Man, one of the biggest pet peeves I have when I am out at stores and restaurants is when someone does not give me the correct change back. I’ve been doing this series all about the different strategies kids use for subtraction, and giving change back is one of the real world applications of these strategies. So, in this series, we’ve been looking at what the strategies are and how they help kiddos with solving time, money, and measurement word problems.
I’m Christina Tondevold, The Recovering Traditionalist, and in today’s video, we’re tackling that big one. We are looking at solving money word problems with subtraction 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:
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
6 Types of Subtraction Strategies (1st video in this series)
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 (This Video in series)
Free Training – Components of Number Sense in PreK-2
Free Training – Components of Number Sense in 3rd-5th
Now, the warnings on these, if you haven’t seen any of these videos yet, you can go back to the very first video I did in this series, which I’ll link below this video, to get more in-depth understanding of these warnings, but just to summarize, because just in case this is the first one you’re seeing, I want you to have these warnings.
#1 I’m going to show you the 6 different strategies, but do not teach these 6strategies to your students. If you teach all 6, all that happens is your kids get thoroughly confused. So, this video is to just inform you of what the strategies are, what they potentially might look like, so that when your students are doing them, you can recognize it and say, “Ah, that’s what you’re doing.”
#2 I’m going to give them names because that’s what helps me. You might have different names for these strategies, and that’s okay. Just look at the mathematics that’s happening. That’s what we’re really wanting to pay attention to.
#3 Now, the other piece that I like to warn you about is that how I write it down on paper is not exactly how your students will do it. So, theirs might look a little different. They might even combine strategies. But the idea is to just be aware of these strategies. Again, we’re not directly teaching them, so it’s okay if they’re combining strategies, if theirs looks different than mine. I just want to inform you of potential ways that your students could possibly solve these problems.
We are going to use one story problem to take a look at the 6 different ways that kids could solve money word problems with subtraction.
So, the problem we are going to use for this video is:
Jaeger has $120.06. He went to the store and bought a few DVDs and some candy, and he spent $48.85. How much money does he have now?
Let’s take a look at how to solve that one problem with these 6 strategies. And remember, don’t teach your students all 6 of these. That’s not the point. Just so you can see what possible ways your students might have for solving that one problem. All right, let’s take a look.
Traditional Algorithm
The traditional way of solving a problem like this is just to stack it and do the traditional algorithm like we were taught. We start with the 6 minus five and we get one. The problem occurs when kids have to subtract where they need to regroup, and especially when it’s regrouping across a zero. So like, I don’t know about you, but when I was a kid, this was like the worst for me. And even now, I really have to pay attention to what I’m doing and think about like, which one becomes a 9? Which ones are tens? It’s a very confusing area. But if we can build this idea of the number sense and place value for our students, that helps them better understand this algorithm.
Now, the algorithm in and of itself is devoid of any place value. We as teachers help bring that in for our students, and help them understand what’s really going on as we come through and do all of this regrouping here. There we go. Then when they go to subtract here, going to get to this spot where it’s another little tricky spot where I see kids showing me that they don’t really understand what’s going on, and they’re just kind of going through the motions. When the kids get to this point where they see the 1 and they have to subtract 4, so many students come over and regroup. They’ll cross out this one and put a one there. When really, it was there already. If we have built place value and the students have developed this place value understanding, when they get to that last step there, they see it as 11 and they can subtract 4 from that, instead of needing to regroup. They really understand what the numbers in the problem mean. They’re not just individual columns that they need to subtract from.
It’s not that the algorithm is bad. It’s just kind of devoid of place value and number sense unless we as the teachers bring it in. And these other strategies that we’re going to talk about really do involve lots of number sense and place value. And so if we’ve built that for our students, they will naturally develop these other strategies because, dare I say it, I think they’re a whole lot easier way than our traditional algorithm to figure out giving the change back and subtracting with money.with this algorithm.
Compensating
With the compensating strategy, that’s when the kids look at the amount they are supposed to subtract, and they look at that and they say, “Man, $48.85, that is super close to $50. “If it was just $50, it would be so much easier.” So yeah, subtracting $50, it’s like rounding, and it would get me close so I could estimate about how much he has left over. But if I want the actual answer, I have to come back and compensate for what I did.
