With our textbooks and standards asking students to use multiple strategies to solve subtraction problems, I’m doing a little series all about what these strategies are and how they can actually be helpful when students are solving time, money, and measurement problems around subtraction.
I’m Christina Tondevold, The Recovering Traditionalist, and today’s video is all about solving measurement word problems with subtraction strategies in our quest to build our math minds so we can help 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 the series)
Solving Elapsed Time Subtraction Problems (2nd video in the series)
Solving Subtraction Money Word Problems (4th video in series)
Free Training – Components of Number Sense in PreK-2
Free Training – Components of Number Sense in 3rd-5th
Now, in all of these videos, I start off with a little bit of a warning. But I’ve done a lot of these videos. I’ve had a whole series around addition, so I’m going to summarize my warnings. If you need more in-depth understanding of why I warn about these things, go back to the first video in this series, I’ll link to it below this video.
#1 I want you to remember, as you watch this video, is do not teach all 6 of these strategies to your students. These strategies are for our knowledge so that we can understand where students might potentially go when solving problems. So if you were like me, I had one way to solve the problem. And it was the traditional, kind of way that our teacher had taught us. I had no other thinking strategies. So when my students did something else, I just kinda looked at them like, ah, that’s cool. And if they got the wrong answer, I didn’t know how to help them. So it is just for our knowledge.
#2 is that I have certain names that I call these strategies, but might call them something different. So, don’t go by the name. Look at the mathematics that’s happening in the problem.
#3 is your students might not do it the same way that I am showing. Just because I’m showing it a certain way, they might choose to break apart or think about different kind of benchmark numbers. I’m just showing you the general idea and your students might do it a little bit different, and they might even combine a couple of these strategies to create their own invented strategy.
We are going to use one problem and look at the 6 different subtraction strategies using that one measurement word problem. And here’s the problem we’re going to use:
Sierra has a piece of wood that is 6 feet, 6 and 1/4 inches long. She wants to use it to create bookshelves for her room. She cuts off one shelf that is 2 feet, 10 and 1/2 inches. How much is left for her second bookshelf?
Let’s delve into those 6 strategies and see what they look like on that one problem.
Traditional Algorithm
Okay, let’s start with the traditional algorithm. She has 6 feet, 6 and 1/4 inches. And they need to take away 2 feet, 10 and 1/2 inches. So right off the bat here, when kids look at this, they say well, I don’t have enough, right? The half, hopefully they know that a 1/2 is bigger than the 1/4. We gotta combat that first, if they don’t. But when they come over here to regroup, often, what we see kids do, is they will make that a five. They know that goes down. But they think, well, I just add one here. No. No, no, no, no, no.
So again, this is another prime example where we really need to focus on building student’s sense of the numbers that are in there. It is not just about steps and procedures. Even with the traditional algorithm. They need to have a sense of numbers before they will ever be able to do these strategies. So, if we do help them understand that yes, you are regrouping 1, but what does 1 look like? Well, 1 is 4/4, so really, this becomes 5/4, and 1/2 they need to instantly be able to know that 1/2 is the same thing as 2/4. So 5/4 minus 2/4 gives us 3/4 here. Same thing happens when we come over here. And kids need to regroup. They need to bring over 1 from here but what is the 1? It’s not just regrouping a 10 this time, right? They need to know that there are 12 inches in that foot and we are bringing over 1 foot. So we’re bringing over 12 inches, so this becomes 17 inches in that spot, okay?
Now they can go through and do all of the subtraction there. But there is so much involved in this traditional algorithm, and that it is not just about the steps and procedures. They really need a sense of these numbers. And the other strategies that we’re going to talk about here help build that. So again, it’s not that we don’t want kids to be able to do this strategy, we do, but we want them to develop some flexibility and number sense along their path to getting to be fluent with this algorithm.
Compensating
Okay, with the compensating strategy, the kids start out with the original amount. But when they go to subtract, when they look at that 2 feet, 10 and 1/2 inches, a lot of kids look at that and say, man, that is really close to 3 feet. If it was only 3 feet, it would be so much easier. Well, they can. They could subtract 3 feet. Right, they’ll get 3 feet, 6 1/4 inches. But they really weren’t supposed to subtract 3 feet. So they have to come back and compensate for what they did.
