Math is Figure-Out-Able!

Ep 242: Integer Multiplication

Pam Harris, Kim Montague Episode 242

One of our most requested topics! In this episode Pam and Kim discuss integer multiplication without mismatching rules or randomly switching signs. 

Talking Points:

  • Whole number multiplication strategies + meaning of integers leads to integer multiplication.
  • The meaning of integers (See episodes 169, 181, 182, 183, 186)
  • An integer multiplication Problem String and how to model 
  • Important noticings about modeling and context
  • The importance of distance, reflection and opposites
  • The importance of equivalence and commutative property


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Linkedin: Pam Harris Consulting LLC 

Kim  00:00

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather. 

 

Pam  00:09

And

 

Kim  00:09

And I'm Kim, a reasoner who now knows how... I was thinking about the fact that you said "I'm Pam," not I'm "Pam Harris," and I was like, "I knew she was going to do that!" I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

 

Pam  00:26

improve this intro to this podcast every time. Ya'll, we know that algorithms are amazing human achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. And I am the prime example of that. 

 

Kim  00:45

Yeah. In this super fun podcast, we help you teach math, building relationships with your students, and grappling with mathematical relationships.

 

Pam  00:53

And thanks for joining us to make math more figure-out-able. 

 

Kim  00:58

Hi there. 

 

Pam  00:59

Hey, Kim. What's up? You know, all the good mathy things. Some math, some math teaching. 

 

Kim  01:04

Yeah. I have the best review for you today. 

 

Pam  01:07

Oh, I love good reviews! Go, go, go! 

 

Kim  01:09

Okay, so MNMadre said, "LOVE IT!" All caps, five stars. And here's the best part, "I learned about this podcast from Christina Tondevold when she was speaking about lifting up other women."

 

Pam  01:23

Ya'll, we love Christina Tondevald in so many ways,

 

Kim  01:26

I don't remember if it was an email or something. Maybe it was her podcast that she's talked about this. Anyway, I thought it was brilliant and wonderful. I loved it. So, MNMadre said, "I am an interventionist who works with students from K-5, and I'm very conscious of where they are headed math wise..." Love it! "...and how important connections can be to higher math. I love your StratChats and all the episodes I'm trying to keep from binging, so I have time to really digest each one. Keep up the great work!" Well, thanks so much, and we are not opposed to you binging, but then it'll be slow and sad. It's one a week.

 

Pam  02:02

You get all caught up. Yeah, we really appreciate Christina Tondevold and the excellent work that she's done. Thanks for listening both to her. We highly recommend her work. And yeah, thanks. I wonder if it's MinnesotaMadre. 

 

Kim  02:16

Oh, maybe. 

 

Pam  02:17

Mom from Minnesota. 

 

Kim  02:18

Maybe I don't know. Okay, so listen. Today, we are going to do something we've been asked about 127 million times.

 

Pam  02:27

Let's do it. 

 

Kim  02:27

And we are going to talk about integer multiplication and division.

 

Pam  02:33

Every middle school teacher in the world just sat up tall and said, "Yes! Bring it on! Bring it on" 

 

Kim  02:38

Yeah.

 

Pam  02:39

Alright, so we've done a little bit of work in the past, and we'll put these episodes in the show notes, about the meaning of integers. And maybe I should have started with saying, I'm going to kind of give you a progression, and it's kind of a high level of progression of some things I would do. And then we'll kind of dive into integer multiplication and division. 

 

Kim  02:59

Okay.

 

Pam  02:59

The very first thing in this high level progression is kids have to understand whole number multiplication strategies.

 

Kim  03:07

Mmhm. 

 

Pam  03:07

So, that doesn't mean that you stop the presses, and you back up, and you do a bunch of fact stuff that they should have done, that you're repeating that they did in third grade.

 

Kim  03:16

Mmhm. 

 

Pam  03:17

But it does mean that you could do some work with whole number multiplication that leads into integer multiplication. And the kind of work I'm suggesting is that kids are exposed to and learn the major multiplication strategies.

 

Kim  03:31

Mmhm. 

