5 Types of Addition Strategies

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With textbooks having students use different strategies to solve addition problems, I wanted to do a short series of videos to share with you what those strategies are, and how they help kids not only do normal addition problems, but also time, measurement and money addition problems.

I’m Christina Tondevold, The Recovering Traditionalist. And today we are going to do the first video in this series that will take a look at the 5 types of addition strategies in our quest to build our math minds, so we can build the math minds of our students.

 

Watch the video or read the transcript below:

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4 Videos of this series

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Now, a couple of warnings before we get started. 

#1 I’m going to talk about these strategies, but they are not strategies that you should directly teach to all of your students. I want you to be aware of these strategies because these are possible ways that students think about solving addition problems, but we should not be directly teaching them to our students. There are certain times when they will use them and certain times that they won’t. It’s not like kids need to use all 5 of these to solve any addition problem they ever encounter. 

#2 is that you might call them by different names. You might have different names for these strategies. When I first started doing this in my classroom when a student would use the strategy, we would name it after the student. “This is Sierra’s strategy. Isn’t this awesome the way that Sierra solved this?” 

One of the articles that I read in the “Teaching Children’s Mathematics” which was published by NCTM, I can’t remember the title of it, I can’t remember who authored it, but I remember the gist of the article was that we should stop naming them these cute names and really name them after the mathematics that is happening. So I’m gonna be using names that I use to talk about the math that we are doing in the strategy. You might call them something different, but these are the names that I personally use for them. 

Now, the other thing I want you to be aware of is that to me there are 5 distinct strategies. You might end up seeing kids who will combine or have variations on these strategies, but these are the main basis of the 5 strategies. So one of the things that actually ends up happening is that we will see these strategies, but in different formats. A kid might use the same strategy, but they were doing it using base ten blocks, whereas another kid just wrote out the equation. So just because they’re using manipulatives or an equation or a number line to show their strategy does not make it a different strategy. They’re using a different model to show that strategy. So how they notate it does not determine the strategy. We need to be paying attention to what is the mathematics that they are doing for this problem. It doesn’t matter how they’re showing that mathematics. The strategy is based upon what they are doing with the numbers and what’s the root of the mathematics. All right, so let’s dig in to those 5 types of addition strategies. 

Compensating

Okay, we’re gonna start with the strategy I like to call compensating. And for all of these, we’re gonna use the same problem, 399 + 456. And I like to call this one compensating because to me compensating is when you do something you’re not supposed to do and you need to compensate for it. So the 399, this is when you will see kids say, “Well that’s almost 400 so if I just add 400 to 456 “that would be 856.” But you weren’t supposed to add 400, so we need to compensate for what we did. So this strategy only works if kids understand the relationship between the numbers that are in this problem and how they’ve changed the problem. So they need to look at that and understand that they added 1 more than they were supposed to so they need to come at the end and subtract 1 of those away to get to their answer of 855. 

Now, all of these strategies, the reason I love these strategies is that they work no matter what kind of addition problem you are working with. So even when kids are learning just their basic facts, you will see kids who will do these strategies. You’ll hear kids say, “Well 9 + 7, “that’s like having 10 + 7 which is 17, “and then I just need to take away 1 to get to the 16.” And then it can extend even further into work with decimals and so on. 

The reason that I love helping kids develop these strategies is that they last kids beyond just what they’re doing with that one problem. When they really understand how numbers work, they will use these strategies no matter what size of numbers they are working with.

Give & Take

One that is really similar to compensating is one that I call give and take. So it’s the kids who want to use 400 because they see that that would be nice. But instead of just rounding it to the 400, they decide, I’m gonna take 1 from the 456 and give it to the 399 so that I have 400, but then I’m only left with 455 in that space now. So they can get directly to their answer after they have kind of done this adjustment. 

They’re still moving things around, but they do all of the adjustments before they go to actually solve the problem. So with our young kids, that might look like this with the 9 + 7. You might see a kid who says, “Well, I’m gonna take 1 from the 7 “and give it to the 9 so I’ve got 10, “and then I’ve got 6 leftover, “so 10 + 6 gives me the 16.” So again, these are really related, but it’s like what are they seeing as the relationships? How are they going about the problem? So that’s what really makes a difference between the two strategies.

