January 09, 2024
Pam Harris
Episode 186

Ep 186: Subtracting Integers - Add The Opposite

Math is Figure-Out-Able!

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Math is Figure-Out-Able!

Ep 186: Subtracting Integers - Add The Opposite

Jan 09, 2024
Episode 186

Pam Harris

Minus-minus, plus-plus, or is integer subtraction actually figureoutable? In this episode Pam and Kim use a Problem String to build real understanding of integer subtraction.

Talking Points:

- The importance of having multiple contexts
- Seeing the negative symbol as "opposite of"
- Using distance
- Understanding takes time, but is so worth it!

See Episodes 181-183 for more about Integer operations.

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

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Minus-minus, plus-plus, or is integer subtraction actually figureoutable? In this episode Pam and Kim use a Problem String to build real understanding of integer subtraction.

Talking Points:

- The importance of having multiple contexts
- Seeing the negative symbol as "opposite of"
- Using distance
- Understanding takes time, but is so worth it!

See Episodes 181-183 for more about Integer operations.

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

**Pam **00:00

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

**Kim **00:06

And I'm Kim.

**Pam **00:07

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 mentor students to think and reason like mathematicians did when they were students. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be. Hey, Kim.

**Kim **00:35

We don't want to do that, right?

**Pam **00:36

That's right.

**Kim **00:37

When we could... Hi! How are ya?

**Pam **00:39

I'm crazy. How are you?

**Kim **00:41

You know... Yeah. Living my best life.

**Pam **00:45

Whoo!

**Kim **00:46

Can we tell people that we aren't in the same room? Is that a thing?

**Pam **00:49

Well, you know, it's funny because when we were NCTM, people were like, "You're not?!" You know, we chat about it a little bit. Yeah, we're not. Yeah. We know each other well, though. Maybe too well, sometimes. But yeah, we've... How long have we known each other? My son had you in third grade, and he's now 20...

**Kim **01:07

I think 21 years.

**Pam **01:09

Yeah, it's been a hot minute. A hot minute.

**Kim **01:12

Hey, you know I like to find a review every now and then. And, like people have given us some really good ones, so I keep throwing some in here, but.

**Pam **01:20

We sure appreciate it.

**Kim **01:21

Yeah, it is super fun. So, the one I want to share with you is super short, but it's from I'mLovingMentalMath. That's the handle.

**Pam **01:29

Nice.

**Kim **01:30

And I don't know if it's a... I don't know who it is, but I'm going to say "they", so, "I love this podcast and recommend it to all the math teachers I work with at our small school. Over the summer..." Oh, I do know. "Over the summer, my husband and computer engineer has started listening with me too. Keep up the excellent work."

**Pam **01:49

Oh!

**Kim **01:50

Isn't that fun?

**Pam **01:50

Fun!

**Kim **01:51

This person got their husband to listen. And I wanted to tell you. Did I tell you that my dad started listening every once in a while?

**Pam **01:58

No!

**Kim **01:59

Is that hilarious?

**Pam **02:00

Oh, that's awesome.

**Kim **02:00

Yeah, they were on a road trip. And he said, "Hey, I listened to your podcast." Ohhh... Okay.

**Pam **02:07

Oh!

**Kim **02:08

Super sweet. Anyway.

**Pam **02:09

You stood up proud and said, "Sweet!"

**Kim **02:12

Oh, it's super fun.

**Pam **02:12

Yeah.

**Kim **02:13

You know, he's kind of mathy.

**Pam **02:14

That's cool. Well, family math. That's the best. Let's...

**Kim **02:16

Super fun.

**Pam **02:17

We encourage everybody to listen as a family or whatever. Yeah. Cool.

**Kim **02:22

Thank you, I'mLovingMentalMath.

**Pam **02:24

Yeah.

**Kim **02:24

Okay. So, we recently did a couple of episodes about integers. Right? That was episodes 181 to 183. And...

