Well our standards mention that students need to have strategies when they solve addition and subtraction problems. In a recent series of videos that I did, I talked about the addition strategies. And some people kind of think that those are a waste of time.  Trying to get kids to solve addition all these different ways takes a lot of time to develop them when really we should just teach them the standard algorithm and go from there. 

Now, not that the standard algorithm is bad, but these strategies build so much more in our students. And when you really start to see it is when they start to use these strategies when they’re solving subtraction problems. Because if you have ever taught elementary kids, you know subtraction is so difficult for them. And if they have this flexibility with numbers, oh, it makes subtraction so much easier. 

I’m Christina Tondevold, The Recovering Traditionalist, and I hope you stick around because today we’re going to take a look at the 6 subtraction strategies that kids could use when they’re solving subtraction problems. This is in our effort to build our math mind so that we can build the math minds of our students. 

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog. (Some may be affiliate links which just means that if you do purchase using my link, the company you purchased from sends me some money. Find more info HERE about that.)

5 Types of Addition Strategies video

Solving Elapsed Time Subtraction Problems (2nd video in the series)

Solving Subtraction Measurement Word Problems (3rd video in the series)

Solving Subtraction Money Word Problems (4th video in series)

Number Sense Course

Free Training – Components of Number Sense in PreK 

Free Training – Components of Number Sense in 3rd-5th

Now if you watched those addition videos, you’ve already heard my warnings. So these warnings are for people who have not seen those prior videos. 

#1 Do not directly teach these strategies to your students. I am talking about the possible ways kids could do it. My philosophy is, though, and it’s backed by the research of cognitively guided instruction, is that we do not teach these. Kids naturally develop them when they see connections between numbers. So that’s one of the biggest things that we can help develop as teachers is build our students’ number sense. I’ll link to some other videos below. And I have a full course about how to build number sense. 

These videos are to help you understand what kids potentially might do when they’re solving subtraction problems if they have this foundation of number sense already built. You’ll start to see kids do these other ways. So we’re not directly teaching them. This is just so that you know, what in the world are these kids doing? Because when my students were doing them, I didn’t know any of these ways. And I was like, I don’t get what you’re doing. The only way I knew how to do it was the traditional algorithm. So I just want to expose you to these other ways so you know what in the world your students are doing. Now I have certain names that I call these strategies. 

#2 You might have something, a different name, I should say. You might have a different name for these strategies. And that’s okay. I want you to look at the mathematics that’s involved in the strategy. That’s how I determined what its name is or even what strategy it is. You might call it something different. But look at the math. It’s not about the model. I can use the number line on a lot of these strategies. The number line is not a strategy. It’s a way to model my strategy. 

You’ve got to look at what’s the mathematics that they’re doing in that strategy, and then that’s what tells you whether or not it’s a different strategy than somebody else has. Now my third one is that kids will often combine some of these. Like, sometimes it doesn’t look exactly the way I’m going to show you. They will take bits and pieces and kind of move things around. These strategies are personal to kids. That’s another reason why I say don’t directly teach them. Kids will see relationships. They’ll see what they want to do with the numbers and they will try to make sense of it on their own without us directly teaching them. 

So let’s take a look at the 6 different ways that you can solve a subtraction problem. And the problem that we are going to use is 543 – 289.  Now it’s one example I’m going to be showing you. I’ll probably show how it kind of extends or goes down lower. But I want you to think about what kind of number sets are you giving your students, and maybe you can play along and do that problem while I’m showing how to do it with this problem of 543 – 289.

Compensating

All right, let’s jump in. Okay, we are going to start off with a strategy that a lot of kids and even adults really love, which is basically they want to make one of the numbers friendly to subtract. So when they see 543 – 289, they’re like, man, if that 289 was just 300, it would be so much nicer. because they can do that easily. 243. The problem happens when they try to figure out, okay, I know I wasn’t supposed to subtract 300, so how do I fix it? Now instinctively, even as adults, we look at this and we say, well, we added 11 to get to 300, so I should subtract 11. But it doesn’t give us the right answer. So why doesn’t that work? So this idea of using this strategy with subtraction, we really have to understand what’s happening in the problem. 

