Pam Harris  00:01

Hey fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam.

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim Montague  00:41

In this episode, we plan to continue the conversation around decimals, and begin to answer some questions around decimal multiplication. So we got an email from a listener, Willow. And she said, "I was hoping that you ladies could do a podcast on multiplying decimal numbers less than one. And a way to model or help students figure out what's happening when you multiply 0.3 times 0.7. Halves are easier to explain, but the concept does not always transfer because 0.3 is not 1/3."

Pam Harris  01:13

So it's always funny to me when you say 'zero point three' instead of just 'point three', but it's okay. It's because you're reading it off of the email. So we don't, like we don't usually say 'zero point three'. We usually just say 'point three' or 'three tenths'.

Kim Montague  01:25

Do you not?

Pam Harris  01:26

I don't.

Kim Montague  01:27

I sometimes do. Yes. It's so funny that you say that. I do say 'zero point three'. 

Pam Harris  01:32

I don't think I've ever in my life said 'zero point three' unless I was writing on the board. Okay, maybe not important. I don't know. Alright. However, it can be important that we say 'three tenths' sometimes not just 'point three', but I will often have middle school teachers say to me, "Don't say 'point three', you have to say 'three tenths'." I will say, "Okay, common language is 'point three', and kids need to hear both." So I think they need both, not only one, but they need to hear both.

Kim Montague  01:57

Have you ever talked to a fourth grade teacher? They're cringing right now. They're absolutely losing their minds about me saying 'point' anything.

Pam Harris  02:04

Because they want you to say 'three tenths', not 'point three'? 

Kim Montague  02:07

Never say 'point'. Uh huh. 

Pam Harris  02:09

Well, I mean, same thing with a sixth grade teacher. So they're in the same, they're in the same boat. And we're going to disagree with both of them, we need to say both, kids need to hear both. Now in today's podcast, we are going to say 'point three' and not 'three tenths' because for communication's sake, on the podcast, we want you to sort of hear what students are seeing when because we're in this sort of verbal kind of a place. So we might use both a little bit, but for the most part, we're gonna say 'point', in an attempt to make sure that we're all talking about the same representation. So that's why today you're gonna hear us, though, I don't know that you're ever going to hear me say 'zero point three'. But it's all good.

Kim Montague  02:52

I'll have to think about that. Okay.

Pam Harris  02:54

Alright. So today's episode, we are going to answer Willow's question, kind of in a couple of ways. Or maybe I should say, two parts. So we want to suggest that it is really important that students actually understand and feel the relationships involved in a question, like point three times point seven first, before we get very sophisticated and efficient at solving that problem. Not because it needs to be concrete, and then representational and then abstract. It's not about that. It's much more about we actually have to feel the relationships that are happening. And then we can get more sophisticated, which doesn't necessarily mean just symbols, it means then actually thinking using more sophisticated relationships. What do we mean by that? Well, let's get at what first it means to multiply by 1/10 or point one. What does it mean to say something times point one? To do that, we're gonna do my favorite instructional routine ever, a Problem String. Kim, I'm asking the questions today. You get to answer. Here we go. 

Kim Montague  03:59

Can't wait. 

Pam Harris  04:00

First problem of the string. What is 8 times 0.1, 8 times 0.1.

Kim Montague  04:08

Okay, so that is, I know, 0.1 is like a dime. So it's 8 dimes. So 80 cents, 0.8.

Pam Harris  04:15

You're saying it's 80 cents, okay, 0.8. Cool. I might represent that on the board, I might write the problem 8 times 0.1 equals and she just said 80 cents, so I might write 0.8, and now on the board, I'm going to write '0.80'. And the zero on the board that precedes the decimal is all about readability. I want that decimal to pop on the board. And so I'm going to write that leading zero. And then I might also write '0.8'. And I'm also going to represent that as the dimensions of an array. And you didn't say this Kim, but I might ask students, "Can anybody think about that as the area of a rectangle that is 8 by point 0.1? What would that look like?" And I might ask students, "What would a rectangle look like?" And then I might represent that as a long 8 by a data tiny 0.1, right? I mean, like 8, and I'm probably not going to even do that on the board, because it would like take up too much space to be exactly proportional. But I'm going to have a long 8 by a little 0.1, so we're gonna have a really tall, skinny rectangle, I'm gonna put those dimensions up there. And the area inside, I'm going to write 0.8 as the area of an 8 by 0.1. Cool. Next problem. What is 8 times 0.3? Or 8 times three tenths?

Kim Montague  05:39

Okay, so that, Yeah, that's going to be three times as much as what you just asked for. So it's going to be two, so 8 thirty cents is $2.40.