**Pam **02:33

Hey, Kim.

**Kim **02:34

Yeah?

**Pam **02:34

I have to tell you. So, I was talking with one of my kids. I believe it was the kid who was here for Thanksgiving, so maybe I just won't tell you which one it was.

**Kim **02:42

Oh, okay.

**Pam **02:43

Anyway, and he said... Okay, now, you know it's one of three. He said, "Why do you say 'integers' when you are talking about negative and positive numbers?" And I was like, "Well, that's the name of the set." You know, I got all kinds of teachery. "That's the name of the set of numbers when it's negative and positive." He goes, "No, I know that. But when you're talking about positive integers, you're also talking about integers." I was like, "Yeah, whatever, snot nose." Yeah. But I said...

**Kim **03:09

I love your mathy kids.

**Pam **03:10

I know, but then I said, "I'm using the more specific set because when I'm only talking about positive integers, then I can call those the natural numbers. Or if I'm including 0, I can call them the whole numbers." Ha! Anyway. Okay, carry on. Yes. Yeah, so we did (unclear).

**Kim **03:24

We did some about negative integers (unclear).

**Pam **03:28

I mean, we had both. They were negative and positive, but yeah. Okay.

**Kim **03:30

Oh, yeah they were actually. Okay. Well, anyway, we thought we would follow up with one more for you because there's always more to do, right?

**Pam **03:38

Mm, absolutely.

**Kim **03:39

So, we are going to give you a heads up that we would not do these strings... We're going to do some strings, and we would not do these if we had not already established first what integers are, and then done some work on subtraction as difference versus removal. So, if you haven't heard the last episode, this isn't going to make a whole lot of sense, so stop right now and go back and listen, especially to Episode 183.

**Pam **04:04

Yeah, because it's really important that there's some sort of ground work, things that we would do with students. That doesn't mean that they've like owned them completely, but they at least have these ideas floating around. They're playing with some different notions about like what does it mean to be an integer? Integer. Alright, Craig. What does it mean to be a negative number? And how does that relate to a positive number. And distance from 0? And all that kind of stuff. And then, also the idea that subtraction can be thought of as two ways, like you said, as difference is the distance between numbers. But it can also be thought as removal or minus. Which is typically the only one that many of us ever have with subtraction because that's the typical meaning that we pull into the algorithm, that we're always subtracting these single-digit integers. But we need to know that subtraction can also have this idea of the distance between numbers. And you might be like, "I don't know. You know, really? Because my kids have got all the rules down the rhymes, or raps, or mnemonics." Or like, for example, I was working with a group of sixth grade students who had already. They had done some meaning of negative and positive numbers. And the teacher was very clear. You know, "I talk about how it works, and why, and meaning. But then, we do get to the rule." For example, they had Pizza Steve. They had one that was minus, minus, plus, plus. As I was walking around, and kids were solving problems, I heard a lot of kids saying "minus, minus, plus, plus." And a few times I would say, "You know like, when do you do that?" They're like, "I don't know. When you see two minuses, then you just do plus plus." And I was like, "Like, anytime?" And they're like, "Yeah. Anytime you see two minuses." I was like, "Anytime? You know like, "Addition, subtraction, multiplication, division? Minus, minus, plus, plus?" That's a bit of a problem if it is any operation. Anyway, so they have some sort of all these rules kind of floating around. And it became really clear that that's what was happening was there was all these rules floating around. So, it's not about rules. We're going to do a string today, Kim, on the podcast, where it's not about turning this into a rule. It's about understanding what's happening and generalizing some patterns. We need one more thing before we do this particular string. In... I think it was episode 183. I think. We tried to establish that another big idea. Which is, if we are subtracting, and I say to you, "Hey, Kim? You know like, you've been doing since second grade. You've got a number, and then we're going to subtract something smaller than it." So, like, I don't know. 8. And then, we subtract something smaller than it, 2. And if I say 8 subtract 2, then you would obviously say?

