Multiplying in parts and the standard algorithm

I haven't blogged for a while but I've been thinking about this topic for a little while now. It is your multiplication algorithm, also called long multiplication, or multiplying in columns. I also happen to be writing a lesson about it for my upcoming LightBlue series 4th grade book.

The standard multiplication algorithm is not awfully difficult to learn. Yet, some books advocate using so-called lattice multiplication instead. I assume it is because the standard method is perceived as being more difficult. But let's look at it in detail.

Before teaching the standard algorithm, consider explaining to the students multiplying in parts, a.k.a. partial products algorithm in detail:

To multiply 7 × 84, break 84 into 80 and 4 (its tens and ones). Then multiply those parts separately, and lastly add.

So we calculate the partial products first: 7 × 80 = 560 and 7 × 4 = 28. Then we add them: 560 + 28 = 588.

If you practice that for one whole lesson before embarking on the actual algorithm, how much better prepared the kids will be!

Next, they will see the standard way of multiplying:


2 
84
× 7

588


Obviously, the steps here are the same. You multiply the ones first: 7 × 4 = 28, write down 8 of the ones, and carry the 2 of the tens. then you multiply 7 × 8 = 56, add 2 to get 58 and write that down in tens place.

What about this way of writing it down?


84
× 7

28
+ 560

588


It uses a little more space, but the underlying principle of multiplying in parts is more obvious.

It works with two two-digit numbers as well:



 84
×   47

28
560
160
3200

3948


Now, the individual multiplications are 7 × 4, then 7 × 80, then 40 × 4 and lastly 40 × 80.

Lastly, I'll touch on lattice multiplication. It uses the same exact principles; however I am not sure if it makes the underlying principle any more obvious to the students than the standard algorithm (and it does take more time and space).

8 4
+---+---+
|5 /|2 /|
| / | / | 7
5 |/ 6|/ 8|
+---+---+
8 8

Answer 588.

Check out Lattice Multiplication to learn how it's actually done; it's hard to explain without images.

Either way, you NEED to explain multiplying in parts to the students. In this case it's not enough just to be able to go through the motions of an algorithm, because multiplying in parts is so needful in everyday life, and later in algebra (distributive property).

Consider for example these mental multiplications you might encounter while shopping:

5 × $14.
Just do 5 × $10 = $50 and 5 × $4 = $20, and add those. Answer $70. I'm sure most of us are quite used to doing such simple products mentally.

4 × $3.12. Go 4 × $3 = $12 and 4 × 12 ¢; = 48 ¢, and add. Answer $12.48.
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