It's time for subtraction! In this episode Pam and Kim name and give examples of the four main subtraction strategies students need to own to solve any subtraction problem that is reasonable to solve without a calculator.
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Pam Harris 00:01
Hey fellow mathematicians, welcome to the podcast where math is Figure-Out-Able! I'm Pam.
Kim Montague 00:07
And I'm Kim.
Pam Harris 00:08
And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. But it's about thinking and reasoning; about creating and using mental relationships. Y'all, we take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?
Kim Montague 00:38
And we are answering that right now. For the last couple of weeks, we've been in the middle of a series about answering the question: If not algorithms, then what? Last week we tackled addition and shared the four major addition strategies that we believe students need to own and have experience with. This week, we're going to tackle subtraction.
Pam Harris 01:02
Yeah, everybody's favorite, right?
Kim Montague 01:04
Yeah, everybody's favorite.
Pam Harris 01:05
Not so much. In fact, Kim, true story. The second workshop I ever did with elementary teachers - I had done tons of professional learning workshops with secondary teachers, it was mostly centered around graphing calculators - but the very first one that I did for elementary teachers, or the second one I did, first one was kind of prescribed and I had to do certain things. But the second one I did was like me, you know, like, hey, come work with our teachers. And I said, "Great, happy to do that. What are your pain points? You know, what do you want me to do?" And they said, "Oh, so glad you asked: subtraction across rows." And I was like, "Absolutely, we will do that. What does that mean?" I had no idea. Like, what is that? As a secondary teacher that didn't even make any sense to me. So subtraction is a big deal. Like it's a big pain point for teachers. And let's talk about subtraction today. What are, and we're gonna say there are four, what are the four major subtraction strategies that students need to own, that we need to build in students so that we can give them then any subtraction problem that's reasonable to solve without a calculator, and they will be successful and fluent and efficient.
Kim Montague 02:11
You ready for me to name them?
Pam Harris 02:12
Kim Montague 02:13
Okay. All right, we've got 'Remove a Friendly Number'. We've got 'Remove a Friendly Number Over', 'Remove to a Friendly Number', and 'Constant Difference'. And now everyone owns those right?
Pam Harris 02:28
You got them! Alright, we're done. Thanks for joining us today. It doesn't quite work that way. Like, our names are not important. Other people have different names. If you're familiar with Cathy Fosnot's work, you can see the influence of her work on my work. I give her a lot of credit with her Landscapes of Learning and helping me sort of ferret these out. But I think in a huge way, we've taken sort of a step further from that and said, what are the major ones that kids need to own in order for them to be where we can give any problem that's reasonable without a calculator, and they can be successful and fluent and efficient. So you might have noticed that those four strategies might have sounded familiar. If you have not listened to last week's episode, you might want to go listen to the prior episode to this before you listen to this one, because there's going to be a lot of similarities. And we're going to kind of build on that. We're going to build from last week's addition episode into today. We're not going to maybe go as in-depth in some things, because we can build on those. So for these subtraction strategies, Kim, our problem of the day that we're going to use is 71 minus 37.
Kim Montague 03:41
Pam Harris 03:42
I'll bet you a Dairy Queen blizzard that you just wrote that problem down. Am I right?
Kim Montague 03:51
Pam Harris 03:52
We're just trying to make - no this is a good thing. We're just trying to make the point that mental math does not mean you did it all in your head. Mental math means you do it with your head, I just quoted Cathy Fosnot there. It is perfectly okay for you to not have to hold that in your head. You can free up that working memory so that you can actually think about what you want to think about, not just hold numbers. Totally cool, just made that little point right there. Alright, I'm gonna start. Would you please solve that problem? No, you have me start because then I want you to do the over one.
Kim Montague 04:20
Okay. Alright. So you start by Removing a Friendly Number.