**Kim **06:34

6.

**Pam **06:34

6. And it's a positive number, right? If I have a number, and from that number, I subtract something smaller than it, 8 subtract 2, then you still have stuff. So, for example, if I've had money. We had $8.00, and I give you $2.00, we still have $6.00 bucks. Or if we're at 8 feet above sea level, and we dropped down 2 feet, we're still above. We're still 6 feet above sea level. What's another one?

**Kim **07:00

Money.

**Pam **07:00

Money. If you have $8.00 bucks, and you spend $2.00 bucks, you still have $6.00 bucks, right? We're still in the black. We still are positive if you've removed something smaller than what you started with. And that makes sense to everybody because you've been doing that since second grade. But then, we also developed in 183 that if you remove something larger than you started with. So, you started at 3 feet above sea level, and you dropped down 5 feet. 3 subtract 5. Well, now you've gone through the 0, right? You've dropped down below sea level. And so, 3 subtract 5. Where are you, Kim? 3 subtract 5.

**Kim **07:33

Negative 2.

**Pam **07:33

You're at negative 2. So, drop 2 below the sort of sea level. You can do the same thing with temperature. If we were at... What numbers did I just use? 3. If we're at 3 degrees above 0, brr it's cold, but it's not freezing yet. And then, it drops 5 degrees, you're 2 degrees below. So, if you're at 3, and you subtract something more than, then you have to drop through the 0, right? If you have a number, and you subtract something more than it. Here's another one. I've got $10.00 bucks, and I pay you $12.00. That's kind of funky with money.

**Kim **08:05

You owe me $12.00

**Pam **08:07

Well, if I have if I have $10.00, and I pay you $12.00, then I can hand you $10.00 of them, and now I owe you $2.00, right? Yeah, so as soon as I subtract more than I have, I'm in debt. I'm below sea level. I'm below the 0 on the temperature. Okay, so we kind of have to have both of those things happening in order for this string to make sense. So, I may have taken too long to get all that. If this is the first time you've heard any of this, really, maybe go back and check out those other episodes. But here we go. Alright. So, Kim, we play a lot of football here in Texas. Yeah?

**Kim **08:41

I don't, but a lot of people do.

**Pam **08:43

There you go. We don't actually at my house either. I can't get anybody to watch it with me. Ah! Makes me crazy. Anyway, in American football, we have this thing, this line of scrimmage, and when you are playing the line is kind of where you measure everything from, and you have four tries to get past that line. And so, the other night, there was a football game on, and the team had gotten pushed back 8 yards behind the line of scrimmage. So, that's not good, right, because now you've got not only the 10 yards you need to get past that line, but now you're 8 yards back. So, you're really... Oh, it's terrible. Everybody's sad and stuff. And then, the announcer says, "Oh, no. The team got pushed back 7 yards." That's terrible.

**Kim **09:24

Yeah.

**Pam **09:24

So, as I would say that, I would... With kids, I would like sort of act that out a little bit. So, can you picture, Kim, where did we start?

**Kim **09:32

I actually drew a number line when you were telling the story.

**Pam **09:35

Tell us about it. Yeah, like what are you picturing?

**Kim **09:37

So, as soon as you said we started 8 yards back, I drew 0 on my number line, and then I put negative 8 to the left.

**Pam **09:47

Mmhm. That's where we start. Yeah.

**Kim **09:49

Mmhm. And then, when you said we got pushed back 7, I drew a jump back of negative 7.

**Pam **09:56

Okay.

**Kim **09:56

(unclear) negative 8 minus 7, and then I landed at negative 15.

