10mathtips

10 Math Tips to Save Time and Avoid Mistakes

[Total: 10    Average: 4.4/5]
Exam situations are always situations of stress. It comes with our endeavor to be as good as possible together with our fear to fail. Some students handle these situations better than others. But there are some tricks I encountered over the years tutoring young students. I’m sure everybody has developed their own ways to get along with tests, so the following has to be a personal view of mine. It resulted from my experiences when I considered the mistakes my students had made. They are also of help on the daily business to do homework as quick as possible, save time, and avoid mistakes.

Preparation

Preparation is more important than one might think, and I don’t mean the learning process here – this is self-evident. I mean the physic preparation for an exam or homework.

  • Have some scrap papers at hand. You might want to do a calculation for control purposes, a long division, a quickly done graphic, or some plausibility checks which don’t belong to your final answer.
  • Get your writing tools ready: pencil, eraser, sharpener, ruler, compass or whatever is needed and allowed. Have some additional pencils and maybe colored ones, just in case.
  • Don’t forget your calculator or book of formulas if you’re allowed to use them. Check the batteries.

Now you can begin. But before you dive in the first problem, get an overview of all questions, just a short impression of what they are about and whether they match your expectations. You don’t need to read them in detail here, but you should do a triage, i.e. sort them in mind from easy to difficult. Maybe you even mark them with A-B-C for this reason. And this rating should be an individual one: your own perspective and none in an objective sense or whatsoever.

Clock Management

  • Don’t panic. I know this is sometimes easier said than done, but it is important. Panic of any kind rarely is helpful. And although different people have different methods to deal with it, your preparations can help here, e.g. you can sort your writing tools, so this manual process can be used to channel your emotions and distract you.
  • Don’t think … for too long on the same problem! You can change your rating, and if it takes too long, upgrade the difficulty and tackle the next question. If you’ll have time left at the end, come back on those questions.
  • This means as a general rule: easiest questions first.
  • If you’re stuck, you could easily invest the entire available time on one certain problem. However, you’re here as a gatherer, not a hunter. You want to gather points. And this means sometimes many small steps instead of a few big ones. So in case your working flow is interrupted: switch to the next hurdle. If your passing play doesn’t work, you’ll have to run for the yards. Of course a Hail Mary is more attractive than three runs for a new first down, but this doesn’t count here.
  • In case you’re left with plenty of time and have solved even the hard problems, there is a risk that you correct right things wrong. To go over your solutions again can be useful, but normally tension has left your body and you start to make mistakes. So it might be the better choice to end here.

After play

This part maybe the most difficult one. Yes, you might feel relieved because it is done, but chances are high that it still is haunting your thoughts. But done is done and you shouldn’t waste time on what cannot be changed anymore. I know this is easily said. But it is true, get used to it. Instead it can be helpful to concentrate on new lessons. But time will come when you receive your results. And what should it have been good for, if there cannot be learnt from mistakes. So although it is over, it is also necessary to reconsider it. It’s this balance, what makes the time after a test a bit difficult. So the best thing would be to deal with it the most analytical way possible. The task is clear: to learn from mistakes. To do so it is helpful to reconsider the wrong answers and eventually write a correction. Figure out where you have been mistaken. You should also check them against the list of tips and tricks below, which is meant as a guide on how to solve individual questions, resp. problems.

1o Tips and Tricks

 

1. Read it twice

Read the question in one step once: from start to end. This way you know what it is about. Maybe you’ll already get an idea how to solve it. After that read it again for the second time. But this time use your scratch paper to extract the numerical information given and mark the quantity which is asked for by a question mark. Now you can tackle it: Huddle – Line Up – Start.

2. Claim, Conditions and Dependencies

It is important to keep given conditions and assertions apart. I’ve seen them confused a lot of times. Even if the result might be incidentally correct, it will be difficult to impossible to read your answer. This will be bad not only for your readers, but also for yourself, if you want to check it for mistakes. Part of it is to keep track of the right direction. Sometimes you’ll need to start at the claim to get an idea. It is somehow running first backwards, before you know the right way to go forward. Do this on your scratch paper and don’t forget to reverse it again, when you leave the box, run and pass the ball. Typical examples for such a procedure are ##\varepsilon – \delta## proofs or limit calculations.  Also mark dependencies whenever it is helpful. For all ##\varepsilon## there is an ##N##should actually be better For all ##\varepsilon## there is an ##N_\varepsilon##or ##N(\varepsilon)## because the choice of ##N## depends on the choice of ##\varepsilon##. The same is true in physics: A velocity might be dependent on time, so ##v=v(t)##. Use ##v(t)##. It is at almost zero costs in terms of time, and it makes sure you don’t overlook internal dependencies, like inner derivatives. It is important, because the same situation could as well be ##v=v(x(t),t)## with the position ##x=x(t)## at a certain time and an entirely different situation. Keeping track of those dependencies automatically helps to avoid mistakes.

