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  1. High school maths is uses very different thinking process from primary school maths. I see a lot people who were good at primary school maths wonder why they are struggling at high school mathematics, basically you must put in renewed effort to understand algebra and geometry (and later trig).

    Here are some tips: Get a book where you right down all of the theorems and definitions, do this every week. When doing an exercise refer back to this book. When studying for a test, first make a summary of this book. The summary should include every theorem that you cannot remember at the snap of a finger.

    [Rant: If A = B, then you may always replace A with B. Do not write A = B, if you may not replace one with the other. For example if you are asked to simplify 1 + 2 + 3 + 4 + 5 + 6. Do not write 1 + 2 + 3 + 4 + 5 + 6 = 1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15 + 6 = 21, rather write 1 + 2 = 3, and then 3 + 3 = 6, and then … . Or write 1 + 2 + 3 + 4 + 5 + 6 = 3 + 3 + 4 + 5 + 6 = 6 + 4 + 5 + 6 = 10 + 5 + 6 = 15 + 6 = 21. If you write A = B, then you must mean I can replace the whole of A with the whole of B at any time. End rant]

    Everything is unknown until taught/proven otherwise. If you do not know whether you may do something, then you may not do it. Memorise what you have been taught you may do (i.e. the theorems), treat everything else as something you may not do; thus do not try to memorise what you may not do, (except when you keep making the same mistake repetitively). For example do not try to memorise that sqrt(36 + 25) is not equal to sqrt(36) + sqrt(25), rather say I do not know whether sqrt(36 + 25) = sqrt(36) + sqrt(25), hence I may not replace sqrt(36 + 25) with sqrt(36) + sqrt(25) and must look for another rule to use to simplify sqrt(36 + 25). (I cannot draw the square root sign, I am using sqrt(x) to mean square root of x). There is a rule that states 36 + 25 = 61 and there is a rule that states if two things are equal, then I may replace the one with the other to get a new expression and the new expression will be equal to the old expression, thus sqrt(36 + 25) = sqrt(61).

    You may not make up your own rules. You may only use the rules that you have been taught. At each step of your work, you must mentally refer back to a theorem that you have learnt. For example do not make up the rule that (x+y)/(a + b) = x/a + y/b because you do not know whether this rule is true. Most trick questions test whether you are making up your own rules.

    Every part of a theorem is important until proven/taught otherwise, if you do not understand one part of the theorem, then you cannot use that theorem. If you change a theorem in the slightest, then you are making up your own rule and you may not do that. For example you will be taught that “if a is not equal to 0, then a^0 = 1”; that “if a is not equal to 0” is important until taught otherwise. The value of 0^0 is unknown to you (it is actually unknown to me too). The value of 10^0 is 1 because 10 is not equal to 0, hence we can apply the second part of the rule. I am using x^y to represent x to the power of y.

    Study the theorems and then look at examples to see whether you understand the theorem. Do not look at examples first to figure out the theorems, if you do this then you run the risk of making up your own rules. If you do not understand why they did something in an example, then do not make something up. For each part of an example you must ask which rule did they use. For example if the teacher writes [h(a + 1)]/[h(b+1)] = (a+1)/(b+1), go an find out which rule they used rather just saying “what is at the top always cancels with what is at the bottom”. It is unknown whether [h + (a+1)]/[h + (b+1)] equals (a+1)/(b+1), hence you may not do that.

    Do not study past exam papers first. First study the theorems, then use the exam papers to test whether you can apply the theorems. If you do not know what to do, first look at the theorems to see if you can use any of them before looking at the memo. If you used the theorems that you were taught, then you will not get a question wrong, the memo might use different theorems to get to the same result (or very rarely it might contain a mistake). Some theorems get to the result faster than others, some theorems are more useful than others, all theorems are correct; use the memo to know which theorems are the most useful. Do not use the memo to learn theorems, study the theorems first.

    In summary: All theorems are correct. Everything is unknown until taught/proven otherwise. Create a book containing what you know to be correct. Do not try to memorise what is incorrect. If you do this you will be able to get more than 90% (if you do not make silly mistakes, you can get 100%; most people make less mistakes than me and I would get 99%).

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