Ep 119: Elapsed Time on a Number Line

Pam  00:01

Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim  00:07

And I'm Kim.

Pam  00:08

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim  00:36

Okay, so last week, we talked about time, and we talked a little bit about elapsed time. But today, we're going to actually talk about solving problems with elapsed time. Okay, so I remember. It's been a while. It's been quite some time. 

Pam  00:52

You're going to tell on me. 

Kim  00:52

I remember. What?

Pam  00:54

You're going to tell on me. 

Kim  00:55

Oh, I was just going to say "how long ago", but.

Pam  00:59

[Pam laughs].

Kim  00:59

It was quite a few years ago. Anyway, so we were working together.

Pam  01:04

Just a second. Just a minute ago. It was just.

Kim  01:06

Just a minute. Just a second.

Pam  01:07

Just a little time. You know, just a couple years. 

Kim  01:10

Just a few years ago.

Pam  01:11

I'm poking on Kim because in the last episode we talked about being specific about time. Now I'm not.

Kim  01:17

It was 20 years ago.

Pam  01:18

Dang it! I didn't want you to be specific here, but okay. Alright. Kim's going to be specific. 20 years ago.

Kim  01:23

Yeah. Okay. 20 years ago. 

Pam  01:24

Neither of us are that old.

Kim  01:25

And I was teaching third grade. And I remember that you were working in the district that I was teaching in. And you came into my classroom one day, and you said, "Hey, so some of these teachers have been talking to me, and they're asking me about elapsed time. And how do we find elapsed time? Like, how do I help my kids get better with elapsed time?"

Pam  01:47

Yeah. And this is the proof that I liked you and trusted you because in that moment, I was still the high school teacher who was supposed to be the God of math. And so, I was like, "I can't look like I don't know what I'm doing." Right? So, in that moment when they asked me, I was like, "Oh, yeah, yeah. We'll do that. Sure. You bet. We'll do that. I'll absolutely help you with the elapsed." So, I just gave that knowing look with that smile. And then I went to you. And I was like, "What? What does that even mean? Elapsed time?" Like, that was not even a phrase as a high school math teacher. I haven't really. I mean, I kind of knew what it meant, but I was like, "That's a thing?" And you chuckled. You're like, "Yeah, that's a thing." And I was like, "Alright. Alright." Like, "How do we do that?"

Kim  02:23

Yeah. Well, and I want to. I do want to give you a lot of credit in this because I. You know, we've talked about this before, and how I mess with number and what not. But I didn't really have a great way to model, a lot of times, what I was thinking about. And so, I just did a lot of equations, and that kind of like chicken scratched a bunch of stuff on my paper. But you brought number lines, open number lines to my life. And at that point, as soon as I saw an open number line, I was like, "Oh, this is a great model for lots of things that I kind of do naturally, and this will be really helpful for my kids." So, as a third grade teacher, I did teach elapsed time. And I remember very clearly being like, "Oh, this is going to be great because all you have to do is stick the elapsed time on a number line.

Pam  03:13

Sort of unwrap a clock. 

Kim  03:15

Yes. 

Pam  03:16

Right? Kind of like, unwrap that circle to be a straight line. And bam! You've got time on a number line.

Kim  03:22

Yeah. And so like, the start time would be at the beginning of the number line, and the end time would be at the end, or whatever pieces you're trying to find. And the only real difference was that this time on the open number line because we're talking about time, the benchmarks are 60 minutes or an hour. And so, instead of landing on, like, friendly numbers being 100, then we just talked about friendly numbers being the whole hour, or 60 minutes at a time. Yeah. Nice. And I know you just said that you can put the start time at the one end and the end time at the other. And totally, we can do that. I wanted to mention a little bit of a more general look, which is you can actually kind of put your schedule on a number line. Like, often visual schedules in classrooms, I see as a rectangle. So, it'll have like a rectangle of, I don't know, construction paper that will say we start at 8:00 am. It might have the picture of an analog clock, and it'll say "reading time". And then, it'll the next. You know, so there's kind of these almost bulleted versions of your visual schedule, and that's fine. But I think there's a nice activity exercise to do with students where it's like, "Put your day on an open number line. When do you leave your house? When do you get? When do you wake up? Put that time as your start time." And then, "When do you leave your house? Or when do you eat breakfast? Put that time." And then kind of like arrange it, all of that, on an open number line. So, like, "When do you leave school, and you walk home? Or, this is your bus ride." Or, like, whatever it is. Yep.