Now, I’m not going to go too in-depth on this explanation, but essentially, the idea is they subtracted more than they were supposed to because they were only supposed to subtract $48.85, so they have to come back and add back in the amount extra that they took away. Now of course, they do have to be able to look at that $50 compared to the $48.85, and know instantly how much extra they took away. And again, this is part of number sense.
One of the components of number sense is understanding benchmark numbers and being able to tell quickly how numbers relate to benchmarks. So, I have some videos out there all about the components of number sense. We’ll link to those below so you can take a look. It’s not about money, but you can take the ideas that are talked about around number sense and apply that, how do they apply to money? And this is a big one, the benchmark. So, how do numbers relate to benchmarks? So if they can do that quickly, they’ll use this strategy, and then they come back and compensate, and then they can find the actual answer, not just rounding to get close. They can actually use that rounding strategy to get the exact amount that he has left over.
Constant Difference
In the constant difference strategy, this is where you see those kids who want to change that $48.85 to $50. They want to say, “Man, if it was just $50, “that would be so much easier.” But then they aren’t sure how to adjust the other amount. If I changed one, they have this sense already that they need to change the other one. They need to do something to it, showing it on a number line is a great way to help kids see that if we adjusted this $48.85, if I’ve shifted that over, I also have to shift this one over the same amount so that this difference stays the same. Because the whole idea of subtraction, or one of the big ideas, really, there’s two ways to think about subtraction.
You can either view it as take away, or you can view it as the difference. How far is it between those two numbers? And you can figure out that distance any way that you want, and one of those ways is to keep the distance the same, but shift it on the number line. So that’s what kids will do. They’ll see that they added $1.15 to get to that $50, so now they have to add $1.15 to this one, and then this difference tells them how much this difference was. It’s the same difference. I’m just sliding it up or down the number line to make it a little easier to do the subtraction.
If kids are viewing it as finding the difference, another way that kids will do it is instead of shifting the numbers, they’ll keep the numbers the same on the number line. Like visually in their head, they’ll see this number line. And we’re just showing it on paper here. But all they’re doing is counting up, kind of adding up, from the $48.85 up to the $120.06.
Adding up
Now, most people add up, but really, you are just finding the difference. And I could start at the $120, and work my way down to the $48, but most people add up. So I call it adding up. So, what you’ll see is you’ll see a lot of kids start with the 15 cents because that, they know that that would get them to the next dollar. And then they’ll add in that dollar to get to the $50. Now, kids might do this completely different. I’m just showing you one of the ways to get there. Once they’re at that $50, then you might see them do a $50, and then a $20. I’m going to do the big leap of adding $70 because I know that I can do 50 and 70, it makes 120. Your students may not be able to do that big jump. But once they’re at that 120, now all they have left is the 6 cents.
There is an additional step here because they don’t know what their answer is. It’s there, they just now need to do a little bit of something extra. Their answer is in that difference. They need to figure out what that difference was. But the hops are usually fairly friendly. I can look at that pretty quickly and say, it’s $71, and then there’s 21 cents there.
So, this idea of finding the the difference will help create this adding up, or even the last one we did, the constant difference. Oh, and I forgot to tell you. I jumped straight ahead into this next strategy, but the last strategy of adding up used to be called the cashier’s algorithm because that’s how cashiers used to count back change. When they didn’t have the cash register to just tell them how much change to give back, they would start at the amount that you owed and then count up to the amount that you had. That was how they would figure out how much change to give you back. So that one was also known as the cashier’s algorithm.
Decomposing
All right, moving on to decomposing. Decomposing is where you keep one of the numbers the same, and then the amount that you’re subtracting, you take it away in chunks or in pieces. You break it down into smaller parts. That’s why I call it decomposing.
It’s a kid who knows they have to subtract $48.85, but they want to do it in friendlier chunks. So you might see a kid who says, “I’m going to subtract $40 right off the top.” Now, some of your kids might not be able to subtract the whole $40. They may need to do $20, because that gets them to $120. And then another $20. But I can do the $40. Some of your students may be able to as well. So I’m showing that way. They may break it up into smaller chunks. But this is the idea is that they’re breaking it down into chunks that are friendly for them to subtract.
