Hard Questions and test preperation

[Karate Chop]

Just wondering how you guys study for the hard questions that occur in tests. I'm in college and what they put in the final test is mainly stuff you haven't done before but apparently have the skills required to do the problem.

Do you use any specific books for maths?

At the moment I'm doing two lines of math, in one were doing calculus and in the other were doing complex numbers and trignometric proofs.

Do you guys suggest any books that have quite a lot of difficult questions in them along with the theory. If so could you please let me know?

and also if you guys could give me any problems which you think might occur in a year 11 mathematics exam, as a discriminator, could you please just type them out so i can have a go at them?

thanks for any assistance in advance.
john

 

[Zorodius]

I'm in college and what they put in the final test is mainly stuff you haven't done before but apparently have the skills required to do the problem.

That might be true, but such a test is a bad one. A student should be expected to make thoughtful extrapolations based on the material throughout the class, rather than suddenly dunked into it while under the abnormally stressful conditions of a test, particularly a final.

You should, hopefully, be able to get information from the instructor on where best to spend your studying time. Different instructors will emphasize different portions of a subject, and they're the only ones who really know what they're going to put on the final.

 

[uart]

I'm in college and what they put in the final test is mainly stuff you haven't done before but apparently have the skills required to do the problem.

That might be true, but such a test is a bad one.

I know what you're saying but there is a fair bit of scope in exactly what "stuff you haven't done before but apparently have the skills required" means to different people.

Personally I don't think it's always a bad idea, as long as sufficient time is given. I know that many examiners tend make tests that have very targeted and straight forward questions (the type you can easily regurgitate), but each question has to be done in a ridiculously short period of time. In such tests having a fairly good overall subject knowledge is often not sufficient to get a good mark, and "swatting up" tends to be more important.

On the other hand an exam that has fewer questions but of the type that are not so likely to be easily "swatted and regurgitated" will often favor the type of person who has really learned the material over someone who can just reproduce the most likely candidate questions very quickly (and often forget them just as quickly a few weeks after the finals).

Oh BTW, to answer the original question. Getting hold of some past years exam papers (if possible) is one of the best targeted ways to study. You should really try and work towards being able to pick an old exam and complete most of it without reference to books or other material. Even if your progress is initially much slower than required to compete the exam in the allocated time don't be tempted to just go looking for a pre-worked answer to memorize but instead try to persevere and complete as much as you can unaided, it's breakthroughs during this type of work that really lead to deeper and more long term understanding. Once you can do this much then it's a good idea to run through the “likely” problems to aim for more speed and the possibility of having some of the most likely stuff pre-worked in your head for the real exam.

 

[Karate Chop]

thanks for that guys. I do have a few past exam papers on the subject I'm learning at the moment and they all seem pretty easy, which is what I'm a bit worried about, because i know the one I'm going to have is going to be a lot harder. I think this because it happened to me in physics, i had two previous tests from a few years back, and i completed all of the problems without much difficulty, so i was pretty confident that i would do extremely well in the one i was going to take, however this was not the case.

I just want to prevent this sort of thing happening again, i would rather be over prepared than under prepared, which is why i need some assistance in finding some challenging problems with complex numbers and also calculus. I don't know where to find questions like these, most of the books I've looked in have basic questions and just the theory, and I'm not very good at making up questions. Have you guys got any questions which you think might be of help to me or any questions you've had difficulty with in the past?

 

[uart]

Post what grade/level maths you're studying and then perhaps someone could provide you with links to extra resource material.

 

[arildno]

Karate Chop:
How do you think the exam format will be?
Examples:
a)Multiple choice
b)More "wordy" exam, for example be required to prove some theorems (this is what I think many would regard as a particularly nasty exam format..)

 

[Karate Chop]

well i asked the teacher and he said that around 60% of the test will be just the standard questions which we have done in class, whereas the other 40% would most likely be the proofs and tougher questions which involve quite a lot of knowledge in the subject area. He said these questions would advantage the brighter students, especially time wise, since they put in questions which have shortcuts in them. So if you can spot the shortcut you will save a lot of time. I was told that the calculus exam was also going to be the same, however there were a lot more worded questions from which you had to extract functions.

So yeah, just basically looking for challenging, but not impossible, questions on complex numbers. Probably mainly the proofs and some questions which will help me fly the factorisation of unreal polynomials.

