50ss Guide to Mathematical Methods

General Overview

In one word, this subject is about functions. You begin by investigating different types of common functions—parabolas, polynomials, square roots, rectangular hyperbolas, the truncus, semi-circles, and all the circular functions. In Term 2, you learn Differential Calculus to investigate how fast a function is changing. Then, you learn the opposite—Integral Calculus—to find the area under a graph. Finally, you put these functions to use in modelling probability distributions.

This can all sound scary, but I promise it doesn't have to be. If you can work with functions like you can add, subtract, multiply, and divide, then this subject becomes very manageable. Indeed, when you don't have to worry about how to sketch a parabola, you can actually focus on some of its fascinating properties instead.

Term 1 sets you up for the year. At a time when SACs seem far away, and you haven't yet been drained of energy and motivation, make sure to drill two things:

• Graphing all the common functions accurately and quickly.
• Understanding the topic of transformations.

These two skills are the bread and butter of Maths Methods—they are like addition and multiplication. In an ideal world, you should be able to do 95% of these textbook questions without much stress or outside help.

The bottom line is that you need to do work to succeed, and that the amount of work you do closely correlates with how well you do. However, what work you do, and how you do it, can be optimised according to your goals.

The following sections explain how you should use your time if you are short on time and want to achieve proficiency as efficiently as possible, and what to do if you have excess time and simply want to maximise your exam score.

Good luck!

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How to Maximise Efficiency

In this section, we shall say that you have achieved mastery when you can do 90% of questions without help (and quickly).

• Start with your textbook. You should always start here. If it is easy for you, excellent! If it is not, the textbook questions provide a lovely introduction into the topic. You must master the fundamental skills before you can do application questions, and the textbook is great for that. Once you are cruising through the chapter and have achieved mastery, move on. The textbook only becomes a waste if you mindlessly churn through questions—actively assess your familiarity with the topic, and the textbook becomes very useful.

It should be mentioned that consistency wins 100% of the time here. If you are serious about making this subject as pain-free as possible, stay on top of things from day 1. This amounts to simply doing the textbook questions and learning the basic skills so that you can follow along in class when more advanced concepts depend on those simpler ones.

That is, if you don't do Methods for a week, you will not have the required knowledge to understand what your teacher is talking about in class the next week, and so all that time will be wasted. Of course, if you have already done this and are falling behind, make an effort to catch up, and once you have caught up, don't fall behind again.

The best way to maximise efficiency is to do 15–30 minutes of Methods a day.

• As SACs roll around, you should aim to complete all the material that your school gives you. This provides you the best bang for your buck—almost everything you do will be highly relevant for the SAC. If you are short on material, the Chapter Reviews in both Cambridge and Jacaranda textbooks are worth doing. There is also lots of material online for you to discover, but don't stress about that unless you are really keen.

SACs matter, but they don't matter that much. Use them as a gauge for how you are progressing. Of course, everything in this guide should not be taken as gospel, but as a recommendation for how to structure your time. Personalise it to what works for you. Maybe you need closer to 1hr to work through the textbook. That is ok. Make that adjustment.

• If you have been doing textbook and SAC questions throughout the year, both Exam 1 and Exam 2 should feel comfortable and accessible to you. Start doing these after you have finished the course. Aim to do 10 sets of exams, all sat in exam conditions.

The key thing to note when doing exams is that the only thing that contributes to your study score is the number of marks you get, not which questions you did, or how pretty your working out was, or what order you did the exam in.

Go on to any Examiner's Report for Exam 1 and look at Question 1. An absolutely predictable and routine differentiation question sees a success rate of about 60%. This reveals how you get ahead of everyone:

Get all the easy questions correct, and you will beat everyone who rushed through the exam, boasted about solving hard questions, but made heaps of careless errors.

Your exam revision plan should focus on consolidation of what you already know. Don't make a mistake with integration or differentiation. Draw your graphs carefully. Create a table of contents for your Bound Reference so you don't waste time in the exam looking for stuff.

Do not stress about hard questions. Stress about getting the marks you deserve. This means doing the easy questions correctly.

Do all this, and you will have a fun time. Good luck!

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How to Maximise Score

It goes without saying that you should do everything that is written above, as this leaves you with the largest amount of time to sharpen and really excel.

A lot of this just comes down to doing more work. The most important thing is to be really cognizant of where that work should be directed towards, i.e. don't waste time doing more questions on Calculus if you know you really struggle with hard Composite Functions questions.

Being honest with yourself about your ability is critically important. It allows you to better identify and improve your weak areas.