Now, in this video, I’m not going to go in-depth with how we help kids understand that. I talked about that in the first subtraction video where I just showed all of these strategies using whole numbers. So you can go back to that one. But the idea is that they took away more than they were supposed to. So they have to add back in that extra amount. So they would need to be able to figure out how much extra they took away and then they can come back and add that in. And again, this is really reliant upon kids understanding their benchmark numbers, being able to fluently see that when it’s 1/2, that they know that that is also 2/4, right? All of that is just underlying number sense that kids need about the numbers that are inside of the problem.
Decomposing
Okay, on this strategy, kids again start with the original amount of the length of the board. And they are going to subtract out that 2 feet, 10 and 1/2 inches, but they realize they can break that number apart and subtract it into chunks that make sense for them. So that’s why I call this decomposing. Because they’re breaking it apart into smaller chunks that make sense for them. So you might see a kid who says, well if I can subtract 2 feet, that’s a whole lot. Just do the 2 feet, that’s a whole lot easier. We still have the 6 and 1/4 inches, okay. But then, now they’ve gotta subtract that 10 inches and the other 1/2 of an inch. But how they do that is kind of up to them. Whatever makes sense to them.
Again, kids might do this differently, but you might see a kid who says something like, well, I’m going to just subtract the 6 and 1/4 inches because that’ll get me to that nice friendly amount of four feet, okay? But then they have to think, okay, I’ve subtracted those 6 inches and that extra 1/4, but what do I still need to subtract? So, you might see a kid who says, well, I know I need to subtract an additional 4 inches, so that would get me to 3 feet, 8 inches. And again, think of all the number sense that’s built into this. They have to be able to instantly know how much taking away four inches is, right? That there’s 12 inches in that foot and taking away four from that would leave the eight inches. And then they have to be able to look a that and say, okay, I’ve subtracted 10 and 1/4 inches, how much more do I need to subtract? Well, I’ve only got one more 1/4 of an inch to take off of there, so I get my 3 feet, 7 and 3/4 inches.
Number sense is a really big deal. Doesn’t matter if they’re doing with fractions or whole amounts, this whole strategy is based upon kids being able to chunk out some numbers that make sense to them and whatever makes sense to them is based upon the number sense that they have built.
Adding up
The adding up or also known as finding the difference strategy, it really is kids needing to see it as finding the difference. We have this board that is 6 feet, 6 and 1/4 inches. And they cut off a piece, right, I may not be proportional here, that is 2 feet, 10 and 1/2 inches. Now, the whole idea is our goal, even in this problem, is how long is this piece right here? Well, to be able to find that, I could add up to do it or really, it’s just finding the difference. I could start at the 6 feet area and go backwards down to the 2 feet 10 inches, but I prefer to add up instead of subtract. And that’s the way a lot of people do. So it’s commonly known as adding up but really, you’re just finding the difference.
So you might see a kid who adds 1 and 1/2 inches right here. Right, actually, I’m going to undo that, because I’m going to take out the add sign, because it’s really just 1 and 1/2 inches. Whether you’re going up or down, it’s 1 and 1/2 inches. Because that gets me to the 3 foot mark. The 3 foot mark makes it a little friendlier to work from here, right? Then I might hop up 3 feet because that will get me to my 6 feet. And then I just have this hop here of 6 and 1/4 inches.
Again, mine is not very proportional, you can see that. My hop for 1 inch is about the same as my hop for 6 inches. So, I’m sorry it’s not very proportional on this iPad. It’s hard to do when I’m writing on the iPad. But now, the only problem is, well, and really, I don’t see it as a problem. Because adding is way easier than subtracting. To actually find their answer, kids need to add up what’s here. But again, if they have built some number sense, this addition piece right here is super quick and easy. It’s easy for kids because they understand really quickly, that 1/2 is the same as 2/4, so they put their fractions together. They got 3/4, their inches are 7, so it’s 3 feet, 7 and 3/4 inches. So, even though it looks like a lotta steps, kids naturally do these, very, very quickly. And if they have the number sense built, they can combine these amounts very, very efficiently.