So, I’ve subtracted my $40. I also need to subtract the $8. That one’s still easy for me, so I’m going to keep it as the $8. I can subtract $8 really quickly. But now I have 85 cents that I need to subtract. 85 cents is not friendly to subtract right here. But 6 cents is because that gets me to $72. Now, the only hard part here is if I’ve subtracted 6 cents, I still need to figure out how much left over do I still need to subtract. And here, I’m using some of my number sense to figure out that I have 79 cents that I still need to subtract from that and to get me to the $71.21.
Again, this is a lot of number sense. So these are, again, why we don’t tell you, do not directly teach these to students. If they don’t have number sense here, they’ll never be able to do these strategies, and they’ll just get frustrated. So we want them to be able to use these strategies if they see ways that they can break apart the numbers and make it easier, but we don’t need to force them to do it.
Place Value
Our last strategy is one that I call like values. Some people will call it expanded notation because they will write it and expand out the numbers. Some people call this partial differences. The idea is that kids naturally look at this and want to subtract out chunks that make sense, and they will typically do it along place value because they do it with addition, so they naturally want to do it with subtraction.
So like with addition, they will want to add their hundreds with their hundreds, and their tens with their tens. Well in subtraction, you can do the same thing. So a kid might look at this and say, “Okay, 100 minus, “well, I don’t have any hundreds to subtract, “so I’ve still got 100 there.” And even if you try to picture this like, with Base 10 Blocks, we don’t go into it in this video, but a lot of the hands-on manipulatives things, the visuals that you can build with kids, helps build these strategies for them.
If they’ve got 100 sitting over there and they don’t have any other hundreds to subtract, they’re still left with 100. If they have two 10s, or 20, and they need to subtract 4 10s, well they can subtract two of their 10s, but they still need to subtract two more 10s. And if they’ve got 100 sitting over there, a lot of kids are doing it as they go along, and they’re sitting here thinking, off to the side, I’m going to just do a continuous thought bubble over here to show you what kids think as they go along. Well, if they’ve got the 100 and they need to subtract 20, they’re sitting there to themselves saying, “Okay, I’ve got 80 left, I got 80 left.” Then they come over here and they’ve got zero minus 8. Well, I still need to subtract 8 because I didn’t have anything that I could subtract. So off to the side over here, now they’re not at 80. They’re at $72. And then they go into their cents area, and they have zero minus 80. Well, I still need to subtract 80 cents. I didn’t have any there to be able to subtract. So if I subtract 80 cents, now I’m at $71.20. And then the final one, 6 – 5 gives me 1 cent left. So now I’m at $71.21.
They’re doing the subtraction along the way. If you’ve never thought this way, I personally had not because I’d always been taught to do subtraction and addition from right to left, but our students do it, a lot of times, from left to right because that’s the way that they’re taught to read, so they naturally look at those big numbers first, like their hundreds and their tens, and then they move down the line basically. So, this is a way to be able to notate their thinking if that’s the strategy that they’re using on this problem.
Kids don’t need to do every strategy
Okay, now that you’ve seen all 6, I just want to remind you again, don’t teach all 6 of these to your students. Give them a problem and see how they approach it. And one of the things that will help with how they approach it is first building their sense of money. You need to build number sense for your students, and specifically, around money so that they can easily be flexible with their understanding. Break apart numbers, put them back together, think about benchmarks.
I’ve got some videos that I’ve done all about what is number sense. Even though it’s not directly at money, but just these general ideas of number sense that kids have to have before they will ever do those strategies. I’ll link to those below.
Now, don’t forget that the context of the problem, the numbers that we use in that problem, will also determine whether or not they will use these strategies. If I had told you that Jaeger had $122.86, and he spent $21.50, you wouldn’t have used any of those strategies, nor would your students. They would just stack it and probably subtract because they don’t need any strategies. It’s a really easy problem to solve. But when things get complicated, that’s where we want kids to have that flexibility to be able to think fluently, think flexibly, about the numbers and decide what is my best course of action based upon this problem and the numbers that are in it? And the way that we do that is laying a really solid foundation of their number sense and how they think about how numbers relate to each other. I hope that this video has 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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