**Pam **10:01

So, like the football team is terrible. Like, we're clear back at negative 15. So, I would want to establish that with kids. Like, what really happened. And then, I would say, "So, we started at negative 8. We were pushed back 8." And I would write negative 8 on the board. So, not number line or anything. Just the number negative 8. "And then, we got pushed back." And I would write a minus sign, a subtraction sign. We got pushed back 7. So, now on my paper, I've got negative 8 minus 7. And I would say... So, I'm saying the thing in context. "We started 8 yards behind the line of scrimmage. We got pushed back minus 7. And you guys all just..." And then, equals. And I'm writing equals, as I say, "And you guys just said we landed at negative 15." And then, I would write negative 15. So, right now, on the board, I've got minus 8, negative 8, minus 7 equals negative 15. Does that make sense?

**Kim **10:49

Yep.

**Pam **10:50

But it was totally in context. Cool. Alright. So, then I would say, "Hey, then later in the game, that defense they were just... Man, they just kept pushing us back. We got sacked 8 yards back from the line of scrimmage." So, it felt like we've been here before. "And then, the announcer says the team now has a loss of 7 yards. Where are they" And most of the kids at this point..." Well, I guess I could just say, "Kim, where are we?"

**Kim **11:19

Same place.

**Pam **11:21

Because most of the kids are kind of like the same, almost intonation, "Same place." And I'm like, "We'll say more about that," because then I'm going to write on the board, we got sacked. We're back 8 yards. And right underneath the equation that we just had, I'm going to write negative 8. We got pushed back, was sacked 8 yards. And then, the announcer said, "We have a loss." And so, as the announcer says "have", I'm going to put a plus sign, and then I'm going to put negative 7 for the loss. So, we've started back negative 8. I get negative 8. Plus negative 7. "And you guys are saying same place. Negative 15." So, now I have two equations on the board. Probably your same number line on the board. You know like, if we had the 0, we were back negative 8, we got pushed back 7. But the two equations I have on the board, the first one, negative 8 subtract 7 equals negative 15. And the second one, negative 8 plus negative 7 equals negative 15. And then, I'm just going to go "huh," and then I'm going to go on to the next question. Cool. So, next question. Let's say that you've got $9.00, and you owe me $12.00. And I'm standing in front of you. So, I'm going to say you have a debt of $12.00. So, I just wrote on the board, 9... Or on my paper. 9 plus. Because you have a debt. 9 plus negative 12. And you're standing in front of me, what are you going to do?

**Kim **12:47

I'm going to give you the $9.00 bucks back.

**Pam **12:48

Okay.

**Kim **12:50

On my number line, I'm subtracting 9. Now, I'm at 0. I'm going to tell you, "Hey, I'll get you next time. I owe you $3.00." Subtract 3, and now I'm at negative 3.

**Pam **13:01

Cool. So, I've got on my paper 9 plus that debt of negative 12. So, 9 plus negative 12, equals, you're in debt, negative 3. Cool. Next problem. What if we've got $9.00 bucks, and you hand me $12.00? Like, how do you do that? I don't even know.

**Kim **13:24

Yeah, yeah.

**Pam **13:24

So, $9.00 bucks. And you're like, "Here's $12.00." And I'm going to go, "Yeah, but you only had $9.00?

**Kim **13:31

Yeah, so I'm at negative 3. Can I suggest that like that one, I almost want to do a different context. Like, if it's temperature would be a nice word for that one.

**Pam **13:42

Ah. Alright, go for it. What would your temperature sound like?

**Kim **13:44

So, if I'm at 9 degrees and it drops 12 degrees, then I'd be at negative 3.

**Pam **13:50

Bam. Oh, I like that. We could also maybe even do elevation. If we're 9 feet above sea level, and we fall 12 feet?

**Kim **13:59

Yeah.

**Pam **13:59

Then, you would be 3 feet below. So, let's say what that equation looks like. 9. And then, fall or drop.Is that minus, subtract?

**Kim **14:08

Yep.