Pam Harris  05:53

So I kind of heard a couple different thinking. You could, I heard the three times as much. So you could think about what three times point eight or three times 80 cents is. And you're saying that's $2.40. So I just wrote down, I already had written down eight times zero point three equals and then I just wrote down 2.40, because you said $2.40. So I'm going to write it like money, $2.40. I could think about it as 2.4. I'm going to also then ask that three times as much, three times I might write, I might scale that rectangle. Is anybody thinking about the rectangle? If an eight by point one is this tall, skinny rectangle, then what would an eight by point three look like? And sure enough, it'd be, that dimension would be three times the length of the eight by point one. Now that point three is three times, is three of those point ones, right? So I might put that up on the board. I might be like, okay, eight by point three. And you're telling me the area of all of that is 2.4, or $2.40. I also might have started this Problem String by saying, we're trying to deal with eights here. And now we know we're dealing with eights, because we have two problems with eights. And so I might also have a ratio table that says, I'm dealing with eight. So I've got one, eight. So I've just seen my ratio table, I've got a one and I've got an eight. And then I might say that first problem asked for a 10th of eight. And so then in the next slots, I'm doing a horizontal ratio table, I've got one and then eight on the bottom. And then next to that, I've got point one, or 1/10. And I might say, "So how do you get from one to 1/10? You guys are telling me that's like divided by 10. So can I also think about that eight divided by 10, and eight divided by 10. I can think of it a couple different ways. I can think of that as the fraction eight tenths. But I can also think about that as point eight, sort of divided by 10. And it's kind of a place value shift." But then that third or second problem, eight times point three, you said was three times. So I can in that ratio table, I could scale that point one times three. So I've literally just drawn an arrow and I've said from that point one times three, then I'm going to have that point eight times three. And that's the 2.4. So I kind of have a ratio table now with three entries 1 to 8, 0.1 to 0.8, and 0.3 to 2.4. What are you laughing about?

Kim Montague  08:21

I'm laughing because it's exactly what I wrote. 

Pam Harris  08:24

Nice. 

Kim Montague  08:24

You just describe like, what is on-

Pam Harris  08:26

Your paper? Okay, very cool. Next problem, what is 8 times 0.05, 0.05. Eight times five hundredths. Or, as Kim would say zero point zero five.

Kim Montague  08:43

Okay, so I already have eight times point one. And so it's going to be half as much. And so I divided by two or times by half. And so, and I, in my head, I thought, "Oh, that's like eight nickels as well." So I was confirming with money, because I considered dimes at the very beginning. And so half as much would be a nickel. Half of 80 cents is 40 cents.

Pam Harris  09:06

Going back to the 80 cents. So if that nickel is half of a dime, then half of 80 cents is 40 cents. But you also then said you're checking that with knowing that you've got eight nickels. Eight nickels is also 40 cents. Cool. And then I might on the board as a teacher take that eight by 1/10 of that eight by point one rectangle, and I might cut it in half. So now I have an eight by point five. So I had to be careful knowing I was going there. That what I drew on the board at that eight by point one, it was pretty tall and pretty skinny. But it wasn't so skinny that I couldn't then cut it in half and get an eight by point five. Oh then therefore half of what we had as point eight now is point four. Cool. Next problem. What is eight times point one five. Or as Kim would say eight times zero point one five.

Kim Montague  10:04

You're gonna catch me every time on that. You know, okay, so I know, I have the answer, but I'm trying to decide which way I want to share with you, because I've got a couple of different ways to get there. 

Pam Harris  10:14

So share slowly share the first one. 

Kim Montague  10:16

Okay. So because it happens to be the previous problem that you asked, the nickel, I noticed that it's going to be three times as much. So 15 cents is three times as much as five cents. So three times as much as the 40 cents will be $1.20. So it's right next to it on my ratio table, you had just ask eight nickels. And now you're asking 8 fifteen cents, and so it's going to be three times as much.

Pam Harris  10:47

And I literally on my ratio table drew the arrow and said times three of the five nickels is 15 cents. So times three, the 40 cents is $1.20. Cool. Cool. 

Kim Montague  10:56

But I look back a little bit further. 

Pam Harris  10:58

Yeah. 

Kim Montague  10:59

Oh, you want to write that on an array? 

Pam Harris  11:01

No, go ahead. No, no. 

Kim Montague  11:01

So I looked back a little bit further. And I noticed that you had also- 

Pam Harris  11:04

Yes, I am going to, I'm sorry. Okay, sorry. I'm sorry, I can't stand it. Because when you say that strategy, I want to go find the point zero five, the eight by point zero five rectangle and I want to redraw it on the board. And then I want to scale it times three. And say okay, so once we had this, and you're saying I can say and I just want to sort of show that happening times three. Okay, now, sorry. Now do your other one.

Kim Montague  11:28

When I looked back to previous problems, I remembered that you had asked me about eight times point three. And it's going to be, the product is going to be half as much as eight by 0.3. Which was 240. And so now 1.2.

Pam Harris  11:43

Because fifteen cents is half, Sorry, I'm interrupting. 

Kim Montague  11:48

Fifteen cents is half of 30 cents. 

Pam Harris  11:50

Cool. And so the corresponding, the 30 cents corresponded to $2.40. So you're saying you could do half of $2.40 to get the $1.20? Yep. That's totally cool. Should I admit that when I prepared this today, I didn't see that relationship? I had not thought about that one, Kim. Good job. Good job. Nice. Nice. That surprised me a little bit. I thought you were gonna do another one. You got one more? 

