> I left school convinced that all the so-called Laws > of physics were produced by theorists who envisaged a result and fiddled > their results to make it work.
You would not have been wholly wrong.
(The other half were expecting A, and were completely surprised when it came out B)
>>> On Tue, 13 Nov 2012 13:15:45 +0000, the Omrud <usenet.om...@gmail.com>
>>> wrote:
>>> >I had no choice but to learn my tables by rote - we chanted them in
>>> >class when I was about five or six. Thank goodness for that, I say.
>>> Absolutely. And children usually like learning things by heart,
>>> anyway: their play and socialisation require it.
>> I don't recall that most of the children I was at school with enjoyed
>> learning their times tables. (I knew mine before I went to school, but
>> then I was a precocious little brat.)
>> Here's something I've never understood. We don't teach addition tables
>> by rote, and yet children usually seem to pick them up all right. How's
>> that?
>Do they? I wasn't aware of that. We certainly did learn which >combinations of two numbers made ten - I can remember us learning that >when we were about eight or nine. Later, I quickly learnt the >combinations for fifteen for crib.
Perhaps children could be started in learning combinations that result
in a number by learning how to shoot craps and then progress to
cribbage.
> On Wed, 14 Nov 2012 09:14:48 -0800, Snidely <snidely....@gmail.com>
> wrote:
> >You people make me sick.
> Me too. Particularly as it's just twenty thirteens plus seven
> thirteens.
Well of course it is, but how easy is it to do that in your head? Most people don't know their thirteen times table.
> Mountains are here being constructed from molehills.
I don't know what you mean. I presume we can all do long multiplication the traditional way, but it doesn't lend itself to rapid mental calculation. You need to find other ways to do it.
Also, knowing more than one way of doing a calculation is useful as a cross-check. I'll frequently perform a multiplication two or three different ways to ensure I haven't made an error.
And it's fun. In these days when everyone uses calculators, mental arithmetic is a dying art. I enjoy it - it helps to keep that part of my brain active. You're making it sound like a chore.
"Robert Bannister" wrote in message news:agiq7tFmcfU2@mid.individual.net...
> On 14/11/12 1:29 PM, Guy Barry wrote:
> > Here's something I've never understood. We don't teach addition tables
> > by rote, and yet children usually seem to pick them up all right. How's
> > that?
> Do they? I wasn't aware of that.
Well, children certainly manage to internalize things like "7 + 5 = 12" without rote learning, otherwise they'd have to calculate it again from scratch every time they did an addition sum. I appreciate that the numbers involved are simpler than with the multiplication table, but you still need to learn them somehow.
> We certainly did learn which combinations of two numbers made ten - I can > remember us learning that when we were about eight or nine. Later, I > quickly learnt the combinations for fifteen for crib.
A very useful skill - and also knowing what card to play to take you up to thirty-one!
>Well of course it is, but how easy is it to do that in your head? Most >people don't know their thirteen times table.
13*27 = 260 + 13*7 = 260 + 70 + 21 = 260 + 91 = 261 + 90 = 351.
Trivial. Why is everyone going on about it? (Why do you, in
particular, have to go on and on and on and on and on and on and on
and on about everything? Can you not recognize when a thread is
played out?)
-GAWollman
-- Garrett A. Wollman | What intellectual phenomenon can be older, or more oft
woll...@bimajority.org| repeated, than the story of a large research program
Opinions not shared by| that impaled itself upon a false central assumption
my employers. | accepted by all practitioners? - S.J. Gould, 1993
> In article <OD0ps.494126$it2.187...@fx22.am4>,
> Guy Barry <guy.ba...@blueyonder.co.uk> wrote:
> >Well of course it is, but how easy is it to do that in your head? Most
> >people don't know their thirteen times table.
> 13*27 = 260 + 13*7 = 260 + 70 + 21 = 260 + 91 = 261 + 90 = 351.
> Trivial. Why is everyone going on about it?
Because some of us find it interesting to come up with different ways of performing a calculation. There's no dispute about the correct answer.
> (Why do you, in
> particular, have to go on and on and on and on and on and on and on
> and on about everything? Can you not recognize when a thread is
> played out?)
Would you care to choose a new topic then? If you're no longer interested in the thread, kill it.
>> We certainly did learn which combinations of two numbers made ten - I can >> remember us learning that when we were about eight or nine. Later, I >> quickly learnt the combinations for fifteen for crib.
>A very useful skill - and also knowing what card to play to take you up to >thirty-one!
Not as useful this close to Las Vegas as knowing how big a number will take you
over twenty-one....r
> > In article <OD0ps.494126$it2.187...@fx22.am4>,
> > Guy Barry <guy.ba...@blueyonder.co.uk> wrote:
> > >Well of course it is, but how easy is it to do that in your head? Most
> > >people don't know their thirteen times table.