**Pam **14:09

12. And then, we're saying equals negative 3. So, now we have 2 equations on the board. One of them is 9 plus negative 12. That was where we added a debt of negative 12. 9 plus negative 12 equals negative 3. And 9 minus 12 equals negative 3. And then, we notice again. "Huh. What are you guys thinking about this?" And then, we would have a conversation with students about what they're thinking as this is happening?

**Kim **14:34

Yeah.

**Pam **14:35

Alright, next problem. What if I were to just put on the board 3 subtract 7, and then ask, "What kinds of contexts could you guys come up with where we actually would ever have 3 minus 7?" So, listeners, you guys think yourselves 3 subtract 7. If you were doing elevation, or debt, or temperature, or football? What could you say that could represent 3 subtract 7. Okay. Kim, do you want to choose one or do you want me to go first?

**Kim **15:05

You can go first.

**Pam **15:06

Okay, so I'm going to have a temperature of 3 degrees, and it's going to drop 7 degrees. So, if I start at 3, and now I'm doing a vertical number line. I started 3, and I drop 7. I'm going to drop through 0. So I've got a jump a 3 to 0. But I have to jump 7, so I'm going to jump 4 more, and I'm landing at negative 4. So, 3 subtract 7 lands me at negative 4 degrees. Brr. Okay.

**Kim **15:31

Yeah, I would bet that my number line looks just like yours except I actually am not on a horizontal, I'm going to vertical number line because I'm going to be elevation, and I'm going to start at 3 feet above sea level, and I'm going to jump. So, I'm hitting sea level at 0, and then I'm in the water, negative 4 feet.

**Pam **15:58

Down negative 4. Nice. So, a couple different contexts that could work for that. What then if I put up 3 plus negative 7. And you want to go first or me? Choose a context that would make sense for that problem.

**Kim **16:17

I'm going to go money, I guess, on this one.

**Pam **16:19

Okay.

**Kim **16:21

So, I have $3.00 in my wallet, but I also owe you $7.00.

**Pam **16:29

So, you have a debt of $7.00.

**Kim **16:30

Yeah, I have a debt of $7.00.

**Pam **16:31

3 plus negative 7. Mmhm. Nice. Nice. Nice.

**Kim **16:34

So, overall, I have a negative balance of negative 4. I have a balance of negative 4.

**Pam **16:43

Yeah, nice. Nice. And I might go football on this one. I've gained 3 yards, so I'm 3 yards in front of the line of scrimmage. And the announcer says, "Oh, no, now they have a loss." "Have". That's why I'm... I know it's kind of tricky. People are like, "Really?" Right. It's not like it's super clean. But we can make some sense of 3 plus negative 7. I started at 3 yards in front of the line of scrimmage. Now, I have. There's the plus. I have a loss. Negative 7. 3 plus negative 7. So, I'm back. I was 3 ahead. But now, I've gotten pushed back 7. So, now, I'm horizontal again. And I'm back 4 from the line of scrimmage, so that's where the negative 4 comes in. And again, now I have on my paper 3 minus 7 equals negative 4. And below it, 3 plus negative 7 equals negative 4. Did you want to say something?

**Kim **17:29

Yeah, I don't know if there are more problems. But as you're saying these, it feels like there's some, you know, there's some pairs here. And for one problem in each pair, it feels to me a little bit more like I'm starting at a location and movement happening. And then, for the other type of problem, it feels like two separate events happened or two separate things have happened. And...

**Pam **17:59

You're kind of putting them together?

**Kim **18:00

You're kind of putting them together. Yeah. So, for that, it feels like maybe I would change context between the two problems.

**Pam **18:08

Ah. Very nice.

**Kim **18:10

The very first one when you said football, and then another football, it felt too similar to me. But I wonder if the football thing worked for me with the negative 8 minus 7. But then, I would change a context for the second problem of that pair.

**Pam **18:25

So, I hear you saying that the subtraction problems feel like movement, like you're removing, you're falling, you're being pushed back, you're spending money. And the adding a negative problems feel like you had debt of $8.00 and now you have another debt of $7.00. So, negative 8 plus negative 7 feels like, "Well, now I've got two debts." And they're kind of...

