Math is Figure-Out-Able with Pam Harris

Ep 183: Integer Subtraction

December 19, 2023 Pam Harris Episode 183
Math is Figure-Out-Able with Pam Harris
Ep 183: Integer Subtraction
Show Notes Transcript

Can integer subtraction feel intuitive without rote memory? In this episode Pam and Kim delve into a Problem String and explore integer subtraction relationships.
Talking Points:

  • Integer chips?
  • Students reason about subtracting integers
  • Distance versus removal 
  • Problem String subtracting integers in context
  • When context is less helpful
  • Don't go to rules, stay in reasoning

See Episodes 149 and150 for more about distance vs removal  

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  00:07

And I'm Kim.


Pam  00:08

And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But, ya'll, it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can actually mentor students to think and reason like mathematicians. Not only are algorithms really not helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be.


Kim  00:33

I'm super excited because it's December 19th! 


Pam  00:37



Kim  00:37

(unclear) And you have birthday, not too long. (unclear)


Pam  00:40

I do have a birthday coming up. It's true. The whole world celebrates my birthday. I mean, really. In fact, I was just talking to some people the other day, and the gal said, "My birthday is on the 31st!" December 31. And she said, "My sister's birthday is on the 30th. And I looked her, and I was like, "Stinks to be her." Because on the 30th, people are just getting over like all the things that were happening before that, right? And then New Year's is the next day, so nobody's going to have a party on the 30th.


Kim  01:05



Pam  01:06

But the entire world celebrates New Year's Eve, so there's always a party. You can always find a party. And I just pretend that everyone is celebrating my birthday, and it works great. It's fantastic.


Kim  01:15

I know several December 31st Birthdays. 


Pam  01:17

Do you? 


Kim  01:17

It's wild. I do.


Pam  01:19

Okay. We're all cool, I hope.


Kim  01:21

Mmhm! Alright, I got another review for you. 


Pam  01:26

Oh, nice.


Kim  01:26

I won't do it every week, but... 


Pam  01:27

Okay, not every week, but (unclear).


Kim  01:29

But it's our podcast, so I can if I want to.


Pam  01:31

Ha! And it's been kind of fun. It's fun. Because you check more often than I do. Yeah.


Kim  01:33

Yeah. For sure.


Pam  01:34

Alright, what do you got?


Kim  01:34

Okay, Kendra619 says, "I love listening to Pam and Kim every week. I teach fifth grade math and each podcas--each podcast..." Good heavens. I can not read.


Pam  01:46

You can do it! You can do it!


Kim  01:47

I'm going to try again. "I love listening to Pam and Kim every week. I teach fifth grade math and each podcast teaches me something new. Pam and Kim are so relatable and definitely help me make math more figure-out-able for my students. I have recommended your podcast to anyone who will listen."


Pam  02:01



Kim  02:02

Thanks, Kendra!


Pam  02:03

Yeah. Thanks, Kendra. Appreciate that. Thanks for the recommendation. Thanks for helping us spread the word. And definitely thanks for teaching your fifth grade kids so well. That's excellent.


Kim  02:11

Yeah. That's awesome. 


Pam  02:12

Nicely done. Cool. (unclear)


Kim  02:13

Alright, we...


Pam  02:14

Thanks for the ratings and the reviews, everybody. We appreciate it.


Kim  02:17

Yeah. Alright, we talked integer addition last week. 


Pam  02:21



Kim  02:22

So, it's time to turn our attention to subtraction today. So, listen, there's lots to do, as you know. So, we're acknowledging upfront, listeners, that we're just starting the conversation. 


Pam  02:32



Kim  02:32

And you'll have to let us know if you want more. 


Pam  02:34

Yeah, absolutely. 


Kim  02:35

You will. We know. We'll put it in somewhere.


Pam  02:37

Hey, and before we started recording, Kim reminded me that last week we had promised that we would also talk about integer chips. 


Kim  02:43



Pam  02:44

So, I'm going to maybe start there today. So, what do I think about integer chips? I think integer chips are...


Kim  02:55



Pam  02:56

(laughs). So, what do I mean. They are those round things. 