Now this is just a bare problem, but if this was put into a context, like I had $543, and I went to the store and bought my groceries. It came out to be exactly $289. And I’m paying in cash, right? Well, most of the time you probably don’t have $289 exactly. But in your purse you might have 300. So if you paid 300 but you were only supposed to pay $289, what would you expect to happen after that, right? You’re not going to give them more money. They need to give you back $11. To get to the 254. I also really like not just putting it into a context, but creating a visual image that goes with it. 

So a number line for me is really helpful on this one. You may want something different. But if I’m at 543 and I subtract out 300 from that, I know it’s going to get me to 243. But I wasn’t supposed to subtract 300. I’m only supposed to subtract 289. So where would that land in relation to where I was?  It should be up here and it should be back 11 spots. 54. Right? 

So a visual, a context can help us really understand what’s happening in this problem. Now remember, I’m showing you how to solve this with multi-digit numbers. But even when we are helping kids basically learn their facts, kids are doing these strategies. So let’s take a look at what it would be like if we were doing 16 – 9, and a kid was doing this compensating strategy. You might see a kid who says something like, well, 16 – 10 would be 6. And then they need to do something after that. But the tricky part is that a lot of kids don’t know what I need to do, right? Well again, put it into a context. Have them model it. Get out the 16 blocks and take away 10 and talk about what should I have taken away and how do I need to fix it. That’s why I call this compensating, is because you did something you weren’t supposed to do, so now you need to come back and compensate for it. I subtracted 1 more than I was supposed to, so I need to give that 1 back to compensate for what I did.

Give & Take–>Constant Difference

So if you watched the addition video about the addition strategies, there was this strategy where kids who want to make that 300, they want to make a friendly number, but they want to do it right away before they do any of the addition or subtraction. And so naturally, when they get to subtraction, they’ll do the same strategy. They’ll say, man, if I could only make that 289 a 300. So they’ll want to make this a 300. And they’ll say oh, okay, well I have to give 11 to the 289 to make that 300. So I’m just going to take 11 from the 543. And then we subtract. And we already know we solved it one way, and that’s not the right answer. But it worked when the kids did addition. But now all of a sudden it doesn’t work when they’re doing subtraction. So why doesn’t it work? 

So here’s the problem. With subtraction, there’s two ways to think about subtraction. And there’s the way that is known as take away, where we’re viewing the problem as take away. 543 and we need to take away 289. That’s one way to view it. The other way to view subtraction is actually what the answer to a subtraction problem is called. The difference. What you’re really trying to figure out in subtraction is the difference between the subtrahend and the minuend. Between the 2 numbers. That’s the simple way to say it. 

The idea is if we plot these on a number line, we’re really looking at how far is it from 289 to 543. Well, there’s lots of ways I could solve that. But a kid who’s doing or trying to do this give and take method, we kind of have to switch up what it’s called. It’s no longer give and take. It’s a little bit different when you get to subtraction. We can’t call it give and take. So let’s take a look at what’s actually happening here. Let’s say the kid does want to change the 289 to a 300. Okay? So we plot that point on the number line. Well, if I do what that kid said and I, hey, if I added 11 to the 289, that means I need to subtract 11 from here. 532. Well this is not the same distance as this. Those are not the same difference. So that’s why that idea doesn’t work. 

Instead what we have to think about is, and if I shifted, if I changed this 289 to 300, what can I change the 543 to so that I keep the distance the same? So this strategy, I kind of, this was the name that I called it personally. This is not something I shared with students. I called it the holy shift. Yes, that is with an F in there. But I have since leaned that some people also call it constant difference. You’re keeping the difference the same, but you can shift those numbers any way you want. So if I shifted the 289 up 11, I also need to shift the 543 up 11 to keep that distance or difference the same. This distance has to stay the same as this distance. And you can shift it anywhere you want. 

Now the problem has become 554 minus the 300. It’s not really give and take anymore. It’s kind of give and give. If I give 11 to the 289, I have to give 11 to the 543. Or let’s say I wanted to take some away. You could take some away from one, but then you’ve got to take some away from the other. But the real reason why you have to give and give or take and take is because you need to keep the difference constant. Okay? 

So I think that that visual really helped me internalize what was happening in that strategy. Now with our small amounts, when kids are learning their facts, right? This might look something like this. If it was 9 and 16, again, 16 – 9, we could shift this up to the 10. A kid might want to make that a 10. But then what do I have to shift the 16 to to make that difference stay the same?  I shifted up one both ways. I gave one and then I gave another one to the other side and that keeps the distance the same.