 

Pam  03:32

So, the fact that they can chunk and do smart partial products, sort of clever chunking. Or that they can do an Over Under strategy with multiplication. Or a Five is Half of Ten multiplication. Even flexible factoring. Or Doubling and Halving is probably the one I should have said first. So, those whole number multiplication strategies are going to come into play in integer multiplication. So, I think that's a really good sort of background basis. If I was a middle school teacher, I would work on those, leading up to when we start working on integers. So, there's a first thing. Secondly, we need to have the meaning of integers down. Yeah. Ya'll, I remember the day when I was at a... Texas kind of led the way, mostly good, in kind of standards development, and what it meant to have a standards progression. There were some other states working on it kind of simultaneously, but Texas was definitely working on that here is a set of standards that you should be really working towards in these grade levels. And there was kind of this progression.

 

Kim  04:37

Yep. 

 

Pam  04:37

And I remember one of those very first kind of, "Here they are!" outlaying to leaders across the... TASM is the leadership organization. Kind of gave put it out to their leaders. And leaders said, "Why in the world is there a standard in sixth grade that just says, "Understand the meaning of integers. We just need kids to be good at the operations." And I'm going to strongly suggests that we actually do need quite a bit of work on students understand.... Quite a bit? Is that the right way to say it? There is work to be done to help have students actually understand the meaning of integers. What do I mean by that? Now, we're going to put an episode in the show notes that will just talk more about that. Like, if the temperature is 6 degrees above 0, then where's a temperature of -6? And putting a number line up and marking those on there. And if I'm at -15 below 0, where is that? And how does that relate to 15 above 0?

 

Kim  05:11

Mmhm.  Right.

 

Pam  05:14

This idea that there's distance involved and there's direction involved. And I like to even do some addition and subtraction problems with integers, before kids are then held accountable for those operations to get at the meaning of integers. So, I might say something like, "If I'm 10 feet above sea level, and I fall down a hole 15 feet, where am I? 

 

Kim  05:56

Mmhm.

 

Pam  05:57

So, that's kind of could be an addition problem where I'm adding a negative... What did I say 15? Adding a -15? So, 10 add -15. But it could also be subtraction problem where it's 10 minus 15. And where do you end up if you do that? And then I want to draw a vertical number line, and I want to have that kid fall through 0. Then bam, if I'm at 10 and I fall 15, I've landed at -15. Yeah, go ahead. 

 

Kim  06:21

going to say, I don't remember if this was in Investigations when I was teaching third grade or...  I was Or if we just went there with this conversation. But I remember my students talking about like elevators, and like below the ground floor. They can absolutely understand below 0, if it's in context.

 

Pam and Kim  06:27

Yeah. 

 

Pam  06:27

Yeah. Yeah. And I think temperatures fantastic for that, sea level, elevation is fantastic for that. Debt is interesting, but it can be good in some circumstances. American football is not too bad if you use the line of scrimmage as 0. But definitely temperature and elevation can be fantastic contexts. I think your elevators is kind of like elevation. To get at this idea of two major things. That a negative number is the same distance from 0. You can't just tell kids that. They have to kind of experience it.

 

Kim  07:11

Mmhm. 

 

Pam  07:12

Same distance as the positive number is from 0. And that has everything to do with the direction from 0. And so, I like to use both horizontal and vertical number lines. Okay, that was super quick. Go listen to the other episode to get more on that?

 

Kim  07:23

Yeah. 

 

Pam  07:23

So, first, whole number of multiplication strategies. Second, we need to know the meaning of integers with the distance and opposite. Oh, and I got to say one more thing about that. The word "opposite" should come into that meaning (unclear).

 

Kim  07:36

Yeah. 

 

Pam  07:37

So, that if I'm saying something about, "Hey, if I'm 15 feet below sea level, where's the opposite of that?" And the opposite that's going to be 15 feet above sea level. And if you could see my hand, I just totally took my hand and I kind of reflected. I'm like... How do I do this? I'm spanning a distance with my fingers, and it's as if my fingers are on 0 and my thumb is down at -15, and then I kind of reflected my hand up, so now my fingers are 0, and my thumb is at positive 15. So, it's like this, I'm reflecting, and so I've got that vertical reflection, but also if I'm over at, what, 13, the opposite of 13 is -13. And when I write that negative, I'm going to say the word "opposite" 

 

Kim  08:19

Yeah, so important. 