Kim  02:59



Pam  02:59

There's red on one side, yellow on one side. And teachers often use them to help students kind of think about 0 pairs. And they model some stuff with integer operations using integer chips. I'm just going to flat out say I think you can do without them. So, are they bad? No. Are they wrong? No. Are they super duper, duper helpful? Eh... They're helpful. But are they helpful enough? Mmm...


Kim  03:01

Yep.  Mmhm. 


Pam  03:02

So, I'll give you a quick for instance. When we were at NCTM, NCSM. I'm sorry, I don't remember you were. Whoever you were, you made me think. I just don't remember who I was having a conversation with. Said, "Integer chips. Like, Pam, like this problem." And we did a couple of problems. She goes, "They're so important because..." And then, we did a subtracting a negative. And she subtracted a negative incorrectly because she used the integer chip model and couldn't quite think about how. Because it's insane. To subtract a negative... I know right now people are thinking, "Pam, I can clearly tell you how to do it." Yes, you can. You can clearly tell me how to subtract a negative using integer chips. But it's not very intuitive. And it's not very... What is a good word? (unclear).


Kim  04:12

Yeah, I got words for it.


Pam  04:12

You got words. Ha. It's not the thing that I'm going to pull up in my head when I'm doing integers later. 


Kim  04:17

(unclear).  Right. 


Pam  04:20

And what were you going to say? What did you say?


Kim  04:21

I said, it's procedural, and it's one at a time. Which...


Pam  04:24

Super procedural. And I had to tell. Like, hardly anybody comes up with how you can model subtracting a negative with integer chips. I know there's a few of you out there thinking, "No, it was super easy for me." It's not for everyone else. Can we just like... It's just a very intuitive. So, I think we can get more mileage out of a model that we're going to use more often. So, if you want to do integer chips, I'm not going to tell you not to. But I am to say maybe don't spend too much time there. Maybe get out of there. Why not? What's a pitfall of integer chips. So, quickly. I'm already spending a little more time on this than I was intended, so quickly a pitfall of integer tips. Definitely subtracting a negative. But also, the counting happens. If I say...


Kim  05:03

I jumped your gun. Sorry. 


Pam  05:04

No, it's okay. If I say, you're going to do something like 6 plus negative 7. Show me with integer chips. Well, most kids can think about 6 plus negative 7. Especially if you're just thinking on a number line. Not actually even drawing the number line, but just thinking on a number line. Especially if you do it in a context. But if I make you do it with a model, now you've got to count out those 6 positive integer chips. And then, you have to count out those 7 negative integer chips. And then, you line them up, so that you've got the 6 negatives and positives that cancelled a 0 pair. I shouldn't say "cancelled". Caught myself. That add to 0. Those 0 pairs add to 0, and you've got 1 leftover. And what color was it? Okay, it was negative. Notice that when you have that 1 leftover, you're kind of almost reading off the answer. 


Kim  05:46



Pam  05:47

You're doing less reasoning, and you're doing more, "I do this. I do that. Then, I read off the answer." If you read off the answer... Thank you, Cathy Fosnot for this. If you read off the answer, you were probably following a procedure and you weren't involved in the relationships. You weren't mathing. Too much counting. Too much reading out the answer. Those are my reasons for de-emphasizing integer chips. Now, notice, I didn't just say you couldn't do then at all. But if it was me, I wouldn't. I don't think it's worth it. I think we can get the bang for the buck doing other things. Like, modeling things on an open number line. And you might be like, "Well, Pam. You're going to have to do stuff on an open number line." Yes. But that model lives on. We need open number lines through calculus, right? We just put two of them together, and now we have a coordinate axis. So, open number lines are super helpful. We need kids... What's the word? Fluent on number lines. That's super important. Okay. So, Kim, if we were going to think about subtraction at all? 


Kim  06:52



Pam  06:53

We have some episodes that we have talked about before for whole numbers, where we have talked about that there are two main ways to think about subtraction.


