I’m sick of doing multiplication, division, all the basic arithmetic, in the way I learned in school. I know there are alternate strategies out there: many of them faster and easier to do in ones head, but I can’t seem to find any that aren’t some sort of two hundred dollar mega-math programs. Are there any recreational math fans out there who know what other strategies there are.
Basically, for multiplication, I know the standard
12
x 13
36
120
156
And for addition and subtraction, the basic column approach where you carry the digits from right to left or, “borrow” tens places for substraction right to left.
One thing that I’ve personally found helpful is to divide larger problems into smaller problems based on tens.
To use your earlier example, you might not look at 12 x 13 as 12 x 13, but as 12 x 10 plus 12 x 3. This is then 120 + 36 (as above), or 156.
Addition and subtraction I do the same way – I try to look at things as multiples of 10. 675-283 is really (675-300)+17, or 392. 432+567 is (430+560)+(2+7), or 999.
When doing percentages, the same rule applies. If you want to do a 20% tip at a restaurant bill of $75.42, you take off the last digit (rounding up as necessary) to calculate 10%, and multiply that by 2. On the $75.42 bill, the tip would be $7.54 x 2, or $15.08. (I did this as ($7.50 x 2)+($0.04 x 2).) 15% is similar – $7.54 + half of $7.54 ($3.77) would give you $11.31. (I did it as $7.50+$3.00+$0.70+($0.04+$0.07).)
One book that I’ve read that gets into some very specific tips and tricks is Rapid Math Tricks and Tips by Edward H. Julius. While this book deals heavily with specific tips that one might use in specific situations, there were a few in there that I’ve been able to use in real-life situations.
** Please note that I’m just a recreational math fan and have no special expertise in the area. My math education was limited to what I took in high school and I don’t work with any of this regularly, so take this with a grain of salt as need be.
I’ll definately try out those two books. Of course, this feels like a problem that should be public knowledge, on the web somewhere… but I don’t know how to winnow things down in a search engine.
—To use your earlier example, you might not look at 12 x 13 as 12 x 13, but as 12 x 10 plus 12 x 3. This is then 120 + 36 (as above), or 156.—
But this is essentially what I do, going through which each “place” and then adding. The problem is that this is lousy for estimating (it’s backwards), and it’s slow: it’s hard to just “see” the answer.
I’m definately interested in the “left to right” version that the mathemagics book seems to start talking about. Maybe I’ll be back to answer my own questions when I next venture outside my room.
I can already tell that this left to right method for addition is working better for me, and I wonder now why I couldn’t just have figured it out on my own.
When I was a kid in the 70’s, I remember 60 minutes doing a piece on “Chisenbop” a kind of Korean fingermath. They had these 5 year olds using their fingers like an abacus and calculating these fairly difficult problems. I have no idea if the method is practical or not, but you can find quite a few websites which mention it- Google Chisenbop or Chisanbop.
Yeah, I know a bit of that: mostly the x times 9 trick. Basically, you hold out your two hands, and provided you have all ten fingers, think of them as being numbered 1-10 from left to right. Then, when thinking about any single digit number X times 9, you bend whatever finger corresponds to that digit. The answer is found by counting the fingers to the left of the bent finger, and then to the right. The fingers to the left represent the tens digit, and the fingers to the right represent the unit digit. Of course, if you know your multiplication tables up to at least 9, you wont need your fingers at all.
The thing is, there are a lot of simple tricks like this that work for very specific situations. But I’m trying to find general thought models along the lines of the very basic multiplicaiton sequence I listed above: that work for ANY numbers. Even those that get “close” are interesting. The mathemagic example of how to add left to right is definately better for me than the standard right to left method.
I’ll have to find out about it, but the procedure for multiplication that is being taught at my niece’s primary school is not the standard ‘borrow tens’ method, it looked utterly alien and somewhat confusing to me; I’ll try to post back later when I have the details.
I was going to suggest that method, as we’ve taught it to grade-schoolers, and some catch on quickly, especially for two-digit multiplication. It takes some practice.
http://hucellbiol.mdc-berlin.de/~mp01mg/oldweb/Tracht.htm
The Trachtenberg system looks interesting too, though it seems pretty complicated and definately is requiring some practice. The addition part might be much faster than normal for adding on paper, but I’m not sure how it could be used to do mental calculations, what with having to remember two values (a digit, and # of dots) for every place you count, then remember all of that while to add in the end.
I do like that it allows for easy checks too. However, it’s unclear, also in the addition part, what you are supposed to do when you end up with a 10 at the end of the column. Where does the carry digit go? How does it fit into the system of dots and digits for the final adding?
Step one is to square the right digit. Write the right digit of the answer in the rightmost column, and carryover any left-hand digits of the answer to the next column. 7*7=49
So far, we have:
4 ||9
Of the original number, multiply the two digits together, and then double the answer. For this problem, that’s 4*7=28 and then doubled is 56. Add the carryover to 56 to get 60. Again, put the rightmost digit of that answer in the middle column.
6
_|0|9
Finally, square the lefthand digit of the original number, and add it to the leftmost column. 4*4=16
6
16|0|9
Add the carryover, and you’re done: 22|0|9 is 2209.
It sounds really complicated at first, but when you get what’s going on, you can speed through it almost instantly. Neat. I just wish it worked for all squares, instead of just two-digit squares.
Unfortunately, that didn´t come out quite right. The spaces got edited out. The 4 carryover digit should appear over the middle column in step one, not the leftmost column. The 6, in step three, should appear over the other 6 in step three.
Apos, since your trick is based on (10x + y)[sup]2[/sup] = 100x[sup]2[/sup] + 2*10xy + y[sup]2[/sup] with the 100 and 10 implicitly handled by the columns, you could just multiply out (100x + 10y + z)[sup]2[/sup], but I doubt it would be as easy to remember.
And being the nerd I am, in the time it took for the preview to come up, I worked it out:
10000x[sup]2[/sup] + 21000xy + 2100xz + 100y[sup]2[/sup] + 10yz + z[sup]2[/sup].
If you can find any use in that, godspeed.
Odd how many of these I’ve used. I’ve even tried explaining my 15% tip trick to people (same as lel’s) to people, and have gotten weird looks.
New trick (kind of weird, mostly useless):
Hold your two hands out with the palm towards you. From the bottom number each hand from 6 (pinkies) to 10 (thumbs). To multiply, touch the two appropriate fingers together. Count the fingers below and include the touching fingers. That’s the ten’s place. Then, for the fingers above (but not including) the touching ones multiply the number of fingers on the left with the ones on the right. Add that result to the first step.
So for 8 x 7 (which I can never remember), touch left middle finger to the right ring finger. This gives you 50 (two pinkys, two rings, and one middle finger). Remaining on your left hand is your left hand is a thumb and index, on the right is thumb, index and middle. 2 x 3 = 6
answer: 56
We’ve talked about the divisibility by three and nine tricks. Here’s one for seven that my daughter brought home a couple years ago: Take any number, double the last digit and subtract that from the rest of the number. If the result is divisible by seven, so is the original. Keep going until you can tell whether seven divides it or not.
Example: 3592. 359-2x2=355. 35-2x5=25. Obviously not divisible by 7, so neither is 3592.