Decomposing

Another strategy that kids will devise is that they will see that 456 and they’ll say, “I don’t wanna add it all at once,” all right? They will want to break it apart into chunks that make sense for them. And this is what I call decomposing because it’s just like decomposing in science. It’s breaking it down into smaller pieces. 

You might see a kid who, this might show up on a number line. They might show it on a number line. I don’t really like to show this one on a number line ’cause it’s hard to be proportional. So I’m doing this on an iPad so don’t judge me if my hops are not totally proportional. I know they should be. 

But you might see a kid who says, “I wanna do add 1 here “because it’s gonna get me to the 400.” Sounds similar to what we’ve been doing already in these out strategies, right? But it looks a little bit different from this point on, is that once they get there then you might see a kid who goes, “Okay, now I’m gonna add 400. “I’m gonna do a big chunk of it, and then I’m gonna add a 50,” all right? But the hard part here is that they need to know how much they’ve added. They have to keep in mind that they need to add the entire 456 but they can do it in chunks. But they have to keep track of what those chunks have added up to. 

So along the way, they need to be holding in their head that they’ve done 450 and 1, and then think about how much more do they need, right? They need to know once they’ve done 451 how much more is that to 456? So they need to know that there’s 5 more. See there, my hop for 5 looks the same as hop for 50. And then they would get to 855. 

There’s a lot of number sense that goes into play here. They need a lot of knowledge about how these numbers work. All right, the general idea is that they are keeping one of the numbers the same, the 399. And then they’re breaking apart the other number into smaller chunks. And that doesn’t matter if they do it on a number line, or one of my favorite ways to show this is just using arrows to show what I am doing, so that I don’t have to worry about being proportional. 

So I’m gonna do 399. I’m just doing a little arrow. I’m not using the equals sign because you can get this big mathematical run-on sentence is what I like to call it. So I just use arrows instead of equal signs here. So I’ve got 399 plus 1 gets me to the 400. And then I’m gonna add another 400, it gets me to 800. And then maybe this time I might add the 55 all at once. It doesn’t matter what their chunks look like as long as the general idea is that they’re breaking one of the numbers down into chunks to make it friendlier to add. 

And again, it doesn’t matter if they do that on a number line or they show it with these arrows or they make different equations for every single step that they do. The idea is that they’re just breaking it down into smaller chunks to make it friendlier to add. Kids will do the same thing with 9 + 7, right? They might do 9, add 1 gets me to 10, add the 6 gets me to 16. Now, this looks a lot like give and take, but it’s because there’s not a whole lot of ways to break up the 7, right? 

You may even see a kid who maybe can’t add the 6 all at once so they add 1 more and they get to 11, they add 1 more, they get to 12 and so on it goes. They just add 1 by 1 by 1, right? That’s the extreme form of decomposing is doing it one by one by one. But even with these upper amounts, you might see a kid here at that 400 stage, they might not be able to add the 400 all at once. They may add 100 then another hundred, then another hundred, then another hundred, right? The idea is they’re breaking it down into friendly chunks that are friendly for them.

Place Value

Another strategy where kids are breaking apart numbers is where they break apart by place value. Some textbooks will call this the hundreds, tens and ones strategy. I don’t really like that strategy because it limits kids to thinking they can only do it with hundreds, tens and ones. When they get into decimals, I want them to still be able to use this strategy. 

The general idea is that they’re breaking it down, and they are adding the values together in each place value and then they’re putting those together. Some of you may see this as expanded form. So again, it doesn’t matter how you model it. It is the idea of what’s the mathematics that we are doing? 

I’m going to do a really shortened version of this here. And you could have it out in expanded form, but we’re doing the same thing. The idea is that I’m adding my hundreds together. I have 300 and 400, that gives me 700. I add my tens together, 90 plus 50 is 140. And oftentimes what you’ll hear kids do right here is they’ll be thinking off to the side here that that’s already 840. Then when they add their ones they get the 15. And then they can add that to 840 and they get their 855. All they’re doing is adding the chunks by place value, and this, kids will naturally do if they have a lot of work with place value. 

