Composing And Decomposing Numbers

Why do kids need to know how to compose and decompose numbers? Composing and decomposing numbers makes math problems so much easier because it helps kids to make numbers friendlier.

Today, I would like you to stick around as we investigate the power of composing and decomposing numbers as we work to build our math minds so that we can build the math minds of our students.

Watch the video or read the transcript below.

What Is Composing & Decomposing

What exactly is composing and decomposing numbers? Well, it's basically just being able to break apart numbers and then put them back together. Other people might call this "part, part, whole,” understanding that you have a whole amount and you can break it into its parts. This is especially how we talk about it in the early grades in elementary, but as kids progress up we start using the terms composing and decomposing. Now, this is a way to take an amount and be able to chunk it into friendlier pieces.

So let's say we're doing 37. 37 to a lot of kids is just 37. They might understand that 37 is thirty-seven individual pieces, but that doesn't often make it friendlier when you go to add, right? So we want them to understand that they can break 37 apart into friendlier chunks. Now the issue is “what's friendly?” Oftentimes, we start out with helping kids understand 37 as 30, three 10s, and then seven 1s. That is helpful, it's a big idea around place value, but decomposing numbers is so much more than breaking it apart at its place value.

I'm going to show you how to decompose numbers. And we're not just talking about whole numbers here. We're going to delve into fractions, and maybe even some decimals, to look at how this idea of being able to break apart numbers lasts kids all through elementary school and is so powerful when they are learning to add and subtract and even multiply and divide.

Why It’s So Important

Let's start off with understanding why we want kids to be able to compose and decompose numbers. The first thing is that because it makes it easier and it allows a stepping stone between the formal way of knowing an answer, or getting to an answer, and just having to count one, by one, by one. It starts off in the early grades when kids are doing problems like 9 plus 5. Oftentimes we will teach kids how to count first and then the next stage we tend to move them towards is just knowing it, but if they don't know it, they go back down to counting one, by one. So, instead, having some alternative ways to think about the problem can pay off not just for facts like this, but as we progress through. When there are symbols on a piece of paper like this it doesn't often come right to them about how to decompose to make the problem friendlier. So we start off with the young kids, giving them images to help them visualize how to break apart, or decompose one of the numbers, and then put it together and compose a new problem.

So depending on the kids, they may see this differently. You might have one kid who wants to break apart the 9 into 5 and 4; they might see 5 dots and 4 dots in the 9. Then they may want to put the 5 and the 5 together to make a 10. Other kids, may want to break apart the 5 and move one of those dots over with the 9 to make a 10. They're breaking apart, or decomposing, one of those numbers and then putting it together and composing their answer through different ways.

As students progress they begin working on more complicated problems and we end up moving them towards the traditional algorithm with problems like 87 plus 38, but in between this, there is so much number sense and place value that a student can develop. We can use the idea of decomposing and composing numbers to help them build their number sense and their place value understanding. Some see 87 and they realize it's really close to 100, and so, they might want to break apart the 38 to make it nicer. This is not the only way to decompose, and other kids may see this differently, but this is just an example of how breaking apart a number can make the problem friendlier.

Students could also break the 38 into a 13 and a 25. It depends upon the problem as to what you're going to break it into. A lot of times we talk about 38 and it's just a 30 and an 8, but guess what? The number can be broken up into lots of different ways, and it just depends on the problem you're using.

When kids develop these concepts of being able to break apart numbers and seeing how numbers relate, it really opens up strategies. It doesn't even depend upon grade level, it depends upon their number sense and their understanding of this idea of breaking apart numbers and using them to make the problem friendlier.

In this vlogs video above, I share a clip of a student, in the Spring of his first grade year solving problems well beyond a first grade level. However, when students understand and have built some number sense and they can comprehend the idea of composing and decomposing numbers. They can solve problems in ways that are sometimes unimaginable to kids at such a young age. So jump ahead to the 7:12 mark of this video and come back and finish reading.

Wasn't that cool? Again, this is not an expectation of first graders. However, the thinking and number sense and just the powerful mathematics that this kid is building is so incredible. Oftentimes people will say, “that's a really long and drawn out process”, but he thought of that way quicker than what he was able to explain. And yes, through his explanation, he realized he was off by 1, but he is building powerful thinking skills and it's thinking that will last him through so much of what he's doing in elementary mathematics.

So this idea of composing and decomposing numbers is not just for addition, we can move into doing problems with subtraction. If I think about 16 minus 8, and oftentimes we say well kids should just know that, but if they are solving that through the idea of decomposing, understanding that 8 could also be a 6 and a 2, and I could subtract the 6 and then subtract the 2. This idea progresses as they start moving into multi-digit stuff and even larger amounts when they're doing subtraction.

Now again, in this last one, there are often people who will think, “well that's a lot of steps. There's a lot to keep track of”, but let's just for a moment, just for fun, let's compare. This was basically three steps: At 346 I take away 10, I know that that's 336. I take away 6 that gets me to 330, and then I take away 2 so it's 328.