Kim  07:01



Pam  07:02

So, if you're listening to this podcast right now, and that doesn't ring for you. You're like, "What do you mean two different ways of looking at subtraction?" Well, we can look at subtraction as the difference between the two numbers in the subtraction problem. We can also look at as removing the one number from the first number, the second number from the first number, in a subtraction problem. Most often, especially if kids have been drilled in a traditional subtraction algorithm, kids are always thinking about minus, subtracting, removal. They're only using that one meaning of subtraction. They're not thinking about that we can consider subtraction as the difference between two numbers. That often comes to bite us when we're doing comparison problems because we say, "Look, it's a subtraction problem." And they're like, "No, this is a comparison problem." Because they haven't ever thought about that we can label comparison, how far apart numbers are, as subtraction. So, if that's new to you, maybe we recommend that you go... We'll put the podcast episodes in the show notes. Or just go search. You know, ya'll, just "Math is Figure-Out-Able podcast, subtraction," and check out the ones where we talk about the two meanings of subtraction, difference and removal, because we're going to need both of them when we subtract integers. Okay. So, to be clear, we're not going to develop difference versus removal right now. You need to have had some background there for us to use it here. Let me just say, maybe, when we just filmed... Kim, we often film us doing Problem Strings or expert teachers doing Problem Strings. In fact, I'm so excited that we have some new teachers we're going to film (unclear).


Kim  08:36

I'm excited too.


Pam  08:37

We film these expert teachers in real classrooms with real students. You always tease me. "It's not fake students. They're real students."  I don't know why I say "real students". I know. So, where we do Problem Strings, and then we put those Problems Strings. We have a couple out on the website that anybody can view, and then every month we put a new one in our Journey program, so you can see it happening with real kids. And my goal is to build up this bank of Problem Strings where teachers can go look at tons of different things being taught using Problem Strings with kids across the grade bands. When I did that, when I went into a sixth grade classroom not too long ago, and we filmed subtraction of integers. We filmed a bunch of integer stuff and subtraction was one of them. The first day that I went in, I did Problem Strings with kids with whole numbers, where we talked about difference and removal. And I got them thinking about how we can view subtraction as difference, and we can view it as subtract, minus, take away, removal. And so, that was kind of a starting place for kids. And then, the cameras came in the next day. And then, we dove into integer stuff. So, I practice what I preach. We give kids kind of that sort of sense. If you don't have that sense, go check out a podcast on it. We'll put that episode in the show notes. Come on back, and let's talk about subtracting integers. Alright. So, Kim, let's do a quick Problem String...


Kim  08:43

I do. Okie dokie.


Pam  08:48 get at integer subtraction.


Kim  09:58

I was almost a really smart aleck. I don't know why.


Pam  10:00

You were going to be smart? You? You were going to be smart alecky. 


Kim  10:03

I was.


Pam  10:03

Kim, you're never smart alecky. Not. What's the opposite of that? You're always smart alecky. 


Kim  10:09



Pam  10:10

I don't know if you're always smart alecky. I wouldn't say there's a day that doesn't go by (unclear). 


Kim  10:15

(unclear) snarky and smart aleck? I feel like smart aleck is probably negative. I'll just not be that. 


Pam  10:21

Okay. Let's not be that. You are snarky. That's true. 


Kim  10:24

For sure. 


Pam  10:24

So, you're saying snarky is positive?


Kim  10:26

I don't know. Probably not to anybody but me. Witty? I don't know.


Pam  10:29

You like being snarky.


Kim  10:31

I do. Alright, moving on. I'll do (unclear).


Pam  10:33

Sometimes I'll meet people, and I'll go, "Oh, Kim. You'll like them. They're snarky."


Kim  10:36

You do!


Pam  10:37

She instantly lights up. She's like, "Oh, yeah! I want to meet that person!" So, everyone who's not smarky, Kim... Smarky? Everyone whose not snarky...


Kim  10:46

I like them too!


Pam  10:46

...Kim will also like you. Yeah, she like you as well. She just is pretty confident she'll like if you're... Okay, moving on. Oh, goodness gracious. Neither of us got sleep last night. Moving on. 


Kim  10:55

know (unclear). Except we're busy. But we love it. I


Pam  10:59

Okay. If I were to ask you, Kim. Now, the cadence in this Problem String is going to be very important. So, maybe help me a little bit because I'm going to spend some time talking about a problem, but not because I would with kids. I'm going to talk about why it's the problem there. So, listeners...


Kim  11:16

Should we do the Problem String, and then? Or do you need to talk about (unclear).


Pam  11:19

And then, backup?


