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Topic: Russian multiplication (Read 2374 times) |
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Noke Lieu
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Russian multiplication
« on: May 11th, 2007, 12:00am » |
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another thing that has peaked my curiosity.
The fable of russian peasant multiplication.
Take your two hands (assuming you haven't lost fingers in corn threshing accidents or some such)
Thumbs uppermost (looking at your palms), they have the value of 10. Then working you way down- index=9, middle=8, ring=7 and little=6.
To multiply say, 7 and 6, touch finger seven on your left hand to finger 6 on your right.
From those fingers down, is the place value for 10's (as in each finger is worth ten).
In this example, there is three, so that's 30.
Then the next step is how many fingers above the joined fingers? 3 on left hand, 4 on the right. Multiply those together, and add it to the tens you figured out.
so 30+12=42. Yay.
Why does this work though?
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jollytall
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Re: Russian multiplication
« Reply #1 on: May 11th, 2007, 1:25am » |
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You have a and b.
a*b you want to calculate:
a-5+b-5 fingers below
10-a and 10-b above.
So (a+b-10)*10+(10-a)*(10-b) is the number you get.
10a+10b-100+100-10a-10b+ab = indeed ab.
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Grimbal
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Re: Russian multiplication
« Reply #2 on: May 11th, 2007, 2:17am » |
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While I find these tricks fascinating (it feels magic), I would absolutely recommend not to teach that to a child before he completely masters the multiplication table. If when hearing 7*8 the answer 53 doesn't pop up immediately, it is a handicap in mental calculation. It is very difficult to unlearn bad habits.
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| « Last Edit: May 11th, 2007, 2:19am by Grimbal » |
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Grimbal
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Re: Russian multiplication
« Reply #3 on: May 11th, 2007, 2:17am » |
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Grimbal
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Re: Russian multiplication
« Reply #5 on: May 11th, 2007, 2:37am » |
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on May 11th, 2007, 2:17am, Grimbal wrote:
I must have missed a finger somewhere.
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| « Last Edit: May 11th, 2007, 2:39am by Grimbal » |
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jollytall
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Re: Russian multiplication
« Reply #6 on: May 11th, 2007, 4:18am » |
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I lost my other thumb, so it is 54!
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ThudnBlunder
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Re: Russian multiplication
« Reply #7 on: May 13th, 2007, 10:59pm » |
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ON BEHALF OF Noke Lieu, WHO IS PRESENTLY INCOMMUNICADO
on May 11th, 2007, 2:17am, Grimbal wrote:| While I find these tricks fascinating (it feels magic), I would absolutely recommend not to teach that to a child before he completely masters the multiplication table. |
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( I should add that perhaps you've retracted this when you recounted? but it's got me thinking)
This is an interesting point.
I found out recently that kids around here are no longer formally taught any times table higher than 10. I blame the French.
Perhaps I am gettting grumpy as I age, but it seems that the proportion of students actually genuinely capable of mental arithmetic is dwindling.
Adding is surviving, just, but even subtraction is wavering. (to be fair though, I was just at a school where the 12 year old students knew that , say 19x3 was much harder to do in you head than (20x3)-(3x1))
I remember being taught when I was 12 about how to square halves in my head (round up, round down, multiply, add a quarter). The whole class was taught, but only those with an aptitude for maths actually took it in.
I'd like to claim that I took it further, but that didn't happen for many years.
Maybe I agree with your statement, if you'd let me tweak it a little?
"not to teach that to a child until they are expected to have mastered multiplication".
I suggest this because it gives those who have mastered something to chew on, it imparts some of that magic and mystery sensation (in my mind, that's some of the beauty of maths). It could even be used as an investigation as to why it works.
Then, and importantly, it gives those who haven't mastered mental mulitplication a leg up. I concede that it's potentially a harbour from which they'll never emerge, and that would be an unfortunate event, but better than leaving them out in the ravages of a storm.
perhaps *real* teachers (instead of folk like me) have an insight to this? I'd be most interested.
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| « Last Edit: May 13th, 2007, 11:02pm by ThudnBlunder » |
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Noke Lieu
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Re: Russian multiplication
« Reply #8 on: Jun 11th, 2007, 10:50pm » |
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Thanks for that.
Something went funny with my remote connection.
Strange that IM worked, and nothing else.
Having experimented with this little thing, the reaction (sample size of approximately 2000) is mostly (and possibly thankfully) that it's far too complicated and that just knowing 8x9 is easier.
Huh. maybe there's a paper in that?
Most folk prefer the "drop the nth finger to do 9n" thingo.