For calculus, just looking for challenging worded questions.

and answering arildno's questions

a) Might be about a page of multiple choice questions in the calculus exam, but none in the complex numbers one.

b) Only the calculus exam will have worded problems. In the complex numbers one I'm guessing it's just going to be plain old mathematical proofs and deductions. I'm not sure if there will be multiple choice ones in this, I've never seen one involving complex numbers, if you know of a good worded question please do let me know.

cheers, I'm in year 11 by the way and am doing the specialist maths course.

 

[FlatlineLemon]

Well, first off...good luck on your exam. Hope you do well. When i have no idea what types of questions will be asked, i simply do everything. I pace myself and do every problem extensively. several times if i think its difficult. With the classes i take, it always seems like those are the ones that end up on the test. Of coures by now, everyone knows that there is no way to read the teachers mind. So basically just overstudy and say a prayer. Good luck again.

 

[Zorodius]

For calculus, just looking for challenging worded questions.

I found this tricky:

Sketch the circles x²+y²=1 and y²+(x-3)²=4. There is a line with positive slope that is tangent to both circles. Find the points at which the tangent line touches each circle.

The answer is (-1/3, (2√2)/3) and (7/3, (4√2)/3).

 

[mathwonk]

It is hard to give advice without knowing what sort of school you are at, whether it is an Ivy league school, European school, or a state college in the US, and what type of course you are in, honors level, math major, etc.

My experience is mostly with average non honors calculus at a state college. Most of my students do not seem to prepare as well as you are trying to do, so maybe my advice is bad for you.

But I think it is actually extremely easy to "read the teachers mind" about what will be on the test, as the teacher almost always tells you in class exactly what to learn for the test. This happens also at top schools. The problem is many students ignore what they are told and do the least amount possible hoping to pass.

So first of all listen closely to your professor. He/she will write the test and hopefully grade it.

As to a source of problems, first of all know how to do every single problem that has been given on homework or worked in class. Also know how to work the examples that are worked out in the book. And know every single problem that has appeared on a previous test from this same instructor in this same course.

It is depressingly easy to stump a class simply by giving the same test over again next time.

The teacher is telling you what to learn by everything he teaches or emphasizes, or assigns on a test or homework or reading assignment. Going into a test without learning the answers to old questions is ignoring the hints you are being given for reading his mind.

In my experience it is also trivially easy to stump most classes simply by giving exactly the same problems that were worked out in examples 1 or 2 in the chapter, or by picking homework problems from the first 10 or 20 at the back of the chapter.

Furthermore almost any standard calculus book, like Stewart, or Edwards and Penney, has a huge array of problems at the back of every chapter, of all levels of difficulty, ranging from trivial computations to theoretical worded questions and proofs.

I would be surprized if your book does not contain such questions, including some more difficult than the standard ones.

It is hard to imagine a class in a typical college today where the student could not get an A+ by preparing by working all (or half of) the problems in the book.

So I suggest studying:

1) what you were told to study.
2) what you were given as homework
3) what you have been tested on before
4) what was worked as examples in the book
5) harder problems at the end of the chapter

On another level of difficulty, when I was a senior level student taking a grad course in real analysis and measure theory, and wanted to be sure I got an A, after learning all the material in our course, (every theorem and every proof), I bought an alternate book on the subject and read that too, and then worked out all the old tests given by the same professor that I could lay hands on.

On the final, sure enough he asked a question that was a standard theorem we had never seen but that was proved in the alternate book, and I got it easily.

I also knew cold the theorems that he had a habit of asking on old tests in prior years.

Even though I had never handed in a single homework all semester, I had a perfect paper and got an A. (Not all professors grade like that, my buddy got a B+ on his final in honors calculus and got a D- for the course. Read the grading policy or ask.)

In another class, I looked up in advanced books the proofs of theorems our teacher had taken for granted in our course, after learning all our own material.

Then on the final, he asked 7 questions and said to work any 4. I worked all 7 and asked him to grade any 4. I also added the extra proof he had left out of the course. I got an A, after getting a D on test 1.

Oh yes after the wakeup call on test 1, I bought a Schaum's outline series problem book and worked as many problems as possible. They seem to have dumbed those books down a bit now though, and they do not contain as many hard problems as before.

So, after doing 1-5 above,

6) go to the library and get alternate books on the subject and read those.