Adding Up (Finding the Distance)

Another one that’s related to this idea of viewing it as the difference is basically just find the difference. And a lot of ways that kids will do that is by adding up. So they’ll plot the numbers on the number line. So they have this visual. And the idea is you need to find that distance or difference between the 2 numbers. 

A lot of kids will start here and work their way up. Because adding is a whole lot easier than subtracting. Again, ignore, I know that it’s not proportional. That’s why doing number lines on an iPad is not very easy to keep this proportional. And then to hop up 43. Now basically I added up to find that, but really, I’m just finding that difference. I could have started at the 543 and worked my way down to the 500, right? It doesn’t matter which side they started that. They’re finding the difference here. 

Now if we go back to our 16 – 9, again, it’s about finding that difference. And I could start here at the 9 and I would probably hop up one to get to the 10. And then I could do a big hop of 6. And so the difference of 7. And up here your difference, your answer is in those hops. And you have to add those up to figure out what your answer is for subtraction.  A lot of people think that’s a lot of work. But man, addition is just easier than subtraction. So a lot of kids find this strategy a whole lot easier than subtracting. 

Decomposing

Another one we see kids do is what I like to call decomposing. And yes, it’s just like decomposing in science. It’s breaking it down into smaller chunks. So basically they start with the 543 and they’re going to take away 289. But you don’t have to do it in just subtracting 289. You can break it into chunks that make sense. So a kid might start out by saying, well, I’m just going to subtract 200 because that’s easy for me.  

So I can just subtract the 200 really quickly and get 343. Well now I need to subtract the 80, but 80 is not very easy for me to subtract right here. So I might chunk that 80, or decompose that 80, into a 40 because I can do that pretty quickly. And then another 40. 263. 

Now, the hard part is though, they do have to keep track of what they’ve subtracted. And that’s why kids oftentimes will think about the hundreds, then the tens, then the ones. But they don’t have to subtract, like, the 9 all at once. We want them to be able to see that they could chunk the nine into something that’s more friendly like a 3 and then a 6.  Now what this would look like when kids are doing 16 – 9 is they’ll say, well, 9 is not very friendly here. But hey, a 6 is. So that would get me to a 10, and then I could take away the remaining 3 to get to the 7. The idea is that they’re breaking it down into friendlier pieces for them to subtract.

Place Value/Like Values

Okay, another popular one is when kids will want to basically break it apart by its place value. So I put up here it’s place value, but I’m going to tell you why I prefer the name like values as we go through this. So it’s a kid who sees that 543 is really a 500, a 40, and a 3. And I need to take away a 200, I need to take away an 80, and I need to take away a nine from that. There’s lots of ways a kid might go about it from this point. Now what our textbooks typically try to teach kids is they will call this, like, expanded form, partial differences. But the idea is basically they teach the kids to do the algorithm. So they start here at the 3 and they can’t take away a 9, so they come over here and they regroup a 10 and so on. 

So I could model that here. That is definitely a place value or like value strategy. The idea is that they are looking at trying to subtract like values. Like they’re subtracting their ones from their ones, their tens from their tens, their hundreds from their hundreds. This progresses through all that they’re going to be doing in subtraction. We want kids to be able to see that they have to have like values. You can’t just add halves with fourths. You need to have fourths with fourths. You have to have like values as you progress through. 

They can regroup to be able to do this, but you can also just subtract. And I’m going to actually start from left to right, because that’s the way kids read and that’s the way a lot of kids naturally want to do this. 500 minus 200 gives you 300. 40 minus an 80. Well I could take away 40, but I still need to subtract a 40. And guess what? I got 300 over here to be able to subtract that 40. And off to the side, the kids are over here thinking already, 300 minus that 40. That’s 260. And then they come over here and they’ve got 3 – 9. Well, I could subtract 3, but I still need to subtract 6. Or depending upon the grade level of your kids, you might go into this idea that it’s a negative 6. Well I’ve got 260, and then I need to subtract the 6. So put those together and I’ve got 254.  