3. Think or Write?

Write instead of think. It sounds silly at first glance and I remember some very gazing views. But it is true. Writing is several times faster than thinking. What I mean is of course not that you should stop thinking. What I mean is this:

How many yards does a car get, when I’m distracted by my mobile phone for 10 seconds at a speed of 70 mph? 

So instead of trying to solve

$$ 70\, \frac{mi}{h} \cdot 10 s $$

by calculating with 3,600 seconds an hour and 1,760 yards a mile all in your mind on an imagined big quotient, write it down, the more steps the better, because it means less risk and is easier to check:

$$70 \,mph \cdot 10 \,s = \frac{70 \cdot 10}{1} \cdot \frac{mi\cdot s}{h} = 700 \cdot \frac{1,760\, yd \cdot s}{3,600 \cdot s}\\ = \frac{700\cdot 1,760}{3,600}\, yd = \frac{7 \cdot 1,760}{36} \,yd = \frac{7 \cdot 440}{9} \,yd = \frac{3,080}{9}\,yd\\ = 342.2 \,yd $$

This might be a bit over-the-top but it is still a lot, lot faster than doing it in mind, unless you are a savant. And the only risk of mistakes here is by writing, not by concept. It actually saves time, although it doesn’t appear so.

4. Units are your friends

As my example already has shown, keeping track of the units is important. It is an automatic control for free and therefore reduces the risk of mistakes. If you are asked about yards as in the example and you still have miles or even seconds in your final result, then something went wrong. Units are your friends. Use them all the time, even if you have to invent them: If a woodcutter chops four trees an hour, then write ##1 W \triangleq 4\cdot\frac{T}{h}##. It might look funny, and if also work occurs in the problem you should chose another letter, but it definitely helps you to keep track on what you’re doing. Remember the catastrophic failure of the Mars Climate Orbiter.

5. Fractions

There is a lot that can go wrong with fractions: minus signs get lost, primary and secondary fractions are confused, if either the nominator or the denominator or both are quotients themselves, or units miraculously change from one side to the other, division by a quotient results in the wrong multiplication, the distributive law is forgotten – so basically every possible mistake is twice as likely if it is part of a quotient. So there is only one real advice: be careful and distinguish primary and secondary quotients by their length and position. Another misbehavior which seems to last forever is a notation like ##1\frac{1}{4}##. Personally, I almost hate it, because usually, if we leave out a sign, it will be a multiplication dot as in ##5\,m## which is ##5 \cdot m##, or ##2\pi## which is ##2\cdot \pi##. But when people write ##1 \frac{1}{4}## it is all of a sudden a plus which is dropped, since ##1\frac{1}{4}= 1+\frac{1}{4}=1.25\,##. Even worse is the horizontal expression: Is it ##1\,1/4 = \frac{5}{4}## or ##1\,1/4 = \frac{11}{4}\,##? And what does ##1\,\,1 / 2 + 2## mean?

So either use the decimal expression ##1.25 ## or even better than this, write ##\frac{5}{4}##.

6. Parentheses

Use parentheses and if in doubt, use them. Sometimes this will result in quite a few nested parentheses but the solution to this is not to omit them but to use different ones: ##\{\; [\; (\;)\;]\;\}\;##. The most common mistakes are sign errors like

$$ – (1+ x) = -1 + x $$

and lost factors by disrespecting the distributive law like

$$ – 2 \cdot (1+ x) = -2 – x $$

Needless to say that these are completely unnecessary. But without parentheses things are even worse.

7. Accuracy

A common question is: how many digits should I use? Well, it obviously doesn’t make sense to answer a question like:

How long is the diagonal of a room which is ##4.9 \,yd## long and ##8.7\, yd## wide?

by

$$9,9849887330932927798322882923393\, yd$$

just because the calculator says so. However, the problem with precision starts normally earlier than at the final value. What to do in between? A golden rule is to calculate as accurate as possible and only adjust the answers to the questions, whether in between or at the end. If we calculated instead with ##5\;yd## and ##9\;yd## in the example above, we would give as an answer ##10.3 \;yd## whereas the rounded answer would have been  ##10 \;yd##. This might be a small difference, but it shows the problem: calculation errors pile up, so this rule makes sense: calculate correct, answer meaningful. This means roughly to use as many digits as the input values or measurements have.