But mainly learn to understand the principle behind the methods you are using to work problems. Do not be like the student in my class, who said he had learned how to maximize the volume of a box with a closed top as I did in class, but he thought it unfair for me to ask on a test for him to maximize the volume of a box with an open top.

That's just plain stupid. Every professor will expect you to apply the ideas you have learned, to related but different situations. Otherwise there is no point taking the course. It is hopeless to learn to solve every problem in the world individually (Ionesco's hopeless student notwithstanding, in "The lesson").

You have to learn general methods that work on whole classes of problems and then be able to work any reasonable problem of that type. Try to see what all problems of a certain type have in common.

After learning to work problems, you should also learn the theory behind the material. This means first of all learning carefully the statements, and understanding the meanings, of the definitions and the theorems.

Learn to use the theorems to prove the important corollaries, and finally learn the proofs of the theorems.

Then practice explaining things to your friends and classmates, and set up a study group to compare levels of understanding and help each other. Trying to do it all alone is usually hard except for unusual students.

On another level, and for future courses, one of my professors said when he was a student he read the material at home before it was sacheduled rto be rpesented in class so he could compare differences of emphasis. Another student I knew sat in on a hard course all year the year BEFORE he took it, and did the work, so when he took it for a grade it was the second time.

I myself sat in on one like that once, but after learning it I did not see the point in sitting through it again, and never took it for credit. I learned more from that course than many others I actually took for credit.

Anyway, you are on the right track, good luck!

 

[Karate Chop]

what are some of the types of questions you would put in tests that would most likely give them a bit of a hard time? we've just learned the product, quotient and chain rule and have done maximum and minimum problems, integration and differentiation aswell. not very advanced stuff on it though. is it possible to give me a few examples of the questions so i can see how i go?

 

[Alkatran]

In my grade 12 and 11 physics, grade 12 through 10 maths, ETC the teachers always had "new material" on the tests that made a lot of people get bad notes.

Of course, by "new material" I mean new "problems", which required a slight amount of logical thinking to solve. People would memorize all these formulas then hit the test and have no idea where they went! It must've have been hilarious to correct them, considering some of the things I'd heard.

This, of course, made the tests easier, and the exam was something along the lines of "copy paste" from the book. No I am NOT kidding. I got a 97 because the exam gave us questions we'd alreay answered. There's a note I deserved.

 

[mathwonk]

The stuff I put on that throws people is the stuff that requires thinking instead of just rote behavior. For instance I emphasize that not all integrals can be computed by antiderivatives simply because you do not always know the antiderivative. Nonetheless you can always estimate the integral by fitting the graph under a known graph and using the principle that a smaller graph has a smaller area.

So for instance I will ask them to prove that the integral of 1/(sqrt(1+x^4)), taken from 0 to 1, lies between 1/2 and 1. Well no one ever gets it because they have memorized how to compute integrals by antidifferentiation, and this integral does not have an antiderivative that anyone in class has ever seen.

Even though I have emphasized that you can estimate integrals, no one notices that this function's height is always caught between 1 and 1/2, so its area is also between 1/2 and 1, (since the base has length 1.)

the problem seems to be that many people are not thinking at all about what an integral represents, just what is the rule to get the answer.

So this question should be much easier than actually antidifferentiating a hard function, but no one gets it because they have to use what they know, and not just repeat what they have done over and over.

But this is not what your prof is going to ask, because she/he will try to test the knowledge that they have taught in your class.

Another thing very hard for many students is to justify the answers they give. I.e. most will memorize that to find a maximum you just set the derivative equal to zero and solve. But if you ask them to explain why it is a maximum, they will not say any more, just that the derivative is zero.

They forget or ignore that that is the same procedure for finding a minimum, so how do they know which one they have? or maybe it is neither?

Any kind of proof or reasoning is very hard. For example if I explain with a picture why the fundamental theorem of calculus is true, i.e. why the area under a curve is given by the difference in values of the antiderivative, no one can ever give me back this explanation on a test.

Or if I prove that the function L(t) defined by the integral of 1/x from 1 to t, satisfies the law L(at) = L(a) + L(t), no one can ever reproduce this proof. Many will just say something like: "Well that's the log function and that is the law logs obey". This misses the whole point that one does not know this integral is a log function until one has proven it obeys the same laws as a log function, so this reasoning is backwards.


Most students lose points by not reading or following directions. If I say "do not use any properties of logarithms" in answering the previous question, this has no effect.