 

Pam  08:20

Yeah. And I'm also going to talk about reflecting, so that that brings in geometry. So, I'm reflecting, and I'm going to use the word "opposite" kind of along with the word "negative". So, I might say "negative five", and I might say the "opposite of five" when I write the symbol minus and five, the opposite of five. So, those are important things to sort of start with. Third, I think we need to have kids do some operations with addition and subtraction on number lines with integers. We've done episodes on that. Go check out those episodes for more on that. Fourth, for multiplication and division. I'm going to start with multiplication. Ideally, kids have done some work on number lines with addition and subtraction strategies. Though, I don't think you had to have. But ideally, you would have because you would have wanted that for your addition and subtraction with integers. But now (unclear) multiplication. When I say to a kid, "Hey, what is 5 times 8?" And we're going to do a Problem String, Kim (unclear).

 

Kim  09:17

Okay. Yep. 

 

Pam  09:18

So, if I say, what does 5 times 8 mean, what might you say? Just throw out anything. 

 

Kim  09:23

I'm going to say that's five 8s. 

 

Pam  09:26

So, if that's five 8s, I could represent... And you might also say 40. But I'm going to represent that, and I'm just going to choose a horizontal number line for this one. I'm going to start at 0, and I'm going to make five jumps of 8. 

 

Kim  09:41

Yep. 

 

Pam  09:41

And so I'm going to painstakingly do this the first time, Kim. I'm not going to do this very often because I don't want to really promote all this additive thinking, but I am going to do it once. 

 

Kim  09:49

Yep. 

 

Pam  09:49

So, I'm going to make a jump of 8, land on 8. Make a jump of 8, land on 16. Make a jump of 8. Every time I say "jump of 8", by the way, I'm making a jump and I'm writing 8 above it.

 

Kim  09:49

Okay.

 

Pam  09:49

So, jump of 8, land on 24. Jump of 8, land on 30. How many did I say? Five? 

 

Kim  10:03

Yeah. 

 

Pam  10:04

One more jump of 8, and I land on... Wait, where am I? 24, 32. Jump of eight, and I land on 40. I think I said 30. I meant 32 Okay, so I now have a number line that has 5 jumps with 8 written above each jump with these landing marks of 8, 16, 24, 32, and 40. And then I'm like, "You guys just told me that it was 40. Sure enough." So, five 8s is 40. Cool. What about five times -8? Now, I might put this in context where I'm saying like, "What if I had 5..." I might have started the first problem with if I had 5 jumps of elevation of 8 feet. So, I'm at sea level, and I've jumped up 8 feet at a time. Or, sorry, I've jumped up... Yeah, 8 feet at a time, and I've done it 5 times.

 

Kim  10:51

Mmhm. 

 

Pam  10:51

Then I might say, "Now, I'm jumped down 8 feet at a time, and I've done it 5 times." Now I just said that, but I might say like, "If this, if it's 5 times -8. What's happening here? Oh, well, you jumped down 5 times but 8 at a time. So, now... You know what? I did this wrong on my paper. I would have to erase this on the board right now and slide it over to the right because I didn't leave myself any room on the left like a dork.

 

Kim  11:16

I knew that we were going to do some negative stuff, and so I put my 0 in the middle. 

 

Pam  11:20

So, you left yourself room.

 

Kim  11:21

I did (unclear). 

 

Pam  11:22

Alright, I need more sleep or something. So, yeah, the 0 should have been in the middle of my board. And then have those 5 jumps of 8 to the right. And now, I would put a new number line. I'm going to do a new number line, but I'm going to put the 0 in the same spot, same lined up, horizontal spot. And now, I'm going to make 5 jumps to the left. And again, for the last time, I'm going to make a back, and I'm going to put a jump of -8. So, jump, -8 above it, land on -8. And then I'm going to say, "Hey, what if I make another jump of -8, where am I landing?" And they're going to say, "-16." And now, notice how I'm, again, continuing to get at the meaning of negatives.

 

Kim  11:58

Right. 

 

Pam  11:58

If I go back 8, where do I land?

 

Kim  12:00

Mmhm. 

