Well, with textbooks and standards, saying that our students need to be using strategies when they are solving subtraction problems, I decided to do a little series about these strategies and how they can be helpful when students are solving problems that involve time, and money, and measurement.

I’m Christina Tondevold, the Recovering Traditionalist, and today’s video, we’re going to look at solving elapsed time 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 Subtraction Measurement Word Problems (3rd video in the series)

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

Number Sense Courses

Free Training – Components of Number Sense in PreK-2

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

Now, I always start these videos off with some warnings, and I’ve done a lot of these videos already because I did them all about addition strategies, and I’ve done the base one about the 6 strategies for subtraction, so if you want really depth stuff about my warnings, go back to those videos. But just to kind of get into the content quickly, I’m going to summarize my warnings. 

#1 Do not teach all 6 of these to your students. I’m doing these videos to help you see what these strategies are, so that you can understand them when you see your students doing them. 

#2 I call them by certain names, you might call them something different. So just because you might have a different name, I really want you to look at the mathematics that I’m doing, and that’s what you look at and base the strategy off of what the student is doing mathematically, then what we call it doesn’t really matter. 

#3 You will often see students do a combination of these strategies. It doesn’t always look exactly the way that I show in these videos.

Let’s get started, and we’re going to use 1 problem to look at all 6 of these strategies, but I’m going to warn you, actually in this one, this is a prime example of why you don’t want kids solving 1 problem 6 ways, because this problem I’m going to give you, there are a couple of the strategies that make no sense to use on this problem, and I’ll talk about it as we go through these strategies. 

Here’s the problem we’re going to take a look at:

Camden’s baseball game ended at 2:22 p.m.. The game lasted for 3 hours, 50 minutes, what time did the game start?

Traditional Algorithm

So let’s take a look at the traditional algorithm and why this is kind of so hard when it comes to solving time problems. So if the game had ended at 2:22, and they needed to subtract 3 hours and 50 minutes to figure out when the game started, well, the first one’s fine, it’s 2, right, but the moment they get here, and they see 2 – 5, so many students come over here, they’ll drop that to 1, but they’ll just carry 1 over, right? Because that’s what we do with the traditional algorithm, they don’t really understand what’s happening here. 

Let’s say they actually do understand what’s happening here, we are really borrowing, or regrouping, 1 hour. . So it’s not a 1, but 1 hour is really 60 minutes, so this becomes 80 minutes here, and then we can subtract and we get our minutes. But now when we come to the hours, it’s 1 minus 3. And so many of our students will just say, “Ah, 2,” right, because there’s nothing to come over and regroup from when it comes to time. 

So there’s a whole lot of this time understanding that they need to really get that one o’clock is kind of like 13, right? I think that’s why some of our military families have a little bit of an advantage here, because military time helps us better understand this idea. But if you’re just looking at that traditional algorithm, so many of your students will try to generalize what we did with whole numbers because it looks exactly the same. But we’re dealing with different units here. And that makes this a whole lot more difficult for our students.

Compensating

The compensating strategy is the kids who will see that they need to start at the 2:22 and subtract the 3 hours and 50 minutes. But as soon as they see that, they say, “Wait, 3 hours and 50 minutes, “that’s really close to 4 hours. “So I’m just going to subtract the 4 hours “to get to the 10:22.” Now, I know I skipped over a whole lot right here, because there’s a whole lot of understanding that kids need to know in order to subtract those 4 hours, and it’ll come from the next one we’re going to talk about, I’ll show you a little bit of how kids might end up getting to this strategy so quickly with it being able to subtract 4 hours really quickly. 

But in their head, basically what they might be thinking, here’s what I think, personally, is, well, I know if I took away 2 hours, that would get me to 12:22, and then another 2 hours would get me to 10:22.  So they’re at 10:22am. But the game didn’t last 4 hours, the game only lasted 3 hours and 50 minutes. So what do I need to do to compensate because it really didn’t last 4 hours? Well, I’ve got to add back in those 10 minutes that it really didn’t take to get to the 10:32. And again, we’ve talked about how to help kids really understand what’s going on. A lot of kids will want to do this, but they aren’t quite sure what to do here, with do I add 10, do I subtract 10? So again, if you go back to the first video I did about the subtraction strategies, we talked a little bit more in depth about helping kids understand what to do at this point. 

