Math is Figure-Out-Able!

Ep 173: The Most Sophisticated Subtraction Strategy

October 10, 2023 Pam Harris Episode 173
Ep 173: The Most Sophisticated Subtraction Strategy
Math is Figure-Out-Able!
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Math is Figure-Out-Able!
Ep 173: The Most Sophisticated Subtraction Strategy
Oct 10, 2023 Episode 173
Pam Harris

Some subtraction strategies can only get you so far, but with a little more sophistication complex problems become really nice to think and reason about! In this episode Pam and Kim build and discuss the Constant Difference strategy for subtraction and use it to solve a really gnarly problem.
Talking Points:

  • Equivalence strategies are more sophisticated
  • Hallmarks of sophisticated stategies
  • Modeling Constant Difference
  • Relational Thinking Routine

Check out the Instructional Routine called Relational Thinking at:

Check out our social media
Twitter: @PWHarris
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC

Show Notes Transcript

Some subtraction strategies can only get you so far, but with a little more sophistication complex problems become really nice to think and reason about! In this episode Pam and Kim build and discuss the Constant Difference strategy for subtraction and use it to solve a really gnarly problem.
Talking Points:

  • Equivalence strategies are more sophisticated
  • Hallmarks of sophisticated stategies
  • Modeling Constant Difference
  • Relational Thinking Routine

Check out the Instructional Routine called Relational Thinking at:

Check out our social media
Twitter: @PWHarris
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC

Pam  00:01

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


Kim  00:08

Sorry, I'm having a hard time. I almost took a drink right as I was  supposed to say my name. I'm Kim, ya'll. You did. 


Pam  00:14

You forget that you say your name in the middle of the opening. Alright. And you found a (unclear)... Hey, take a drink right now where. I just said (unclear). Dang it. Take a drink, while I say the rest of this. 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. And drinking when necessary. We can mentor students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students...and it kept me... from being the mathematicians they can be. Woah! Alright, we got through that.


Kim  00:53

I'm so glad that you took the time to become such a great mathematician because I sure do like having chats with you.


Pam  00:57

Absolutely. I'm glad you were my kids teacher. That was the start of a fruitful relationship.


Kim  01:04

It's been great. Okay. So, ya'll, not that long ago, we asked people to leave a review. I don't know. It's been a little while. But I got to tell you. I opened up the... I don't know. The Apple podcasts. Whatever. And there were so many reviews! Yeah.


Pam  01:04

Cool. Aww!


Kim  01:18

I just got excited. I was in the car, and I was like scrolling and reading them to my husband. He was like, "Okay, I get it." 


Pam  01:23

Okay, like you weren't driving? 


Kim  01:25

No, I was riding. 


Pam  01:27

Be safe, Kim. Be safe. 


Kim  01:29

I was just looking it up. And it was so fun to read. It was so, so fun, and it really made me happy for the people who are experiencing math differently, right?  I mean, it's nice you like it. But the fact that people are growing is super exciting and super fun. So, I called Pam and I said, "Don't read any of them because I want


Pam  01:37



Kim  01:47

to read them to you. 


Pam  01:48

I have not read.


Kim  01:49



Pam  01:49



Kim  01:50

So, you're going to have to bear with us because I want to celebrate some people. We actually got this one as an email. And it was Kaylene who said, "Pam and Kim, I heard about you and started listening to your podcast about a month ago. I absolutely love your philosophy and methods. I teach eighth grade math intervention, and my students struggle with the most basic principles. I have always worked on developing number sense, but it wasn't till I had tried my first Problem String that I started seeing real magic happen. 


Pam  02:18

Aww! Oh, wow. Nice, nice.


Kim  02:19

I know. I love it. She said, "I've been doing I Have, You Need with my students for the last few weeks, and we're already moving on to partners of 1,000. Their eyes got huge today, when they realized they had figured out the partner to a 1,000. After class, I was asking one of my students what he thought about I Have, You Need, and he said that it is his favorite part of his day. I thought he was going to say of my class, but nope. It's his favorite part of his day!"  I know that's so cool. "Their number confidence is soaring, and they are more willing to try and think about what we're learning. I can't wait to see what this year brings. Thank you."


Pam  02:54



Kim  02:54

I mean, gosh, that feels good, right? 


Pam  02:57

Hey, I want to point something out that people might be like, "Oh, you know, you did something, and they felt better." Like why is I Have, You Need so important? So, weighty? So... What's the word I want? Impactful. That's the word I want. Because it's almost maybe the first time your students get a chance to go, "This is how I'm thinking about it. Like, all I have to do is think. And I can do it. And I feel powerful. And you're listening to me. And it just is like a different experience. It's just enough." Anyway.


