Solving Elapsed Time Problems with Addition Strategies

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With textbooks having students use different strategies to solve addition problems, I’m doing a short series to share with you what those strategies are, and how they help kids solve time, measurement, and money addition problems.

I’m Christina Tondevold, The Recovering Traditionalist, and I hope you’ll stick around because today, we’re going to take a look at solving elapsed time problems with 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 in this series

Number Sense Course

Free Training – Components of Number Sense in PreK-2

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

Now, I just want to start off with these warnings. 

#1 is don’t teach all of these strategies to your students. These are possible ways that students think about solving problems, but we do not need to directly teach them. If you directly teach them, what ends up happening is you get kids who see all these 5 ways to solve a problem and they don’t really understand any of it and become thoroughly confused. 

I’m going to share with you what the strategies are in relation to how you solve time problems. If you’re not familiar with the 5 strategies, I’ll link to the first video I did that just explains the 5, but now we’re going to show how they apply and what it looks like when kids are trying to solve elapsed time problems. 

#2 is that you might have different names for these strategies. I like to use the names that I’ve created because it helps embody the mathematics that kids are doing. The strategy name helps me to know what they’re doing with the numbers, hey. The other thing to know is that some kids will do combinations of these strategies. So there are 5 base strategies but you will often see kids who will do part of one and part of the other. And a lot of them are really similar. So they look a lot alike, because hey, we’re solving the same problem. 

It looks really similar because we’re solving the same problem, they are connected. Now, the problem that we’re going to use, we’re going to use the same problem over and over for all 5 of these strategies. But the problem really does matter. I picked a problem that I could use all 5 of these strategies to solve but there are oftentimes problems where you wouldn’t use any of these strategies. And I’ll explain more about that after we look at all 5 of them. 

So the problem that we’re going to be using today is:

Camden’s baseball game started at 10:47 a.m.. The game lasted for 3 hours, 35 minutes. What time did the game get over? 

Traditional Algorithm

So let’s start with using the traditional algorithm. Just set it up, stack these numbers, and let’s talk about why this problem is so difficult. Right, and we’re going to look through it through the lens of the traditional algorithm to start with, and then we’ll take a look at how kids can use their sense of numbers to help them devise some other ways to think about this problem. Okay, so when we set up the problem, kind of in the traditional fashion, we have the 10:47 start time, and basically we’re adding the hours and we’re adding the minutes.

So kids who are used to adding in this traditional way, will say, “Well, 7 and 5 is 2, “we carry the 1 over, we get 8, and then we get 13.” But that’s not a real time that we talk about. So this becomes super confusing because basically, inside of a time problem, it’s not a base 10 system anymore. Forever when kids have been adding you carry over something when you have 10 of something, it moves to the next value. But when you’re dealing with time, it’s not just a base 10. 

Yes, there is a base 10. Let me go back here a few steps and think about this, that the 7 and 5 does make 12. So we would put one up here and the 2 there. And now when we add up our time here, we have 80 minutes. And so kids are used to, like I don’t have 10 so I don’t carry something over. But really, we’re working with minutes here, and 60 minutes is when you carry something over. So it’s no longer a base 10 it’s really a base 60, and so once you get to the 60 point, anything over that is when you create another hour. So we do have 80 minutes, right? But that is 20 minutes more than what we need and 1 hour. So 1 hour and 20 minutes, right? And then, it doesn’t even stop there with the base system, because once I come here, I have 14 hours, but our hours aren’t a base 10, they aren’t a base 60, it’s a base 12. After you get 12 hours, we start over again. And what’s even more confusing is you don’t carry anything over. 

This whole transition to help kids really understand what 13 hours and 82 minutes really is, is a really lengthy and in depth process to help them understand how to get it, to be a time that is understandable, right?  We could go into this in military time, and so on, but if you want them to really understand what’s happening, what’s happening is that in our hours after 12 hours, we start over, it’s a new base, we’re working on a base 12 so it starts over. With our minutes it’s a base 60. So and that’s why we transfer over one here because we had one hour and 20 extra minutes. Here we have 14 hours, but at 12 hours, that’s when it starts over so we have two hours left over. And so really, it’s 2:22 p.m.. 