Pam Harris 04:24
Okay, so I'm going to Remove a Friendly Number. Remove a Friendly Number. So I'm going to start at 71. And I'm going to remove. I'm going to subtract. I'm going to minus 37. Yuk! I'm going to minus 30. I'm gonna remove or subtract 30 first. I can do that. 71 minus 30. No problem, that's 41. 71 minus 30 is 41. I've removed a friendly 30. But I was supposed to remove 37. So I've got to remove seven more. I'm gonna subtract off seven more. So I'm at 41. I got to subtract seven more. I'm thinking about that as one to 40 and six more to 34. So 41 minus seven is 34. So yeah, so the entire problem 71 minus 37, remove 30 first, then get rid of the seven, and I land on 34.
Kim Montague 05:14
And like we said last week, if you are familiar with some of the strategies, you could easily pick up a pencil, because that's what you write with a pencil -
Pam Harris 05:24
Or a pen! I got a pen in my hand, do you literally have a pencil?
Kim Montague 05:28
Pam Harris 05:28
I have a pen, always.
Kim Montague 05:30
And you could model what Pam and I are saying to give yourself some more practice, just modeling thinking.
Pam Harris 05:37
And you might look at the model that you just drew, and were your jumps the same height? I mentioned that because we have somebody right now helping us do some computer work where we are turning hand drawn models into computer generated models. And some of the bigger jumps like a longer jump are taller than the shorter jumps. No, don't do that. Make your jumps the same. Like all the jumps are the same height, they're just shorter and longer, not taller. That's not important. Okay. So that was 'Remove a Friendly Number'. Kim.
Kim Montague 06:15
Pam Harris 06:15
I would like you to solve the same problem. 71 minus 37. But this time, I want you to Remove a Friendly Number Over.
Kim Montague 06:21
This is good for me. Okay, so, I need to subtract 37. But I don't want to do that. So I'm going to subtract 40. Because 37 is close to 40. And 71 minus 40 is 31. But I subtracted too much. And I asked myself how much too much. And because I was only supposed to subtract 37 I subtracted three too much. So now I'm going to add three back on to 31 to get 34.
Pam Harris 06:51
Nice. Those of you that are practicing modeling, take a look at your models right now. Are your 71s lined up from the first strategy to the second one? Are your 34s lined up from the first strategy to the second one? Bam. That's a thing to look for. Because we want to represent the strategies in such a way that the relationships are a bit more visible, they're a bit more apparent. So we want the number lines to be in relation. Okay, so not what this episode is about. Sorry, moving on. Okay, so we've Removed a Friendly Number, we've Removed a Friendly Number Over now let's talk about 'Remove to a Friendly Number'. Can you keep going? Because I want to do the next one.
Kim Montague 07:31
Sure. That's fine. Alright. So Remove to a Friendly Number. I'm starting at 71. And I need to remove 37. But I'm going to remove just one to get to 70. And then I still need to remove 36. And actually, it's not really maybe fair to the listeners, because I play 'I Have, You Need' a lot. And so I know what the next hop is. So I'm going to do that. Can I just make the next...?
Pam Harris 07:59
Say it and then maybe we can pick back up.
Kim Montague 08:00
So then because I need to remove 36 still, that I know that I'm going to land on 34. Because I know the partner of 70 and 36.
Pam Harris 08:14
The partner of 36 to 70, you're saying, is 34. Like, you play 'I Have, You Need' with 70?
Kim Montague 08:21
No but I've played with 100 a lot and with 10s and 20s a lot. So I feel like any 10s number, any multiple of 10 for me is not a bad combo.
Pam Harris 08:34
Hmm, that's interesting. Alright, I'll work on that. Because then I can say I'm like you. Because I want to go up and be like Kim. Alright, totally cool. Alright, so 'Remove to a Friendly Number' was all about starting with that first number, the 71. And then not removing a big friendly something, but removing to that friendly 70 and then getting rid of the rest of it. So that could have looked like, if it was more like me, once you remove the one to get to 70 and you have to remove 36, you could have removed 30 to get to 40 and then remove the six to get to the 34. That could have been how you do it.