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rmsgrey
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Re: Russian multiplication
« Reply #9 on: Jun 12th, 2007, 9:12am » |
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on May 13th, 2007, 10:59pm, ThudanBlunder wrote:
ON BEHALF OF Noke Lieu, WHO IS PRESENTLY INCOMMUNICADO
[...]
Adding is surviving, just, but even subtraction is wavering.
[...]
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Many years ago, I missed getting full marks on a UKIMC paper (25 5-choice questions requiring mathematical insight, but no knowledge beyond the GCSE syllabus) by a very narrow margin - I realised at the last moment that my solution to the final question was inconsistent, so declined to answer rather than taking a 1-in-5 chance of being right with a penalty for failure balanced for 1-in-3 odds of being right. Some hours later, during a spare moment in another lesson, I took the time to analyse my solution and discover where I went wrong - it turned out that I'd failed to get a "simple" subtraction correct.
Addition, multiplication and exponentiation are relatively easy, but their inverse operations are (in my opinion) hard.
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JiNbOtAk
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Re: Russian multiplication
« Reply #10 on: Jun 12th, 2007, 5:57pm » |
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on Jun 12th, 2007, 9:12am, rmsgrey wrote:
Addition, multiplication and exponentiation are relatively easy, but their inverse operations are (in my opinion) hard. |
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I guess that goes back to how you are introduced to the topics. Most of the students here ( myself included ) would rank addition as easiest, followed by subtraction, multiplication, division, and so forth, simply because that was how we were taught back in school.
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Noke Lieu
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Re: Russian multiplication
« Reply #11 on: Jun 12th, 2007, 6:10pm » |
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There might be something in both of those.
FOr simple operations, I'd possibly rate it A,M,D,S. What do I mean?
Well, disturbingly, I still feel my brain going through the algorithm for subtraction and I feel clumsy (possibly embarrassed) for it, where as with division, it's deemed to be hard, so I do't mind having to think about it.
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Grimbal
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Re: Russian multiplication
« Reply #12 on: Jun 13th, 2007, 2:34am » |
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on Jun 12th, 2007, 5:57pm, JiNbOtAk wrote:| I guess that goes back to how you are introduced to the topics. Most of the students here ( myself included ) would rank addition as easiest, followed by subtraction, multiplication, division, and so forth, simply because that was how we were taught back in school. |
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Maybe it is the other way round, the topics are introduced in order of increasing difficulty.
Multi-digit multiplication is bound to be more difficult than addition because part of the multiplication is an addition.
Division is the most difficult because it is less automatic. You have to find the largest multiple of the dividend that still fits and subtract it. Some multiplication is required for multiple-digit dividends.
Another reason why some operations are considered harder could be how much one uses them. In everyday life I add much more than I subtract. I seldom divide by hand.
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Aurora
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Re: Russian multiplication
« Reply #13 on: Jun 13th, 2007, 8:00am » |
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on Jun 12th, 2007, 9:12am, rmsgrey wrote:
Many years ago, I missed getting full marks on a UKIMC paper (25 5-choice questions requiring mathematical insight, but no knowledge beyond the GCSE syllabus) by a very narrow margin - I realised at the last moment that my solution to the final question was inconsistent, so declined to answer rather than taking a 1-in-5 chance of being right with a penalty for failure balanced for 1-in-3 odds of being right. Some hours later, during a spare moment in another lesson, I took the time to analyse my solution and discover where I went wrong - it turned out that I'd failed to get a "simple" subtraction correct. |
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Having taken the UKIMC paper and qualifying for the kangaroo one earlier this year, I can safely say that rmsgrey isn't the only one to make silly mistakes like this, especially with subtraction.
Personally, I'd say A,M,S,D, although I'm more likely to make a mistake in subtraction.
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| « Last Edit: Jun 13th, 2007, 8:03am by Aurora » |
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rmsgrey
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Re: Russian multiplication
« Reply #14 on: Jun 13th, 2007, 8:11am » |
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on Jun 13th, 2007, 8:00am, Aurora wrote:
Having taken the UKIMC paper and qualifying for the kangaroo one earlier this year, I can safely say that rmsgrey isn't the only one to make silly mistakes like this, especially with subtraction.
Personally, I'd say A,M,S,D, although I'm more likely to make a mistake in subtraction. |
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It was particularly annoying because it was the last question - I answered all the questions in half the time allotted, then spent the rest of the time going over it again to check my answers - and the only mistake I found was on the very last question, without enough time left to find and correct the error...
