Tag: maths extension 1

Maths Extension 2 tips from our top Maths tutors

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:

  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).

Explanation of HSC Scaling

HSC scaling is a popular topic to HSC students and parents, and is often an area that is commonly misunderstood. Scaling is important as it affects all students aspiring to get into university after the HSC.

Scaled marks versus HSC marks

A commonly misunderstood concept is the relationship between HSC marks and scaled marks. HSC marks are the marks the Board of Studies awards you, and appear on your Record of Achievement. These marks determine which performance band you fall in (e.g. Band 6 or E4) for each of your HSC subjects. These marks measure how well you did according to the subject’s requirements. E.g. if you received a Band 6 in English Advanced, it means your performance satisfied all the criteria required by the HSC English syllabus to achieve a Band 6. However, in any year, any amount of HSC students can get a Band 6. For example, in a particularly smart year, a higher proportion of students may receive Band 6 in English Advanced. It is not how well you do in your subject, but rather, how well you do relative to other students which determine your UAI. Here’s where your scaled marks come into play.

HSC Scaling Your scaled marks will NOT be shown to you at the end of your HSC, as you will only be shown your HSC marks (aligned marks, to be precise). Ironically, it is your scaled marks which are the most important determinant to your UAI. Scaled marks are calculated by the UAC (not the BOS) under a totally different process. Basically, these marks measure your performance relative to other students. (For a more technically accurate discussion on scaled marks and what they mean, as well as the mathematics behind UAI calculation, please read our article on the mechanics of HSC scaling) Remember, your HSC marks are a measure of how well you did in your subject, but your scaled marks measure how well you did relative to other students. It is your scaled marks which are used to calculate your UAI, not your HSC marks.

Through the process of scaling, the UAC converts your raw examination marks (the actual marks you received in your external and moderated internal assessment) into scaled marks. These scaled marks are then added up to arrive at your aggregate mark (students refer to this as your ‘aggregate’) out of 500. The UAI is simply a percentile rank of your aggregate, which is the total of your scaled marks in your top 10 units.

How can knowledge of HSC scaling help me?

Understanding the process allows you to plan your HSC, to an extent, in such a way as to make scaling work to your advantage. For example, if you enjoy maths, you should choose Maths Extension 2 in order to take advantage of its enormous scaling effect. Similarly, if you enjoy science, you should take Chemistry and Physics, as they scale relatively well.

In other words, comparing subjects in terms of their scaling effect can assist you with your decision as to which subjects to take for your HSC. In order to quantitatively compare the scaling effect of different courses, you will need to get familiar with reading statistics published by UAC. The rest of this article will highlight the important things to note.

Reading ‘scaled means’

Firstly, what are ‘scaled means’? The scaled mean for each subject is the average scaled mark received by all students who took that subject for that year. For example, in 2008, the scaled mean for Maths Extension 2 was 43 out of 50. This means that among the Maths Extension 2 students in 2008, the average of their scaled marks was 43 out of 50. This subject has traditionally been one of the highest scaled subjects available for the HSC. In terms of reading these scaling statistics, generally the higher the scaled mean, the higher the scaling effect.

Each year, the UAC publishes a scaling report which contains important scaling statistics for all HSC subjects eligible to contribute to a UAI. For more information, read about UAC scaling statistics. In the report, there is an important section called Table A3, which is a table setting out the scaled means of all subjects.

To illustrate the effect of scaling, in 2008, a Maths Extension 2 student only needs to be in the top 46% out of all Maths Extension 2 students to get a scaled mark of 45 out of 50 (or 90/100). A Maths (2 unit) student would need to be in the top 3% out of all Maths (2 unit) students in order to achieve the same result. These facts are read off the UAC scaling report. In the 99th percentile, a Maths (2 unit) student receives a scaled mark of 46.1 out of 50. In the 75th percentile, a Maths Extension 2 student receives a scaled makr of 46.2 out of 50. Arguably it is easier to be above average in Maths Extension 2 than to be near the top of the state in Maths (2 unit). This is the main benefit derived from choosing high scaling subjects.

Effect on UAI calculation

Simply put, the higher the total of your scaled marks, the higher your UAI will be. Sometimes when students choose subjects with lower scaled means, do spectacularly in their HSC (e.g. receive Band 6 for all of their units) but receive a UAI that is lower than what they had expected.

For example, if you did English Standard, IPT, Legal Studies and Biology, and scored 90 in all of your subjects, your UAI would be around 94 in 2008. While this is in no way a poor UAI, if you received the same HSC (aligned) marks for English advanced, Maths Extension 1 & 2, Chemistry and Physics, your UAI would be in the vicinity of 99. Again this is because of the scaling effect across different subjects. While all subjects are different and some will be more difficult than others, the best approach to dealing with HSC scaling is to choose the subjects you are interested in, while giving consideration to the scaling effect of your choices. (For more information, read our article on HSC subject selection)

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