We just love the Over strategy! In this episode Pam and Kim go through a Problem String to develop the relationships used to think about the Over strategy for subtraction.
Pam Harris 00:01
Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.
Kim Montague 00:08
And I'm Kim.
Pam Harris 00:09
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all 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 keeps students from being the mathematicians they can be.
Kim Montague 00:36
Last week, we dove into a new series about a strategy that we both love and find so incredibly valuable in everyday life. It's the Over strategy. We also shared a handy download that we think you will love that model represents what it might look like to use this strategy. So if you want, you can pause the episode and go find that link now, so that you can put a mental image in your mind and how you might use the Over strategy for subtraction. You can find that link in the show notes or at mathisFigureOutAble.com/over. Okay, so Pam, last week, we tackled addition, let's dive into subtraction.
Pam Harris 01:16
Let's dive in subtraction with your favorite strategy, which to be clear might be becoming my favorites.
Kim Montague 01:22
Oh, all right.
Pam Harris 01:23
Just maybe. I'm finding-It's such a good one. And here's what's partly good about it. It's so good to work with, what's a good word neophyte, is a big word, like new people, like anybody who's never thought about numbers before. If we do just a little Over with them, all of a sudden, they're like seeing things differently. And they're like, big wide eyes. And they're like, "What, like, this is a thing we can do that?" I'm like, "Yeah, we can do that." And all of a sudden, then we can begin, like to dive in. So diving into subtraction today, with maybe our favorite strategy, I don't know if it's our favorite, but it's a super, super good one.
Kim Montague 02:01
You can have it as a favorite, also.
Pam Harris 02:05
Favorite doesn't mean it's the only favorite. Right? We can have multiple favorites?
Kim Montague 02:08
Pam Harris 02:08
Okay, that's important. Okay, so definitely one of my favorites. Absolutely. That's part of my work, right? Where I looked at the body of knowledge and said, if it's not about teaching kids, the subtraction algorithm, what do kids need to know? What does anybody need to know to be able to solve any problem that's reasonable to solve without a calculator? And in that research, we decided what the major strategies are, the major relationships that we need to develop in order to be thinking additively about subtraction, really reasoning with subtraction. And the good news is the Over strategy made the cut. The Over strategy is one of those major important relationships that if kids own, they don't need any stupid algorithms, I mean, algorithms. All right. So for today, let's start off with a problem. And we're going to ask you to solve it, listeners, and then pause the recording and come back and we'll do a Problem String together and see if maybe it changes the way you might be thinking about that problem. So the problem is 8201, 8201 subtract 1987, subtract 1987. So 1201 subtract 1987. Pause the podcast, give it a solve, and then come on back and see what we're doing. All right, so you've taken a look at the problem. And let's do a Problem String to see how maybe we could get your mind sort of pinging in a direction that's kind of over-ish. Alright, So Kim, first 47 minus 10.
Kim Montague 03:46
Pam Harris 03:47
Thirty-seven. What is 47 minus 9?
Kim Montague 03:52
Pam Harris 03:55
Kim Montague 03:56
Because back ten was 37. But I don't need to go back as far so not a whole 10. I just need to go back nine. So I actually have a pen in my hand, Pam. Is that weird?
Pam Harris 04:08
Where's your pencil, Kim?
Kim Montague 04:10
I don't know. I don't have one handy. But I'm...
Pam Harris 04:12
To be clear, I have a pen in my hand.
Kim Montague 04:14
I'm sure you do. So I actually um, you know it was interesting for me, because over subtracting, I think a little bit more than I have to think for over adding.
Pam Harris 04:28
You have to engage a little more.
Kim Montague 04:30
I do. So back 10 would be 38. And then I went...
Pam Harris 04:34
Back ten was 30...?
Kim Montague 04:35
Yeah. 37. And then I added one more to get to 38.
Pam Harris 04:40
But we're subtracting so why are you adding?
Kim Montague 04:42
Because subtract 10 was too much. So then I added one back on so the distance between my two numbers 38 and 47 was only nine.
Pam Harris 04:52
So I actually as you were talking I wrote a number line and I started at 47 and I jumped back a jump of 10 and labeled that 10 and landed on 37. And then when you said, "But I didn't subtract as much, just nine," I drew a shorter jump of nine, and it was just one shorter than the 10. And I've labeled that nine. And so then it's obvious to me that I would need to look at on the number line, what is just one to the right of that 37. Sure enough, that's 38. Yeah, because you didn't subtract as much, cool. Next problem. How about if I gave you something like, oh, my goodness, I have a standing desk. And let's say it wasn't locked. So it just began to stand up. Okay, I'm alright here. We're back to podcast land. Here we go. So how about 82 minus 20? Eighty-two minus 20?
Kim Montague 05:42
Pam Harris 05:44
Sixty-two. And how about 82 minus 18?