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JiNbOtAk
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Re: Russian multiplication
« Reply #15 on: Jun 13th, 2007, 5:51pm » |
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on Jun 13th, 2007, 8:11am, rmsgrey wrote:
It was particularly annoying because it was the last question - I answered all the questions in half the time allotted, then spent the rest of the time going over it again to check my answers - and the only mistake I found was on the very last question, without enough time left to find and correct the error... |
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In other words, if not for that one error, you could have taken two UKIMC within the stipulated time, heh ? 
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| « Last Edit: Jun 13th, 2007, 5:52pm by JiNbOtAk » |
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Aurora
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Re: Russian multiplication
« Reply #16 on: Jun 14th, 2007, 7:57am » |
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Of course, there's always the possibility of using the wrong operation, although I don't know who would possibly do that... 
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CapriRS302
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Re: Russian multiplication
« Reply #17 on: Sep 4th, 2007, 10:15am » |
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I disagree with grimbals assumption that children would be at a disadvantage if they didn't learn their multiplication tables and have 56 (53 with missing fingers  ) "pop" into their heads when they see 7 * 8. I am a 30 y/o math teacher who has yet to "learn my multiplication tables" by such standards. However i do feel that multiplication should be mastered to a degree.
For instance, if you drill a student in their tables they may very well memorize 7*8 and many other goodies, but memorization is the evil factor of this equation. They are then stuck learning how to use their new memories to multiply numbers that are bigger then what was on their chart, and if you lead them to algorithms and memorized basic facts there is a good chance that no understanding is taking place whatsoever.
On the other hand, if certain devices or problem solving skills are used instead, those same skills can be applied to all numbers.
For instance when you ask me what is 7*8 I think 7*7 + 7, or 6*9 is 6*10 - 6. In this fashion students (or I in this case) calculate the answers each and every time (unless of course they are imbedded due to repeated calculations, which is different from straight memorization)
by doing this I only have to know how to add, come to a very few answers (a number times itself or ten, which are important base 10 skills anyway) to be able to solve any table question quickly.
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Grimbal
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Re: Russian multiplication
« Reply #18 on: Sep 4th, 2007, 3:02pm » |
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That's exactly what I mean. I have trouble with 6*7 and 7*8. I always stop and check if it ought to be larger or smaller than 49 (which I know by heart) to make sure whether it is 42 or 56. It slows me down, possibly making me forget an intermediate result. I also figured that multiples of 9 are easy. Take n minus one for the tens and 10-n for the units. As a result, I never learned them.
Also, when for instance adding 18 and 5, I mentally cut the 5 into 2 and 3, place the 2 with the 8, and end up with 20 and 3. I would prefer that 8+5 becomes 3 + carry unconsciously.
I guess I always could get away by working the answer out quickly enough with such tricks, so I never learned the multiples as perfectly as I should have, and I don't compute as fast as I should be able to.
But then, on the other side, when computing 36, I translate it to 93 which becomes 81·9, and I put together 72 (with the trick above) and 9 to get 729. So it can pay to do it smarter than fast. But still, knowing the basics is important, and all the tricks that make it easier to remember, in the end, just slow you down.
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| « Last Edit: Sep 4th, 2007, 3:24pm by Grimbal » |
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Eigenray
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Re: Russian multiplication
« Reply #19 on: Sep 4th, 2007, 4:46pm » |
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What do you get when you multiply 6 by 9?
I don't divide by 4; I divide by 2 twice. I mean, I could if I wanted to, and it would probably be faster with practice, but I've usually finished halving the number before I realize what I'm doing. It takes a conscious effort for me to divide by an even number other than 2.
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TenaliRaman
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Re: Russian multiplication
« Reply #20 on: Sep 4th, 2007, 7:34pm » |
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on Jun 14th, 2007, 7:57am, Aurora wrote:| Of course, there's always the possibility of using the wrong operation, although I don't know who would possibly do that... |
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Does writing 4*2 = 6 count as a wrong operation?
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TenaliRaman
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Re: Russian multiplication
« Reply #21 on: Sep 4th, 2007, 7:41pm » |
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on Sep 4th, 2007, 4:46pm, Eigenray wrote:| I don't divide by 4; I divide by 2 twice. |
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Me too. Infact, i even do subtractions in a similar way.
For example, if asked to do 133 - 67, my thought process would be along the lines,
1. 130 - 60 = 70
2. 70 - 7 = 63
3. 63 + 3 = 66
The break up of 133 into 130 + 3 is completely arbitrary. It changes with the numbers. Sometimes, i feel that i am more comfortable with some numbers like multiples of 10,100 etc than i am with others.
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