How to Teach Multiplication and Division

How do we teach multiplication and division with understanding?  A friend of mine got ahold of me because she was concerned that her 4th grade daughter was struggling with division.  Once I dug in a bit I realized that her concern was that her daughter didn’t know how to do the standard algorithm for division.  So, I explained that the standard algorithm for division isn’t until 6th grade and up until that point we should be helping kids develop an understanding of division.

I hope you will stick around as we investigate how to teach multiplication and division with understanding in our quest to Build our own Math Minds so we can Build the Math Minds of our kiddos.

Watch the video below or read the transcript.

Multiplication

If you were anything like me, I could perform the algorithms, but I really had no sense of what I was doing. It was hard for me to even know if my answer was correct, I could perform an algorithm, but I had no sense of numbers. Let's dig in to how we teach multiplication, and then we're going to dig into how to teach division, all in a way that leads towards understanding the standard algorithm, but builds kids' number sense, and place value sense, along the way.

Okay, so let's start off with multiplication and the traditional way that we all learned this. So let’s say you have 24 times 17. We learned to do the 7 times the 4, which gives us that's 28, and you have to put the 8 below, and carry the 2. Then we do 7 times 2, which is 14, and then add the flying 2, so now we should have 16. And then we go over to the 1, and then we do 1 times the 4, and half the kids want to put the 4 below the 8, but our teachers will teach, no, you just move the 4 over, under the 6. Then we go to the 1 times the 2, that's just 2, but half the kids will think that they’ve got to do something with that flying 2 up there, but if they do it correctly, they will end up with 408. (refer to 2:19-3:21 of the video)

The hard part is, a lot of our kids can do this, but they do it with no real understanding, of what they're doing. Why do we do that though? Most of us just learned that's what you do. But, let's take a look at some ways that are being taught how to help kids understand this algorithm.

One of the things we’re trying to develop in kids is to think about it as groups of, that basically, we need 24 groups of 17. So, as another example, let's just say that we have 4 groups of 17. Some kids, when we're starting with young kids, a lot of times they’ll start out with just repeated addition. They’ll do 17 plus 17, plus 17, plus 17, but that becomes really inefficient as we start moving into bigger amounts like 24 groups of 17. But, kids might know something like 10 groups of 17 is 170. Then, if 10 groups are 170, 20 groups of 17 is going to be double that, which is 340. Now I've got my 20 groups, but I need 4 groups. One of the things that kids might know is that 2 groups of 17 is 34, so 4 groups of 17, is going to be 68. So, to get 24 groups, I just need the 20 groups and the 4 groups, and I have to add those together. (refer to 3:53-5:38 of the video)

Now, this seems more lengthy like there are a lot more steps than in the algorithm, but we're helping to build an understanding of what 24 groups of 17 is.

When we've got 20 groups of 17, which gives us 340, and 4 groups of 17 that gives us 68, where does that 68 show up in the original algorithm? I'm going to show you another strategy that should connect the first algorithm a little bit clearer.

Before kids learn the algorithm, another way that we want them to understand multiplication, is that multiplication is connected to area. When you're solving area, you're using multiplication, and vice versa, when you're thinking about multiplication, it can be the length and width, or rows and columns, however, you want to approach it.

Basically, you are wanting kids to see a box that measures 17 x 24. I don't know right off the top of my head, what 17 times 24 is, but, I can slice this area model into pieces that are friendlier. A way that kids will often slice this up, is with the idea of 17 being just 10 and 7, and using this knowledge, they’ll slice the length of the model. The other side could be seen as 20 and 4, then this side would also be sliced. Now I've got four areas to figure out, but the area of those rectangles are a whole lot easier for me to solve. I know the area of a 20 x 10 rectangle is going to be 200. I also know a 4 x 10 rectangle is going to be 40, 20 x 7 is going to be 140, and then the last one, a 4 x 7, is going to be 28. (refer to 7:08-8:52)

Now I've got to add those all together. Plenty of people will say that that’s a lot of adding, but guess what, even when they're doing multiplication, kids still need a lot of practice in addition. This just gives them another opportunity to become a little bit more fluent and flexible with their addition. Now, before we dive into actually adding this all up, I want you to think for a moment, how this strategy connects to that traditional algorithm?

