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Our teacher loves to test us on pure concept based questions and test if we really know what we are doing when learning a particular lesson.

For example, when we first started learning about trigonometric identities, he once put a question on a test that was something along the lines of $$\sin18 \theta = \frac12$$

Looking back on this question now, it's extremely simple, but at the time of the test, I was perplexed because I had never seen this question before and never ran into it.

I understand what is going on in school and never fail to do my homework, but always end up missing one or two questions on the tests that pertain to pure conceptual questions that we have never seen before.

My question is: How can I prevent making this sorts of conceptual errors on these tests when running into them. Is there a website or certain textbook I can refer to, or some exercises I can do, in order to prepare for this sort of conceptual testing?

Shrey
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  • why do you consider this to be extremely simple? – Quality Apr 12 '15 at 05:05
  • @LearningMath This was the first lesson taught in the beginning of the year. Since then, I've reflected on question $sin18\theta = \frac12$ and I know how to solve it now. The question I have is for future math problems and concepts, not this one. I was just giving an example. – Shrey Apr 12 '15 at 05:07
  • Is the class trigonometry? What are you currently learning in class? – Christopher Apr 12 '15 at 05:17
  • This is an advanced trigonometry/pre-calculus class. It covers all of trigonometry (identities, polar equations, etc) as well as pre-calculus (parametric equations, sequences and series, limits, derivatives, etc). – Shrey Apr 12 '15 at 05:22

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As someone who is mostly self-taught and your age, my advice is you must first get comfortable with abstraction and proofs. You should be arguing with your teacher either mentally or verbally. I built up a great relationship with my trigonometry teacher this way because he welcomed my pressing curiosity, although there is no guarantee this can work for you.

I assume you do hold some mathematical curiosity (after all you're on Stack Exchange and your other questions suggest so), so a good place to start is with your current textbook. Skim the chapter ahead of time for certain theorems, and become adept at reading the theorem and proof. Exercises can only improve the time at which you work. So if time is an issue, then you should do the exercises before the test. Also, although Wikipedia and sometimes Google are frowned upon, they are usu. good sources to read. MIT OpenCourseWare and Khan Academy as well. Over the course of learning, you will learn what it means to cleverly manipulate equations rather than do procedural math, do your own rigorous proofs (besides simple algebraic derivations), and to ask and answer many questions to arrive at a solution.

Most importantly however is that you model whatever schedule to yourself. I don't force myself to do what I consider banal and boring. I may study math for 3-11 hours per day, but I do so because I enjoy it. And hey, what you might think to be boring might not be after you see some potentially cool connections.

Christopher
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