Kim Montague 05:50
It's gonna be two more, so it's 64.
Pam Harris 05:53
Two, why two more?
Kim Montague 05:55
Because I'm not subtracting as much. I'm not subtracting 20. I'm only subtracting 18. So it's going to be two more than my original answer.
Pam Harris 06:04
So that first problem when you said 82 minus 20, is 62. Then when you said comparing that to 82 minus 18, that's too much. You subtracted too much. So when I look at that on the number line, 82, jump back 20 land on 62, that 20 is too big. It's too far. Wow, way backed too far. So I have to like bump back up, and 62 and two is 64. And I know I'm kind of repeating what you did. But I'm trying to repeat it in a way so that our listeners can kind of see what I would be making visible on the board when I share it on the board. And, teachers, I would be doing that. So if I was doing this with a real student, because Kim is a fake student. If I was doing with a real student, as the kid would be saying it, I would be drawing it. I wouldn't necessarily be repeating it. But I wouldn't be drawing it. Kim, land on this problem with me for a second. So 82 minus 18. It's almost like I said to you, "Hey, you owe me 18 bucks. Give me 18 bucks." And you said, "All I have is..."
Kim Montague 07:07
A $20 bill,
Pam Harris 07:08
You handed me a 20. And I said, "Oh man, I gotta give you back two right?" It's almost like you're like, "I'm not 18 I don't have 18. Here's 20. Then I'm gonna have to give you back, even though we're subtracting 18, I would have to give you back that two, because we subtracted too much of the 20.
Kim Montague 07:26
Oh, my gosh, you're making me think of playing Monopoly with my kids. And how at a very young age, so Cooper's three years younger than Luke. And that happened all the time. Luke would say things like, "I'm not counting out $18. Here's a 20. Give me change." And Cooper very young would say, "I'm not giving you some of my money." And I can remember when he would start to go, "Oh, okay. I'm still, I'm getting more, I'm giving you less back." I remember that very clearly.
Pam Harris 07:56
Well, brilliant. Well done, Luke, and well done to support that idea of that and then can turn into Over. Supper.
Kim Montague 08:02
Pam Harris 08:03
Okay. Next problem. How about 467 minus 200.
Kim Montague 08:06
Two hundred sixty-seven.
Pam Harris 08:10
So I've just drawn a number line with 467 on the right. Then I've jumped back this big old jump of 200. And that is you said that was 267. Is that correct?
Kim Montague 08:22
Pam Harris 08:22
I'm pausing for a second because I'm thinking about the Problem String. And I'm wishing I would have tweaked it slightly. So maybe ask me that at the end. Okay. So the next problem is 467 minus 195. What do you got?
Kim Montague 08:37
Two hundred seventy-two.
Pam Harris 08:39
Kim Montague 08:40
I actually just built off of the problem, I had written 467, I made a jump back of 200, to get 27. So at that point, instead of drawing something new, I just added five from 267. To get to 272. Just made a jump kind of four to five.
Pam Harris 09:00
And you might have had to think about that 267 plus three to get to 270. And then two more to get to 272. Nice, you just had to tack on that five because you subtract a bit too much. Two hundred was too much. You only wanted to subtract 295. Cool. Nice. Next problem. How about 8201 subtract 2000.
Kim Montague 09:20
Six thousand, two hundred one.
Pam Harris 09:22
Six thousand, two hundred one. Next problem 8201 subtract 1987. Oh, it almost makes me want to play a little I Have, You Need.
Kim Montague 09:34
Mm hmm. So then I did the same thing as I did before. I used the problem that was already there. And I just added 13 to 6201 to get to 6214.
Pam Harris 09:48
Thirteen? Kim, where'd that come from?
Kim Montague 09:51
Yeah, the distance between 2000 and 1987 is just 13.
Pam Harris 09:55
So you're supposed to subtract 1987. But the previous problem had you subtracting too much, you subtracted over, over subtracted 2000, So you had to give back that difference of 13. You said 6201 and 13 was 6014. So that was our problem that we started the day with was that 8201 minus 1987. That's not the only strategy you could have used. In fact, I'm sure some of our listeners out there might have used a Constant Different strategy, which would have been a fine strategy. But we intentionally tried to get your brains going in such a way that it would ping for you to think about using that minus 2000. To think about subtracting 1987. Yeah?
Kim Montague 10:40
Pam Harris 10:41
Cool. Anything else you want to say about that problem? Just curious.
Kim Montague 10:46
Pam Harris 10:47
Yeah, I liked that you thought. That's good. That's it.
Kim Montague 10:52
Wondering if there's anything else I have to say about it.
Pam Harris 10:54
Kim is thoughtful, y'all. We're a thoughtful pair. We like big thought. Alright, next problem. Wait, there's another problem? Oh, yeah, we're never done. How about 18.2 or 18, and two tenths, subtract 10.