 

Pam  12:00

And it's weird that the numbers get big as I go to the left because that's never happened before when we've been on number lines. So, this is an experience, again, where we're kind of getting used to that. So, now I've got these 5 jumps with -8 above each jump and those landing points. And you're telling me that then I would land on -40. Okay, cool. Then I've written the equation 5 times -8 is -40. And then we might have a conversation. Like, what are you going to think of in the future when you see something like... You've seen 5 times 8 for years, but now you're going to see 5 times -8. Ah, you can think about it as jumps the other direction. Okay, cool. 

 

Kim  12:03

So, can I just say that the look of what you have on your board right now...

 

Pam  12:40

Mmhm.

 

Kim  12:40

...is a beautiful look for the idea of reflection and opposite because these literally look like it's reflected at that 0.

 

Pam  12:49

Yeah. Like for example, if I look at where my 8 is, and I reflect that over 0, that same distance over 0, I'm landing on where I put -8. So, as a teacher, I'm really trying to make these jumps proportional, so that I do get that look, so my -40 is exactly the same distance on the left of the 0 as my 40 is on the right of the 0. 

 

Kim  13:09

Yep. 

 

Pam  13:09

Cool. Nice noticing. Next problem in the string. Ya'll, what is 6 times 7? Now this time, I'm going to do a vertical number line, and I'm going to put 0 right in the middle, and I'm going to make 6 jumps of 7, but I'm not going to painstakingly make them all. I'm going to make them real quick. I'm just going to go 1, 2, 3, 4, 5, 6. So, now I've just got 6 lumps, and I'm going to say, "Where is that everybody?" And they're going to say, "42," and I'm going to put 42. Now, you could tell me that your students might need one more example of seeing each of the jumps and where they land because they're really still working on building multiplicative reasoning, and this is an opportunity for you to connect additive thinking, where I'm adding each of the factors to multiplicative reasoning. I might believe that. I don't know that you have to at this point, but I could. Know your content, know your kids. If you know your kids, maybe I would do one more kind of detailed number line. Especially because I just changed the direction of the number line. Now, it's vertical. So, I can hear that I'm not going to today. Kim doesn't need that. And then below the... So, then I'm going to write 6 times 7 is 42. So, 6 jumps of 7 lands me up there at 42. Cool. Next problem. What is 6 times -7. I might put this in... I might have put both of these in context. I'm going up the hill, and every time I go up, I go up 7 feet at a time. Maybe this is the elevator. Every time I go up a floor, I go up 7 meters. That's kind of tall. 7 feet is too short. 7 meters too tall. Whatever. So, now we're dropping, right? Six -7s. I might just stay on the same vertical number line at this point for 6 times -7.

 

Kim  14:51

Mmhm. 

 

Pam  14:51

Kim, what is 6 times -7? 

 

Kim  14:53

-42.

 

Pam  14:55

And how are you thinking about that? 

 

Kim  14:58

Well, we just did 6 times 7 is 42, so I thought about 6 and 7 again, and then said it's going to be the opposite."

 

Pam  15:05

Ah. So, one way we could think about it is you've got that 42 up there, and you're like, "I know it's going to be the opposite of that. I can take that same distance vertically." If you could see my hand, I just made kind of that span from 0 to 42, and then I reflected it over the 0, and now I'm down at -42. A fine way of thinking about that. Did anybody think about it as six -7s, where you kind of had these 6 jumps and each of them were -7. And bam, that would land... Six of those -7s would also land you on negative. So, kind of two ways of looking at it? Interesting. I wonder how you guys might think about 9 times -3. 

 

Kim  15:43

Mmhm.

 

Pam  15:44

Alright, Kim, how are you thinking about 9 times -3?

 

Kim  15:47

Same as the last one. I think about 9 times 3 is 27.

 

Pam  15:53

And I'm going to go ahead and model that. This time, I'm going to go back to a horizontal number line. I'm going to have 0 right in the middle this time. 

 

Kim  15:59

But it's the opposite, so you would reflect to get -27.

 

Pam  16:03

Yeah, so this time I'm going to the left, and I'm going to do 9 jumps of -3. I'm making those 9 jumps just really, just la, la, la, la. Like I'm not painstakingly putting anything above them. Just, "Look, there's 9 jumps of -3. We know what 9 times 3 is, so nine of those -3s is also going to lane me on 27. But hey, I'm over here to the left. Cool. 