So, yes, kids can regroup, but they don’t have to. Kids will naturally want to do this strategy and they will make sense of it on their own based upon the numbers and number sense that they have, place value understanding that they have. And when it comes to our fact that we’ve been working on, it’s not really a strategy here, but it kind of can be. I’m going to actually rewrite that here and talk about it here. Because there’s not really, like, place values here. We only have one number that’s got a tens in it. But some kids will see it like this. It’s a 10 and a 6. And I need to subtract a 9. Well, I don’t want to take the 9 from the 6, because there’s not enough there. I’ve seen so many kids do this, where they will take the 9 from the 10 and they get 1, and then just bring down the 6, and their answer is 7.  So they’ll break it apart and then find ways to be able to match these up and subtract it in a way that makes sense to them.

Traditional Algorithm

Our last strategy is the traditional algorithm. And it’s called that because it’s the traditional way that we kind of all learned it. And it’s an algorithm because you can program a computer to do it. It is solved the same way every single time. It’s the same steps that you go through to solve the problem. We start in our ones. There’s not enough ones here, so we come over and regroup or borrow. The reason we call it regrouping is because we’re just taking some of the number and regrouping it into something that makes a little bit more sense. So this becomes a 13. This becomes a 3. So I could subtract here now. But now when we get here, I don’t have enough tens to be able to subtract. So we come over and we regroup again. 

All kids see are just these flying numbers, basically. But what’s really going on here is an important piece to connect to. And it connects to our past one that we just showed. So I’m going to space this out a little bit just to show it really quickly. Plus a 3. That what really happened here is when we took 10 from this, this became a 30 and this became 13. And one of the things we really need kids to understand here is we still have 543. 

That’s why we call it regrouping. I didn’t borrow it and take anything away. There’s still 543 in that top amount. I’m just changing how we have it grouped. And then when I come over here and change this to a 4, it’s really a 400 and I gave this 100. So the 400, the 130, and the 13 still make the 543. We’ve just regrouped it to make it easier to subtract. That’s an important piece for kids to understand with the traditional algorithm when we get there. We don’t want them to go through the steps and procedures and have it be devoid of meaning, devoid of place value or number sense. So I know a lot of people have built this in. But back when I learned it, we just learned the steps and procedures and not the underlying meaning behind it. So as they go through this, then they get that 254. And we want them to really understand what’s happening, not just going through the steps. 

I guess if we want to come back to our fact, this becomes the point where hopefully they just know it. Hopefully we do not get kids who do this. Because I gotta tell ya, I’ve seen middle school and high school kiddos do that. They did the traditional algorithm on a problem like this and they don’t even realize it did nothing. It’s still 16 – 9. We want them to be able to either just know it or use some of those other strategies that we’ve talked about in this video.

Kids don’t need to do every strategy

Now that you’ve seen all 6 different ways to solve a subtraction problem, I want to remind you yet again. Do not directly teach these to your students. These are for you to know what are potential ways your students will solve these types of problems if they have a foundation of numbers first. We’ve got to build that foundation of numbers, of place value, and then these strategies come naturally to students. They will derive these strategies on their own. You do not have to directly teach them. But yes, we’ve got to build that foundation of number sense and place value, and then kids will do this. 

Another reminder. Like, the other reason you don’t want to teach them all these ways to solve it is because some problems you should not be solving these ways. You don’t have students solve 1 problem 6 different ways. It doesn’t make sense, that you want to look at the problem first and then decide the best way to approach that problem based upon the numbers that are in there, the context of the problem, and my own personal knowledge of the numbers. 

If I had given you something like 579 minus 236, you might have just stacked that and solved it, right? You would have used the traditional algorithm. There wasn’t much thinking you would have had to have done. There’s no reason to round it. There’s no reason to break it apart. But yet sometimes our textbooks make kids do that. They will say solve this problem using whatever strategy it is that they want them to practice when I would never in my real life use that strategy. 

We want kids to be to look at the problem first and then see what makes sense to them. It’s about the strategies making sense based upon the context of the problem and their own understanding of the numbers. If they don’t have a sense of the numbers, none of these strategies are going to make sense. So having them solve it 6 different ways is really going to confuse them. So we don’t want kids to be having to solve it 6 different ways. Okay, it’s just a reminder for you to know what the possible ways are that they could solve it, not that they have to solve it this way. All right. I hope that this helped you build your math mind so you can build the math minds of your students. Have a great day.

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