Okay, compare that to doing the algorithm. The first thing the kids have to do, number one, they've got to write it down and line it up. Then they have to look at that and say, okay 6 minus 8, I don't have enough there in the 1s so I'm going to regroup and I'm going to take one from here, but really it's not a 1, right? It's a ten and then I have to add that there, and so I've already done about 5 steps, and I haven't done any subtracting yet.

Yet, in the first one I'm subtracting along the way with each step, and I'm also building a powerful number sense. Don't get me wrong, eventually yes, we want to get kids to this algorithm, but the point is that we need to stop and let them understand numbers first and develop this idea of composing and decomposing numbers. Because even in that traditional algorithm we are breaking apart numbers and putting numbers back together. We are composing and decomposing numbers.

Now, the thing that I really love about this idea is that it's not just for whole numbers. We can use this idea when it comes to adding and subtracting with decimals. If I chunk off .01 it makes that problem so much nicer. Same thing when it comes to fractions.

3 and 3/4 are really close to 4, all I need is 1/4 more. So I'm going to chunk off a 1/4 from the 5 and 1/2 because 5 and 1/2 are really like having 5 and 2/4. So I take off one of those 1/4 and I'm left with 5 and 1/4. That makes that problem a whole lot easier to add and dare I say fewer steps than the traditional way that we teach our students.

Even as we move into multiplication, when kids are trying to solve 6 times 7, they don't instantly remember their multiplication facts. But if they knew how to break one of those numbers apart it might make it easier. Again, you may break this apart differently than how I'm about to break this apart, but I chose to break this 7 into a 5 and a 2. Because multiplying by 5 and multiplying by 2 is often a lot easier for people.

So if they know what 6 times 5 is, and what 6 times 2 is, they've decomposed and then they can compose that back together to see how it comes back together to make the original 6 times 7.

One of the things that our kids really struggle with is division, especially as we start getting into multi-digit, long division situations. On a problem like 4,016 divided by 4, we often see kids struggle and get the wrong answer when trying to solve it using the traditional algorithm. Kids will set this up, and then they'll ask how many times does four go into four? How many times does four go into 0? And they'll forget to put their 0 there, but instead if we think about helping kids break the number apart to make it easier. If I have 4,016 and I'm dividing it by 4, how could I break that apart to make it easier and friendlier to divide? We start off with numbers that are hopefully easier for kids to break apart. 4,016 is really 4,000 and a 16, and we can divide each one of those numbers by 4. 4,000 divided by 4 is 1,000, 16 divided by 4 is 4. So, the answer would be 1,004, we've got to compose that back together. I decompose to make the problem friendlier, and then I'm going to compose those answers back together to get my answer of 1,004.

Now, even as we start getting into ones that look a little bit trickier, like 6219 divided by 3, I want to help kids understand and look for decompositions that can make the problem nicer. Is there any other way to break this up without using 6, 2, 1, and 9, like the traditional algorithm teaches us, or is there another way I could break that number apart? Can I break apart 6,219 into pieces that are friendlier to divide by three? Here's how I thought about it. It would be way nicer if I thought about it as 6,000, 210, and 9. Because I knew that I could divide each one of those by 3 fairly quickly and easily in my head. So 6,000 would give me the 2,000, then I would get 70 from 210, and a 3 from 9. So the answer to 6,219 divided by 3, is 2,073. I'm able to do that all in my head, once I'm able to compose and decompose numbers quickly.

When To Use

The last piece that I really want to hit home is talking with your students about when. There are a lot of activities in textbooks that have kids just practice decomposing numbers. What are all the ways that you can break apart 24? What are all the ways that you can make 56? But there's not a whole lot of times when they actually ask the kids to talk about and reflect upon when you would use those decompositions.

There are lots of different ways that we could decompose 16. A lot of times kids will know the 10 and 6, they'll do the 8 and 8, and then we could keep going from there. Those are the top two, but really there are lots of different ways we could break apart 16. The way that I should break apart 16 is dependent upon what the purpose is, why am I breaking apart 16, to begin with? So if they’re doing the problem 24 plus 16, they can think about how they might break apart 16 to make that problem easier. Now here's the deal, I'm not going to tell you how I would break it apart, but what I want you to think about is how you would break apart 16 for that problem, would you break it apart the same way for 498 plus 16? When students are solving 498 plus 16, are they going to break apart the 16 in the same way? My guess is no. I know that I didn't, I broke the 16 apart differently based on the problem that the 16 was in. So I want kids to, yes, understand that you can break apart 16 in all these wonderful ways, but the real reason is being able to use that decomposition to make mathematics friendlier. Now again, here's another one. Even when they move into multiplication, talk about the different ways that you could break apart 16, and wouldn't that be helpful when you're trying to multiply it by 24? Would that change if I was doing 16 times 356? I don’t know, right? Have that discussion, that's the whole point.

One of the big takeaways I really hope that you got out of that, was that it's not just about getting kids to be able to break apart numbers, but it's helping them understand why. Why and when should I break apart numbers? Because there are a time and place for it. I’m not going to want to break apart numbers every problem, there are times when I want to leave the numbers whole. So helping our students understand the power, the why behind breaking apart numbers and putting numbers back together, and then understanding when are the two essential pieces to helping your kids be able to use this idea of composing and decomposing numbers. I hope that this has helped you build your math mind so that you can go build the math minds of your students.