**Kim **18:52

Yeah.

**Pam **18:53

There's not like some action happening. You've got this debt. You've got this other debt. How much debt do you have? Well, combining them, I've got a debt of 15.

**Kim **19:00

Yeah.

**Pam **19:01

That kind of idea?

**Kim **19:02

Yeah. And I feel like maybe that's why it's so important to have several different contexts in mind, like be able to change between different contexts, because sometimes I think we force one context into a situation and it doesn't really work as well. You know, so when we only have one context in mind, and they're like, "Well, I told my kids think about money," or we say, "Think about money." Maybe that doesn't fit as well for a particular problem.

**Pam **19:30

Mmhm. Yeah, and I've got one for you in just a second. So, once we have these three sets that you said there were like in pairs on the board. So, negative 8 subtract 7 and negative 8 plus negative 7.

**Kim **19:44

Mmhm.

**Pam **19:44

And then, we had the 9 and 12s and the 3 and the 7s. Once we have those pairs on the board, I would step back, and I would say, "Just looking at the numbers, what patterns do you see?" So, Kim, got any patterns that you're noticing?

**Kim **20:02

(unclear).

**Pam **20:02

And I know I'm putting you on the spot here.

**Kim **20:04

No, it's okay. I mean, they're all negative.

**Pam **20:09

All the answers were negative?

**Both Pam and Kim **20:10

Mmhm.

**Pam **20:10

Yeah, yeah.

**Kim **20:13

Every... Not every time. In some of them, you're starting with a larger number, and you're subtracting a smaller number. In the second and third pair.

**Pam **20:26

Okay.

**Kim **20:29

Oh, that happens in the first two. Okay, so you're starting with large numbers and subtracting a smaller number.

**Pam **20:42

Give me an example. Where?

**Kim **20:43

9. 9 minus 12.

**Pam **20:46

So, you started with 9, and you subtract something...

**Kim **20:48

What did I say? So, you're starting with a larger number and subtracting.

**Pam **20:52

From a number. From a number. From a number (unclear).

**Kim **20:55

So, your starting with 9, and you're subtracting 12. So, you're subtracting something larger.

**Pam **21:01

Which is why you kept getting negatives.

**Kim **21:03

Did I say that backwards?

**Pam **21:04

You did the first time, but it's okay. So, like from 9, you subtract something larger than it? (unclear).

**Kim **21:09

Yes, which is why you're ending with a negative.

**Pam **21:11

Which is why ended with a negative. Or even negative 8, you subtracted 7. 7 is much larger than negative 8. Which is why you get negative 15. 3, you subtracted 7.

**Kim **21:21

Yep.

**Pam **21:22

That's something larger than 3, so then that's why the answer is negative 4. Mmhm. Okay. Yeah, that was consistent.

**Kim **21:27

Okay.

**Pam **21:28

And if you look at the pairs, one relationship that I would want to bring out is, if you're subtracting a number, can you also think about that as adding its opposite? Does that track? So, like, if we look at the (unclear).

**Kim **21:44

So, for like 3 plus negative 7?

**Pam **21:47

Yeah. So, we had 3 subtract 7, and you said that was negative 4.

**Kim **21:50

Yeah.

**Pam **21:51

Can you also think about that problem as 3 adding the opposite of 7?

**Kim **21:56

Yeah.

**Pam **21:56

Which is negative 7. Yeah. And we could see that if you were... You know, this is a podcast. I have no idea what people are envisioning right now. But on my paper, I have 3 subtract 7, and underneath it, I have 3 plus negative 7, the opposite of 7.

**Kim **22:08

Right.

**Pam **22:08

So, sometimes people will say, "Well, I can subtract, but I can also think about it as adding the opposite. I can lose money, but I can also think about it as adding a debt."

**Kim **22:18

Right.