Kim  11:20



Pam  11:20

I can try. 


Kim  11:21

Okay. Well, it's okay if you can't.


Pam  11:22

You know, I'm bad at that.


Kim  11:23

I know. That's why I maybe thought. I wondered (unclear).


Pam  11:26

It's interesting, podcast listeners, when Kim and I present together, I'm often the example of a string with lots of teacher talk all the way through it. And Kim is the example of Problem String. That goes straight through. You do it. And then, she backs up and does like, "Let's talk about why I did what I did." So, for that reason, we work well together. I have gotten better at doing some.


Kim  11:50

You have. 


Pam  11:50

Yeah, I'm better when someone times me. If I say, "time me", then I'm kind of... I don't know, somehow that tells my brain to not stop and do teacher talk. Let me see. Let's see. Let's see if we can do it. Okay. So, Kim, what is 5 minus 3? 


Kim  12:03



Pam  12:04

And I know that's hard. We're not going to pause very long here. 2. I'm just going to make a quick picture of your thinking, just so we can kind of refer to it after. You didn't have to think about that. But I'm just going to say, can we think about 5 minus three as 5 subtract 3? We can. But can we also think about it as the difference between 3 and 5?


Kim  12:20



Pam  12:20

We've done some work with that. So, on the board, I would put 3 on a number line. I would put five to the right of it. I would draw a little jump above it and put 2. And you're saying the difference between 3 and 5 is 2. Cool. Next problem. What about 3 subtract 5? 


Kim  12:35

Negative 2. 


Pam  12:37

And how do you know?


Kim  12:42

For this one, I started at 3 and go back 5. 


Pam  12:46

And if you do that, you kind of skip over the 0.


Kim  12:48



Pam  12:48

And you kind of landed on negative 2. So, I've now written on the board, 3 minus 5 is negative 2. And I'm going to pull out... Kim, see I think I have to talk about as we go. I don't think it will make sense if I don't.


Kim  12:48



Pam  12:49

So, I'm going to pull out of kids that negative 2, and then I'm going to say, "Could you reason about that with elevation?" So, Kim, can you reason about that with elevation?


Kim  13:08

So, I'm at 3 feet above sea level, and I drop 5 feet, so I'm at negative 2 feet below sea level. 


Pam  13:17

There you go. Okay, cool. And how about temperature? Can you reason that with temperature?


Kim  13:20

Yeah. It's 3 degrees, and it drops 5, and so it's really cold. Negative 2 degrees.


Pam  13:24

Brr!  And can you do football?


Kim  13:27



Pam  13:28

No? No, you don't have to football? Football is kind of a...


Kim  13:30

I'm 3 yards ahead of the line of scrimmage, and I get pushed back 5 yards. 


Pam  13:36

Nice. And so where are you? 


Kim  13:37

I'm 2 yards behind the line of scrimmage. (unclear).


Pam  13:40

Bummer. We're back. That's terrible. And what about... What am I leaving out?


Kim  13:45



Pam  13:45

Debt. Yeah. Money.


Kim  13:46

I have $3.00. I owe someone $5.00. So, I'm in debt $2.00. Negative 2.


Pam  13:51

Cool. Alright. So, then, I'm going to say, "Hey, I'm a little curious. Because we just said on the first problem, 5 minus 3, that we could think about that as the difference between two numbers. But every scenario you just told me about, you were kind of falling. The temperature was falling 5 degrees. You were going down below sea level. It was removing. You were going in debt. Like removing money from you. What did I leave out? Debt. Elevation. Falling. Football, you were being pushed back 5 yards. All of those were kind of a subtraction removal context. Could we look at the difference between 3 and 5? Can we let the subtraction? Because right now I'm pointing at the subtraction symbol. 3 minus 5. I'm pointing at that subtraction in between the numbers, and I'm saying, "If that means difference. That's what we've said. We said with whole numbers it can be difference. Then, I should also be able to look at the difference between 3 and 5. Ooh, but the distance is 2. But you just told me the answer was negative 2. Oh, crumb! I guess we can't use distance to think about subtraction. Or can we?" So, I would have... For both problems right now, I only have one number line on the board. We talked through those other contexts when you were removing 3 minus 5, but I've really only got the one number line, and that is from 3 to 5 is this distance of 2. And so, I'm going to say, "Could we use distance? Could we find out how far apart the numbers are, but then think about removal? To say, "Yeah, they're 2 apart. But is it going to be positive 2? Or negative 2?" The distance is 2. But from 3 feet, you fell down 5 feet. Where are you? Ah, you're 2 feet below sea level. From 3 degrees, you fell. So, we can actually use distance to find out the magnitude of the answer, the absolute value of the answer. But then, we have to think. We have to use removal to decide is it going to be positive or negative? I wonder if that will work. Let's try another problem. So, Kim, what if I said 7 subtract negative 2?