> > 13*27 = 260 + 13*7 = 260 + 70 + 21 = 260 + 91 = 261 + 90 = 351.
> > Trivial. Why is everyone going on about it?
> Because some of us find it interesting to come up with different ways of
> performing a calculation. There's no dispute about the correct answer.
Isn't there? 595 is a perfectly correct answer to - in base 14.
> On Nov 15, 11:08 am, "Guy Barry" <guy.ba...@blueyonder.co.uk> wrote:
[13 x 27 again - apologies to Garrett Wollman!]
> > Because some of us find it interesting to come up with different ways of
> > performing a calculation. There's no dispute about the correct answer.
> Isn't there? 595 is a perfectly correct answer to - in base 14.
Don't you mean 307 (in base 14)? The answer is 595 in base 10, but presumably you'd want to express the answer in the same form as the problem.
> > On Nov 15, 11:08 am, "Guy Barry" <guy.ba...@blueyonder.co.uk> wrote:
> [13 x 27 again - apologies to Garrett Wollman!]
> > > Because some of us find it interesting to come up with different ways of
> > > performing a calculation. There's no dispute about the correct answer.
> > Isn't there? 595 is a perfectly correct answer to - in base 14.
> Don't you mean 307 (in base 14)? The answer is 595 in base 10, but
> presumably you'd want to express the answer in the same form as the problem.
So you agree, there is dispute about the right answer!
I'm not sure why I'd want to express the answer in the same form as
the problem, apart from consistency, clarity and convention.
> On Nov 15, 12:39 pm, "Guy Barry" <guy.ba...@blueyonder.co.uk> wrote:
> > Don't you mean 307 (in base 14)? The answer is 595 in base 10, but
> > presumably you'd want to express the answer in the same form as the > > problem.
> So you agree, there is dispute about the right answer!
Well, if you're determined to manufacture it...
> I'm not sure why I'd want to express the answer in the same form as
> the problem, apart from consistency, clarity and convention.
I am reminded by this thread of the old pantomime routine where one person "proves" to another that 13 x 7 = 28, using a variety of dishonest techniques. (I'm sure some people here will remember it.)
"Lewis" wrote in message news:slrnkaaare.2e6d.g.kreme@mbp55.local...
> In message <OD0ps.494126$it2.187...@fx22.am4>
> Guy Barry <guy.ba...@blueyonder.co.uk> wrote:
> > Well of course it is, but how easy is it to do that in your head? Most
> > people don't know their thirteen times table.
> You don't need to.
> Ten thirteens are 130, so 20 are 260. Since ten are 130, then 5 must be
> 65. 65 and 260 is 325. All you have to do now is add 2 thirteens (26)
> and Bob's your uncle!
Not bad. Maybe we should have a vote on the best method.
One method I haven't seen mentioned so far is what I call "cross-multiplication". It usually works OK for two two-digit numbers but gets complicated after that. You work out the hundreds, then the tens by imagining a cross drawn between the two numbers (e.g. for 13 x 27 that would be (1 x 7) + (3 x 2)), and finally the units.
Hundreds: 1 x 2 = 2.
Tens: (1 x 7) + (3 x 2) = 7 + 6 = 13.
Units: 3 x 7 = 21.
Total = 200 + 130 + 21 = 351.
> >I don't know what you mean. I presume we can all do long
> >multiplication the traditional way, but it doesn't lend itself to rapid
> >mental calculation. You need to find other ways to do it.
> It depends on what you consider the "traditional way." If you mean the
> asinine backwards methods taught in schools, then the very first thing
> one learns is to never, *ever* solve a math problem that way.
Yes, that's what I mean by "long multiplication", as described here. (I hadn't seen the "boxes method" before though.)
Schools should really teach a variety of methods, and let students choose what works best for them. But calculators seem to have made the learning of arithmetic sadly redundant.
Since I know 13*13 is 169, my first solution was to parse the problem into 2*13*13 + 13 = 338+13 = 351.
>Trivial. Why is everyone going on about it? (Why do you, in
>particular, have to go on and on and on and on and on and on and on
>and on about everything? Can you not recognize when a thread is
>played out?)
>> > Here's how I'd do 13 x 27 in my head: it's (20 - 7) times (20 +
>> > 7), which is the same as 20 squared minus 7 squared, which is 400
>> > minus 49, which is 351.
>> Oh that's good.
> Do many people use the "difference of squares" method for
> multiplication? I can't remember where I learned it - I was taught
> that (a+b)(a-b) is a^2-b^2 quite early on, but I don't remember being
> taught it explicitly as a method of calculation. I remember I once
> had to set a question for the rest of the class and asked them to
> multiply something like 993 x 1007 without the aid of a calculator. I
> don't think many of them got it.