**Pam **22:18

Yeah. Cool. So...

**Kim **22:20

Well, but do you think that... It seems to me like there's a lot of missed opportunity to thinking about the (unclear) symbol as opposite as well. I think for so many years, so many years in elementary and as we're entering middle school, they only think remove, remove, remove, minus, minus minus. And so, when somebody sees minus 7, negative 7, whatever you want to call it, I don't know that kids go, "Oh, that's the opposite of 7." Like, I think there's some really early foundation stuff that we can do very simply to say, this means the opposite of 7.

**Pam **23:03

When you write a negative sign in front of a number, it means the opposite of seven. Yeah. When we started the episode, one of the things we said is you really want to dive into the meaning of integers. And that's part of it. Yeah, absolutely. And when you get to a string like this, you're going to have to lean back and kind of... "Review" is not the word I want. But you're going to have to like acknowledge that kids are going to need a reminder of that, you know. That this is a perfect place to not say, "Okay, everybody. We've got to get this prerequisite skill down before we can move on." No, no, no. That in the midst of it, you could go "Wait, as we're looking at this negative 7, what does that mean? Oh, that's the opposite of 7." Yeah, sure enough. Here's a place to help make that even more generalized, more concrete. Yeah, for kids. Good. Nice. So, if we could maybe. If I had just given you a problem like 3 minus 7, you're saying you could have thought about it as 3 plus negative 7. So, what if I gave you a problem like 4 subtract negative 7? And I've written that as 4 subtract. The big subtraction, minus sign. And in parenthesis, the opposite of 7. 4 subtract negative 7. And I'm wondering. This is a subtraction problem as well. Could we think about subtraction as adding the opposite of the number in this case?

**Kim **24:24

Yeah. So, if I'm thinking about it as adding the opposite, then I'm also going to write beside it 4 plus 7.

**Pam **24:36

And why 7?

**Kim **24:38

Because the opposite of 7 is 7.

**Pam **24:43

It's hard to say. Yeah. So, I've got two equations written on my paper right now. 4 subtract negative 7. And then, 4 add the opposite of that negative 7, which is 7. So, you're saying 4 subtract negative 7, we could think of as 4 add 7, which is 11. And maybe the answer to both of those is 11.

**Kim **25:06

Yeah.

**Pam **25:07

Cool. Could we wonder if we could reason about. We've sort of made this generalization that we can add the opposite. But let's make sure reasoning another way. So, I'm looking at that first problem again. 4 subtract negative 7. And I'm going to think about that subtraction symbol now as the distance between the two numbers in the problem. Why are you laughing?

**Kim **25:28

That's what I actually did the first time.

**Pam **25:30

Oh, that's how you reasoned about it the first time. Okay. Which we would actually expect because we're suggesting this comes after that, right? So, can you tell us how you reasoned about 4 subtract negative 7 using distance?

**Kim **25:41

Yeah, you know, what I actually did is... So, I drew like a little arrow, but on like the distance subtraction.

**Pam **25:52

Okay.

**Kim **25:52

Like, I thought to myself, "I want this to mean distance between 4 and negative 7." So on the number line, I drew 4 and negative 7, and I found the distance between those was 11. And then, I knew I had to say, "Is it positive or negative?" So, I found the distance, but I need to know is that a negative 11 or positive 11.

**Pam **26:13

(unclear). Mmhm. Yep.

**Kim **26:14

Mmhm. So, I went back to the problem, and I thought to myself, "I'm subtracting..." I started with 4, which is bigger than negative 7.

**Pam **26:22

Okay.

**Kim **26:23

So, I'm starting with the bigger number, and I'm subtracting something smaller than it. Negative 7 is smaller. So, then, I knew that it was going to be a positive 11.