Kim  14:56

I just drew a number line, and I put negative 2. And then, I also put 7. And so, I...


Pam  16:02

Where are they? Can you orient us? Because nobody can see.


Kim  16:05

Yeah. So, negative 2 would be to the left of 0. I didn't have a 0, but I'll put it right now. So, negative 2 is to the left of 0. And then, 7 is to the right of 0. So, the space between those is 9 units. 9 whatever.


Pam  16:22

Okay. I've got that negative 2 to 0. I've got the 0 to 7. You add those together. You got this. So 7 and negative 2 or 9 apart. Okay. Okay.


Kim  16:31



Pam  16:32



Kim  16:32

And so, the way I think about it is I asked myself is it going to be positive or negative by thinking about the context. So, if I'm at... I have $7.00. No, I don't think I actually do. I don't...


Pam  16:48

Yeah, this is tricky.


Kim  16:49

I think, I think about if I remove a number that is smaller or larger. Like, am I removing something that is less than what I had? Or am I removing something more than what I had?


Pam  17:01

Well, and let's pause for just a second and look at the first two problems that we did. Because we had 5 minus 3 was 2, and 3 minus 5 was negative 2. So, you just kind of said bigger, smaller numbers. And this is exactly what we're going to do with kids. We're going to kind of dive in. How did you know the distance between 5 and 3 and 3 and 5 was 2? What was the size of numbers involved? 


Kim  17:23



Pam  17:23

Your whole life, up until we started bringing in integers. Like, when you were subtracting at grade two. Grade two. I've been talking to Canadians lately. Can you tell? In second grade. Where was the big number? Where was a small number? In most of the problems you were ever subtracting? 


Kim  17:39

You were always subtracting the smaller number. 


Pam  17:41

Yeah, so from a big number, you were subtracting something smaller than it, then you would get a positive answer, right? Like, you had money, and you subtracted just a little bit of money, you still had money. If you were above sea level, and you fell down just a little bit, you were still above sea level. If the temperature was whatever, and it dropped just a tiny bit. You know like, you're not close to 0, so just dropped less than getting you to 0, then you're still above 0. So, all of those contexts help us think about if from the first number, you just subtract something smaller than it, bam, the answer is positive. But what happened in that 3 minus 5? How does that? Go ahead. 


Kim  18:22

You're subtracting the bigger number now.


Pam  18:24

Yeah. So, from 3, you're subtracting something bigger than it. Oh, from $3.00. I'm spending $5.00. I'm in debt. From 3 feet above sea level. I'm falling down 5. I'm 2 feet below sea level. From 3... (unclear) Contexts. Sea level. Temperature. Did I just? From 3 degrees, that temperature dropped 5 degrees. Brr, I'm below 2 degrees. All of those contexts help me think about from a number, if I subtract something bigger than it, the answers got to be negative. So, what does that mean for 7 and negative 2? From 7? 


Kim  19:03

Oh, sorry. You're asking me? 


Pam  19:04

Yeah. Sorry.


Kim  19:06

So, I think about those as from 7, I'm subtracting a number smaller than it. So, then my answer is going to be positive.


Pam  19:16

And that is the clincher of comparing those numbers. From 7, are you subtracting? So, as I've written on my paper. I have 7 subtract. And I wrote in parentheses negative 2. So, from seven, you're subtracting something smaller than 7. So, you actually have to get kind of in your head. How are 7 and negative 2 related? Negative 2 is smaller. They're 9 apart. Since it's smaller. I'm removing something smaller. That's like what I've done since second grade. Bam! The answer is going to be positive. Cool. Well, so then what if I just turned that problem around on you and said, same numbers. Negative 2, subtract 7. 


