Order of Digits in MultiplicationDate: 03/27/2001 at 09:33:35 From: Hannah Jones Subject: Multiplication place value order Why do we learn to multiply starting with the least significant digits in the ones place value instead of starting with the larger place values? If I had 325 pieces of gum and I had 300 in one hand and 25 in the other, I would choose the 300. I don't understand why we start with the smaller place value when multiplying. Wouldn't it be easier when we have a problem such as 325 x 54 to start multiplying by the hundred's to get 1500? If you can answer this I would greatly appreciate it. Thank you Hannah Jones Date: 03/27/2001 at 12:37:22 From: Doctor Peterson Subject: Re: Multiplication place value order Hi, Hannah. I've just been teaching my daughter how to multiply, and she's found that you can do it in any order you want, as long as you put everything in the right place. There are two reasons for always starting at the right: it's a good idea to do it the same way every time so you don't get confused; and starting at the right allows you to do a bit less writing and avoid erasing. There are some older methods that were just right for use on an abacus, where there's no writing or erasing, or on a sand table or chalk board, where erasing was easy. What we teach now seems to work best with pencil and paper. The basic idea behind all these methods doesn't depend on order. What's happening is that you multiply each part of one number by each part of the other. In your example, we're really multiplying: 300 + 20 + 5 by 50 + 4 and we do so by multiplying 300 by both 50 and 4, then 20 by 50 and 4, and so on. (For example, if I had 5 bags, each containing 50 red marbles and 4 blue marbles, I would have 5x50 red marbles and 5x4 blue marbles, which is the same as 5x(50+4) marbles.) We can see it this way: (300 + 20 + 5) x (50 + 4) = (300 + 20 + 5) x 50 + (300 + 20 + 5) x 4 = 300 x 50 + 20 x 50 + 5 x 50 + 300 x 4 + 20 x 4 + 5 x 4 = 15000 + 1000 + 250 + 1200 + 80 + 20 I can add these together in any order I want. But since the ones column will carry into the tens column, I usually add starting at the right so I don't have to erase the number I already wrote for the tens, after I add the ones. That makes me tend to do everything from the right. Here's one way you can write this down: 325 x 54 ----- 20 <-- 5 x 4 = 20 8 <-- 20 x 4 = 80 12 <-- 300 x 4 = 1200 25 <-- 5 x 50 = 250 10 <-- 20 x 50 = 100 15 <-- 300 x 50 = 15000 ----- 17550 The usual way "telescopes" the first three lines into one, by writing down a digit and carrying the rest: 325 x 54 ----- 1300 <-- 5 x 4 + 20 x 4 + 300 x 4 = 20 + 80 + 1200 1625 <-- 5 x 50 + 20 x 50 + 300 x 50 = 250 + 1000 + 15000 ----- 17550 Within each line, we started from the right so the carries would work neatly; but there's no particular reason why we have to multiply the 4 first, then the 5. We could have done this: 325 x 54 ----- 1625 <-- 5 x 50 + 20 x 50 + 300 x 50 = 250 + 1000 + 15000 1300 <-- 5 x 4 + 20 x 4 + 300 x 4 = 20 + 80 + 1200 ----- 17550 The only problem with this is that it can be hard to keep everything lined up right; it's easier to remember that the first line starts at the right, and each line after that moves one place to the left. Here are some answers in our archives that deal with the same issues a little more completely, including some about an old method called lattice multiplication where you multiply all the digits in any order you want: Learning to Multiply Three-Digit Numbers http://mathforum.org/dr.math/problems/jonathan7.3.98.html Lattice Multiplication http://mathforum.org/dr.math/problems/susan.8.340.96.html Lattice Multiplication Explained http://mathforum.org/dr.math/problems/durham.10.20.99.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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