RHS much easier.

Think of these types of questions as requiring you to make LHS and RHS meet, but there’s a valley in the middle. Instead of pushing LHS all the way through the valley (down the valley, then up the valley), push LHS all the way down, then push RHS all the way down, so they meet at the bottom.

• Don’t be afraid to use graphs as part of your answer

Sometimes, graphs are appropriate as part of a mathematical proof. For example, if you’re required to prove some inequality, you can use a graph (and some calculus of course) to show that a line is tangential to a curve, in order to support your inequality.

• REMEMBER the definition of the log integral:

int dx/x = ln |x| + C

Remember that when you integrate 1/x you get the log of the ABSOLUTE VALUE of x, not just x by itself. Although you won’t lose a mark for not including the absolute value signs, some questions with definite integrals (e.g. requiring you to find the area under a curve) will result in logs of negative numbers and hence impossible to evaluate unless you remember to include the absolute value signs. Don’t get tricked!

• Strategies for ‘hence or otherwise’ questions

In multipart questions, the last part is usually either a ‘hence’ or ‘hence or otherwise’ question. When you have ‘hence’, you have no choice but to use the previous result(s) to do the question. When you have ‘hence or otherwise’ you have an option either to use your previous result(s), or take a wholly new route to the answer.

Here’s the tip: if you can see that the question reduces to anything you recognise, its often actually FASTER to use your ‘otherwise’ option. For example, in tricky Extension 2 question 8 type questions, you are often required to show LHS = RHS, or LHS > RHS, or LHS < RHS. If you can re-formulate the equation into something you recognise, then it’s just a matter of writing out your proof for that thing you recognise, then reshuffling it back into the required form.

The reason why this is a better approach is because for harder questions, the amount of time you could sit there potentially thinking (on how to do it using your previous result(s)) is highly variable (could take a very long time), and risky (you may not even see the answer after spending plenty of exam time). If you can reduce it to a recognised form and write out a memorised proof for it, even if it’s not the most elegant / efficient proof, you will score full marks, and the time you take is only dependant on how much you need to write out.

Study year 11 and 12 maths topics together

The syllabi of HSC mathematics is integrally linked with the preliminary (year 11) syllabus. This applies to all levels of HSC maths, from General to Extension 2. There is no sudden identifiable transition between preliminary topics and HSC topics. In contrast to HSC sciences (such as Chemistry and Physics), their syllabi are clearly split into preliminary topics and HSC topics.

In mathematics, topics you learn in your preliminary year, or even going back to year 10 (e.g. the sine and cosine rule are sometimes used in year 12, even in Extension 2) are unavoidable when you need to study for HSC topics. For example, we all need to know how coordinate geometry works, and how to find the equation of normals and tangents, before we can understand the Conics topic in Extension 2, or parametrics in Extension 1. The key point here is that there is no clear distinction between year 11 and year 12, for mathematics.

One approach to maths tutoring or teaching at schools is to teach topics according to their relationship with each other, instead of whether the actual syllabus categorises them as preliminary or HSC topics. For example, we can teach year 11 Extension 1 probability, up to the harder permutations and combinations normally studied in year 12. This approach in studying is also advantageous, as it helps you consolidate and group relevant topics together.

An extreme example that may work for some is the anecdote of a private maths tutor that is reputed to teach year 7 geometry, then for the entire year, progress to harder and harder geometry topics, finishing off with Extension 2 style circle geometry. While we can see this approach may work for some students, the extreme case is not recommended for most students. Instead, we recommend students to study the relatable preliminary and HSC topics together. For example, the reason why the Fitzpatrick series of books (the yellow book for 2 unit, the green book for 3 unit, and the pink book for 4 unit) is split according to 2, 3 and 4 unit reflects this fact about HSC mathematics. The writer did not choose to split his books according to preliminary and HSC as he correctly identifies that it is more convenient and advantageous to student learning by making them learn year 11 and 12 topics together, where they are very related.

Recommended approach for HSC sciences

HSC sciences, unlike mathematics, have topics that are clearly divided as preliminary and HSC topics. For example, in Preliminary Physics, you learn about waves and communications devices in The World Communicates, resistors and using Ohm’s law in Electrical Energy in the Home, vector addition and movement in Moving About, and some basic astrophysics in The Cosmic Engine. Now, if we look closely at the topics taught in the Preliminary year, and compare them to the HSC topics, there is very little direct overlap. The main value in Preliminary Physics is for students to gain a solid grasp on the physical principles that are relevant to the HSC.