 

Kim  16:25

So, it feels to me like when you draw the the jumps back, back, back.

 

Pam  16:30

Oh, I just thought of my strategy, not yours, huh?

 

Kim  16:34

Oh, yeah, I don't care about that. But that is more of an additive way of thinking about it, which might be what students do early on to just like really hang on to that it's nine -3s. But it feels more multiplicative and like knowledge of integers to say, "I'm going to focus on the multiplication, and then add on the integer understanding of it being opposite. 

 

Pam  16:58

Yeah,

 

Kim  16:59

Okay, alright. 

 

Pam  16:59

Did you have something else? "adds" a little tricky there. You're going to think about the multiplication. I know 9 times 3 is 27. To model your thinking, I would have put the 0 in the middle, marked 27, and then reflected it over to -27. And I agree with you that I think that would be a more multiplicative way of thinking about it. And I'm going to ask for both of those strategies as we go. Mmhm.

 

Kim  17:14

I was going to ask if in this particular string, you would want... You know, if this is... Is this like a first string that you would do with students? 

 

Pam  17:31

Mmhm, yeah. For multiplication of integers. Yep. 

 

Kim  17:34

Would you want both of those strategies is what I was (unclear).

 

Pam  17:37

Yes. 

 

Kim  17:37

Okay.

 

Pam  17:37

Yes. Yes, absolutely. Yep. 

 

Kim  17:39

Alright.

 

Pam  17:39

Yep. And so, notice that in this Problem String, I gave 5 times 8, 5 times -8, a pair. 6 times 7, 6 times -7, a pair. The next problem I just said 9 times -3. And then I'm going to follow that with the last problem, which is -7 times 4. And wait, the way that doesn't follow the pattern. So, how are you thinking about -7 times 4, Kim?

 

Kim  18:04

Well, this one's not bad for me because I'm not thinking about jumps backwards. So, I think 7 times 4 is 28.

 

Pam  18:10

Mmhm.

 

Kim  18:11

And then the opposite of that, which is -28.

 

Pam  18:14

So, as you were talking, I drew a vertical number line. I put 0 in the middle. And when you said 7 times 4 is 28, I marked a distance up of 28. And you said, "Then I'm going to take the opposite of that," and I made motions with my hand to reflect over the 0. And I'm like, "Oh, so down here at -28." That would definitely be a great strategy. Yep. I'm not done. Yeah, go ahead. 

 

Kim  18:18

Mmhm. Well, I was going to say, I think this might be where teachers, students run into a little bit of a tricky situation. Because if they're thinking additively, how do you do... 

 

Pam  18:51

-7s of anything. 

 

Kim  18:52

Mmhm. 

 

Pam  18:53

Yeah. Yeah, absolutely. So, if they're thinking additively, I am going to try to ask questions that nudge them towards four -7s. The problem was -7 times 4.

 

Kim  19:04

Yeah. 

 

Pam  19:05

And I'm going to maybe say something about the commutative property. Can we think about this as four -7s? 

 

Kim  19:09

Yeah. 

 

Pam  19:10

But I'm going to also really think about this reflection part.

 

Kim  19:14

Right, right.

 

Pam  19:15

Yeah. So, can we really think about this as 7 times 4 and the opposite of that? Yeah. And at the end of this string, I might write that -7 times 4 as negative... Probably -1 times "parentheses", times a quantity, 7 times 4.

 

Kim  19:34

Mmhm. 

 

Pam  19:34

And I know a lot of teachers do this, but I think they kind of do it first. I'm going to invite you to do it after you've done the kind of experience that we just did. So, that's a way of modeling algebraically what Kim's kind of been thinking about. She's thinking about -7 times 4 as the opposite, that -1, times 7 times 4. And so, I'm sort of thinking about the 7 times 4 first, and then I'm taking the opposite of it. And we're also adding in this maybe new idea that we're reflecting. If we know where 7 times 4 is, we're reflecting that over the 0.

 

Kim  20:09

Yeah.

 

Pam  20:09

But we've built that kind of from the additive way of kind of reasoning about the four -7s, and also connecting that to this idea of thinking about the opposite and the negative. Yeah, of the positive product. 

