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.

Comments

Maria said…
I remember seeing this method on a YouTube video about Everyday Mathematics (or TERC or one of the other reformed math curriculums). When I was looking for something else, I saw it on the www.mathisfun.com, which has some really neat animated examples that show you step by step how to do it.

I was going to show this to my 4th grader just for entertainment to show her that numbers are fun to work with, but she still thinks the words "fun" and "math" are complete opposites. ;)
Rachel said…
I like teaching the multiplying in parts. Many texts teach this as part of a "mental math section" and it is very helpful.

I was wondering if there would be any confusion when you move to multiplying two two-digit numbers? Usually the second line is used for multiplying the tens digit. Or would you move away from having two lines in your answer before you tried teaching multi-digit numbers?
Maria Miller said…
I would say that yes, there is a possibility for confusion. You might have to tread carefully.

You can of course use the multiplying in parts algorithm with two two-digit numbers, as well. See above; I've updated my blogpost to include an example.
Lostcheerio said…
Interesting... I hadn't seen the lattice method before. As someone who learned by going through the memorized motions to get the correct answer, without ever understanding what the heck I was actually doing, I appreciate the need to create understanding not just memorization! :) And I'm finally learning math, too. A bonus.
Maria Miller said…
Lostcheerio,

I'm glad you're gaining understanding. That's what I strive for, in the materials I write.

This algorithm is fairly simple to understand (some others might not).

You can use it to even teach the very idea of an "algorithm": a step-by-step way to calculate something, so that it always works.

The addition algorithm is another example. With it, it can be quite empowering for kids to realize that they can learn to add ANY two numbers, now matter how big they are.

Of course the same is true of the multiplication algorithm, but the calculations get kind of lengthy if you choose too big numbers.
Kathy said…
I love the example you gave for multiplying in parts. That is new to me. I can't want to use it with my students. I teach in public school, but hopefully you will
not mind me joining your blogs. I am always looking for new ways to teach math and homeschooling moms have novel ideas.

I have a great idea to share with your readers regarding learning math facts. Go to www.AMatterofFacts.com
to find a web-based product that will teach students the rapid recall of math facts using arcade games and a tracking system to identify an individual's trouble facts. Our school subscribes and our kids love it. They even use it at home & public libraries!
For what it's worth: all multiplication algorithms are grounded in what I would call expanded notation. The so-called standard algorithm, however, is a compression of that idea, and when an algorithm compresses (generally by taking advantage of place-value), information is "lost" (or at least hidden) from the user. That's okay, if the learner is shown the underlying relationship between expanded notation (e.g., that 36 * 24 = (30 + 6) * (20 + 4)), and the standard algorithm, as well as the reason we can get away with using shifts in that latter algorithm to make things come out correctly. What the lattice method does is force all the digits to align correctly on the diagonals, without having to worry about shifting (just the occasional carry), and it really is easier for a lot of students. If they know their single-digit multiplication facts, it's hard to go wrong. However, it's still possible not to have a clue as to what's going on or why it all works, which I think should not be allowed: teachers need to know and be able to show kids what's what with every method).

Finally, there's nothing new about the lattice method. It wasn't invented at TERC or by one of the other progressive reform projects. It was in wide-spread use until about 1500, when the spread of the printing press caused its demise (the lattices were hard to reproduce in print, given the limitations of the printing presses of that era), and the current algorithm gained in popularity. This change wasn't grounded in anything mathematical, and the lattice method is not an inferior (or superior) approach. It is, however, one that a lot of kids like, given equal presentations of the standard method, the lattice method, and a few others, without bias in one direction or another. It also does a nice job of laying the ground work for polynomial multiplication. I suspect that no one who knew lattice multiplication with 3-digit numbers times 2- or 3-digit numbers, would be much in need of the FOIL mnemonic and would probably cruise through multiplying expressions with two or more terms.

Unfortunately, like a lot of other things in mathematics these days, the lattice method has been politicized. It's a perfectly reasonable method, it has a longer history than the current standard algorithm, and it applies quite nicely to similar algebraic tasks. It is not more time-consuming, it doesn't really take up more room, and pretty much every objection I've heard raised against it seems grounded in some desperate need to make one-size-fits all education justified. I don't adore it; I don't despise it. I only wish it were taught conceptually by any teacher who chooses to present it, as should all methods, and that it be shown as connected to all other methods, as in fact it is.

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