Kim  19:53



Pam  19:54

So, same difference, right? Same distance between the numbers. Okay, but? 


Kim  19:57

But this time, I'm starting at negative 2, and I'm subtracting something larger than what I have. So, it's going to be negative... What did I say? Negative 9. 


Pam  20:07

Negative 9. The distance is 9. But we're removing something bigger than we started with. Bam, the answer has to be negative. So, this is not trivial. Like, we have to like. Let's see what happens in another problem. So, then, I might say to kids, "Okay, what if I start at negative 5, and I'm going to subtract negative 3? Negative 5 subtract negative 3? How are you thinking about that?


Kim  20:08

So, I wrote down negative 3 and negative 5 on a number line. 


Pam  20:14

Yeah. And how do they relate? Which ones which? 


Kim  20:18

That's a good question. So, the negative 5 is to the left of the negative 3? 


Pam  20:24

Because it's?  Yeah.


Kim  20:38

Negative 5 is bigger than negative 3. (unclear). Negative 5 is bigger than negative 3.  Negative 5 is...


Pam  20:54

Ooh, careful. Negative 5. It's a bigger debt.


Kim  21:00

Oh, yeah. It's bigger debt, so it's smaller. Wait.


Pam  21:02

But we would actually say it's smaller. Yeah, that's super tricky. So, as you go to the left (unclear).


Kim  21:08

Yeah. What did I say?


Pam  21:09

You said negative 5 is bigger than negative 3.


Kim  21:11

Oh, it's smaller than negative 3.


Pam  21:13

Okay. Alright, just checking.


Kim  21:14

I'm sorry. 


Pam  21:14

Well, hey, and maybe maybe you meant to say that or not. But lots of... This is a thing that (unclear).


Kim  21:19

Yeah, for sure.


Pam  21:19

(unclear) negatives and positives. Sometimes we have to like stop ourselves and just think. "Okay. Wait, what does that mean?" It is a bigger debt, which is... That's confusing, right? Yep. I tell you, Kim, when I used to do limits in calculus, I would say, "As you go to the right, and x is getting bigger..." Well, that's true as you go to the right. And then, I would say, "As you're going to the left, and x is getting bigger in the negative direction..." I had some very nice people. Thanks, Diane McGowan and Susan Mae, who were like, "You cannot say bigger negatives." I was like, "Why not?" And they're like, "No. Those numbers are smaller." So, I had to really work on that. Yeah. Okay. So, I've got negative 5 and negative 3 on my paper. Kind of like you said, negative 5 is to the left. The negative 3 is to the right. Okay.


Kim  21:23

Yeah. Yep. So, the distance between those is 2. 


Pam  22:03



Kim  22:04

But because I'm subtracting something larger than what I currently have, it's negative 2. 


Pam  22:09

Because negative 3 is larger.


Kim  22:11



Pam  22:11

And the problem was negative 5 subtract negative 3. Negative negative 5 subtract something larger than it, you're like the answer has to be positive, 2. 


Kim  22:19



Pam  22:20

Okay, so what if we flip those around? What if it was negative 3...


Kim  22:23



Pam  22:23

Subtract... Oh, sorry.


Kim  22:26

Did you say was the answer is positive 2. I think you just said positive 2.


Pam  22:29

I did. But I should have said negative 2, huh?


Kim  22:31

Yeah. Yeah, yeah. 


Pam  22:32

Ha! I don't know why I said. Okay, because you're subtracting anything bigger than it, it should be negative.


Kim  22:37



Pam  22:37

Right. And maybe I'll stay in context to help me make sense of that. If I've got some money, but I spend more than I had, I'm in debt. If I'm at sea level, and I drop more than I was, through the 0, then I'm at negative. If the temperature is, you know, whatever, but I dropped more than that, then I'm below 0.


Kim  22:59



Pam  23:00

Yep. Okay, cool. Thanks for that.


Kim  23:03



Pam  23:03

Alright, now we're flipping it around. Negative 3 subtract negative 5.


Kim  23:07



Pam  23:08



Kim  23:09

So,there's still the same distance apart negative three is still 2 away from negative 5. Yep.