 

Kim  20:26

Yeah. 

 

Pam  20:26

I could talk Cool. Alright, so that's a Problem String to get kids kind of thinking about at least a positive times a negative. 

 

Kim  20:34

Right. 

 

Pam  20:35

Alright, so golly this is going a little longer than I was hoping, so let's see how this next one goes. So, Kim, let's do one more string.

 

Kim  20:43

Ooh, a follow up string. Nice.

 

Pam  20:44

Follow up string.

 

Kim  20:44

Two for one, teachers.

 

Pam  20:46

There you go. So, I might start the string with, "Hey everybody, 8 times 3." When they say 8 times 3, then I'm going to go ahead and model it as just 8 times 3 without the jumps. 

 

Kim  20:56

Okay.

 

Pam  20:56

Depending on the class that I have, if I think I need to do the jumps one more time, I might. But today, I'm just going to say, "Hey, if you have eight 3s, you guys are telling me that that's landing on 24." So, then I mark the 24, finish the equation, equals 24. Then I'm going to say, "What is 3 times -8?"

 

Kim  21:13

Oh, that's the opposite of... It's 24. 3 times 8 is 24. And then the opposite of that.

 

Pam  21:19

The opposite of it. And so, I'm going to, again, ask for both strategies because I think I'm going to have kids that are also thinking about three -8s. And so, I'm going to say, "Alright, if I have a jump of -8, -8, -8, I would land on -24 Oh, and I also have this sort of reflection opposite thing happening like we talked about last time. Cool." So, what if I have -3 times 8? Not going to spend a lot of time in the podcast on that, but I would have the same conversation that we just had where we had -7 times 4. Can you think about that as opposite? Can you also use the commutative property? Go ahead. Well, no, go ahead and finish. I want to say one thing.  Okay, so two major things coming out. Can we use the commutative property to think about 8 times -3 or eight -3s? What would that look like on a number line? Where would we mark it? Can we also use sort of the Kim strategy of thinking about just the opposite of 3 times 8? So, I would want to make sure both of those come out. Go ahead. 

 

Kim  22:12

Yeah. So, the two problems you put in a row, I really like because if you're thinking about the fact and then the opposite, then you have 3 times 8. The opposite of it when you have it with -8. 

 

Pam  22:27

3 times -8.

 

Kim  22:27

It's the same thing, right? So, the two problems 3 times -8 and -3 times 8 are equivalent.

 

Pam  22:34

Yes.

 

Kim  22:34

But the negative is with each of the factors.

 

Pam  22:38

Yeah, nice. And then I might write that algebraically as with that -1 outside the 3 times 8. 

 

Kim  22:45

Yeah.

 

Pam  22:45

The quantity of 3 times 8. Cool. And then you'll notice that we just whip through that because we're having the same conversations that we would have had in the previous string. So, I wouldn't necessarily do this in the same day. This is me coming back tomorrow and saying, "Hey, remember what we did yesterday?" That's why I'm kind of getting back into the conversation. Next question in the string. What is -3 times -8? Now, I'm going to pause on this one a little bit...

 

Kim  23:09

Yeah.

 

Pam  23:09

...and wonder if we can think about that. Hey, there's probably one more thing I should have said in the problem before. Now, that I'm here. When we had -3 times 8, and we wrote that -1 outside the 3 times 8, I wish I would have said, so we really are agreeing now that we can say if we've got negatives involved, we can think about the product, and then we can think about the opposite of that product.

 

Kim  23:33

Mmhm. 

 

Pam  23:33

Okay, so now that we have -3 times -8, I'm going to let kids think a little bit, and I'm going to wonder if we can use this idea of opposite.

 

Kim  23:45

Mmhm. 

 

Pam  23:45

Kim, what are you thinking about?

 

Kim  23:48

I'm thinking about 3 times -8 is negative 24.

 

Pam  23:52

Okay.

 

Kim  23:53

There will be 3 of them. 

 

Pam  23:54

Mmhm.

 

Kim  23:54

And then I want the opposite of that product because I have another negative in 3.

 

Pam  24:01

So, it's almost like you could pull out a -1, but only one, right? 

 

Kim  24:05

Yep. 