For example, in The World Communicates, knowledge of waves and how they propagate is important to many topics in the HSC. However, knowledge of mobile phones, fax machines, GPS and CD/DVD technology is irrelevant to the HSC. So the point here is: understand the physical principles (waves, electrical resistance, Ohm’s law, vector addition, forces, momentum etc) but don’t pay too much attention to the specifics (e.g. you’ll never be asked to calculate the resistance of a circuit in a HSC question, and you don’t need to know about Red Giants / White Dwarves if your school does not do the Astrophysics option module).

Ideal approach to studying HSC Physics and Chemistry

The ideal approach here is to learn the preliminary course as usual, paying close attention to the physical principles that are involved with the content. However, remember that you will not be tested in your HSC year on the specifics of the preliminary course. For example, you will not be required to know how to calculate resistance in series and parallel circuits in the HSC Physics course. In fact, the HSC assessments and exams will only test what is in the year 12 HSC syllabus. Therefore, you will definitely need to know the specifics of each dot-point in the HSC syllabus, but not the specifics of the preliminary syllabus.

A good approach is to start your learning early. Cover the preliminary topics as quickly as you can, (with the help of Chemistry tutoring or Physics tutoring, or from your school teachers) and move onto the HSC topics as quickly as you can. This leaves you with the maximum amount of time to study the content that is directly relevant to your HSC. Remember, only the content of the year 12 syllabi will be examined, so use this fact to your advantage when studying HSC sciences!

Maths is so universal that it will be a useful skill if you end up doing Commerce, Business, Medicine, Science, Engineering, Pharmacy, Physiotherapy – just about any commonly chosen university course you can think of.

Maths Extension 1 and 2

We also recommend students who are good at maths to enrol in Mathematics Extension 2, to benefit from the subject’s large positive scaling effect. Students often have a hesitation about signing up for Maths Extension 2 when they need to decide near the end of year 11. The issue is, most students find Maths Extension 2 – and rightly so! It is not a subject that can easily mastered, and requires the most practice to familiarise among the different types of questions that can appear in an exam.

However, the scaling benefit is massive – even if you end up scoring the average raw mark for Extension 2, it is equivalent to the top 10%-15% for relatively high scaling subjects like HSC Physics, Chemistry, English Advanced, or Economics, or the top 10% for Biology. This is not including the added benefit of having Maths Extension 1 count for 2 units, instead of 1, which in itself is a huge benefit to your final aggregate score.

Similarly, Maths Extension 1 has a large positive scaling benefit in its own right. Its scaled mean of 40.0 in 2008 continues the trend of it increasing over the past few years. Currently, this places the scaling of Maths Extension 1 equal to that of scoring in the top 15% for English Advanced.

Other advantages of mathematics

There are other less direct advantages of choosing mathematics for your HSC. Firstly, because it is so common, you will find there is an abundance of good textbooks available for the subject. There is also an abundance of free notes and materials on the internet. Also you will find that if you ever need assistance outside of school, HSC maths is one of the most commonly offered subjects when looking for a maths tutor.

However, maths can be challenging at times. It is a subject which requires plenty of practice to master, as much of what goes into making a top maths student comes down to experience. For example, as mentioned in the previous article, How to do well in HSC maths, it is a subject that requires you to literally sit down and do thousands of questions before you gain enough experience for the top band. The main thing you will gain through practice is the ability to see overarching patterns and connections between seemingly unrelated topics – but also after doing so many questions, you will come to a point where you are familiar with every type of maths exam question that can be asked.

Mathematics tutoring

Somewhere down the line as you go through the Preliminary course and into the HSC, you may consider whether to seek maths tutoring. The advantage of choosing maths is that so many places offer mathematics tutoring, students have a nice selection of maths tutors to choose from.

Students will also have to decide about whether to seek out a private tutor for maths, or maths tuition classes. Each means of maths tutoring has its own advantages and disadvantages, and there are situations where one is appropriate and the other is not. To illustrate, generally speaking, class tuition is not suitable for students on either extremity of the ability spectrum – those that are exceptionally advanced and those that cannot follow on in a class environment. Those students may benefit more from a private tutor.

On the other hand, students that fall within the majority of the bell curve can benefit greatly from a class environment due to some or all of the following factors:

· Structured environment: reputable maths tutoring providers will always provide learning materials, homework, feedback and deliver their program in a structured way. This is the main thing private tutoring lacks.