Kim  03:25



Pam  03:25

Way to go Kaylene. Good job. 


Kim  03:27

Yeah, it's super cool. So, last week, we talked about the most sophisticated strategy for addition, and often people say, "What about subtraction? Can you Give an take for subtraction?" Let's tackle that today.


Pam  03:37

Let's do it. Alright, cool. Hey, Kim, do you mind before we dive into subtraction that we just play a little? We just talked about I Have, You Need. 


Kim  03:46



Pam  03:46

Let's just play it a little bit. We're just trying to give all of our listeners an example of do it often, short. Alright. So, Kim, if we had total 100. Total 100. 


Kim  03:55



Pam  03:55

If I have 45, what do you need? 


Kim  03:57



Pam  03:57

How do you know?


Kim  04:01

Because I know 50 and 50 is 100, and you asked me 5 less than 50, so I'm going to go 5 more than 50 is the other addend. 


Pam  04:10

Cool, cool. What if I changed the total and we needed 1,000? So, total 1,000. If I had 545.


Kim  04:18

Then that's 455. Those numbers are all mixed up. Yeah, 455


Pam  04:25

Yeah, there's like lots of fives and four digits going on in there. Okay, so when I said 545, how did you? What we're you thinking about?


Kim  04:25

I think kind of greatest to least on these, and so I thought about the 500 and the 400 being 900. And I want it to hit 900 Because I know what's left needs to total 100. So, the 45 that you gave me and the 55 that I'm going to supply to you make that last 100.


Pam  04:52

Cool. So, 545, you need?


Kim  04:54



Pam  04:55

455. Cool. Alright, so just a little I Have, You Need. Totally un-related to what we're doing today. Hey, Kim, I wanted to tell you, remind you of a story. You weren't there. I was... This is early when I was working with the my kids teachers in the district. And you were teaching fourth grade, I believe, at this point. Maybe third. So, you taught third, fourth, and fifth while I was in that school.


Kim  05:20



Pam  05:20

You weren't teaching fifth because I know I was working with all of the fifth grade teachers in the district. So, we had like thirty-five fifth grade teachers. And we were all in the room. And I was pretty new. I was still figuring out a lot of stuff and just kind of diving in, doing the work with teachers. I sometimes think that was still the better way to do it because I honestly didn't know what was going on, you know, and people would rise to the occasion and kind of, "Oh, here. Let me help you." Anyway, one of the teachers said, "You know, like all the stuff you're doing is kind of interesting, and trying to get us to think about, you know, relationships and everything. But what about a problem like this?" And they threw up a problem. And, Kim, in that moment, I said, "Well, let's dive in, and let's start working on it." And I had teachers up at boards, and they were kind of, you know, doing. And it was getting gnarly, and I was starting to sweat. It was a subtraction problem. And you're like, "Subtraction problem, whatever." It was taking a lot of work. A lot of work. Let's just... Oh, and it was a problem from one of our high stakes tests here in Texas. So, it had actually been asked of students, and they had seen it because it was a released item. Texas does that. They release certain items, so you can kind of get a feel for what's on the test. Which actually I'll just say. I don't like the tests, and I think they don't do what the public thinks they do. But I do appreciate the fact that Texas will release items, so that teacher's (unclear) see them. There are states that don't do that. And I don't like that. Don't do that. Release some items. Give teachers an idea of what the students are going to be asked. Anyway, so I'm just going to raise this idea that early in my subtraction there was a problem that I kind of took a deep breath and went, "Oh, dear. Maybe we do need the algorithm for this problem. Let's do some work, and then we're going to return back to that problem at the end. Is that alright? 


Kim  07:06



Pam  07:06

Alright. Okay. So, Kim, you got your pencil handy? 


Kim  07:09

I do. 


Pam  07:10

How many pencils do you have today, Kim?


Kim  07:12

Three. And one of them is dull. The one in my hand is dull. 


Pam  07:14

Oh, sadness.


Kim  07:15

It's okay. It's alright.


Pam  07:16

I have a delightful pen.


Kim  07:18

Listen, I've been thinking about this because we talked about it a little bit more last week. And I think that it might be true that... I'm left handed, and so writing with things. 


Pam  07:28



Kim  07:29

Well, I mean, I just love a pencil anyway, but I don't like a lot of pens, because I get the smear...


Pam  07:36



Kim  07:37

...when I write from left to right.


Pam  07:38

You know, I didn't appreciate that until I married a lefty.


Kim  07:41

It's a thing.


