Wait. That's Multiplication?

When today’s adults were in elementary school they probably learned to multiply in much the same way. Today, however, children in the US are often taught methods of multiplication that their parents have never seen, and, frankly, may not understand. This article is dedicated to all those parents who have looked at their children’s math homework with a puzzled expression and wondered, “What the heck is that?” For everyone else, I hope you find it interesting to learn a few new ways of doing multiplication! A future article will highlight some alternative methods for doing other basic operations.

As a starting point, I’ll walk through a multiplication problem the way that I learned to do it back in the seventies, and this should be familiar to most adults. It’s often called long multiplication, or the standard algorithm.

PartialProducts1.jpg

96 and 32 are called “factors” -- the numbers being multiplied -- and they’re written one above the other, lined up by place value (ones-digits vertically aligned, tens-digits vertically aligned, etc.). To begin, we multiply the ones-digit of the bottom factor (2) by each digit of the top factor (96), and the results are written below: 192. Next, the tens-digit of the bottom factor (3) is used, yielding 288. To the right of 288 a zero is written as a “placeholder.” This zero is necessary because we actually multiplied 96 by 30, not by 3, and 96x30=2880, not 288. Typically, children aren’t taught why the placeholder zero is written; they’re just told to write it (in contrast to the “partial products” method; see below). The partial products of 192 and 2880 are added, and the result is obtained: The final product is 3072. This method is considered very efficient in that it requires fewer steps than many other methods, and its logical connections to our positional, or place value, number system are apparent. However, depending on the math curriculum used, there may be other multiplication algorithms being taught to children, and I’d like to introduce you to three of them.

Partial Products

Some of the newer math curricula, those that follow what is known as a “constructivist” approach, introduce students to the partial products method of multiplication. A benefit of partial products is that it retains more of the place-value meaning of the products obtained along the way (in the example above, 192 and 2880). Some teachers might feel that certain students more often obtain the correct answer using partial products, although this certainly does not apply to all teachers (or all students). A drawback is that the method can be more time-consuming. For comparison purposes, I’ll do the same problem as above. The basic idea is that every digit is taken for what it actually represents in the factor. In other words, 9 is taken to be 90 and 3 is taken to be 30:

If you compare this to the long multiplication problem above you’ll notice that it requires more rows. Typically, that will be the case with the partial products method. To its credit, however, it retains more of the meaning of how multiplication actually works, which is, basically, that every part of one factor must be multiplied by every part of the other factor. This method does become very cumbersome if the two factors have a larger number of digits, and, in my experience, middle school students who rely on this method take longer to solve multi-digit multiplication problems than those who use long multiplication.

Lattice Multiplication

Dating back to perhaps the thirteenth or fourteenth century, and known by many names, lattice multiplication has been one of the popular “alternative” algorithms taught in recent years:

In long multiplication, we multiply each digit of one factor by each digit of the other factor. The same thing takes place here, although it isn’t as obvious. 8x3=24, and in the cell below the 8 and across from the 3 you find 24. The diagonal lines serve to separate the partial products by place values. The order in which you multiply the digits of one factor by the digits of the other factor doesn’t matter; you just need to enter the individual products into the appropriate cells. Similar to long multiplication, after you multiply digits, you must add, although here you add along diagonals (from upper right to lower left) and carry, when necessary, into the next diagonal. This method certainly works for multiplying whole numbers, and with some minor adjustments it can be used to multiply numbers with decimals as well. Students are often intrigued by the visual nature of this method, but it’s certainly less efficient than long multiplication. I’ve watched middle school students use lattice multiplication with mixed results: Sometimes they obtain the right answer, and sometimes they don’t, but in either case it takes longer than using long multiplication, and I have yet to find a student who understands how adding along diagonals leads to a correct answer, or, basically, how this process works. It’s interesting but mystifying.

Russian Peasant Multiplication

Whereas some teachers consider lattice method to be a reasonable alternative to long multiplication (you can probably tell that I’m not one of them), Russian peasant multiplication is generally taught as a novelty, although understanding why it works requires some sophisticated thinking (which we won’t get into here). Based on a method that came out of Ancient Egypt, this mainly requires halving, doubling, and adding; basic multiplication facts aren’t needed for this one. The steps that must be followed are listed below, and a sample problem, 38x52, is worked out below that:

  1. Write the two factors next to each other.
  2. Cut the first factor in half. If it’s odd, subtract 1 first.
  3. Double the second factor.
  4. Repeat steps 2 and 3 until you are left with 1 as your first factor.
  5. Cross off any number in the right column if the matching number in the left column is even.
  6. Add the numbers remaining in the right column.

104 + 208 + 1664 = 1976, and sure enough, 38x52=1976!

While this method is certainly unusual and interesting (for those, like me, who find these sorts of things interesting!), it’s one more method that is less efficient than long multiplication, and it’s hard to see the connections between the process and what’s actually happening when you multiply two numbers.

So?

When I was in elementary school, we learned one way and only one way to multiply whole numbers. Today, however, many students are taught several methods of multiplication, and often their parents are unfamiliar with these methods and can’t help them with their work (or even understand it). I hope this basic introduction to several methods of multiplication serves to help those parents better understand what their children are doing, and for those who don’t have children doing this type of math, I hope this has been interesting and informative nonetheless!