Another source for precision problems is ##\pi##. Calculating with ##3.14## might not be sufficient, so the use of ##\pi \approx 3,14159265359## might be justified, but within an answer it is probably not. Mathematicians usually keep those values as they are: ##\pi \, , \,\sqrt{2}\, , \,e## and leave it to others to put in the values they need. In this sense ##2\pi## has to be preferred over ##6.28##.

8. Inequalities

First of all: Don’t forget to turn the sign if you multiply by a negative number and consider both cases, in case you multiply by a variable which could be negative!

$$2 < n \text{ leads to } -2 > -n $$

and

$$\frac{2}{x} < 4 \text{ means } \frac{1}{2} < x \text{ or } \frac{1}{2} > x$$

The general treatment of inequalities depends heavily on the purpose they are used for. If we want to estimate the maximal run time or space an algorithm needs in a worst case scenario, or if we want to determine the error margin of a numerical solution of a differential equation, then we will certainly try to calculate these quantities as accurate as possible. This means the closest inequalities that are possible have to be taken: ##M=0.01## for ##\frac{1}{n^2}<M## with ##n > 10##. But far more often these narrow margins aren’t needed. If we have to proof a typical limit problem like

$$\lim_{n \to \infty} \frac{n+1}{1-2n}= -\frac{1}{2}$$

we’ll have to show, that for every ##\varepsilon > 0## (any arbitrary small given(!) margin) we can find a natural number ##N(\varepsilon)## such that all following terms of the sequence, that is all ##\frac{n+1}{1-2n}## with ##n > N(\varepsilon)## are closer to the limit ##-\frac{1}{2}## than the given margin. This is the entire idea, and it doesn’t play a role whether we chose ##N(\varepsilon)## as small as possible or not. There is no need here for especially narrow margins. So we can be generous and write

$$\left| \frac{n+1}{1-2n} – (-\frac{1}{2})\right| \stackrel{(n\geq 1)}{=} \frac{3}{4n-2} \stackrel{(n \geq 3)}{<} \frac{3}{3n} = \frac{1}{n} \stackrel{(n \geq \frac{2}{\varepsilon})}{<} \varepsilon $$

and we can simply chose ##N(\varepsilon) = \max\{3,\frac{2}{\varepsilon}\}\,##. Don’t even try to find a smaller value for ##N(\varepsilon)##. It takes more time, bears the risk of more errors and hasn’t the least advantage over a greater value.

9. Guesswork

  • Don’t conclude or guess on assumptions.
  • Don’t make hidden assumptions.

Both of these mistakes sometimes creep into a proof or a calculation. E.g. in my previous example, when an inequality is multiplied by an unknown value, and you are tempted to consider it as a positive quantity, simply because you don’t consider the other possibility. Or a division without checking if the divisor is zero or not. And often there are even more subtle ways for hidden assumptions to enter your calculations at certain points. Take nothing for granted unless it is listed in the question. Also fast conclusions based on intuition rather than facts can be a honey trap. I remember that I tutored an eighth grader, whose method it was to check her answer by looking into my face instead of thinking herself, whether she was right or wrong. It is a quite smart idea. Unfortunately I knew this and looked accordingly and during a test, there wasn’t someone to control. So look out for the hidden things!

10. Plausibility Check

At the end of each calculation, there should be a short consideration, whether the achieved result makes sense or not. Are the units correct? Are the quantities within the expected range? Short: Is the result plausible?

If you started with lightyears, it is very unlikely that your result would be in millimeters. If you’re asked for the volume of a space, something cubed should have occurred in your answer. In earlier – pre-calculator times – students had to use slide rules or tables. They gave some digits as a result and a calculation which quantities had to be used, in terms of significant digits and the position of the decimal point had been mandatory. Times might have changed, the reason to check amounts, numbers, and units hasn’t.

Appendix

The discussion of this article has revealed another important hint which I like to include and give an example how to use it.