> If you memorize the squares of numbers from 1 to 25 (which isn't
> hard), there are simple rules that allow you to derive the squares of
> numbers up to 100. Once you've learned those, you can use "difference
> of squares" to multiply any two numbers below 100, as long as they're
> both odd or both even. If one's odd and the other's even you can
> reduce one of the numbers by 1 and then add the other on. So for
> (e.g.) 27 x 38 I'd calculate 27 x 37, which is 32 squared minus 5
> squared or (1024-25) = 999. Add 27 to give 1026.
>> I'd do 12x24 = 2x12x12 = 288. Plus another 24, plus 13x(27-24) =
>> 39. Total = 351
>> >> "Show your work" to me means just show your answers. "Show your
>> >> working (out)" means explain how you got from the question to
>> >> that answer and prove that you didn't copy it from Perkins.
>> > That doesn't prove you didn't copy it. You could have copied his
>> > working as well.
>> Yes but it's much, much more obvious. Trust me, I've marked
>> undergraduate lab work.
> There's an art to copying. You don't copy it verbatim. You rework
> things slightly and phrase it differently. Sometimes it can even be a
> good idea to introduce deliberate mistakes if you're under any
> suspicion. I've done it a few times and never been found out. (I
> used to let people copy my work as well, so I didn't feel bad about
> it.)
The real problem is when the person you are copying it from got it wrong
in the first place.
> >> >> "Show your work" to me means just show your answers. "Show your
> >> >> working (out)" means explain how you got from the question to
> >> >> that answer and prove that you didn't copy it from Perkins.
> >> > That doesn't prove you didn't copy it. You could have copied his
> >> > working as well.
> >> Yes but it's much, much more obvious. Trust me, I've marked
> >> undergraduate lab work.
> > There's an art to copying. You don't copy it verbatim. You rework
> > things slightly and phrase it differently. Sometimes it can even be a
> > good idea to introduce deliberate mistakes if you're under any
> > suspicion. I've done it a few times and never been found out. (I
> > used to let people copy my work as well, so I didn't feel bad about
> > it.)
> The real problem is when the person you are copying it from got it wrong
> in the first place.
For some of us, that's more a source of innocent merriment.
Dr Nick wrote:
> "Guy Barry" writes:
>> "Dr Nick" wrote:
>>> "Guy Barry" writes:
>>>> Here's how I'd do 13 x 27 in my head: it's (20 - 7) times (20 +
>>>> 7), which is the same as 20 squared minus 7 squared, which is 400
>>>> minus 49, which is 351.
>>> Oh that's good.
>> Do many people use the "difference of squares" method for
>> multiplication? I can't remember where I learned it - I was taught
>> that (a+b)(a-b) is a^2-b^2 quite early on, but I don't remember being
>> taught it explicitly as a method of calculation. I remember I once
>> had to set a question for the rest of the class and asked them to
>> multiply something like 993 x 1007 without the aid of a calculator. I
>> don't think many of them got it.
>> If you memorize the squares of numbers from 1 to 25 (which isn't
>> hard), there are simple rules that allow you to derive the squares of
>> numbers up to 100. Once you've learned those, you can use "difference
>> of squares" to multiply any two numbers below 100, as long as they're
>> both odd or both even. If one's odd and the other's even you can
>> reduce one of the numbers by 1 and then add the other on. So for
>> (e.g.) 27 x 38 I'd calculate 27 x 37, which is 32 squared minus 5
>> squared or (1024-25) = 999. Add 27 to give 1026.
>>> I'd do 12x24 = 2x12x12 = 288. Plus another 24, plus 13x(27-24) =
>>> 39. Total = 351
>> So you think in the duodecimal system?
> No, but I know up to 12x12.
I used to write down two 7-digit numbers and multiply them in my head
without writing down any of the intermediate results -- only the final
result. The hard part is the keeping track of the intermediate results.
On Wed, 14 Nov 2012 22:20:32 -0800, Snidely <snidely....@gmail.com>
wrote:
>Robert Bannister wrote on 11/13/2012 :
>> I left school convinced that all the so-called Laws >> of physics were produced by theorists who envisaged a result and fiddled >> their results to make it work.
>You would not have been wholly wrong.
>(The other half were expecting A, and were completely surprised when it >came out B)
I suppoose it was a.u.e. that taught me that great scientific
breakthroughs are heralded not with "Eureka!", but "That's funny..."
>>>> Here's how I'd do 13 x 27 in my head: it's (20 - 7) times (20 +
>>>> 7), which is the same as 20 squared minus 7 squared, which is 400
>>>> minus 49, which is 351.