· Healthy competition between peers (students know exactly how well they’re doing relative to a sample of above-average students)

· Quality teachers: with private tuition, there’s no guarantee as to the quality of your tutor, whereas reputable tuition providers will always hire high quality tutors as they are experienced in finding and training talented educators.

That’s not to say the more talented individuals benefit less from a structured environment. Often, students find it is of greater benefit to be able to follow a structured study regime which can guarantee a comprehensive coverage of the entire course, rather than leaving it to private tuition, with a teaching approach that can leave gaps in their understanding.

The choice also comes down to economic factors. Private tutors often cost several times the cost of enrolling into a class-structured course.

Whether you do 2 unit maths, maths extension 1, or maths extension 2, doing well in HSC mathematics requires a similar strategy. In this article, we will briefly look at what makes a successful HSC maths student, as well as some exam preparation techniques which would be relevant to students today, as most have their all-important HSC trials and HSC exams coming up in the next few weeks.

Seeing connections between HSC topics
The most common characteristic shared by successful HSC maths students is their ability to see connections and patterns between the various topics of maths. This is important, as many questions are not worded in an immediately straightforward manner.
For example, a 4 unit (Extension 2) question may initially appear to be an integration question, but in part b or c, knowledge of polynomial roots or complex numbers needs to be used. Similarly, such questions involving a mesh of different topics are also common in 2 unit math and 3 unit (Extension 1).

Practice makes perfect

Training for mathematics is much like training for sports. Your core skills ultimately comes down to how much practice you have had. There is a limited number of ways an exam can ask you questions. If you have gone through two or three complete (reputable) HSC maths textbooks, good chances are that you have seen most of the ways questions can be asked.

Therefore, doing well in HSC maths, regardless of what level of maths you do, comes down to simple practice. This piece of advice is the most simple to describe and understand, but the most difficult to implement and follow through. The key is to set yourself an ongoing goal – decide how much exercises or hours you can do every day or week, then persevere.

Build up a habit for the long run and stick to it. Focus on sustainability, rather than studying for the short term. For example, if you can get into a simple habit of studying just an hour, purely dedicated to mathematics, on each school night, this would be so much more useful than being highly motivated for a period of a few weeks prior to exams, but being unmotivated throughout the year.

Convert real facts into a mathematical problem

Longer, more difficult maths questions tend to be phrased as a problem question. There is no rule of thumb as to which topics can be phrased in a long-worded question – any topic can be presented this way. However, some topics tend to have a greater abundance of such worded problems. For example, in maths Extension 1, there’s Applications of Calculus, which includes things like projectile motion and Newton’s Law of Cooling. In maths Extension 2, there’s even more! (Mechanics, volumes, conics, complex numbers and most of Harder 3 unit – to name a few).

Some students find it difficult to convert a worded scenario or problem into a mathematical / numerical problem. The issue is that students are mostly taught to think in terms of numbers and algebraic expressions, but only occasionally (or for some, rarely) get to practice on real-world worded problem questions. A good maths student would have had plenty of practice at synthesising complex worded facts into a numerical problem, especially by the time they need to prepare for their HSC trials and HSC exams.
In terms of good preparation, it is good to pay close attention to questions which are long, have multiple parts and represent mathematics in some real-world application. Doing these questions (and asking your tutor / teacher questions if necessary) will give you adequate preparation.

Avoid over-relying on your calculator

This point is not talked about much, probably because it is not raised often. But I’d like to shed some light on the issue. Pulling out your calculator for every arithmetic operation (e.g. you need to add single digit coefficients together) wastes your exam time, and increases the risk of pressing something wrong. In the end, for the more simple operations (e.g. adding / multiplying single or even double digits) is simply done faster in your head, than with a calculator.

I remember, not long ago while supervising an exam at university, I saw a first-year student take out his calculator and press 2 + 2 =. Maybe I have a strange sense of humour but I found the incident funny and memorable. However this does highlight a current issue for some HSC students. For some students, it has come down to total reliance on their calculator for all arithmetic calculations, even simple ones that ought to have been done mentally without a doubt.

I always tell my students, you can do an entire Extension 2 exam without touching your calculator. Most of the more advanced students know this. To minimise the incidence of human error, again, this comes down to practice. In everyday life, whenever you come across a situation where you need to add / multiply / subtract or even divide, (e.g. when shopping, or on the train, or at school etc) you should do the math in your head. Break the instinct of moving to grab your calculator. Think of the brain as like a muscle – the more mental exercises you give it, the better it will become.