11. When in doubt, draw it out!

Most people (three out of four) are visually driven, i.e. they grasp an idea faster by visual input in contrast to acoustical input. So whenever possible, it can help a lot to sketch a given problem. If you’re well prepared, then you have some paper at hand to do it. Try to include all given information, but only be as accurate as necessary. E.g. I know that straights are given by an equation ##y=mx+b## and that the slope ##m## is a quotient. But was it ##m =\frac{\Delta y}{\Delta x}## or was it ## m =\frac{\Delta x}{\Delta y}\,##? One way to choose it right, is to remember that units are your friends. So for the units we have ##[y]=[m]\cdot [x] + [b]## which shows, that the ##x-##part of ##m## has to be in the denominator in order to get the correct units: ##[y]= \frac{[y]}{[x]}\cdot [x]## and therefore ##m =\frac{\Delta y}{\Delta x}## has to be the correct one. Another possibility is to sketch it with a few lines. The following picture shows two ways, how this could be done:

Now guess what took me longer to draw, the left or the right one? If we have a question about certain numbers, and we want to check their plausibility, then the right one might be of greater use. Here, however, we spoke about to illustrate the slope ##m= \tan \alpha##, and in this case the left one is sufficient. There is no extra award for exact scales, arrows or similar, only costs in terms of time. Therefore be as accurate as necessary, but do not paint a van Gogh if a Miró will do.

 

 

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  1. QuantumQuest
    QuantumQuest says:

    Really good advice and list of tips. From my own experience in the past I'll stress out the third one: Think or Write? adding the benefit of focus to this, as it helps being more focused than when thinking, not only in calculations but in general as someone tackles a problem in the context of the limited time of an exam.
    Also, easiest questions first is something often overlooked under stress conditions but it is really indispensable. This cost me once a generous -30% because I spent a great lot of time on a hard problem as I knew how to solve the other ones. I finally solved it but I didn't have the time to write down all the rest.

    Education, in my view, is a huge interconnected system of chains of stepwise improvements across disciplines and failure is an inherent part of the process. This – as is pointed out in the insight article, has to be taken as an opportunity to learn from the mistakes and go to the next level. There is nothing that can be undone after an exam failure but there is no need to undo anything either.

  2. DS2C
    DS2C says:

    Great stuff. As far as skipping hairy questions, I just experienced this on my midterm. The question made no sense to me at all and I spent too much time on it already. So I skipped it and came back at the end to give it another go. Still couldnt do it. Turns out the question had a typo and could not be solved nor simplified. Glad I eventually skipped it!

  3. WWGD
    WWGD says:

    One good reason to read everything first is that you get the ball rolling on those problems at a subconscious level. When/if you go back to them later, you have (subconsciously) done some of the work, broken down the question somehow./somewhat

  4. WWGD
    WWGD says:

    If I may, this one has been helpful to me recently: I have around 3 options (with in-betweens, but let's keep it simple for now) of what I intend to do with material at a given point:
    1) Thorough reading and getting myself
    2) Skimming, intro reading, breaking material down.
    3)Reviewing

    I try to decide base on the context, including considering my mental condition, which I can do. Under ideal conditions I try 1: enough time, rested and clear-minded.
    Under worse conditions, tired and/or on public transportation,I try just reviewing. I often text myself notes on important things and I just check them while there. The idea is that if you have a clear goal of which of the three you will do, it is easier to get them done, and feel satisfied you did what you set out to do. You may even do this at a more "micro" level and do changes as you read the material. It is not realistic to fully get material that is new when you are not fresh , let alone tired, etc.

  5. Biker
    Biker says:

    This honestly happens to me a lot. How I deal with this is by solving everything again before exams not because I need to understand everything from the beginning or memorize problems but to ensure that my hand and my mind are focused and don't make silly mistakes like forgetting a sign..etc. It just bumps my confidence. Also, If I have not revised before the exam or a quiz, When I solve a question, I always get the feeling that I did something wrong so I go back, check then continue which wastes time.

    I myself like to not read questions. If I am nervous, I don't want to read them and say "This might be difficult" because of how an equation looks. It distracts me while solving other questions. I simply go one by one, If I am not able to solve it or I need more time I leave it for the end.

    Really nice insight

  6. vela
    vela says:

    The third suggestion is really good. I'm amazed at the reluctance of my students to write stuff down. They repeatedly try to do calculations in their head, invariably make mistakes, and ending up spending way more time on a problem than they should have.

    Here are a few suggestions I have:

    Check your algebra as you go. After you write down a line, go back and redo the calculation. You can often catch simple arithmetic errors, sign errors, and the like. It's a lot easier to catch and fix those as you go than after you get to the end and discover your final result is wrong.