Pam  03:53

So, you might then say, "Well, when are you doing homework?" Or, I don't necessarily want to promote a ton of homework, but. "When are you going to martial arts? When are you doing your soccer practice?" Like, whatever it is, like, let's stick those things in there. I think that open number line view of a schedule can be, really, a brilliant way of helping kids just get a better feel for time and elapsed time. Absolutely. 

Kim  05:11

Yeah.

Pam  05:11

Yeah.

Kim  05:11

Absolutely. So, because the benchmarks for time are often 60 minutes. I think it's really valuable to play I Have, You Need with combinations of 60. And I don't know that we've mentioned that before, but you know, maybe in other areas. But I definitely think that if we're going to have kids mess with elapsed time to the minute, then they've got to be familiar with, and get some more experience with, combinations of 60.

Pam  05:40

Yeah, that makes sense. Total 60. So, for example, if you're playing with your students. Well, actually, would you just say, "Okay, guys, today the total's 60. If I have 40, you need? If I have 37, you need?" Or...I'm actually really curious...would you put it in the context of time? 

Kim  05:57

Yeah. 

Pam  05:58

Like, which one?

Kim  05:58

Yeah, I would definitely put in the context of time.

Pam  06:00

Ooh, interesting.

Kim  06:00

I'd put it in the context of time because I feel like if we can connect the dots for kids, then I think that it makes a little bit more sense. And so, I might, or I might. You know, honestly when I think about.

Pam  06:12

You don't mean direct teach, though. 

Kim  06:13

No, no, no.

Pam  06:14

When you say. I know you. I know what you mean. But if somebody can hear you. "If we connect the dots..." They're like, "Yeah, if I can be really explicit, and just tell kids." That's not what you mean. You mean, like. 

Kim  06:22

No, I think I'd probably say, "Hey, today, we're going to work on 60. What do we know that comes in 60s?"

Pam  06:29

Nice, nice. Keep going. What else? 

Kim  06:30

And then, you know somebody's probably going to say time. Or if not, you know, I might look at the clock, or I might look at my watch, or whatever. And someone is going to say 60. And I'll say, "Okay, so cool. Today, we're going to work on combinations of 60." And then, just like I do with I Have, You Need to 100s or 1000s, I'm going to start with multiples of 10. And so, I would do like 30, and 40, and 20, 10. And then, when that is familiar, then I would work on the multiples of 5, and then the multiples of 1.

Pam  07:07

Mmhmm. In that order. Yeah. 

Kim  07:09

Mmhmm.

Pam  07:10

And I would suggest that you start in those three groups, the multiples of 10, multiples of 5, and 1s, that you start with numbers closer to 60. 

Kim  07:19

Yeah. 

Pam  07:20

Just to kind of get a feel. And I also. Here's an advanced tip. Don't start with the half. So, like, if you're playing I Have, You Need total 10. Don't say, "I have 5." And then, when they say back, that you need 5, then it could be like an echo thing. Like, then you say, "I have 6." they say, "6." You're like, "No, no. No, what's the partner to 10?" So, don't start with 30, if you're doing partners of 60. Just don't. That's a little advanced tip there. Just a little bit. Yeah.

Kim  07:50

Yeah, and you might. You know, if I Have, You Need is routine that you do anyway. If it's a routine that you do regularly, you could kind of nestle this in a little bit before you start working on elapsed time. So, that you've done a little pre-work to have kids familiar, rather than saying, "Oh, today we're going to talk about elapsed time and also get good at combinations of 60."