Pam  07:42

If he's writing, he will not write with a pen unless it's a very certain kind of pen because it won't smear. Yeah. And it has to be really smooth for him, which it also has to be for me, but it cannot smear or he's like, "Nope!" So, I think it's funny. He just doesn't write a lot.


Kim  08:00



Pam  08:00

Anyway. That's his way that he gets around. 


Kim  08:03

Give me some math. 


Pam  08:05

Alright, bam. Okay, here we go. Ready? 


Kim  08:07



Pam  08:08

71 subtract 35. What do you got?


Kim  08:12

I like 70 and 35 because I know 35 and 35 is 70. So, I'm sketching a number line. And I started with 35. And I wrote 35 plus 35 is 70 plus 1 more is 71. So, that 35 and that 1 that I added is my answer. 36.


Pam  08:32

That's 36. Cool. 


Kim  08:33



Pam  08:34

So, it kind of looks like you kind of thought about the distance between 35 and 71. 


Kim  08:40

I did. 


Pam  08:40

Is that accurate?


Kim  08:41



Pam  08:41

So, on my paper, I have a number line starting at 35 and going up to 71. You have that nice jump of 35 in there. But when I total that distance between those that's a distance of 36. 


Kim  08:52



Pam  08:53

Cool. Next problem. How about 65 minus 29? And I'm actually going to ask you, if you don't mind, to not use removal on this problem. So, could you find the distance between these two numbers as well? 


Kim  09:07

Sure. So, on my number line, I put 29 to the left. 


Pam  09:12



Kim  09:13

And 65 to the right. And I'm going to find the distance between those. So, from 29 to 30 is just 1. And then, from 30 to 65 is 35. 


Pam  09:25



Kim  09:26

So, that's also 36.


Pam  09:29

That's a distance of 36? 


Kim  09:30



Pam  09:32

And so, you're saying that those numbers are 36 apart. You could have removed right? You're an Over girl, so if I would have said 65 minus 29, chances are high you might have subtracted 30. You know like, doing the over thing? 


Kim  09:44



Pam  09:44

That's kind of why I forced it because I just, you know, wanted to get you out of your groove a little bit, you know?


Kim  09:49

You know your student.


Pam  09:50

I do indeed. Why did you say "also 36" What do you mean "also"?


Kim  09:56

Because I realized that the first problem had a difference of 36, and so did the second.


Pam  10:01

Oh. Ha. Look at that. That was pure coincidence. Okay. Next problem. How about 62 minus 26? And if you wouldn't mind this time, if you wouldn't mind, would you also find the difference? Actually, though, distance. I have to tell you, my pen's dying. This is tragic. So, if you hear me shaking my pen, it's because I'm like, "Ink, get down there. Get down, down, down."


Kim  10:25

So, you have a busted pen, and I have a dull (unclear).


Pam  10:28

Well, I have an old pen. I have a pen that I love so much I've used it a ton, and so I think I'm running out of ink. Darn it. I might have to find another pen. 


Kim  10:36

Okay, I'm going to... Oh, my gosh.


Pam  10:40

I'm shaking my pen. Our editor is like, "Do I keep that sound in?" Yes, keep it in. I'm shaking this stupid pen. I mean, I love this pen. It's going to be very sad when... Okay.


Kim  10:48

Can I tell you something that bothers me about my first two? I'll solve it. But I'm realizing that I just sketched my number line, and it doesn't kind of match up, so that bothers me between the first two. So, I am putting 26... What do you say? 62 minus 26? Yeah. So, I put 26 and 62 on a number line. But... 


Pam  11:15

You put them both on there? 


Kim  11:16

I did. I did, I did. 


Pam  11:17

Why? Why would you start by putting both numbers on the number line?


Kim  11:20

Because I'm trying to find the distance between them, and so...


Pam  11:24

Ah, okay. 


Kim  11:24

Yeah. It's nice to have them both on the number line. 


Pam  11:26

That makes sense.


Kim  11:27

Right at first. 


Pam  11:28



Kim  11:28

And so, the distance between those 26 and 4 is 30. And then, 30 plus 32 is 62. So, there's a distance of 36 also. And that's why I was sad about my number lines not matching up. 


Pam  11:49

What's why? 


Kim  11:50

Because I realized it was going to be 36. 


Pam  11:54

This one's 36 too? 