    Figure out the best way to do the algebra. When I did homework, I wasn't satisfied with simply getting to the correct result; I wanted to find the most efficient or elegant way to get there. After a while, it made a big difference. I learned to do algebra faster and with fewer mistakes than my peers. And it was eco-friendly: while my classmates would turn in 30-page homework assignments, I'd be turning in 12 pages.

    When doing physics problems, don't plug numbers in right away. For one thing, it reduces the amount of stuff you have to write. The less you have to write, the less likely you'll make a careless mistake. And when you do make a mistake (and you will make a mistake sometime), it's a lot easier to find it when you're working with variables instead of numbers.

    Learn to use your calculator. The device is supposed to save you time, not be another source of errors because you don't know how to enter ##5.42times 10^{-4}## correctly.

  7. cccccttttt
    cccccttttt says:

    As a math teacher of many years certainly support the 10 tips.

    But would add an 11th:

    "DRAW IT TO KNOW IT"

    Much of the brain is devoted to visual processing.

    Using words came along a very long time later.

    So have found this an effective approach:

    1. draw diagram to illustrate the theorem, concept, or process

    2. discuss in detail till student says "that's easy, lets move on".

    3. give student a blank sheet of paper

    4. ask them to draw diagram from memory.

    5. compare with original

    6. good match: congratulate them as we "move on" to the next topic

    7 poor match: go back to step 2.

    We all deceive ourselves to some degree as to what we really know.

    Seeing the diagram is a vital feedback.

    The act of drawing uses visual, motor, and kinesthetic senses to cement
    the info into the nervous system.

    Once students experience success, they find it works just as well when studying alone.

    ct

  8. Mark44
    Mark44 says:
    cccccttttt

    "DRAW IT TO KNOW IT"

    This is excellent advice. For some reason, students are often reluctant to draw an image of the situation, thinking that doing so will take too much time. This is often a false economy, as taking a short time to get the wrong answer is not an optimal solution.

    Another possible reason is that, students are often enamored of working with algebraic symbols. Using the visual part of the brain adds some insight that isn't possible with the algebra alone.

  9. fresh_42
    fresh_42 says:
    cccccttttt

    "DRAW IT TO KNOW IT"

    Mark44

    This is excellent advice. For some reason, students are often reluctant to draw an image of the situation, thinking that doing so will take too much time.

    I really have forgotten this, resp. I thought of it afterwards. In my experience, one of the reasons they don't draw something is, that it actually does take too much time. However, the reason is, that they start to draw accurately, chose a scaling on the axis, recognize it doesn't match the situation, take – no search for – an eraser, erase nearly everything and start again at zero, don't find their compass and so on and so on. At least this is what I've experienced. It's hard to get them known, that a drawing is about the principles of a problem and not the statics of a building. I think this should be practiced far more often than it actually is. At least I mentioned the drawings as a reason to prepare for. But you're right, it should have been on a more prominent place.

  10. nepsaol
    nepsaol says:

    Great list! I normally have problems keeping calm during exams and would consider myself a master of "shitty mistakes", especially when it comes to simple algebra that I normally have no problem with. All in all, this article gives you great tools to improve the structure of problem solving. Thank you, this is very much appreciated!

  11. Mark44
    Mark44 says:
    DS2C

    What are some tips on drawing out a problem? For example if you have very large numbers, or imaginary numbers, it can be hard to visualize.

    I don't think problems involving numbers (real or complex) fall into the category of drawing a sketch, although an Argand diagram of a complex number might be helpful. Argand diagrams are used to locate complex numbers in a two-dimensional plane. The number i is located at a point one unit above the origin.
    The kinds of problems I have in mind are various applied problems in calculus, such as find the area beneath a curve, finding the volume of some solid, finding the amount of work done in lifting something a certain distance, etc. I'm also thinking of problems in trig, where sketching a triangle is helpful, or problems in linear algebra, to be able to visualize a plane or line in space.

  12. dkotschessaa
    dkotschessaa says:

    I love the "Write instead of think." step. I tend to think of doing proofs as 'thinking on paper."

    A bit of a modern challenge to the usual idea of doing easy questions first:

    https://lifehacker.com/improve-your-test-scores-with-the-hard-start-jump-to-e-1790599531

    Short version:
    1. Try the hard problems first, but don't waste too much time on them
    2. Do the easy problems
    3. Go back to the hard ones to finish

    By starting 1, you get your brain rolling (subconsciously) on the harder problems while you are doing other stuff. Sometime it takes our brains a bit to pull stuff out of the filing cabinet.

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