But that is a long road to get kids to understand that, when you’re trying to set it up in this traditional way because they’re so used to when you have things stacked like this you are grouping by 10s, because we work in a base 10 system, but with time, that is not the same thing. We have different bases that we’re working with. So we can do this in a traditional way. It’s just you got a lot of stuff that kids need to understand around time to help them really understand what’s going on here. That’s why the traditional way becomes so confusing for them. 

Compensating

As we work into some of these other strategies that we’ve been talking about in the other video, I’m going to start with the one that I call Compensating. Now, it’s not to say that these ones they still don’t need an understanding of time because they do, but it helps them. I feel like it helps them get ready for this understanding of what we just talked about was so confusing with the traditional algorithm, is that when we’re doing these other strategies we are highlighting these numbers sense concepts and being able to help them really understand what matters around time. It’s not just adding up numbers, we really want to focus on what are those benchmark numbers, and why are they the benchmark numbers? 

This idea of compensating is when the kids will do something they weren’t supposed to do like they’re rounding to make the problem easier, and then they need to come back and compensate for what they did. So the original problem again, was it started at 10:47, and the baseball game was three hours and 35 minutes. This is when a kid says, “Well, 10:47, that’s almost like “11 o’clock, that’s really close to 11 o’clock, “So I’m just going to pretend for a moment “it started at 11 o’clock, and I’m going to add “the 3 hours and 35 minutes to that.” Right, and so, then they say that it’s 2:35 p.m.. 

Some of these kids may have counted on their fingers. We’re not talking about what they did to figure out that the 11 o’clock plus 3 hours and 35 minutes got to 2:35. Some of your kids may have counted by each hour, some of them may have done other things, but the idea is they started with the 11 and they knew that they needed to add the 3 hours and 35 minutes to that. And then, they’re like, “Okay, but it really didn’t start at 11 “so I need to come back and compensate for that,” and that’s where it can become a little tricky. They have to understand that we’re starting at a spot that they shouldn’t have and how do they come back and rectify that? Is that they need to take away 13 minutes from that. 

How this looks again, may look different. This is one way to notate it, but it may end up looking something a little bit different as well. It doesn’t need to be written like this. Some kid may show it on a number line.  How they model it doesn’t matter, it’s what’s the mathematics? They started with 11, they rounded, and then they had to come back and compensate for that at the end. 

Give & Take

Okay, another popular strategy is the kids who are who want to do something with that 11. They’re like, “Man, if it was 11 o’clock, “that would have been so much easier to add and figure out.” And so, they will do the 10:47 plus the 3 hours and 35 minutes, but instead of just going ahead and rounding, they’re going to take some from the 35 and give it to the 10:47 and say, “Okay, well, if 13 minutes passed, then we’d already be at 11 o’clock. Right, so this together makes that 11 o’clock. And so, then all we have left is the 3 hours and 22 minutes.” 

So they’ve broken it apart, they want to get to that rounding, that nice benchmark number of 11 o’clock, and then they work from there, but they do it beforehand. 

So there’s nothing that they need to fix after they get to this point, they can go straight to the answer here.  I know that it’s to 2:22. And again, another one that is similar. And all of these are similar because we’re solving the same problem, right? 

Decomposing

Another one that’s similar is that the kids will start with the 10:47, and they will decompose the 3 hours and 35 minutes into chunks that are a bit more friendly for them. You might see a kid who again, does that 13 minutes because they want to get to the 11 o’clock. Right, that’s friendly. They may not do the 13 minutes first, some kids will add the hours first. The point is that what they’re doing here is they’re breaking down the 3 hours and 35 minutes into friendlier pieces for them to add. And what’s friendly for one kid is going to be different for another kid, right? But this is the general idea. 

Now personally, I would probably add 1 hour right here, I would add an hour because I want to get to that 12 o’clock, the noon marker, because at noon, that’s when everything flips over. So there’s a lot of number sense and sense about time that kids need to have an understanding of before they will ever do these strategies. So remember, you cannot directly teach these strategies, we have to do work with what happens at 60 minutes? What happens at the 12 o’clock hour? We really need to emphasize a lot of those benchmarks and then they will start to use those benchmarks when they are doing addition and subtraction strategies. Hey, so at this point, then I might add the 2 hours to get to the 2 o’clock. And then the last thing to do is to add the 22 minutes and that would get me to my to 2:22 p.m.. 