So a lot of kids will naturally want to do this, but then they mess up right in this area. And what I used to do as a teacher was just say, “Well, that’s not working for you, so let’s try my way.” And I don’t want you to do that, I want you to be able to see these strategies understand what’s going on, and be able to help your students make sense of the strategy that they’re trying to use.

Decomposing

Decomposing is when the kids keep one of the numbers the same. So we know that the game ended at 2:22 p.m., and they’re going to subtract the 3 hours and 50 minutes, but they do it in chunks, or they decompose, or break down the number, to make it easier for them to subtract. 

So instead of subtracting 3 hours all at once, we’ll see a lot of kids who will subtract 2 hours. Let me get that to where you can see it. We will subtract 2 hours, because 2 hours is a whole lot easier to subtract here. I know that it gets me to the 12:22, which is kind of one of those benchmark numbers, if you can get to that 12 o’clock hour, it helps kids be able to transition to the rest of the way, right? And then I could subtract that extra hour because I know that’ll get me to 11:22. Then they’ve got to subtract their minutes, and kids might have subtracted the minutes to begin with, again, how they do it might look a little bit different, but the general idea is they’re breaking it down into friendlier chunks. So at this point, they might subtract the 22 minutes to get to that 11 o’clock. The hard part right here is they have to think about how much they still have left to subtract. They have to know that they’ve taken out 22 minutes, but they needed to subtract 50 altogether, so they have to subtract another 28 to get to the 10:32 a.m.. 

Now, again, kids may not do it this way, I’ve seen plenty of times where kids kind of combine this decomposing and the compensating, which is basically one of the ways I was explaining it is that you’ll see kids who will start at 2:22pm, they might subtract the 2 hours again, subtract the extra hour to get to that 11:22, but what happens here, is instead of saying, “Well, I can subtract the 50 into chunks,” they’ll say, “well, 50 is really close to an hour, “so I’m just going to round up and subtract an extra hour. “But I really wasn’t supposed to subtract an hour, “so what do I need to do?” This is the point where they end up compensating to make it a little bit nicer for them. So they need to add back the 10 minutes because they took away 10 extra minutes than they were supposed to, to get to that, right? So this decomposing can look really different, and it may even be combined with some of the other ones.

Place Value/Like Values

With like values, or some people call this expanded notation, partial differences, you basically are subtracting the hours and the minutes, and we need subtract 50 from that, okay, and then you put them together. This is, again, a really difficult, it comes back to when we were doing the traditional algorithm. If you’re at 2 o’clock, do they understand how to take away 3 hours? Well, if they’ve done some of this decomposing stuff where it will, I can take away 2 hours, that gets me to the 12 o’clock, and then one more hour will get me to 11 o’clock, right? All of those things come in, it’s not just natural that a kid’s going to see this 2:00 p.m. and subtract 3 hours and just magically know it’s 11, there’s a whole lot of stuff we need to build around their understanding of time that gets them to that point. 

Then when they get here, they have 22 minutes, and they need to subtract 50. Well, I can’t take away all 50 but I can take away 22 and I would still need to subtract 28 more minutes from our 11 o’clock over here, and that would get us to the 10:38 a.m..

These strategies don’t work for this problem

All right, now for those of you who are paying attention, that’s only 4 strategies, and I skipped 2 of our strategies, I did not talk about Constant Difference or Adding Up, because they make no sense whatsoever. The reason they don’t make sense is because they are built on this understanding of subtraction as the difference, not subtraction as takeaway.  

