Being Fast At Calculation

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Timefly
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Being Fast At Calculation

Postby Timefly » Thu Mar 22, 2012 11:55 am UTC

I have a problem, I'm slow at calculation.

In my maths courses I can understand everything I need to, I can do all the calculation I need to, I just can't do it as quickly as I need to. This makes exams particularly hard for me as my work becomes a rushed mess.

So does anyone know any good ways of becoming fast at calculation?

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Re: Being Fast At Calculation

Postby gfauxpas » Thu Mar 22, 2012 12:41 pm UTC

Do you mean at arithmetic?
If you're allowed calculators on exams, there's a ton of stuff that you can do on calculators, a lot more than just addition/subtraction/multiplication/division. So if you get familiar with your calculator you can save a ton of time. What subject is this in? And can you give examples of problems that you feel you can't do fast enough?

Timefly
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Re: Being Fast At Calculation

Postby Timefly » Thu Mar 22, 2012 4:01 pm UTC

Currently I'm working on linear differential equations. I always get slowed down in things like equating coefficients to find a particular integral. Arithmetic is generally fine as I'm quite handy with a calculator. It's just highly mechanical things.

Edit: Here's an example of the kind of thing we're currently doing. http://www.wolframalpha.com/input/?i=d% ... %3D+4e%5Ex
Don't know latex sorry.

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z4lis
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Re: Being Fast At Calculation

Postby z4lis » Thu Mar 22, 2012 4:42 pm UTC

If you're slow at calculation, you can do what anyone has to do to become faster: practice. Identify what's taking you a long time, make up some problems similar to those, and practice solving them quickly and without errors.
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Yakk
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Re: Being Fast At Calculation

Postby Yakk » Thu Mar 22, 2012 8:56 pm UTC

Timefly wrote:Arithmetic is generally fine as I'm quite handy with a calculator.

Remember when people talked about the harm you do yourself when you rely on your calculator to do Arithmetic? Maybe you don't, but people do talk about it. There is a reason.

Arithmetic is a way to build up a basic working memory and reliability when doing algebraic manipulation. If you do all of your Arithmetic on a calculator, you haven't been practicing doing algebra (on numbers) nearly as long as you would have otherwise.

Try doing all of your arithmetic without a calculator. It will be slower, and less accurate, but you'll get practice at it, and get better at it if you practice it. And the skills transfer: It will give you extra practice at manipulating symbols in algebraic ways.
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Timefly
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Re: Being Fast At Calculation

Postby Timefly » Thu Mar 22, 2012 11:00 pm UTC

z4lis wrote:If you're slow at calculation, you can do what anyone has to do to become faster: practice. Identify what's taking you a long time, make up some problems similar to those, and practice solving them quickly and without errors.

i suppose I just need to knuckle down and get on with things. Thanks.


Yakk wrote:Remember when people talked about the harm you do yourself when you rely on your calculator to do Arithmetic? Maybe you don't, but people do talk about it. There is a reason.

Arithmetic is a way to build up a basic working memory and reliability when doing algebraic manipulation. If you do all of your Arithmetic on a calculator, you haven't been practicing doing algebra (on numbers) nearly as long as you would have otherwise.

Try doing all of your arithmetic without a calculator. It will be slower, and less accurate, but you'll get practice at it, and get better at it if you practice it. And the skills transfer: It will give you extra practice at manipulating symbols in algebraic ways.

Now that you say it, it makes a lot of sense. Thanks for this :D, really helpful.

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Re: Being Fast At Calculation

Postby Xanthir » Tue Mar 27, 2012 3:15 pm UTC

I can only add to what everyone else already said. Fast calculation in the human brain is *not* just a sped-up version of slow calculation. It involves a lot of tricks and estimation that lets you break down the problem into smaller problems that can be solved by rote memory and then recombined.

As a simple example, yesterday I needed to multiply 18 by 12. I did it by, essentially, instead computing 18*10 + 20*2 - 2*2. This was the simplest way my brain knew how, on the spur of the moment, to break it down into calculations that I could solve by memorization rather than computation. Note how, even though I know by memorization that 18*2=36, I still broke that part down further into 20*2 - 2*2 - this happened because it's much easier to add 40 to a number and then subtract 4 than it is to add 36 to a number, because digits > 5 often produce a carry which is hard to do quickly in my head, and because 40 only has one "real" digit for the addition operation to worry about.

These tricks can only be developed by long practice - they're very instinctual, and I suspect very idiosyncratic to my brain. You'll develop your own methods. (Or you won't - it's possible that your brain just doesn't like the particular operations that let you do fast mental calculation. We're all different up there.)
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Re: Being Fast At Calculation

Postby eSOANEM » Tue Mar 27, 2012 10:28 pm UTC

Xanthir wrote:These tricks can only be developed by long practice - they're very instinctual, and I suspect very idiosyncratic to my brain. You'll develop your own methods. (Or you won't - it's possible that your brain just doesn't like the particular operations that let you do fast mental calculation. We're all different up there.)