Pam  08:11

Yeah, absolutely. Hey, we have a blog post if you're. Listeners, if you have not listened to. We're kind of talking about I Have, You Need as if you've done it a lot. So, if you're not familiar with that routine, we have a blog post. I Have, You Need. Just go Google "Pam Harris. I Have, You Need." Or, I think just, "I Have, You Need" will pull it up. Check out that blog post. We have a super download that lists all sorts of strategies and helpful hints for you as you're doing I Have, You Need with kids. So, check that out. Okay, so let's do some elapsed time, like. Let's, like, solve some problems. 

Kim  08:42

Okay. Alright. Do you want to go first, or do you want me to go first?

Pam  08:46

I'm going to give you a problem first.

Kim  08:48

Okay. Sounds good.

Pam  08:49

Okay. So, Kim, if... Mmm... P.E. Let's. Like, Physical Education. If sport class starts at 1:35 pm. And we know that it's 45 minutes long. When do we need to leave to come back to class? Just like, when is it over?

Kim  09:10

Okay, so. When is it over. Okay. 

Pam  09:11

Yeah.

Kim  09:11

So, I just wrote down 1:35 pm. 

Pam  09:14

Mmhmm.

Kim  09:16

And because I've played I Have, You Need, I know that it is 25 minutes to 2:00 pm. So, I just made a jump of 25 minutes to get to 2:00 pm. 

Pam  09:25

Mmhmm.

Kim  09:27

And then, I noticed that I had done 25 of the 45 minutes. And so, then, I just made another jump of 20 to land on 2:20 pm. So, I think that we are done with P.E. at 2:20 pm.

Pam  09:41

Nice, nice. What would you expect a kid to do if they didn't have that I Have, You Need?

Kim  09:48

Yeah, that's a great question. So, instead of the combinations of 1:35 pm and 25 minutes, I might see kids make a jump of 5 to get to 1:40 pm. And then, another jump of 20. Or, you know, if they are brand new to elapsed time, brand new to number lines, then they might make a jump of 10, 10, 5. Or, 5, 10, 10. But we're constantly wanting to nudge towards bigger, fewer jumps. So, I might notice that, take note of it and help a student with that.

Pam  10:24

Nice. Do you ever see... So, my inclination might be, if I didn't know. Well, or if I did. I was actually wondering if you might think 1:35 pm, and then jump 30 minutes? 

Kim  10:35

Yeah, I totally.

Pam  10:36

So, we started at 1:35 pm...

Kim  10:38

I thought you were going to ask that when I saw the 1:35 pm. Yeah, it's. I mean, it's possible. And honestly.

Pam  10:44

Well, what is? Let's. So, 1:35 pm. Jump 30 minutes. Now, you're at 2:05 pm. 

Kim  10:48

2:05 pm. Mmhmm. 

Pam  10:50

But we know it was 45 minutes, so we still have to go 15 more minutes. Is that right? Yeah. So, 2:05 pm. 15 more minutes. Now, you're at that same 2:20 pm end time.

Kim  11:00

I mean, theoretically, you could also shift back a little bit and say, "It's the same distance as if we had to leave at 1:30 pm."

Pam  11:17

Okay, and then what?

Kim  11:20

So, I kind of like take the 5 minutes off. Add the 45 minutes. So, that'd be 2:15 pm.

Pam  11:28

Mmhmm.

Kim  11:30

And then, put the 5 minutes back on.

Pam  11:33

To get to the 2:20 pm. 

Kim  11:33

Mmhmm.

Pam  11:35

That is not natural for me. Nope. That one I have to really think about. Nice. Cool. Alright, your turn. Make me one.

Kim  11:42

You want one? Alright.

Pam  11:43

Yep.

Kim  11:43

Here's one for you. 

Pam  11:44

Okay.

