Math is Figure-Out-Able with Pam Harris

Ep 186: Subtracting Integers - Add The Opposite

January 09, 2024 Pam Harris Episode 186
Ep 186: Subtracting Integers - Add The Opposite
Math is Figure-Out-Able with Pam Harris
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Math is Figure-Out-Able with Pam Harris
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 

Show Notes Transcript

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!