Place Value/Like Values
With this strategy called like values, it’s where kids set this up, and it will often kind of look like the traditional way. If they’re writing it on paper. Kids will naturally do this strategy in their head, but this is how I would write down what this kid was saying.
Kids will naturally want to subtract the pieces that are the same. So they’ll take their feet and they’ll subtract that. And they’ll say, okay, I’ve got four feet. Then they’ll take their 6 inches and 10 inches, and again, this is based on, if kids have this number sense, that if I have 6 inches, and I need to take away 10 inches, well, I could take away 6, but I still need to subtract 4 more inches. Well, guess what? I’ve got these feet over here that I could subtract those inches from. So they’re doing this in their head and they’re saying it, that they have 3 feet, 8 inches left there. Then they come over to the fractional amount, 1/4 and I need to subtract 1/2. Again, they should be fluent with the idea that 1/2 is the same as 2/4. So if I have 1/4 and I need to take away 2/4, well, I could take away 1/4, but I still have to take away another fourth. And then they can come over and do that from our 3 feet 8 inches that are left, and they’ll get their 3 feet, 7 and 3/4 inches.
So again, like values is where they are subtracting each of the pieces together, and then they’re putting those together to get their final answer. Some people call this expanded notation, partial differences. I just like to call it like values because no matter what they’re subtracting, whether it’s feet, inches, ounces, grams, fractional amounts, the idea is that you need to subtract things that are of like value.
Constant Difference
Our last strategy, and I saved constant difference for our last strategy because it actually does not make sense on this problem. And your students most likely will not use this strategy because of the context. Now, you might have kids who loved this strategy and will use it even devoid of the context. Like they get that it really doesn’t make sense in this situation but they’ll still use it. But here’s why it doesn’t makes sense. Constant difference is that idea of we’re seeing it as the difference.
The idea on this problem is, we really need to find the difference between these 2 amounts on our board. That’s what’s telling us what’s left over. The idea of constant difference is that we can move those marks, I guess, on the number line, any way that we want it, to make it a little easier to find the difference. So, the idea is that I could take this 2 feet, 10 and 1/2 inches and I could bump it up a little bit, to get it to the 3 foot mark, to make it easier. But if I bump the 2 feet, 10 and 1/2 inches up, I also have to bump the 6 feet, 6 and 1/4 inches up. So I bumped this one up 1 and 1/2 inches, so this one would also need to bump up 1 and 1/2 inches. 6 feet still, 7 and 3/4 inches. And then this difference, from here to here, is the same as the difference we were trying to find. But it’s just a little bit easier subtraction to do.
So, you can still do this strategy, but it kind of doesn’t make sense. Because the board that we are cutting is 6 feet, 6 and 1/4 inches. So it doesn’t just magically become 6 feet, 7 and 3/4 inches. So a lot of kids won’t naturally go this strategy unless this is kind of already a go-to strategy of theirs, and they are just kind of suspending reality a little bit and pretending like they could move this around.
Kids don’t need to do every strategy
So, hopefully as you saw through that, there’s a reason why I say don’t teach all of these strategies to your students. Because sometimes, that strategy makes no sense to use on that problem. We want kids to look at the problem first and then decide what is the best way to approach this problem. So the context of the problem matters, but what also matters is the numbers that I use in that problem. If I had given you different measurements, you may not have needed those. Like let’s say, the piece of wood she was using was 6 feet, 6 inches, and she cut off a piece that was 2 feet, 3 inches. You wouldn’t have needed all of those strategies, because you could just subtract those whole numbers and we’d be done. So, the numbers we use, the context of the problem, help determine what strategy your students are going to use to solve that problem.
Now, the other thing that helps determine that is their sense of the numbers. They have to be able to look at those numbers and know, man, that’s almost a foot. So before you ever delve into addition and subtraction with measurement, we first have to really build a student’s understanding of the measurements. They need to have number sense around the different measurements. So really stop and build that sense of measurement for your students before you ever ask them to operate with those amounts, okay?
I have some videos that I’ve done about number sense. They don’t specifically talk about measurement numbers but as you watch those videos, you can look at it through a lens of, what are those number sense concepts? And how do they apply when we’re working with measurement, okay? So, I’ll link to those below this video. All right, 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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