Kim  11:45

You're going to go to Art instead of P.E. 

Pam  11:48

Whoa! Art!

Kim  11:49

If Art is 55 minutes long. And we need to be back by 10:15 am. When should we arrive?

Pam  11:59

Okay, so I just wrote down a number line. And on the far right, I'm writing 10:15 am because that's when we got to, that's when we're done. Right? 

Kim  12:07

Mmhmm.

Pam  12:07

And we need to go backwards 55 minutes to see when it is that it would have started because that's the time we have to get there. So, I'm looking at 10:15 am. I'm just instantly going to the friendly number of 10:00 am. So, I know that I've. That's 15 minutes we've already spent. So, now, I'm thinking about, was supposed to be 55 minutes. We've spent 15 minutes. So, that leaves me 40 minutes. So, then I'm backing up 40 minutes from 10:00 am. And that lands me on 9:20 am. So, we better get there at 9:20 in order to spend 55 minutes and be done at 10:15 am.

Kim  12:45

Say that again? What time did you get there? 

Pam  12:47

9:20 am.

Kim  12:48

Okay, yeah. I just. 

Pam  12:49

Yeah.

Kim  12:51

I was also solving while you were solving, but I wanted a different strategy. So.

Pam  12:55

Were you giving me a neutral response there? Is that. You're like. Because you made me rethink. I re-looked at my numbers.  

Kim  13:00

"You think it's 9:20?"

Pam  13:01

Yeah, I was like, "Uh, Yes. Yes. I do." 

Kim  13:06

Way to be confident. 

Pam  13:07

Did you find a different strategy?

Kim  13:09

Well, 55 minutes is really nicely close to an hour.

Pam  13:12

Oh! Sure, enough. 

Kim  13:15

Yeah. 

Pam  13:16

Bam. So, talk that out.

Kim  13:18

So, if we have to be back at 10:15 am, then I backed up an hour to put us at 9:15 am.  Yeah. And then it's really only 55 minutes. So, then, that would be 9:20. 

Pam  13:25

A whole hour. That 5 minute jump. That's nice. Over. I should have thought that you would do an "over" strategy. Nice. 

Kim  13:38

I do like "over". Alright. 

Pam  13:40

Cool. Alright. One more problem. This time for you. 

Kim  13:42

Okay.

Pam  13:43

Okay, if we leave the lunchroom.

Kim  13:45

Okay. 

Pam  13:45

Now, you know, sometimes lunchroom schedules are a little crazy. Kids are kind of. Especially if they're big schools, kind of funny time. So, I'm just saying we have to leave at 11:47, 11:47. Yes, because we have early lunch. I don't know, is that early for kids? Feels early for me. 

Kim  14:00

Some go at 10.

Pam  14:01

Do you ever have this right when you have a schedule, the one year where they're like, "Sorry, but you have the 10:45 lunch." You're like, "Okay." No, you like that? 

Kim  14:08

No!

Pam  14:09

Oh.

Kim  14:09

No, no, no.

Pam  14:09

I thought you just said yes.

Kim  14:10

No. 

Pam  14:11

Bleh. I hate the early lunch. 

Kim  14:12

Afternoon snack time.

Pam  14:15

I totally like the later lunch because it just means if you can make it until then, then the rest of the day goes really fast. Anyway, I digress. Okay. So, if we have to leave the lunchroom at 11:47.

Kim  14:24

Mmhmm.

Pam  14:26

And we... Where did we go? Oh, no, sorry, we leave for lunch at 11:47. I read my problem wrong. 

Kim  14:35

Oh, okay. 

Pam  14:35

Sorry. Okay, so we leave our classroom at 11:47 am. 

Kim  14:39

Okay. 

Pam  14:40

And then, we noticed that when we got back it was 12:16 pm. I'm going to ask my students, "Hey, how long is lunch? If we leave at 11:47 am. We come back at 12:16 pm. How long are they giving us for lunch? Kind of curious how long lunch is." So, what are you thinking about?