Kim Montague 11:09
Eight point two 8 and two tenths.
Pam Harris 11:10
Because you subtract off the 10 off that teen. And that just leaves you with the eight point two, very nice, or eight and two tenths. Cool. Anybody want to guess the next problem? If we had 18.2 minus 10, how about a next problem of 18.2 minus 9.9.
Kim Montague 11:28
Eight point three.
Pam Harris 11:29
Kim Montague 11:30
Because nine and nine tenths is only a 10th, away from 10. So we subtracted a 10th too much. So I'm going to add the 10th back on. To get 8.3.
Pam Harris 11:41
And since the previous answer you had when you subtract to 10 was 8.2, or eight and two tenths. One more 10th would be 8.3, or eight and three tenths. Super cool. All right, y'all so, the Over strategy with subtraction is a pretty nifty strategy. However, let's be really clear that with this strategy, it can be super tricky for students who it's not kind of a natural inclination. If I run into students who are already kind of playing with this idea of over-subtracting, they kind of have this idea that they're subtracting too much. And so they need to kind of adjust by adding back. It's like, when she gave me that $20 bill, but I only needed 18, we're gonna have to give her $2 back. That idea, if you kind of have naturally been playing with, it seems to flow pretty well for kids. If you have not been naturally playing around with it already, it can actually be quite tricky for students, because as they're subtracting, they think they're, they just keep subtracting. So for example, when we did 47 minus 10, was 37. Then when students look at 47 minus nine, often they'll they'll use that minus 10, to get to 37. And then they'll go back one more to 36. And then we'll have to ask is that 47 minus nine? Should we get a smaller answer when we subtract less? Or should we get a bigger answer when we subtract less, and that is a complicated idea if you haven't ever been playing around with it. So we like Problem Strings, like in the format that we just did with you, where we kind of give you that helper problem to use and you can use it for the next problem. But boy, then we got to talk about it. We've got to really make sense of where that smaller jump is, if you over subtracted first. Where's that tinier jump? For 82 minus 20, if we subtracted that big old 20, where would that jump of 18 be? Would it be a bigger jump than 20? Well, yeah, then we'd land behind the 82. But if it's a smaller jump than 20, 18 is smaller than 20. Boy, we got to add back on to that answer we had when we over subtracted. Again, that can be really tricky for students, but so much more Figure-Out-Able if we make the thinking visible. Put it up in front of kids. Draw those open number lines. One of my favorite things to do is to draw that first problem, the number line for the first problem, the helper problem and then when we get to the second helper or the second problem, redraw that helper problem, and then draw in a different color that shorter jump. And make sense of longer jump back, shorter jump back and kind of have color help us sort of differentiate between the two jumps. And then help that sort of stand out and kind of be visible as participants are doing the Problem String. As I do a problem string like this, you might find it noteworthy that for the first few problems in the Problem String, I will model the helper problem. And as the student tells me how they did the harder problem, the clunker problem, I will redraw the helper problem and then put the clicker problem on that same number line. So now, I have one number line with the helper problem, and one number line with the helper problem and the clunker problem together. But I usually only do that for the first few problems. If it seems to be going well, now I work with mostly with adults or older students. So for younger students, the first time you see it, you're probably going to do that for every problem in the string as you go. With older students that it's kind of going well. They're using the helper problem. It's making sense. Eventually, I sort of drop off redrawing the helper problem. And I just draw it once. I draw the model for the helper problem, and then on that same model, I go ahead and add back what needs to be adjusted because we were over subtracting. Does that make sense, Kim? Do we need to clarify?
Kim Montague 15:50
Yes and I'm laughing.
Pam Harris 15:51
Why are you laughing? Don't laugh at me.
Kim Montague 15:53
Well, because I'm used to modeling the same way with students. And so on my paper, as I was solving problems, the first two sets, I made two separate number lines. And then I started doing the same thing, where I just added back originally. So I'm laughing as you're describing because that's what I do too.
Pam Harris 16:11
It's a thing. Yeah, absolutely.
Kim Montague 16:12
Pam Harris 16:12
So we've kind of gotten in this rhythm of that makes sense.