So if we were trying to think about adding up, it’s kind of like you have to start at 3:50 and add up to 2:22 p.m., and it’s like, that doesn’t make any sense. Same thing with constant difference. So the story problem that you are giving your students really needs to match with these strategies. All the strategies that we talked about, and I showed about, were really that idea of takeaway, you can start at 2:22 p.m. and take away 3 hours and 50 minutes, but with constant difference in adding up, it is seeing subtraction as a difference, the difference between 2 numbers, and this story problem does not make sense to do that with. 

So instead, let’s take a look at a different problem involving time that does make sense to see it as constant difference: 

If Camden’s game started at 10:32am and ended at 2:22pm, how long was his game? These make sense to see it as that difference between the 2 times.

Constant Difference

So if I’m doing constant difference, this looks something like this. We know his game started at 10:32 a.m. and it got done at 2:22 p.m.. Well, those aren’t very friendly to be able to subtract, so kids will want to move one of these numbers. Now, these numbers aren’t really like baking me, if it was like 10:50, I would want to shift it to the next hour, but I’m just showing you, for the sake of showing this strategy. Probably wouldn’t use this strategy, but I just want to show it, okay? So let’s say a kid does want to shift this up to 11 o’clock. Well, that shift is 28 minutes. In order to keep the difference the same here, I need to shift this one up also 22 minutes, or 28 minutes, sorry, 28 minutes, to 2:50. I had the 22 on my mind from right here, I was looking at it. 

So 28 minutes, and now this difference, constant difference, that difference is the same as this difference. But finding the difference from 11 o’clock to 2:50 might be easier for your students than finding the difference from 10:32 to 2:22, okay? That difference might be easier to find. So the story problem is helping them want to see it as the difference model.

Adding up

Another way that kids will end up solving those kinds of problems using the “difference” view of subtraction is through adding up, but really, they don’t have to add up, it’s just kind of finding the difference. Again, they might see it on kind of a timeline here, and I need to get to 2:22 p.m., and so they’re just add up to figure it out. 

So I’m going to do 28 minutes, gets me to 11 o’clock, I might jump another hour, because I know that will get me to 12, and then I’m going to jump 2 hours to get me to 2 o’clock. And again, remember, I’m really bad at doing this proportionally sometimes, so don’t judge me. Okay, so now we’re looking at this difference, my answer is up here in what’s the difference, how long is it from 10:32 to 2:22, which is exactly what our problem was asking us. So these 2 models make a whole lot more sense in the type of story problem that we were given in this case.

Kids don’t need to do every strategy

In the first story problem, these 2 made no sense because we weren’t finding the difference we were being asked to take away. So as you see there, this is, again, a prime example of why I say do not teach all the strategies to your students. The strategy needs to make sense in the situation. And a couple of those strategies did not make sense on that original problem, right, so the context helps determine what strategy your students are going to use, the numbers that are in there will help determine the strategy. 

If I had given you the problem that was something like Camden’s game ended at 2:22 but it lasted for 1 hour and 15 minutes, you wouldn’t have needed those strategies, you could have just subtracted them, right, it made sense to just subtract the hours and the minutes and you’d be done.

So we really need to focus on building our students’ understanding of contextual situations, like look to the problem first, don’t just pull out numbers and be a number-plucker and try to solve the problem, really delve into the problem, help kids understand the problem, and then the second thing that makes a difference, is really building their sense of numbers. If they have a sense of numbers, they will naturally break numbers apart to make the problem easier. So we really need to build their understanding of time, because the numbers, as you could see, even those strategies, they need to understand what it means to make the next hour, they need to understand how the hours change after 12 o’clock and what happens there. 

There’s a lot of number sense around time that we need to build for students, and that’s where I want you to start, okay? I have a few videos that I’ve done about number sense, they’re about whole numbers and number sense, but I’ll link to those below. And as you’re watching them, be thinking about how it relates to helping my students build an understanding of time and the numbers that are within time problems? So that’s what I want you to focus on, because if you do that, these strategies come naturally from your students and you do not have to spend time teaching them to your kids, you can give problems and then have them talk about how they solved it, and investigate these different ways in a natural way, instead of you directly teaching the strategies to your students. 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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