Kim  14:58

I'm thinking you got shorted on your lunchtime. Okay. 11:47 am. I know that 47 and 13 make an hour.

Pam  15:07

Ooh. I Have, You Need. You just did that combo right there.

Kim  15:10

Mmhmm. So, 11:47 am plus 13 minutes gets you to noon. 

Pam  15:14

Okay. 

Kim  15:15

And so, then, you said you came back at 12:16 pm. So, that's 16 more minutes. 

Pam  15:21

Yep. 

Kim  15:21

So, 13 minutes plus 16 minutes is 29 minutes.

Pam  15:25

29 minutes. Eh, it's not a bad lunch for kids. It's not. I wish adults had longer. Haven't we all learned to eat fast? I don't know that I can think of a different strategy unless I'm a kid who doesn't necessarily own 47 to 60. 

Kim  15:41

Yeah.

Pam  15:41

That feels kind of important. But then, I could go 47 to 50. And then, 50 to 60 to get the 13. 

Kim  15:49

Yep.

Pam  15:50

And if I'm, I don't know, a student who's really thinking in chunks, I might go 12 to 12:10 and then the extra 6. And now that I have this 3, 10, 10, and 6, I might conceivably add those by adding the 3 and the 6 together to get the 9. And then, the 10s together to get 20. I don't know. That might be a younger strategy.

Kim  16:11

Well, it's a pretty solid, natural starting point, right? Like, we're talking as if we know the combinations of 60 right off the bat because you and I do. But it's definitely a natural starting point, which is why we're suggesting I Have, You Need combinations of 60.

Pam  16:26

Oh, I just thought of another strategy. 

Kim  16:27

Oh, good. Whatcha got? 

Pam  16:29

Well, I was thinking, if I wasn't using a number line, and I was really like in my head, then I would probably think a bit more about 11:47 am is almost quarter to noon. 

Kim  16:42

Mmhmm. 

Pam  16:42

And 12:16 is almost 12:15. 

Kim  16:46

Yeah.

Pam  16:46

I wonder if podcast listeners are like screaming at us right now? "Of course, you would just do that!" Because they might not be writing stuff down. And so, this is an interesting moment where we can have a conversation about how sometimes writing things down might encourage one strategy versus having to hold a little bit more might encourage a different one. So, if I'm thinking about "quarter till" and "quarter after" that's a half an hour, but I'm one minute high on the 12:16. But I'm 2 minutes high on the 11:47 am. So, that 2 minute and that 1 minute sort of cancel out to 1 minute. I'm not doing that very well. Anyway, so I have this 1 minute under 30, which then gets me to the 29. And honestly, if it's 11:47 am and 12:16 pm. It's really just about a half an hour. And then, I could just sort of figure which. A little less than half an hour? A little more than half an hour? And in this case a little less than half an hour. Alright.

Kim  17:46

So, I like these problems that you came up with too because you have. What's actually going to happen for kids is that sometimes you don't know the end time of things, sometimes you don't know the beginning time of things, and sometimes you're wondering how long is this going to last. And so, in any of those situations, plunking stuff down on a number line is incredibly useful.

Pam  18:10

Yeah, I'm so glad you mentioned that because we did. We just did three kind of separate kinds of elapsed time problems where it's a different unknown each time, but they all have the same model. 

Kim  18:22

Yeah. 

Pam  18:22

We're all using this brilliant open number line model and these partners to 60 to just sort of reason. We don't have to have a separate strategy if the duration's unknown and a separate strategy if the start times are. We don't have to, like, worry about that. We just, like, put "What do we know?" We can put down what we know, find what we don't. Brilliant way for kids to just be really confident. All I have to do is use what I know, think about what I know about time. Nice.

Kim  18:48

Yep. So, an open number line is a super model to help students think about and solve problems dealing with time, especially elasped time.

Pam  18:57

Ya'll thank you for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able.