 

Pam  24:05

And then use like, "Well, so 3 times -8. I could think about that. That's three of those -8s." We've been doing that. It's like 24, the opposite of that. But I still have then another opposite happening.

 

Kim  24:16

Right.

 

Pam  24:17

And we can then reflect that. So, we can actually sort of look at there's our three -8s. Go 3 jumps of 8 from 0. But now I have the opposite of that. Where have we all agreed where that is? Bam, that reflects over the 0, and we end up with positive 24.

 

Kim  24:30

Yeah.

 

Pam  24:31

"Huh? You guys think that's going to work every time?" I would repeat that same idea with 7 times 8. So, do something like... And should I say it now? Later? I'm going to say it now, so I don't forget. You might notice that some of the facts that we're using are some of the most missed facts. Now...

 

Kim  24:47

Yeah.

 

Pam  24:48

...I did 5 times 8 at the beginning because, actually, that's not such a missed fact. Kids often can think about eight 5s. But I did it as 5 times 8 because I wanted them to use maybe what they know about 8 times 5. So, 40 is not too hard. But the rest of these are all sort of facts that are not necessarily ones that kids just have on their fingertips. This last set of problems going to be 7 times 8. That's a way, middle school teachers, of helping your kids get better at their single-digit facts while you're doing integers. It's not just saying, "Well, if you know your facts, then this integer stuff be easy." It's like, "No, let's actually give you experience with these often missed facts...

 

Kim  25:27

Right. 

 

Pam  25:28

...while we're learning the integer stuff. So, that's a purposeful move. 

 

Kim  25:31

Yeah. I want to call out, specifically, the progression that happened. 

 

Pam  25:36

Okay.

 

Kim  25:36

Because you started with a basic fact, 5 times 8, and then you let the... You kind of let the relationships unfold. So, you went with one negative where they could think about it actively. Then you thought about another negative that then they would maybe move in towards thinking about opposite. You talked about some commutative property stuff. And then the last layer was with two negatives, -3 times -8. What you didn't do was say, "Hey, Kim. Negative times a negative is a positive. And we're going to give you a worksheet where you just put negatives wherever you want, all over the place, sometimes one, sometimes two, sometimes none. Ready? Go." 

 

Pam  26:19

Play the rule. One negative, it's negative. Two negatives, a positive. Go.

 

Kim  26:22

Right. What you're doing here is Building the big idea of equivalence when we talk about 3 times -8 or -3 times 8. That's a huge idea for kids, that they're equivalent. We're understanding opposite, which will continue to be huge. And, you know, kids aren't going to regurgitate two negatives make a positive because...

 

Pam  26:42

Especially when they're adding (unclear).

 

Kim  26:45

(unclear).

 

Pam  26:45

Yeah.

 

Kim  26:45

-7, not 10. And that's where they start messing things up, right? They they over apply and apply the wrong time.

 

Pam  26:54

Misapply.

 

Pam and Kim  26:55

Yeah, yeah.

 

Pam  26:56

Exactly. Yeah, I hope this gives everybody an example of what it means to actually develop relationships rather than give a rule and practice. 

 

Kim  27:04

Yeah. Okay, big takeaways. 

 

Pam  27:06

Big takeaways. Whole number multiplication is going to be helpful. The meaning of integers is super helpful, where we're going to talk about distance, reflections, and opposite, and do that with addition subtraction, and then make sure that we've got some jumps on number lines, where we're using those reflections and opposite every time we see the negative. And we're and we're using equivalence and the commutative of property and, and again, opposite to make sense of those reflections. 

 

Kim  27:34

If you're listening to this episode and you're like, "That's great, but my kids don't know their facts," then we'd also like to suggest that that work can be done in the midst of these strings. But we also have a free download for you that you can check out. It is mathisfigureoutable.com/factsps. Like fact Problem String. mathisfigureoutable.com/factsps. And also at the beginning of the episode, Pam talked about major strategies for each of the multiplication operations. I don't think you mentioned other ones. But you can find out what the major strategies are with a free download, mathisfigureoutable.com/big because it's a big ebook. 

 

Pam  28:14

It's a big idea.  Yeah, check that one out. Alright, Kim, awesome. Thank you for tuning in everybody 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!

 

Kim  28:16

Yeah.