Kim  11:55



Pam  11:58

And so, lining up the number lines. Like, if you looked at my paper. Written in pen with a couple of them that are a little bit less ink. But I've got one that's kind of shifted to the right a little bit because it's 35 to 71. And then, a little bit to the left of that is 29 to 65. And then, a little bit left to that is 26 to 62. Because those numbers were all kind of shifting. The numbers in the subtraction problems were all shifting a little bit to the left. And so, if we lined up those number lines, they kind of. And they're all 36 apart. Huh, that's interesting. It's almost like you're saying that 71 minus 35 is equivalent to 65 minus 29 is also equivalent to that third problem, 62 minus 26. And in fact, I just wrote on my paper that long horizontal equation 71 minus 35 "equal sign" 65 minus 29 "equal sign" 62 minus 26. Do you agree that those are all equivalent problems?


Kim  12:55

I do. Yep.


Pam  12:56

So, if we have three problems that are all equivalent, and they're all 36. They all have an equivalent answer of 36. I wonder. You had to do a little work to find the answers to each of those problems. Just a little. But you had to do a little work to find the answer those. I wonder if we could shift that 36, shift those numbers along on the number line somewhere, that it would make it so you almost didn't have to do hardly any work to solve that problem. What would be your favorite place to shift?


Kim  13:24

I'm going to shift both of the numbers down to the left. So, down, to the left on the number line. 


Pam  13:31



Kim  13:32

And then, think about the distance between 24 and 60.


Pam  13:36

24 and 60. 


Kim  13:38



Pam  13:39

And tell me how you got to 24 and 60.


Kim  13:43

So, the distance or the gap between 26 and 62. I have a jump between them, and so I just scooted this 62 down to 60. I shifted it 2. But then, I also need to keep the distance the same. So, I shifted the 26 down to 24. So, it's kind of like that bubble, that jump just down 2.


Pam  14:05

Just shifted.


Kim  14:06



Pam  14:07

And you're like, "But it's going to be the same because I shifted them both down 2." 


Kim  14:09



Pam  14:10

So, you're saying right now that I could add on to my equivalence equation there an equivalent of 60 minus 24? 


Kim  14:20



Pam  14:21

And that, to you, you don't have to think about. You're like, "Bam, that's just 36." 


Kim  14:25

I mean, yeah. I Have, You Need right?


Pam  14:28

Oh, because I Have, You Need helps you think about that answer? 


Kim  14:32



Pam  14:32

I got to tell you, Kim. 60 minus 24 is not as easy for me as a different equivalent problem.


Kim  14:41

I have a solution for you for that. 


Pam  14:45



Kim  14:45

This little routine that we play called I Have, You Need.


Pam  14:49

So, you're saying that if I played I Have, You Need more often.


Kim  14:52



Pam  14:52

60 minus 24. Alright, I will keep playing.


Kim  14:55

So, you like a shift in a different way, then, maybe. 


Pam  14:58

I do.


Kim  14:59



Pam  14:59

So, I would take that 62 and 26, and I would shift them up 4. 


Kim  15:04

Yeah, that's nice too. 


Pam  15:05

And so, if I shifted them up 4, then I end up with 66 minus 30. 


Kim  15:10

Yeah, I like that as well. 


Pam  15:11

So, you shifted it, so that the first number in the subtraction problem was a really nice 60. 


Kim  15:17



Pam  15:18

You ended up with 60 minus 24. I shifted it, so that the second problem in the subtraction problem turned into 30.


Kim  15:26



Pam  15:26

So, I ended up with 66 minus 30. And to me...


Kim  15:28

Yeah. I wonder if I went to 60 because I looked at the 62 first, and it was just so close to that. I don't know. Yeah.


Pam  15:38

Sure. Yeah, sure could be. So, you and I have had this conversation in the past, where I'm like, "Kim 66 minus 30 is easier than 60 minus 24." And you're like, "No." And so, I started interviewing people. I started asking people. I would do Problem Strings like this, and I would say, "Come on, tell me. I'm right, right? Like, I didn't even try to. I'm like, "I am so right."


Kim  15:58

"Kim's wrong!" 


Pam  15:59

"Kim is wrong! If you make the second number nice, it's totally an easier problem than if you make the first number nice in a subtraction problem." And I would say most people agree with me, but there is a segment of people who probably play I Have, You Need more than I have, who say no, they're the same. And I'm like, "What?!" Now, will you agree with me, though, Kim. They're the same for you. The 60 minus 24 is not easier than the 66 minus 30.


Kim  16:26

No, no, no. I would agree with you that shifting it to the second number being a friendly number, for many, many, many, maybe most people, would be easier. It's just that I've literally played I Have, You Need for so many times for so long. For so many people. But I think that's maybe is rare. Much as I've done that. So, I would agree with you.


Pam  16:51

But there are others. Yeah, there are others like you out there. And ready? I want my brain to do that. 