Kim Montague 16:15
Pam Harris 16:16
Yeah. Nice. All right, y'all hope you've enjoyed that Problem String. Let's talk a little bit about the Over strategy for subtraction and where it fits in the hierarchy with other subtraction strategies. So we like to start with Subtracting By a Friendly Number, and Subtracting To a Friendly Number first, where students are really messing with like, so if it's a problem, like 47 minus nine, that's not a good one to start with. If it was 82, minus 18, if that's the problem, then we would nudge students using Problems Strings to do 82 minus 10, and then adjust to subtract eight more. That would be Subtracting By a Friendly Number. And we would also sort of simultaneously work on things like 82 minus two, Subtracting To a Friendly Number, and then adjusting, then subtracting the rest of it that's leftover. Those two strategies are about on par, Subtracting a Friendly Number, Subtracting To a Friendly Number. Once those are going fairly well, we've done a few strings for both of them, students are creating those relationships. And the strategies are sort of becoming natural outcomes. Notice is becoming I didn't say anything about cinching and cementing and memorizing. Not that. As that's happening, then we might, then it would be a great time to bring in Removing a Friendly Number Over where we subtract a bit too much. And then we have to adjust up, have to adjust back like, give back because we had subtracted too much. And then after that, we would also want to then help students parse out when do you remove? Versus when do you find the difference? When does it make sense? When did the numbers call to you, sing to you, like make you ping in your head that you're like, "Ha, I'm not removing for these numbers, I'm just finding how far apart they are." or "Ah, I'm not finding out how far apart these are, I'm just going to remove that little second number." And so that would maybe come next after we kind of have done those three strategies, then we would, the three being, Subtract To a Friendly Number, Subtract a Friendly Number kind of on par, those are about the same, Subtract a Friendly Number Over and then we would work on this idea of difference versus removal, distance versus removal. And then lastly, the pinnacle strategy for subtraction Constant Difference. We would wait until we had those other ones really solid, especially kind of difference versus removal. And then we would work on the idea of Constant Difference. So that's kind of the hierarchy and how it falls out. If you'd like to see that, I just talked about it. But if you'd like to see it, you could totally download our free ebook, which we're really quite proud of. It gives a lot of information not just about subtraction, you can find that at mathisFigure-Out-Able.com/big mathisFigureOutAble.com/big. Kim, I want to talk about one other thing, and I think I have time.
Kim Montague 18:58
Pam Harris 18:58
So another thing that is super helpful about developing the relationships so that the Over strategy becomes this natural thing that kids do. Another thing that comes out, that pops out that's so useful is place value, right?
Kim Montague 19:15
Pam Harris 19:16
So as we look at this particular Problem String, oh, I didn't tell you in this episode, so might be helpful in this episode, which sort of already gone by but in the next few episodes, to write down the problem string as we go, because then you can kind of look back at it. As we look at things like 47 minus nine in order to decide what the friendly number is, I'm using place value. I'm not just labeling places, but I'm actually using the place value. I'm having to grapple with what number is nice close to that. And then how does that relate to subtracting nine. Similarly with any two minus 18, I have to think about, I could subtract 10 and then keep subtracting, but I can subtract 20, and notice like I'm messing around with what are the closest tens to a 18. I mean, it almost feels like a little bit of rounding like we talked about in a recent episode, we talked about rounding. So that's a natural outcome of using the Over strategy. When I look at a problem like 467 minus 195. Again, I'm sort of thinking about how does 467 relate to 195? Like, what's happening with those numbers? Ooh, 195 is almost 200. A little bit of rounding, a little bit of, in fact, a lot of our listeners, Kim on the rounding, or the yeah, the rounding episode, we're saying things like, "Well, it really does depend." Well, this would be a great example, where what if it was 467 minus 194? Well, ordinarily, 194 would round a 190. That's not helpful in this problem. Right? I would want to round it to 200 and then just adjust by six, if I subtract 194. So it's a perfect example. Yes, we completely agree with you that rounding depends on the context. And in this context, I would absolutely want to round up to that nice 200. Again, this is an example of how students are messing with, dealing with, grappling with place value as they kind of learned the Over strategy. To be clear, it's not about rote memorizing steps to perform to do the Over strategy. No, no, no. It's about creating relationships in the learner's head so that when they see a problem, like 18.2, minus 9.9, they say to themselves, "Hmm, what do I know? Well, 9.9, that's almost 10, I can subtract 10. I just subtract that 10. And then just adjust that little bit because I subtracted a bit too much." So lots of nice place value can come out of, well lots of things, but one of them being placed value out of messing around, playing with, learning the Over strategy. And today, particularly for subtraction.
Kim Montague 21:50
Yep. And you mentioned, pulling thinking out of kids and helping to make it visual so that they have an understanding of what's actually happening. And again, if you're interested to see what it might look like to model someone's thinking for the Over strategy, you can check out the free download with a variety of examples on appropriate models. You can find that download at mathisFigureOutAble.com/over.
Pam Harris 22:12
And we're really finding that leaders in our support group are finding these downloads super handy to use with their teachers. Every time we talk to them. They're like, "Oh my gosh, I love it. Using it all the time." Super.
Kim Montague 22:24
Pam Harris 22:24
All right. 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 mathisFigure-Out-Able.com. Let's keep spreading the word that Math is Figure-Out-Able.