As an example of idiosyncratic methods:

When doing multiplication in my head I tend to work "backwards" (so I calculate 18*12 as 10*12+8*12=10*12+8*10+8*2), this means that I don't need to keep track of any carries whilst doing another calculation because as soon as a carry arises it can be used.

Also, if I'm converting to binary, instead of counting down in powers of 2 I check if it's odd or even first, this just seems to make the powers of 2 fall out more easily in my head, I don't know why.
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Yakk
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Re: Being Fast At Calculation

Postby Yakk » Wed Mar 28, 2012 2:57 am UTC

And in my case, I'd be tempted to do 1 and a half gross (18 is one and a half dozen, 12 is a dozen, a square dozen is a gross, so we have 1.5 gross), and compare it to 12*(20-2) = 240-24 for a sanity check. Because if I need an exact answer, I should do it at least two ways that are extremely different from each other -- and if I don't need an exact answer, 18*12 is just 2E2.
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Coding
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Re: Being Fast At Calculation

Postby Coding » Wed Mar 28, 2012 3:10 am UTC

240 - 24 would be my go-to method. Subtracting from a multiple of ten is pretty fast for me.

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Re: Being Fast At Calculation

Postby Meteoric » Wed Mar 28, 2012 3:21 am UTC

You could also do that one as (15+3)(15-3) = 15^2 - 3^2 = 225 - 9 = 216

The point is, there's lots of these tricks, applicable in different kinds of situations, and which way is the easiest often depends on the person, but learning to use them - not just prove that they work, but use them faster/more easily than you could type the problem into a calculator - requires practice. In general, you want to turn the problem into a version of something you already know how to do, like I did up above with the difference of squares, or break it apart into problems you already know how to do, like eSOANEM's example. If you want to get better at this, use your calculator only as a last resort when you're not pressed for time, and it can also help to practice outside the context of math homework. Calculate the tip at restaurants mentally (a 15% tip is 10% + 5%, 10% is just a decimal shift, and 5% is half that). Estimate your grocery bill while the cashier rings you up. That kind of thing.
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Re: Being Fast At Calculation

Postby PM 2Ring » Wed Mar 28, 2012 3:43 am UTC

Xanthir wrote:As a simple example, yesterday I needed to multiply 18 by 12.

My brain first chose to evaluate that as 2 * ( 9 * 12) = 2 * 108, but then I noticed it was also (18 * 2) * 6 = 36 * 6 = 6² * 6 = 6³. (I memorized the cubes up to 12 ages ago. I used to know the 5th powers too, but most of those are a bit vague now, since I don't use them very much) . I like it when I have two (or more) short-cut methods, since it increases my confidence that I haven't made a mistake.

Meteoric wrote:You could also do that one as (15+3)(15-3) = 15^2 - 3^2 = 225 - 9 = 216

Another reasonably efficient way is (20 - 2) * (10 + 2) = 20*10 + 2*(20 - 10) - 2*2, but this way isn't as fast as the difference of two squares approach because you have to take care to not mix up the sign of the middle term.

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Re: Being Fast At Calculation

Postby Dopefish » Wed Mar 28, 2012 3:55 am UTC

Lets see, how would I instinctively do 12*18... 12*18 = 12*12+6*12 = 12*12+(1/2)*(12*12) = 144+144/2 = 144+72 = 216.

Perfect squares up to 12 got drilled in to me after most of juinor high/high school, so my strategy for this sort of thing tends to be to pull out the perfect square part, and then just add in the extra bits (which happened to work out nicely here since division by 2 is simple and I could reuse the 12*12 knowledge).

It's interesting to see how many variations there are on that calculation that people use. Some of them I think are pretty nifty (e.g. the difference of squares one), but it's the sort of thing I would never recognise unless I was looking for it. I guess some people are better trained to spot that sort of thing than I.

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Re: Being Fast At Calculation

Postby Meteoric » Wed Mar 28, 2012 6:29 am UTC

The difference of squares one actually wasn't the first that came to mind for this particular example, but it's a handy one in some other cases. 18x22 for example immediately looks like (20-2)(20+2), and depending on how well you know the squares, 20^2 might be easier to remember than 15^2. Technically the trick works for any pair of numbers, but it only saves effort if the numbers are both even or both odd.
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Re: Being Fast At Calculation

Postby Timefly » Thu Mar 29, 2012 4:04 pm UTC

Thanks everyone. I guess practice it is.


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