Kim  16:56



Pam  16:56

I have access to that. I want my brain to do that. So, growth mindset. I will someday be able to say that those are as easy for me. They are not yet. But I am working on it. I will get there.


Kim  17:07

I love that. I believe that it will. 


Pam  17:09

Let me give you another one.


Kim  17:10



Pam  17:10

What if I would have given you the problem 73 minus 37. I wonder if you could create an equivalent problem that would be easier to solve?


Kim  17:19

Well, now I want to know if you want me to go up or down. I'm going to do it your way, and I'm going to shift... I'm thinking about the distance between 37 and 73. But I want to shift those, so I'm going to shift them so that I'm finding the difference between 40 and 76. So, I'm shifting them to the right 3. Both those numbers. So, 37 moves to 40, and 73 moves to 76. And that gap is also 36.


Pam  17:47

Cool. It's interesting when you said that you shifted it to 40 and 76, I wrote down 40 minus 76. And then, I was like, "Oh, wait. No, that's not what she's doing. But it's the distance between them, and so we represent that with a subtraction problem. 76 minus 40, which is also 36. Cool. So, lots and lots of 36s. Alright, Kim, I'm going to give you a fresh problem.


Kim  18:09



Pam  18:09

Could you shift these numbers to make an equivalent problem that is easier to solve? So, fresh problem. What about 132 minus 96? 132 minus 96. 


Kim  18:22



Pam  18:24

Okay, what are you thinking about? 


Kim  18:25

I am going to shift both of those numbers slightly bigger. So, instead of the distance between 132 and 96, I'm going to move them both 4 bigger and find the distance between 136 and 100.


Pam  18:41

And what is 136 minus 100? Bam.


Kim  18:43

That is 36. Yeah.


Pam  18:46

Which is one of my favorite things to do is to say, "Alright, now we're going to like completely move away from 36." And then, actually have. You always want a moment in a string where kids go, "Ha! You got me." You know or, "Good try, teach." You know, whatever. Yeah. So, Kim, how would you describe the thinking that you used to solve these problems?


Kim  19:10

So, I would consider the problem and think that I can find the distance. Right, that's kind of the first thing, is I'm not thinking about removal, I'm finding the distance between the numbers. And I'm saying to myself, "I want a maybe easier problem, so I'm going to shift both either up or down the number line to find a nice problem where the answer just kind of falls out for me a little bit.


Pam  19:37

Bam. And I would say that nice problem is an equivalent problem. Yeah?. 


Kim  19:41



Pam  19:42

Alright. And so, we could call this strategy lots of things. We could call it create an equivalent subtraction problem by shifting the distance. We often call it "constant difference" or "constant distance". Yeah, either one. This strategy of constant difference, we're going to suggest is the most sophisticated subtraction strategy. 


Kim  20:03



Pam  20:03

It involves a lot. Kim just said she had to think, "I'm not going to remove. I'm going to find the distance." And then, there's some anticipatory thinking involved to say, "Ooh, I could find the distance. I could find the difference between these numbers. But I actually don't want to do it here. I'm going to create an equivalent problem." That requires you to kind of get outside the problem and think, "Could I create an equivalent problem that's easier to solve? Bam, I could." And then, kind of some simultaneity to consider, "If I shift that number up, what will happen to the other number? And will that create an equivalent problem that's easier to solve?"


Kim  20:42



Pam  20:43

So, some hallmarks that are involved in these more sophisticated strategies are that they require some anticipatory thinking, that we have to kind of get outside the problem and say to yourself, "Ooh, could I create an equivalent problem that's easier to solve?" So, anticipatory thinking is one of the hallmarks. Another hallmark is this idea of creating an equivalent problem that's easier to solve. So, that just very idea of saying to yourself, "Ooh, can I create an equivalent problem?"


Kim  21:13



Pam  21:14

So, this is also an equivalence strategy. Hey, I think we talked about the equivalent strategy, the most sophisticated strategy, last week of Give and Take. it was also an equivalence strategy, right?  Give and Take, you're grabbing a number from one of the addends to give to the other addend to create an equivalent problem that's easier to solve. In subtraction, we're finding the difference between the numbers, shifting them both up or down the number line, which could be thought of as give, give or take, take. But the idea is to create equivalent problem. So, they're both equivalent strategies. That's interesting, Kim. 


Kim  21:26

Yeah. Yeah.


Pam  21:51

That both of the most sophisticated... Let's see. How to say that? That the most sophisticated strategy for addition and the most sophisticated strategy for subtraction are both equivalent strategies. 


Kim  22:03

And that's a really big idea in mathematics.


