Mathematics is one of the most rewarding subjects for your HSC, especially at the extension levels, where it’s one of the highest scaling subjects you can choose. Each year, the majority of students that score a 99+ ATAR did very well for Maths Extension 1 and 2. For maths extension 2 especially, each year the majority of the 21 or 22 students that score the perfect ATAR of 99.95 get 97+ for maths extension 2 as their HSC mark.
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:
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!
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.
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).
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:
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:
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).
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:
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).
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.
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.
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).
At the time of writing this article, we’re in the middle of term 2. Most year 12 students have their HSC trial exams in early term 3, which means while there’s still (almost) an entire term left, it’s about time students begin to prepare specifically for their trial exams.
Learn to use the HSC Standards Packages
For almost all HSC subjects, the Board of Studies has standards packages publicly available for students to read http://arc.boardofstudies.nsw.edu.au/go/hsc/std-packs/. We recommend students look at the standards packages for their subjects – they will give you an idea of the quality required for a band 6 response. Standards packages are particularly useful for humanities subjects like HSC English where your expression and writing style also come into play. Get a feeling of what’s acceptable expression, and details like general paragraph length, the way literary techniques are referenced, how often a text is quoted and the length of quotes etc. Even if technically your knowledge is as good as anyone’s, a poorly structured essay (where you use poor expression, or reference the text in superficial ways or spend too much time on insignificant points etc) will mean the difference between a band 4/5 and a band 6.
Familiarity with the standards packages will also help with subjects like HSC Chemistry, Physics, Biology and the social sciences (Economics, Business Studies etc). In the case of the HSC Sciences, they give you fresh ideas of novel, acceptable ways of structuring your answer. Some questions can be fully answered in terms of a table or as a dot-point list (e.g. identify / outline questions). Also note the details featured in diagrams / graphs – full marks are given to students that remember details such as labelling the axes, or drawing a line / curve of best fit properly (ignoring outliers in appropriate situations) and being able to justify the choices made.
HSC sciences – always refer to the syllabus
HSC sciences like Chemistry, Physics and Biology, are prescriptive by nature. This means the syllabus tells you exactly what you need to know, content wise, and does a great job at that. While studying for these subjects, it’s always a good idea to have the syllabus in front of you, printed or on your computer screen. The dot-points give you a clear picture of what you need to know, and the scope to which you need to know each aspect of the course. For example, if a dot-point requires you to merely identify the qualitative aspects, this means you only need to be able to name the aspects it’s referring to, and qualitative means you won’t be required to do calculations on them.
Another reason is some syllabus dot-points are worded as if they are paraphrased exam questions. This is particularly true for dot-points requiring you to ‘Discuss the impacts on society of…” or “Assess the environmental impact of…”. You very well might get an exam question, worth around 7 marks, that basically asks you to demonstrate your entire understanding of one of those dot-points if they ask you in a general way.
Be careful for internal assessments however, as school teachers are known to set exam questions that are dubious in terms of whether they fit within the scope of the syllabus, so you must also cross reference your own materials with the notes given by your school teacher to make sure all gaps are covered.
HSC Maths – only do exam questions
When it comes to maths, exam questions and textbook questions aren’t the same. The former type are often are structured as a compound question with several subparts. Exam question are often designed with deeper consideration, and incorporates more unique aspects of mathematics (e.g. in Maths Extension 2 question 7 and 8). In contrast, textbook questions can get repetitive and give you a false sense of security. Because textbook questions lack variation in style, once you master the several types of questions it contains and are able to do its exercises, this does not mean you’ve experienced all types of questions an exam can throw at you, particularly if you go to a school that has a talented maths department.
There’s a limitation on the types of questions for each topic an exam can throw at you. If you do Maths Extension 1 and 2, it also takes great effort and skill to design a truly novel and unique maths question at that level. As an industry insider (yes I’m a teacher) I can tell you that many schools simply take exam questions from past papers of other schools. When I did my HSC Maths Extension 2, I actively sought out past trial papers from top private and selective schools for practice before my HSC trials. What I noticed was in one year, say 2002, there would be a question in school A’s paper, then in the next year, say 2003, there would be an identical question in school B’s paper. So it’s a good idea as a student to use past papers as practice – there’s definitely more exam papers worth doing than your time would permit, that’s why I recommend only do exam papers instead of textbook questions.
I spoke with a teacher who works at a top Sydney selective school about how their teachers set exam questions for their year 12 students – “We get exam questions from schools that are out of NSW – resources we know typical students don’t have access to”. So while exam questions are definitely recycled, they aren’t always from sources you’d expect. But it’s still worthwhile doing exam papers for practice, purely for the sake of familiarising yourself with the general style of exam questions which you can’t get from any old textbook.