**Pam **26:34

Just like because in second grade if you had a number and you subtracted something smaller than it, you still had money, you were still above sea level, you were still above 0. Nice. So, you're saying 4 subtract negative 7 is 11. Both ways of reasoning. We just confirmed we can reason through it using distance and removal. Or we can use this pattern we've just found where we can add the opposite, instead of subtracting. Cool, let's do that with one last problem. How about negative 6 subtract negative 2? And I'm really curious which way you want to reason first?

**Kim **27:09

Well, so I wrote it down, but then over to the to the left, I wrote negative 6 plus 2. So, adding the opposite of negative 2, which is 2.

**Pam **27:22

Okay.

**Kim **27:24

And then, on the right side, I'm drawing negative 2 and negative 6 to find the distance between those.

**Pam **27:33

And would you just mind since people can't see your drawing where negative 2 and negative 6 (unclear).

**Kim **27:37

Sure. Yeah. When you just asked me that, I put 0 on there. So, I have 0 kind of in the middle. To the left of that 0, I have negative 2. And then, further to the left, I put negative 6. And the gap between negative 2 and negative six is a gap of 4. But I'm starting this time with a smaller amount. Negative 6 is smaller than negative 2.

**Pam **28:05

So, from negative 6, you're removing something larger than negative 6.

**Kim **28:09

So. it's going to be negative 4.

**Pam **28:10

It's going to be negative 4. And does that track out what you had said originally where you had negative 6 plus 2, adding the opposite of negative 2.

**Kim **28:17

Oh, yeah.

**Pam **28:18

And negative 6 plus 2 is that also negative 4?

**Kim **28:20

Yep.

**Pam **28:21

Bam. So, we have a really nice way of reasoning using distance and removal that we've developed before to substantiate. It looks like when we're subtracting, we can just add the opposite of what we were subtracting. And...

**Kim **28:33

Yeah.

**Pam **28:33

...that seems to play out. Yeah.

**Kim **28:35

Yeah. And neither one of those are, "Hey, kids. When you see two subtraction signs next to each other..."

**Pam **28:40

"...minus minus plus plus."

**Kim **28:42

Yeah.

**Pam **28:43

Yeah. I mean, you can use that. And I get it, teachers. I really do. In fact... I'm going to go off just a second. We had a teacher in our Journey group the other day. It was not about integers. It was about something completely different. Say. Well, in fact, I think it was about scientific notation. Said, "I've always in the past done this... It wasn't scientific notation. It was... Anyway, it doesn't matter.

**Kim **29:06

Yeah, times 10.

**Pam **29:08

And they had this pneumonic, and it was like this thing to help kids get answers quickly. And the person in Journey said, "I got to tell you, when I just gave kids this rule that they could get an answer quickly, they all got answers quickly by the end of the day. When I'm having kids reason, and think, and understand what's happening, it's not coming as fast. Like, kids aren't all getting the answer at the end of the day." And Kim, I was reminded so strongly...in fact, the middle of night when I couldn't sleep the other day...what Phil Darrow talks about when he says, "Yeah. Like, that's absolutely going to happen because you're building capacity. You're building kids brains to think in a different way. It's not going to be just a quick and easy answers. But it's also going to stick better." So, will it take longer? Yeah. Will it take more effort? Yeah. Like, there's a reason that we didn't develop operations with integers throughout the history of mankind for a while, right? Like, it took mankind a while to develop these complicated things. It's going to take kids a while to really make sense of it and get it down. It's not going to be as quick. But it will stick so much better. And we'll have kids confident that they can reason through math, that it's not just a bunch of tricks that they're supposed to memorize, not understand, spit back out, and flip a coin about which one to use when. That math is actually

**Both Pam and Kim **30:33

figure-out-able.

**Pam **30:33

Bam.

**Kim **30:34

Absolutely. Alright, Pam. We're going to talk about fractions next week.

**Pam **30:38

Hey, I'm so excited. We've been planning to talk about fractions for a hot minute, and we are going to talk about fractions. So, ya'll, tune in next week. And thank you for tuning in today and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!

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