Pam  22:06

Yes, yes. Equivalence is huge. 


Kim  22:09



Pam  22:10

I would also suggest that another hallmark that's involved is this idea of simultaneity. 


Kim  22:15



Pam  22:15

That if we shift both numbers, bam, we end up with a problem that's easier to solve. So, three hallmarks that I think are maybe interesting to consider - simultaneity, anticipatory thinking, and equivalence. 


Kim  22:30



Pam  22:30

All of those are involved kind of in this most sophisticated, constant difference strategy. Nice.


Kim  22:35



Pam  22:36

Hey, Kim, how do we model, how do we represent constant difference strategy? 


Kim  22:41

Yeah, I was actually just thinking about that because last week we said that we represent Give and Take, the simultaneity of that, on equations, right? The number lines not so great for that. But this strategy, we absolutely represent on a number line. And we would put the original problem maybe on there. Both numbers for the original problem where you'd find the distance. And then, shift those numbers, either one direction or the other on the number line. And a number line is really nice because you can visualize the shift that's happening. 


Pam  23:15



Kim  23:16

And I think you for early, early on. I don't know if you still do it. But I think early on, you would kind of picture your hands. Like you would kind of throw your hands up like they were on a number line, and you would like make a hand motion. Like, "Where do I want to shift?" I don't know if you still do that.


Pam  23:29

It's almost like karate chop kind of thing? 


Both Pam and Kim  23:30



Pam  23:30

Absolutely. In fact, I remember the day I was in our local grocery store. H-E-B, if you're in Texas. Whoop, whoop. And I had to... I don't know what I was subtracting, but I had to subtract something. And I had been working on the constant difference strategy. I was bound and determined that I was going to develop my brain to be able to think about subtraction and not just... Ya'll, I was so ingrained in the subtraction algorithm. When I would see a subtraction problem, my brain would just put those numbers lined up. Little ones, and crossie-outies, and zeros, and nines. And my brain would just go there. And it was so hard for me to get my brain to stop going there. I'll never forget. I was in the store, and I literally stopped. I was walking. I was moving. Whatever. And I stopped. If you could see me right now, I have both feet firmly planted. It's almost like I'm going to shoot a free throw. I'm like firmly planted. My knees are slightly bent. I stopped, and I said to myself, "If I'm thinking about subtracting these two numbers, here's where they are on a number line." And when I just said that, I put my two hands out like karate chopping. Like, here's this number. Here's that number. But I didn't put it 1, 2. I put it like, there they are. Both hands down. There they are. And then, I said to myself, "So, I want them to be here," and I moved my hands both up because I was shifting up to make the second number a nice number. "I want them both to be here. What would that be?" And then, bam. The answer just was like obvious. But I had to go, "They're here on the number line, and I'm going to shift them up. Like, bam." And as soon as I made that motion with my hands, my brain was able to like, "Yes!" And then, like, I didn't even have to think about the answer. I just have to shift both of those numbers up 4, or 16, or whatever it was. And then, it just fell out because then it turned into something like 2,342 minus 1,000. Oh.


Kim  25:22



Pam  25:22

Hey that wasn't too. I could hang on to that. 


Kim  25:24

So, last week, you said that the...


Pam  25:26

Hey, sorry, before we move on. Well, actually, I'm not sure what you're going to say. Go ahead, Kim. Go ahead. I'm such an interrupter. Go ahead.


Kim  25:35

Me too. It's why it works. So, last week, you said that Give and Take, the most sophisticated addition strategy, is not necessarily something that you would do on the street.


Pam  25:45

Ah. Yeah. 


Kim  25:46

And you just described that you did that in the grocery store. And I think it's possible. But I also... Maybe we want to suggest that because you're hanging on to a lot of numbers, it might also not be the thing that you want to assume everybody's doing as they're walking down the street. Would you agree?


Pam  26:04

I would. Though, I actually... Yes. I think there are definitely subtraction problems that I'm... Well, how do I say this?


Kim  26:12

I mean, you're thinking about the number that you're... The number is where it was. You're thinking about the amount that you're going to shift both numbers, and then you're thinking about the new problem. So, the new problem that you've created is so nice that you're going to do that on the street. But I don't want to leave people with the impression that if I just think about constant difference, then all of a sudden, I don't have to write anything down.


Pam  26:31

I gotcha.


Kim  26:32

The problem that you are creating, the last problem, you probably don't have to do a lot of work with.


Pam  26:38

Sure. Yes. I agree with you. That problem that I just said. Which I don't remember the numbers were because I just made them up. What did I say? 1,462 minus 1,000.