Kim Montague 09:08
And I think this is why we name it by the first move that you make, because what you just described was 'Remove to a Friendly Number'. And then when you removed 30 that was kind of 'Removing a Friendly Number'. Which is awesome, right? If you own both, then you can you them up.
Pam Harris 09:26
Totally. Yeah. And I actually wondered if you were going to remove 40, once you remove to the 70, I wondered if you were going to remove 40 and then pop back up a 4. I was kind of curious. So there are combinations of moves that you can make. We tend to identify the strategy by that first move because it's your plan of attack. It's like what are you thinking about? How are you kind of handling this? Yeah, so that's kind of how we do that. Alright. So we've done 'Remove a Friendly Number', 'Remove a Friendly Number Over' and 'Remove to a Friendly Number'. Now before we get to the last one, we have to talk about a string, a big idea, that we help develop in kids, a problem string that we do to develop a big idea. We've just done three strategies where we were removing, we were subtracting, we were kind of doing the minus thing. One of the meanings of subtraction. We were doing that. But there's another interpretation of subtraction. And that is that we can actually find the difference between those numbers. Like how far apart are they? So we need to do work with students where they look at a problem like 71 minus 37. And they could say to themselves, I'm putting 37 on a number line. And I put 71, on the number line, and then I'm drawing the jump between those and I'm asking myself, how big is that space? What is the distance between 37 and 71? Now brilliantly, I just bring all of my addition work to bear. Now I can Get to a Friendly Number to find that distance. I can Add a Friendly Number to find that distance. I can Add a Friendly Number Over and then adjust to find difference. There's lots of stuff that I could do to find that distance between 37 and 71. So that's going to be an important idea. In subtraction? Yeah, because one of the ways that I can interpret subtraction is that idea. One is removing, which we just had three strategies based on removing, but also this idea of how far apart the numbers are. What's the distance? So Kim, I'm gonna let you take it. Once we know that I can think about numbers, about how far apart they are, how does that help me with that fourth most sophisticated strategy for subtraction?
Kim Montague 11:33
Constant Difference. Okay, so if I put 37 and 71, on a number line, I can find the distance between those. But I'm actually going to shift those down one. And so I'm going to shift the 71 down to 70. And in order to maintain the same distance between the numbers, I'm going to shift the 37 down to 36. And now I'm just going to find the distance between 36 and 70. And I think I did that again, because I just said, I just know the distance between those is 34. Otherwise, I would have made a different shift a different way maybe.
Pam Harris 12:10
Let's talk about the different shifts in just a second. So by shifting both of those numbers, and you kept the distance between them the same, it's like you created an equivalent problem that was easier to solve.
Kim Montague 12:22
Yep. Because 71 minus 37 is equivalent to 70 minus 36.
Pam Harris 12:29
Yeah, that sounds really interesting. So you could then make a choice. Do you want to solve the problem 71 minus 37? Or do you want to solve an equivalent problem 70 minus 36? And you say, 70 minus 36, you got that. And I say, I don't want to solve either of those. So you mentioned that there was another way to shift, can I tell you which way I would shift?
Kim Montague 12:53
Pam Harris 12:55
I would like to have a subtraction problem, where the second number in the subtraction problem is the nicer number. You made the first number nice, I want to make the second. You made the 71 nice, I'm gonna make the 37 nice.
Kim Montague 13:07
Pam Harris 13:07
So on my number line, I've got 37 and I've got 71. But I'm going to shift them both to the right three.
Kim Montague 13:14
The right up a number line?
Pam Harris 13:15
Up the number line. So I'm going to shift the 37 to 40. And the 71 to 73. And create the equivalent problem -
Kim Montague 13:25
Pause, pause, pause.