Here are a couple of numbers to aim for in an ideal world:

• Start SAC preparation 5 weeks out from each SAC.
• Do at least 5, preferably 10 sets of practice SACs for each SAC. SACs created by other schools have been shared in various places online. Do these, only if they have solutions.
• Start Exam Questions in May.
• Do all VCAA Exams from 2010 onwards, including NHT exams.
• In May, you are unlikely to have done integration and almost certainly haven't done probability, so skip these questions. Also, skip questions that aren't on the study design. Finally, recognise the old notation that was used for transformations (using matrices) so that you can still attempt the transformation questions.
• Aim to do at least 20 sets of exams.

If you are aiming for a very high score, the message from the previous section is the same: you cannot lose marks on easy questions. This idea should underpin your SAC/Exam revision.

Finally, develop your own systems. Find what works for you. Some questions to consider:

• What should I do in reading time?
• What strategies should I have in writing time?
• How can I mitigate the amount of careless errors I make?
• If I do make them, how can I check for them reliably?
• What checklists should I make?
• What is holding me back? How can I improve these areas?

Keep chipping away at it every day. The answers to these questions depend on the individual. But if you figure these out for yourself, you will be a better student as a result (not just in Methods!).

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How to Minimise Careless Errors

I never full marked an assessment in my entire year of Maths Methods. The only thing that changed was that my percentages grew (asymptotically!) to 100%.

• I kept an Error Book, in which I recorded every error I made, the reason I made it, and crucially, something to take into the next exam/SAC.

• It felt terrible making these mistakes. It really sucked knowing that I did not get the marks that I could have gotten. Every time I made these mistakes I would commit to never making them again. I would mull over why I made these mistakes.
• I built systems and heuristics that reduced my most common errors. Here are some examples for inspiration:
• I expanded everything three times. Once initially, then the second time I would check that the numbers were correct, and the third time I would check that the signs were correct (+ or –).
• Every time I typed something in the CAS I would immediately check for typos.
• At the end of every question I would check for domain issues.
• Every time I wrote an integral I would check dx was there.
• I would check that every answer 'made sense.' This is a skill that is developed across the year, as you encounter more questions. This also comes from careful reading, because application questions are designed to make sense. If a rollercoaster was going at 700km/hr I would assume I made a mistake. If I calculated that a ball was thrown -10m into the air I would check as well. This is an advanced skill, but is ultimately the most helpful.

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Exam Technique for Maths Methods

Remember your ultimate goal in Methods: get as many marks as possible.

Reading

Reading time is so important. Broadly, tests fall under two categories:

1. This test is easy for me. I will be able to finish it with time to go. My job is to make sure that I don't make any mistakes, because my attention to detail is what will separate me from people who think this is easy and switch off.
2. This test looks hard. I will struggle to finish it in the given time. My job is to attempt every question and leave no question blank, because I know that an 'easy' mark and a 'hard' mark are worth the same at the end of the day. I will cut my losses and move on if I have to. I will separate myself by finishing as much of the exam as possible while others leave entire pages blank.

Your job is to read carefully and make this decision. Remember: in the real exams you have 15 minutes.

If you aren't at the point where every question is immediately obvious, it can be strategic to do the easy questions first and bank those marks. Ideally, by November you can complete Exam 1 with relative ease.

In short, read, read, read again. Start writing time with a plan.

Writing

Three things to start off with.

1. Read the (entire) question (carefully). To reduce the number of careless errors due to misreading, you should read every question 3 times at least. Once in reading time, once when you do the question, and once after you've done it.
2. Answer the (entire) question (carefully). Answer the question. Is it in the form they've asked for? Have I presented my answer as required? Have I checked for domain restrictions? Correct number of decimal places?
3. Watch the clock! Check the clock after every question. Are you on track? Remember, you have 1.5 mins per mark in the final exams.
4. Don't leave anything blank. It just makes no sense. You have nothing to lose and everything to gain by giving everything a shot, even if that is just writing an equation.

On top of that, here are two strategies you should try. See which works best for you, modify these, or invent your own:

1. Push through the exam at a fast pace, just below rushing. Finish the exam with time to spare, and use this time to check each question as many times as possible.
2. Take your time with each question. Use the full 1.5 minutes per mark, and make sure that your answer is fully correct before moving on.

Some people can work fast, and find that strategy (1) is better. Others find that working fast greatly increases the amount of careless mistakes they make, so choose strategy (2). Either way, you should notice the emphasis that both methods place on careful checking.