So, if I add vertically, the top row gives me 240 and the bottom row gives me 168. If we look at the traditional algorithm, and we come down to the 24, guess what, it's not actually a 24. The way I learned it, we just left that space empty there, but really, there was a zero at the end, so it was actually 240. That’s because of how this is all connected. So, that top row is really 10 times 24, and the bottom row is actually 7 by 24. If we think back to our traditional algorithm, we used to do 7 times 4, and then we did 7 times 2, but that's not really a 2, it's a 20. It's 7 times 4, and then 7 times 20, and that, together, makes 7 times 24, which is 168. (refer to 9:57-11:34 of the video)

The strategies that kids are developing before they learn the traditional algorithm are going to lead into, not only being able to do the algorithm, but understanding the algorithm. We don’t want kids to just compute and get an answer, and too often, we see kids who will do the steps, and maybe they'll make an error along the way, but they performed the algorithm. Yes, they performed it incorrectly, but, performing it incorrectly is due to them really not understanding what's happening in the algorithm.

Division

So let's move into division. Okay, so let's say we have, 4,812 divided by 4. I picked this one on purpose so that we can look at why the long division algorithm doesn't build understanding. When we look at this equation, kids will see that 4 can go into 4 one time, and then they’ll bring down the 8. Then we can figure out that 4 goes into 8 two times. After bringing down the 1, kids know that 4 can’t go into 1, but half the kids forget to put the zero up there, and they just bring down the 2. So they say that 4 goes into 12, three times, and they end up getting 123. (refer to 13:06-14:07 of the video)

They may have done this wrong, but their incorrect use of the standard algorithm is often caused by them not understanding what they’re doing. So instead, if we can build their understanding of division, when they start doing the traditional algorithm, they can look at a problem like this and see that their answer should be in the thousands. Many of our students, if they move straight to the algorithm, they don’t build an understanding or build their place value and number sense along the way. So let's look at some strategies that kids might develop around division before they get to the algorithm.

One of the ways that kids will see this is that they will want to break it up by the amounts that make sense to them. They will see that there's a 4 and an 8, and then make some connections like, 4 and 4, those work nicely together. They'll want to use the things they know about numbers. But, we want to help kids develop the understanding that the 4, in the 4,812 isn’t just a 4, it’s 4,000. So, 4,812, divided by 4. How many times does 4 go into 4,000? The 4 goes into 4,000, 1,000 times, and 1,000 times 4, is 4,000. And then when I subtract, I'm not just bringing down the 8, I'm bringing down all 812, so how many times does 4 go into 812? Well, I know that 4 will go into 800, 200 times, and then I just have 12. How many times will 4 go into 12? Three. So my answer is 1,203. (refer to 15:46-17:06 of the video)

We're wanting kids to see it as a whole, it's not just how many times 4 goes into 4, or how many times 4 goes into 8, it's about how many times does 4 go into 4,812? That's the big picture. If they develop that understanding, they'll get to a point, where on these “easy ones”, they will be able to do that in their head. Yes, I realize that this strategy works nicely, on a problem like this, but we want to develop that strategy, on easy problems, so that even when they get to ones that are more difficult, they're using that same idea.

Let's try 9,876 divided by 7. My first thought is, I need to find out if 7 goes into 9, that's the traditional algorithm way, but we're building an understanding that 9 is really 9,000. So, if I have 1,000, I know that 1,000 times 7 would give me 7,000. So now I'm left with, 2,876, and what I know about numbers is that 4 works nicely with the 7 to give me 28. But it's not 28, it's 2,800. So I'm not just putting a 4, that 4 is really 400, and 400 times 7 gives me 2,800, and when I subtract, I've got 76 left. 10 would give me 70, and then I'm left with a remainder of 6. So, my answer, is 1,410, with a remainder of 6, and if you're working with fractions, you can have your students write that as 6/7. (refer to 19:07-20:15 of the video)

But the coolest part of doing division this way is that it builds the connection to the standard long division algorithm.

The idea is that we're building their understanding. The reason why we leave those zeros off, why the traditional algorithm became that way, is because it takes time to write those extra zeroes, it takes time to subtract all the way down. But, when you lessen the time, you lessen the understanding, you lessen the place value, and you lessen the number sense that they are building. So we're getting more efficiency, and it takes less space to write it, but what we're losing is place value understanding, and number sense, and a reasoning of whether or not their answer is even reasonable. So, again, the algorithm is not bad, but it's not the first thing we should be teaching kids. We want to develop their place value, we want to develop their number sense through these strategies so that when they get to the algorithm, they can make sense of, and make connections between what's happening. That 7 is really a 70, that 28 is really 2,800, and so on. So yes, doing the standard algorithm, that we all grew up learning is really fast, and it's efficient, but, so many students don't really understand what's happening, and will end up with incorrect answers, and not even realize it because they don't have a sense of numbers.

I hope that through those strategies that I just showed you, it helped you see how we are moving towards the algorithm, but along the way, we are building our kids’ sense of multiplication and division, along with place value, and just general number sense. We ultimately want kids to be thinkers, not just calculators. So, I hope that this has helped you build your math mind so that you can build the math minds of your students.