Kim  26:47



Pam  26:47

I'd have to like, 1,462, I just wrote down. Now, I don't have to write anything else down because subtract 1,000, I'm just like, 462. So, it's not about writing or not writing. I agree with you.  You definitely might want, might need to keep track of some of the relationships if you were walking down the street. Yeah, absolutely.


Kim  27:07

That's okay, right?


Pam  27:08

Oh, totally. 


Kim  27:08

I think people maybe think, you know, it's bad or it's whatever. 


Pam  27:15

To have to write?


Kim  27:16

It's less than if you if you write something down. (unclear).


Pam  27:18

Well, please not because that's me. I keep track of my mental thinking a lot. I'll often say in a presentation. Hey, you might see me represent some thinking up here, and you might think, "Oh, good. I'm so glad that she wrote that down. I was able to follow what the person was saying, or doing, or whatever." And I'll say, "Actually, I was writing it down, I was modeling that thinking, representing thinking, so I could follow what was happening." So, yeah. Remember, quoting Cathy Fosnot. Thank you for this quote, "Mental math does not mean you do it all in your head. Mental math means you do it with your head." You're actually using relationships and connections you own. That's totally legal to keep track. Yeah, I'm glad you brought that up. The thing I was going to interrupt you on a minute ago, and I'm glad I took that back, you mentioned that the modeling that we would absolutely model this on a number line. But I also have equations. So, I just wanted to stick in...


Kim  28:05



Pam  28:06

...that with constant difference, we'll use both models. So, we'll use the open number line to really develop the fact that we're shifting the numbers, and then we'll represent the equivalent problems. So, for example, when you did the 132 minus 96. On my paper, I have a number line with those two numbers and the distance between them. But I also have 132 minus 96 equals 136 minus 100 equals 36. 


Kim  28:30



Pam  28:31

Yeah. And it's a brilliant way. I'll just mentioned briefly. As high school teachers, even middle school teachers, when we want students to solve equations., all too often students think that that equal sign means "do it". That's not what an equal sign means. The equal sign means "equivalence". It means that whatever's on the left is equivalent to whatever's on the right. And so, if we can start doing work like writing these long equations like that 132 minus 96 equals 136 minus 100 equals 36. Students go, "Oh." They don't get into the trap of thinking that "equals" means "do it." I must take whatever's on the left, and now do it, and come up with something on the right. No, no. No, no. It's a representation of equivalence. It's not a symbol of something to do. Yeah. And I think that's a nice reason to do it early here. And since this is an equivalence strategy, it is a super strategy to hone and refine by using the instructional routine we call Relational Thinking. So in Relational Thinking, we give you an equation where one of the numbers is missing. And the idea isn't to do a bunch of computation. The idea is to think relationally, how are these numbers related, to fill in the blank. So, Kim, I'm going to give you one. Ready? 


Kim  29:53



Pam  29:54

Okay, I've got 5.2, or 5 and 2/10, minus 3.9, or 3 and 9/10, equals blank minus 4. So, let me say it one more time for listeners on the podcast. 5.2 minus 3.9 equals blank minus 4. What are you thinking about?


Kim  30:18

I just put 3.9 and 5.2 on the number line and made kind of a jump between them. And you said the second problem was subtracting 4, so I'm shifting the 3.9 up to 4. And that's a shift of 0.1. So, I'm going to shift my other number, 5.2, up a 0.1, so that the distance between them is still the same.


Pam  30:41



Kim  30:42

So, it's 5.3. 


Pam  30:43

So, the blank is filled in with 5.3. 


Kim  30:45



Pam  30:45

Super. And then, you can have kids talk about what's happening. And don't go too quickly to kids going, "Oh, you just add, add or subtract, subtract." Nah, like keep it about what's happening. You know like, justifying the thinking and the relationships that are happening. 


Kim  31:01

Yeah, and I want to say real quick that there will be students who want to say that it's 5.1. And so, attaching a number line model can help students see that distance shift and maintaining the same distance. Because often, we see kids with Relational Thinking, say something like 5.1 because they're trying to make sense of how the numbers are related. And that's completely normal.


Pam  31:30

Absolutely. In fact, I had drawn what you said. So, I have on my paper 3.9, and then a jump to 5.2. So, then we could say put the 4 on there. That's going to be to the left of... Or, excuse me. To the right of 3.9. And then, put that 5.1 on there. And now, you've got a shorter space, right? Because that 4 and that 5.1 are in between 3.9 and 5.2, the original two numbers. 