Pam  23:13

Yeah, still 2 apart. Okay.


Kim  23:15

Mmhm. But this time, I am subtracting less than what I had.


Pam  23:19

Can you give me a context where you're making sense? Then, does that mean it's positive or negative? If negative 5 is less than negative 3? You're subtracting something less than you started with. 


Kim  23:31

So, if I was at negative 5... Oh, no. 


Pam  23:39

So, don't do too number-y. Just like...


Kim  23:44

I'm subtracting. I have more debt than I started with.


Pam  23:48

I have to think for a second. Negative 3 and negative 5. So, from negative 3, you're subtracting something less than it. So, you have some money. You're subtracting less than that. 


Kim  24:02

Yeah. You can correct me if I'm wrong. But I feel like the challenge when you are subtracting negatives is that we try to stay in context too much.


Pam  24:13

Yes. I will concur. Yeah. Which is why just said try not to actually use the negative 3 and negative 5. 


Kim  24:20

So, I think... 


Pam  24:20

Be a little more general. Mmhm. 


Kim  24:21

I think we do a really good job, or we attempt to do a really good job, really having kids understand negative numbers in context, and what does it mean to have negative numbers. And then, we rely on their understanding of where numbers fall on a number line, that negative 5 is and where negative 3 is, and how they relate to each other, so that we can use those intuitions when we get to something funky like subtracting a negative.


Pam  24:49

Yeah. Because subtracting a negative doesn't work well in context. 


Kim  24:52



Pam  24:53

It just doesn't. And people try to have it make well, make it make sense.


Kim  24:57



Pam  24:57

They try to do the integer chip thing makes sense where you have to add in the 0 pairs. And like, I get it. I get it. So, I'm just going to suggest that subtracting a negative doesn't work well in context. Which is the reason why we developed integers so late in our history. Like, they came very late in history and making sense of subtracting negatives came super late in history. So, we build some generalities about how we can kind of think about it from a number. If you remove something smaller than it. Well, that's like what we did in second grade. Bam, the answer is going to be positive. From that number, if you remove something bigger than it. Oh, no. Now, we're in debt. Were below sea level. Were below 0 degrees. Once we can make that generalization where kids actually understand. They're not just memorizing a rule. They actually understand that generalization. Then, they can look at a problem like negative 3 minus negative 5, and they could say, "Alright, how do those numbers relate? Well, bam. From negative 3, I'm subtracting something smaller than it, then the answer is going to be positive." And how far apart were they? They were 2. So, the answer is positive 2. Yeah. Yes? 


Kim  25:13

Yep.  Yeah. 


Pam  25:19

So, the upshot of today's episode is, we need both meanings of subtraction. So, if you teach younger students, and you're teaching whole number addition and subtraction, please make sure that you have developed with kids that there are these two interpretations, two meanings, two ways of looking at subtraction. If you work with integers, do some of that work enough that kids can like understand that subtraction can have these two meanings. Help kids really understand integers, feel what's happening, be able to justify. Do it a lot. Don't go to rules. Stay in reasoning. You're going to worry it's going to take too long. Let me tell you what happens when you just give them the rules. Then, we have kids memorizing things like Pizza Steve and minus, minus, plus plus. And then, I'm in there working with kids, and they see multiplication, and they start doing minus, minus, plus, plus. And they see addition and subtraction, and they start doing Triangle Steve. And you're like, "What is Triangle Steve?" See, I'm not even going to tell you because it's a memorizing thing that kids just look up, and they miss apply it, and they stop thinking about stuff. I'm going to suggest if you have kids who are thinking and reasoning and confident that they can make sense of problems, they are going to A, do better on your high stakes tests because they go in confident. B, have their worlds open up to more math because they go in confident. That is our goal. They may not be as fast. You may not quote unquote "cover as much stuff", but they're going to do it better, and they're going to do it with confidence. And we need kids who have reason to fall back on. Whoo! Alright, ya'll. Thank you for tuning in. Hey, what have we not done with integers that you still want to hear about? Let us know. We will devote some more... Some more. Some more future? Can I say more future podcasts?


Kim  27:56



Pam  27:57

To integers. Alright, 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 And thank you for spreading the word that Math is Figure-Out-Able!