Pam Harris 13:27
Okay, hang on... Oh, I was looking at the wrong number line. I'm going to shift them both up three. So I'm going to shift the 37 to 40 and the 71 to 74. Alright, it's real, real live. Here we are recording as we go. So what I've turned now is the problem 71 minus 37, I've turned it into the equivalent 74 minus 40. Bam. That is a problem I would prefer to solve. So I guess I could solve 70 minus 36. But not my first inclination, my first inclination is definitely to shift both up three and create the equivalent problem 74 minus 40.
Kim Montague 14:02
Pam Harris 14:03
Yeah. And we would consider that the most sophisticated subtraction strategy. That's a little fun. Hey, Kim, I'm just acknowledging that if we were to give either of us a subtraction problem, I'm wondering if you have a go to. Like, if I were to give you a different subtraction problem, I don't know, like 82 minus 59, or something like that, what's your inclination? Are you like, all of these are the same? You might like flip a coin, and you're like, ah, I mean, they're all - no?
Kim Montague 14:38
No, I really enjoy doing an over subtraction. So I would do 82 minus 60 to get 22. And then add back one to get to 23.
Pam Harris 14:52
And I would not, but you're the over-girl right now. I own over subtraction. I do. And I could do it with the best of them. But it's not my first inclination. And I'm okay with that. Because I own them all. Ya'll, that's the thing. Once we own them all, once we help students really develop all these, then we let them choose. Then we're like, "Hey, if you own those, I'm good, whichever one." And so for me, I would tend to look at the difference, the distance, between 59 and 82. And I would like to shift that to create the distance between 60 and 83. I've shifted both up one, and now I have the problem 83 minus 60. And that just, I can't not see 23 for that problem.
Kim Montague 15:36
You know what, though? You just said, "Once I own them, it's what I want to choose." But I think I would also wonder in myself, if there's a reason why I don't choose Constant Difference if I never did. So I recognize that in myself, I do choose it sometimes. Because I do want to let the numbers influence. And then I think there have been other strategies for other operations where I've said to myself, "Oh, I almost never use that strategy." So I'm going to force myself for a while to use it so that it's as easily accessible as readily accessible.
Pam Harris 16:11
And once we own them, then don't you and I both find that we tend to lean towards the most sophisticated strategy?
Kim Montague 16:17
Pam Harris 16:18
Because they're more efficient. Yeah. And cool, and slick, and often easier to do. Though, there are times where if I don't have something to record my thinking, there are times I might do a less sophisticated strategy, because I can hang on to it in my working memory a little bit easier. It's a bit more sequential, which also means I can kind of hang on to the relationships when I can't write stuff down. So it's another reason why we want kids to own them all. So that given the circumstances, given the situation they could choose, they have the power to say, "Ooh, in this moment, which one am I inclined to do?" Bam. But if they own them all then we can also have this great conversation about which one do you wish you would have chosen? Now that you've looked at the numbers, you've stepped out of the relationships. It's a great place to be.
Kim Montague 17:07
Yep. Okay, so listeners, if you have not yet already downloaded the file -
Pam Harris 17:14
What are you waiting for?
Kim Montague 17:15
What are you waiting for? It's exactly what I was gonna say. You need to get on over to the show notes or go to this site. You need to find this at mathisFigureOutAble.com/big, where we have a download available for you for free, that has all of the major strategies for each of the four operations, with examples, with problems, with models, super easy to find: mathisFigureOutAble.com/big.
Pam Harris 17:41
Big. B-I-G. Because this is big. You're gonna want this download and then listen in for all these episodes, this series that we're doing, and we'll talk you through all the examples that you're looking at, you get to sort of hear them live, which makes it all the more - more what? Figure-Out-Able? Developable? Digestible? I'm going to stop trying to come up with words. Alright, y'all. If you want to learn more mathematics and refine your math teaching, download mathisFigureOutAble.com/big you're gonna like it so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able Movement and help us spread the word that math is Figure-Out-Able!