But as you can see, I’ve added the 3 hours and 35 minutes all along the top here. That’s not my answer, but I’m circling it to just highlight that I still have added the 3 hours and 35 minutes. I’ve just done it by decomposing it and using friendlier chunks.

Place Value

Okay, so another one, again, that is similar to the last one, Decomposing, is what I call Place Value, or in this instance…in any instance, it’s Like Values. So in the previous video, I talked about how you add your 10s with your 10s, your ones with your ones, your tenths with your tenths. The same is true even with time, like you’re going to add your hours with your hours, your minutes with your minutes, and then you’re going to take those and put that together. 

So it doesn’t matter what order you do it in but the idea is you’re breaking down both of the numbers. In Decomposing, you only broke down one of the numbers, in Place Value or Like Value strategy you are putting your hours together and your minutes together. You’re adding those separately, and then you’re putting them together to create the final amounts.  

So it may look different depending upon how the kids chunk them down, but you might have a kid who’s like, “Okay, well, it’s 13 o’clock,” and so off to the side, you want them to be thinking what actually is 13 o’clock? Well, that’s 1:00 p.m. So they’ve got their 13. Then as they work into their minutes, a lot of kids might even break this down and do the 40 and the 30, and that’s 70 minutes, but again off to the side, they’re sitting here thinking, “Okay, 70 minutes.” We want them to always be thinking what actually is 70 minutes? Hey, so 70 minutes is an hour and 10 minutes. So they’ve already got that it was 1 o’clock, they’ve added another hour and 10 minutes, so off to the side..and hopefully, it’s not that they need to notate this, but this is what we want to encourage them to be thinking aloud or even in their minds eventually, is that I have the 1 o’clock, now an hour and 10 minutes, I’m already at 2 o’clock, 2:10. I was trying to do just the hours first and then the minutes, but I’m at 2:10. And then, once I do the 7 and 5, that’s another 12 minutes. So then I’m at 2:22 p.m.. 

So they aren’t like, once they’ve done this, they’re not over here stacking this and saying, “Oh, it’s 13:82.” No, off to the side, they’re already adding as they go. Once they do their hours, they’re adding that hour and the extra minutes on to that, all as they go. The cool part about the Place Value is they’re not waiting until the end to get to their final answer. You want to encourage them to be thinking off to the side, well, what am I at so far? So that they aren’t doing all the addition at the end. 

Kids don’t need to do every strategy

Okay, so now after I’ve shown you all of these ways, a reminder that kids should not be solving one problem 5 different ways. These are possibilities of how a student might solve it. So if I had given students a different problem, like let’s say that the game started at 2:15 p.m., and the game lasted for 3 hours and 30 minutes, some of those strategies would not make any sense. They would not do those strategies. So don’t make them practice those strategies on a problem like that. If you are having them work on strategies or want them to even come up with these strategies, you first have to start with a problem that elicits these strategies. 

A reminder that these 5 strategies are not things you directly teach. Kids will naturally do these strategies when they look at the numbers in the problem, and they see relationships, and they will use their sense of numbers to help them come up with those strategies. The numbers that I picked on this problem were intentional. A lot of the strategies revolve around students thinking about a benchmark number. That’s why I don’t like to call strategies like make a 10 in the early grades because we weren’t making a 10 we were making a 60 because 60 minutes made the next hour. It’s all about those benchmark numbers and helping kids know what those benchmark numbers are, and then being quick to see how other numbers relate to those benchmark numbers. Understanding the benchmark numbers is a foundational piece in our students number sense. 

So that’s something we go really in depth with inside of my Number Sense courses. But for those of you who haven’t taken the course, because look and the course isn’t open right now, I’ll link to some of the free videos that I’ve done that just give you an idea of what these number sense concepts are, and then you’ll take a look at that and think about how does it apply to when we’re working with time or when we’re working with measurement? What are going to be the benchmarks when we’re working with feet and inches and yards? Does that change and how do we help kids understand those? 

So, again, it’s not that we directly teach these to our students, I want you to be aware of the possible ways kids could solve elapsed time problems, not that they have to solve them this way. 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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