All of these strategies are dependent upon number sense and place value. You cannot directly teach it if kids don’t already have a foundation of number sense and place value. Now, these are things that we go really deep into in my number sense courses, but I’ll also link to some free videos that I have about the basics of number sense so that you can get a foundation of where to start with your kids. Because if they don’t have these ideas, they’ll never be able to really latch onto these strategies.

Now, with 9 + 7, you don’t really have a place value strategy because there aren’t any other place values, so you just add your ones and your ones. When we get into things like 19 plus 17, then you’ve got some place values to work with, and then they’ll add their tens with their tens and their ones with their ones.  But before that there’s no real place values to have to work with here.

Traditional Algorithm

Our last one is the traditional algorithm, or what in the United States we call the traditional algorithm. This is the way that we traditionally learned how to solve addition problems. Set it up and we add our ones, and then we carry it over, add our tens, carry that over and then we’ve got our answer. 

It’s called an algorithm because you can program a computer, a calculator to do it. It is the same steps every single time. There’s no real decision making to have to do here. You don’t need number sense. You don’t even have to have place value here. All you have to do is single digit addition, and then be able to carry the one over, and carry the one over. There’s no real judgment that has to go into this, which is one of the reasons why we use it. 

You do the same steps over and over again and it works every single time. But it is devoid of place value, and it’s devoid of number sense, unless we bring that in as teachers. That’s why in most states, the standards bring in the traditional algorithm at 4th grade. Before that time, we need to be building their number sense and their place value to help them be able to see and build those other strategies that we’ve talked about. 

We want kids to be able to look at a problem and decide the best way to solve it. Because sometimes the traditional algorithm, yes it works, but sometimes it takes longer to solve the problem using the traditional algorithm than one of those other strategies.

Kids don’t need to do every strategy

So the main idea from these videos, what I want you to really take away from it is yes, there are 5 different ways to solve an addition problem, but we do not need to be making kids solve this one problem 5 different ways. 

We need them to be able to look at the problem and then decide for themselves what’s the best way to solve this problem? What makes sense? Because what makes sense for this problem, the 399 + 456, is different than what should make sense for the problem 323 + 456. How we solve these two problems should be different. We should not approach them the same way. 

This one with the 399 is just begging me to round that to a 400 or make that a 400. Nothing in the 323 + 456 says round this, do compensating or give and take, but yet, we could make kids do it. We could make them solve it that way, but it doesn’t make sense to. We want them to have these strategies at their disposal if it makes sense on that problem. And the only way that we help kids be able to do that is by building their number sense and place value, so that they can look at the problem first and say what makes sense with this problem? 

So remember that these 5 strategies are not things that you should directly be teaching to your students. Kids naturally do these strategies based upon their understanding of the numbers that are in the problem. We cannot directly teach these strategies to students. It comes when they see relationships within the numbers and they say, “Oh, I see how I could use this and do this with it.” We can’t directly teach it even though our textbooks try to. 

That’s why you will spend weeks and weeks trying to get them to use these strategies and by the end of it all you really get are students who are thoroughly confused. I want you to be aware of these strategies and be watching for kids who are naturally doing these. And then have them talk about and discuss what they’re doing instead of you directly telling them how to solve using these strategies. 

This video was to help you see the possible ways that kids could solve addition problems, not that they have to solve every problem 5 different ways. I just want you to be aware of these different ways. Because if you were at all like me back when I first started teaching, the only way I knew how to do it was the traditional algorithm. And when kids solved it a different way, I really had no clue what they were doing. I could not tell what they were doing and I would just say, “Uh, don’t do it that way, do it this way.” And I would tell them the way to do the traditional algorithm. The ways that kids are solving these are important pieces, but they needed to be coming naturally from them, not us directly teaching them. 

So, I want you to be aware of them, so that you can be watching for them and pull them out when you see kids using them. All right, I hope that this video helped you build your math mind so you can go build the math minds of your students. Have a great day.

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