>>> Oh that's good.
>> Do many people use the "difference of squares" method for
>> multiplication? I can't remember where I learned it - I was taught
>> that (a+b)(a-b) is a^2-b^2 quite early on, but I don't remember being
>> taught it explicitly as a method of calculation. I remember I once
>> had to set a question for the rest of the class and asked them to
>> multiply something like 993 x 1007 without the aid of a calculator. I
>> don't think many of them got it.
>> If you memorize the squares of numbers from 1 to 25 (which isn't
>> hard), there are simple rules that allow you to derive the squares of
>> numbers up to 100. Once you've learned those, you can use "difference
>> of squares" to multiply any two numbers below 100, as long as they're
>> both odd or both even. If one's odd and the other's even you can
>> reduce one of the numbers by 1 and then add the other on. So for
>> (e.g.) 27 x 38 I'd calculate 27 x 37, which is 32 squared minus 5
>> squared or (1024-25) = 999. Add 27 to give 1026.
>>> I'd do 12x24 = 2x12x12 = 288. Plus another 24, plus 13x(27-24) =
>>> 39. Total = 351
>> In message <OD0ps.494126$it2.187...@fx22.am4>
>> Guy Barry <guy.ba...@blueyonder.co.uk> wrote:
>> > Well of course it is, but how easy is it to do that in your head? Most
>> > people don't know their thirteen times table.
>> You don't need to.
>> Ten thirteens are 130, so 20 are 260. Since ten are 130, then 5 must be
>> 65. 65 and 260 is 325. All you have to do now is add 2 thirteens (26)
>> and Bob's your uncle!
> Not bad. Maybe we should have a vote on the best method.
> One method I haven't seen mentioned so far is what I call
> "cross-multiplication". It usually works OK for two two-digit numbers
> but gets complicated after that. You work out the hundreds, then the
> tens by imagining a cross drawn between the two numbers (e.g. for 13 x
> 27 that would be (1 x 7) + (3 x 2)), and finally the units.
> Hundreds: 1 x 2 = 2.
> Tens: (1 x 7) + (3 x 2) = 7 + 6 = 13.
> Units: 3 x 7 = 21.
> Total = 200 + 130 + 21 = 351.
>> >I don't know what you mean. I presume we can all do long
>> >multiplication the traditional way, but it doesn't lend itself to rapid
>> >mental calculation. You need to find other ways to do it.
>> It depends on what you consider the "traditional way." If you mean the
>> asinine backwards methods taught in schools, then the very first thing
>> one learns is to never, *ever* solve a math problem that way.
> Yes, that's what I mean by "long multiplication", as described here.
> (I hadn't seen the "boxes method" before though.)
> Schools should really teach a variety of methods, and let students
> choose what works best for them. But calculators seem to have made the
> learning of arithmetic sadly redundant.
Back in the days before calculators and when slide rules were expensive, I'm pretty sure my school did attempt to teach a number of different ways, but I'm also pretty certain that most of us realised it much too much bother to remember more than one way and that we would be unlikely to ever want to do complicated sums in our head. We had pencils and paper back then; today we have machines. What wasted effort it would have been.
>>> We certainly did learn which combinations of two numbers made ten - I can
>>> remember us learning that when we were about eight or nine. Later, I
>>> quickly learnt the combinations for fifteen for crib.
>> A very useful skill - and also knowing what card to play to take you up to
>> thirty-one!
> Not as useful this close to Las Vegas as knowing how big a number will take you
> over twenty-one....r
Since you know your first two cards are nearly always going to add up to 12 or 13, you can be pretty sure that there's a reason that 10, J, Q and K make up 36/52 of the pack.
"Dr Nick" wrote in message news:877gpmzgsa.fsf@temporary-address.org.uk...
> "Guy Barry" <guy.ba...@blueyonder.co.uk> writes:
> > So you think in the duodecimal system?
> No, but I know up to 12x12.
I wonder why schools always used to teach tables up to 12x12? You only need them up to 10x10 (or in fact 9x9, since the tens are trivial). Someone told me it was to do with the old imperial system and inches in a foot, but in that case they might as well have gone up to 16x16 (ounces in a pound). Or even 20x20 (fluid ounces in a pint, or shillings in a pound in pre-decimal currency).
"Skitt" wrote in message news:k83po4$bug$3@news.albasani.net...
> I used to write down two 7-digit numbers and multiply them in my head
> without writing down any of the intermediate results -- only the final
> result. The hard part is the keeping track of the intermediate results.
Wow. I'm nowhere near that level of proficiency.
How did you calculate the intermediate results? I used to do it using the "cross-multiplication" technique I mentioned elsewhere, but I never got further than three-digit numbers.