Kim  31:58



Pam  31:58

So, if we want the original jump, we can't make it smaller. And then, similarly, if they would have added, if the problem was different, then we'd have a bigger. And so, yeah. An open number line can be a really nice representation to go, "Wait, wait, wait. Think about this." Like, we're trying to create an equivalent problem, not a smaller distance between the numbers. Oh, I'm glad you brought that up. That's nice.


Kim  32:17

Hey, let's remind everybody that we have... On your website, you have a ton of Relational Thinking problems. And they can find that at That's kind of a mouthful. So, it will be in the show notes. You can click there. And there are a bunch of ones that you can use in your classes.


Pam  32:36

Yeah, nice. Cool. Hey, Kim, I mentioned at the beginning of the podcast that I was in this professional learning situation with these fifth grade teachers, and the teachers were like, "Well, what about this problem, Pam?" And the problem was, "If Alaska has square footage of 656,424, and some small state..." And I honestly don't remember which one "...has the area of 1,545. What is the difference between their area?" And teachers were removing. 


Kim  33:10



Pam  33:11

They were removing chunks that made sense to them. And what got hard was they would remove a chunk that made sense, and then they had to figure out how much more to remove. And then, they would go remove another chunk that made sense, and they had to go figure out. And, teachers, I wonder if you felt that. If you've had students sort of think about removal, that figuring out what you have still left to remove gets clunky, gets arduous. Some of the teachers were finding the distance between it. They started at 1,545, and they were making all of these jumps to get up to 656,424. Well, I'll never forget. Anne Roman. Bless your heart. Anne Roman was one of the leaders in our district, and she walked up to me quietly, and she goes, "Pam?" And I was sweating. I was like, "Oh, no." Like, "What's going to happen? Everybody's mad. They're going to revolt." And Anne goes, "Pam, what about constant difference?" And I was like, "What about it?" I didn't own it yet at all. And she goes, "Well, if you just shift everything up..." And as soon as she said that, I was like, "Oh! Then, it would just be..." So, Kim, how would you think about 656,424 minus 1,545? What would you shift?


Kim  34:23

I'm going to shift them both up 455


Pam  34:30

It's almost like we played a little I Have, You Need earlier with 545. 


Kim  34:35



Pam  34:35

So, if you just shift 1,545 up. What did you say? 455.


Kim  34:40

455, mmhm. 


Pam  34:42

What is that second number in the subtraction problem (unclear).


Kim  34:44

It's going to be 2,000. 


Pam  34:45



Kim  34:46

Which is super nice. Mmhm.


Pam  34:47

So, now, you have to add that 455 to 656,424. How hard is that?


Kim  34:53

Which is so... It is just place value. Like that's the kind of problem that could be like a place value left to right. It's literally just adding 455 and 424. 


Pam  35:08

Which is just 800... 


Kim  35:11

Yeah. 656,879


Both Pam and Kim  35:15

Minus the 2,000.


Kim  35:17



Pam  35:17



Kim  35:18



Pam  35:19

Ya'll, it's just this big, ugly, hairy number.


Kim  35:22

Looks horrible!


Pam  35:23

Minus 2,000. And you're done. 


Kim  35:25



Pam  35:25

Yeah. Now, kids have to think about minus 2,000. And that's great place value. Like, there's so many nice place value things that kids can mess with. I love this problem because if kids have developed the constant difference strategy...which to be clear, they should by fifth grade...then this becomes a non-issue. And, by the way, you might just check out those numbers, listeners, and notice how many opportunities for error a kid would have if they used the traditional algorithm.


Kim  35:51

Yeah, yep. So, this is a most sophisticated strategy. And one way that you can get better at all the strategies that we mentioned on the podcast is to participate in MathStratChat on Wednesday evenings. So, you can notice other strategies that people use to see if they make sense to you. You can examine the models that people use to represent their thinking. And you'll notice that sometimes we do a series of problems week after week, kind of like a Problem String, so that you have multiple experiences. So, if a strategy, you know, you look at the first week, and you're like, "Um..." You get to tinker over time, and take the time to let your brain grow. So, I want to share one more, Pam. Mendoza Math Mentoring. Back in August, (unclear) said, "I'm looking forward to challenging my fourth and fifth graders to look for creative ways to reason out problems with MathStratChats this year. So, that's a fantastic idea. On Wednesday nights, grab the problem,  take it back to your class the next day, and see what they can do Have your own little MathStratChat.


Pam  36:52

That is fantastic. And we would love for you to post pictures of your student's work. 


Kim  36:56

Oh, I'd love that. Yeah. 


Pam  36:57

Yeah, that would be super cool. Alright, 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 Let's keep spreading the word that Math is Figure-Out-Able!