Many do not learn the meaning of a derivative, from the definiton, but try to learn to compute all the specific derivatives in the world. then when I try to explain why the derivative of an area is a height, or of a volume is an area, they are lost.

good luck!

 

[Alkatran]

The stuff I put on that throws people is the stuff that requires thinking instead of just rote behavior. For instance I emphasize that not all integrals can be computed by antiderivatives simply because you do not always know the antiderivative. Nonetheless you can always estimate the integral by fitting the graph under a known graph and using the principle that a smaller graph has a smaller area.

So for instance I will ask them to prove that the integral of 1/(sqrt(1+x^4)), taken from 0 to 1, lies between 1/2 and 1. Well no one ever gets it because they have memorized how to compute integrals by antidifferentiation, and this integral does not have an antiderivative that anyone in class has ever seen.

Even though I have emphasized that you can estimate integrals, no one notices that this function's height is always caught between 1 and 1/2, so its area is also between 1/2 and 1, (since the base has length 1.)

the problem seems to be that many people are not thinking at all about what an integral represents, just what is the rule to get the answer.

So this question should be much easier than actually antidifferentiating a hard function, but no one gets it because they have to use what they know, and not just repeat what they have done over and over.

But this is not what your prof is going to ask, because she/he will try to test the knowledge that they have taught in your class.

Another thing very hard for many students is to justify the answers they give. I.e. most will memorize that to find a maximum you just set the derivative equal to zero and solve. But if you ask them to explain why it is a maximum, they will not say any more, just that the derivative is zero.

They forget or ignore that that is the same procedure for finding a minimum, so how do they know which one they have? or maybe it is neither?

Any kind of proof or reasoning is very hard. For example if I explain with a picture why the fundamental theorem of calculus is true, i.e. why the area under a curve is given by the difference in values of the antiderivative, no one can ever give me back this explanation on a test.

Or if I prove that the function L(t) defined by the integral of 1/x from 1 to t, satisfies the law L(at) = L(a) + L(t), no one can ever reproduce this proof. Many will just say something like: "Well that's the log function and that is the law logs obey". This misses the whole point that one does not know this integral is a log function until one has proven it obeys the same laws as a log function, so this reasoning is backwards.


Most students lose points by not reading or following directions. If I say "do not use any properties of logarithms" in answering the previous question, this has no effect.

Many do not learn the meaning of a derivative, from the definiton, but try to learn to compute all the specific derivatives in the world. then when I try to explain why the derivative of an area is a height, or of a volume is an area, they are lost.

good luck!

That is exactly the problem there was in high school. (I noticed it in grade 11 for the first time, the fact that people just didn't.. GET it)

I have gone into physics tests without studying, NOT KNOWING a few of the formulas, then figured them out on the test. (Mind you, these were easy formulaes) I would just think "Ok, I have charge... and charge/area... and I want area... DUH". Of course this won't work in University physics. :surprise:

Sometimes I have trouble following directions. I've made it a point to read them twice, then again when I'm done... so I can redo the whole thing.

One time of note: I take math in french, and in french the decimal point is represented by a comma... but so is the division between series. On a test there was: Find the equation for the series:

1,0 , 12,5 , ... (you get the idea)

I thought I had to go 1, 0, 12, 5, ... when I only had to go 1.0, 12.5, ...
I came up with a formula for it (which would have been beyond anyone else) and still got 0 points. The whole point of the question was to check if I could figure out the formula for a series... should've gotten partial points... grumble grumble...

 

[mathwonk]

yeah, that reminds me of a physics question in high school where they asked for the percentage of the sun's light that comes to the earth. That was a stupid number that was in the book, like we were supposed to memorize that!

So i figured well, I know how far the Earth is from the sun, so I will compute the area of the sphere of light with the sun at the center and the Earth's distance as radius.

Then I will compare that to the area of the half of the Earth facing the sun.

But I got it wrong because if you think about it, it is not the area of half the Earth that gets the light, but the area of the chunk of the light sphere subtended dby the earth, so that is roughly a disk whose radius is the same as that of the earth, not a hemisphere.


So i got a zero on the only interesting question on the whole stupid, test, and the one on which I made the most intelligent attempt of any test in my high school career.

(I got all the other questions right, and on three tests taken the same day, got two 100's and one 97 for that one stupid 3 point question.)

Ahhh. so what, I have almost gotten over it, 47 years later.