# Ep 125: The Over Strategy - Subtraction

November 08, 2022 Pam Harris Episode 125
Math is Figure-Out-Able with Pam Harris
Ep 125: The Over Strategy - Subtraction

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.
Talking Points:

• Over strategy is a great experience for people new to thinking Real Math.
• The Over strategy is one of the major relationships that students need to own.
• A Problem String to nudge the Over strategy.
• The Over strategy can be tricky, so students need lots of experience and contexts.
• Modeling student's thinking on a number line helps make thinking visiable as they grapple with the relationships.
• Teacher moves to modeling that can help support students' thinking.
• Where does over subtraction fit in the hierarchy?
• Bonus! The Over strategy "uses" place value (not just place labeling)!

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

Sure.

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

Thirty-seven.

Pam Harris  03:47

Thirty-seven. What is 47 minus 9?

Kim Montague  03:52

Thirty-eight.

Pam Harris  03:55

Why?

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

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

Sixty-two.

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

Yeah.

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

Yep.

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

And why?

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

Yep. Yeah.

Pam Harris  10:41

Cool. Anything else you want to say about that problem? Just curious.

Kim Montague  10:46

Nope.

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

Because?

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

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

Yeah.

Pam Harris  16:12

So we've kind of gotten in this rhythm of that makes sense.

Kim Montague  16:15

Yeah.

Pam Harris  16:16

Kim Montague  18:58

Yeah.

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

Yes. Yes.

Pam Harris  19:16

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

Yeah.

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.