Kim Montague  12:13

Yeah, there is one more. And so-

Pam Harris  12:15

At least one more.

Kim Montague  12:16

Adding the dime and the nickel together. At least one more, yeah. So adding the dime and nickel together to get to 15 cents. So adding the 80 cents and 40 cents to get $1.20. 

Pam Harris  12:25

Bam.

Kim Montague  12:26

I'm glad you said at least one more. 

Pam Harris  12:28

Because we never want to limit our students. 

Kim Montague  12:31

Because it's one more than I had seen originally. Yeah, there's, I'm sure there's more.

Pam Harris  12:34

Yeah, there you go. We never want to limit our students by what we can think of. So when you added the 10th, and the, or the dime and the nickel together to get the 15 cents, I would have taken those arrays, those corresponding area models or rectangles and add those together. And then clearly, we could sort of show that partial product. Nice. Last problem with the string. What is eight times 0.35. Eight times 0.35.

Kim Montague  13:01

So my first thought is the 30 cents. The eight times point three was 2.40. And then right next to it was the eight times 0.05. 

Pam Harris  13:15

How convenient. 

Kim Montague  13:16

Right. And so that's 2.40, $2.40 and 40 cents, which is $2.80.

Pam Harris  13:23

And you've literally taken chunks, you know, chunks that you can think about, chunks you can reason about to find the answer to a problem like 8 times 0.35 decimal multiplication problem. Nice. Now, you might be thinking, "Okay, but Pam, only one of those numbers was a decimal." Sure, sure. But we're going to start somewhere. We're gonna start somewhere and we're going to do some Partial Products. And we're gonna do some Smart Partial Products, where we're asking students to actually grapple with these relationships, and comparing to money, and using area to bring those relationships together to help students really think and reason about what's happening. To be clear, we're not going to stop here. We're not going to say, "Okay, so every problem you get, we want you to walk back to eight times one tenth, you know, and then work your way up with all these chunks." No, no, no, we're gonna get more sophisticated. In fact, we're gonna get really sophisticated and efficient fairly quickly, especially if students have already built whole number strategies, because the strategies and multiplication strategies for whole numbers are going to hugely impact the way that we multiply decimals. But we're going to back up at this point, even if our students are really good at multiplying whole numbers. We want to get this sense and this feel for what's happening when you're multiplying decimals. So this Problem String that we just did is a great place to start. I don't know Kim, if you want to say anything before we do the next string. 

Kim Montague  14:51

Oh, no. 

Pam Harris  14:52

Ok, just checking, making sure there wasn't anything bubbling out of you before we head on to the next one. So then how do we get a little bit more efficient? Well, then we wanna, you know, get a little bit crankier where we've got a little crankier numbers. It's not just eight. It's not just, you know, a little bit of decimals, but we might do a Problem String like, so, Kim, here we go, what is 2.3, $2.30, 2.3 or two and 3 tenths times two?

Kim Montague  15:18

Okay. Let's prepared for more complicated.

Pam Harris  15:21

You're like waiting. Not yet.

Kim Montague  15:22

Okay, $2.30 doubled, I like doubles, is $4.60.

Pam Harris  15:28

Cool. And so I'm probably gonna represent this in a ratio table. And so I might say one, 2.3 is, we're first starting with the ratio of one to 2.3. And then we got two of them. And you said that was 4.6. So now I have a ratio table of 1 to 2.3 and 2 to 4.6. I might do that. I also, in a separate, I don't think I would do these two together for this Problem String. I could, and maybe should be. So this depends. Which model am I going to use to model this string? And I have this conversation right now in my head. It depends. If I have students who are well versed in ratio tables, I might now just be using ratio tables. 

Kim Montague  16:05

Yeah.

Pam Harris  16:05

If I have students who are really like, just beginning to think about decimal relationships, I actually would probably model this on a rectangle. I would probably use an area model here first. So I would have a 2.3 on one dimension and a two on the other. And this is going to look very squarish because 2.3 and two are very close together. So almost square, not a square, perfect square, but almost square, a little bit taller than it is wide. So 2.3 by 2 and then you told me the area that was 4.6. Next problem, what is 2.3 times 0.2, or the area of a rectangle that is 2.3 by 0.2 or 2 tenths?

Kim Montague  16:47

Yeah, it's funny that you say that because I wasn't sure where you were going. But I know that 0.2 is a 10th of two. So I just scaled down my 4.6 by 10. By a tenth. I multiply it by a tenth. And it's 0.46. And when you were talking about your area model just right now, I pictured the area model, like scaling down, like it got skinny all of a sudden.

Pam Harris  17:18

Nice, nice. And so I might model that as a 2.3 by this really skinny 0.2. And the area that you just said was 0.6. Look at that. I just said it like you would, point four six, cool. Next problem. What is 2.3 times 4?

Kim Montague  17:40

Okay, it's going to be double when it was 2.3 times two. So 2.3 times two is 4.6, doubled, again, is 9.2.

Pam Harris  17:51

And you double that 4.6 really fast, we might have some conversations about kids doubling the 4.6, you know, we're gonna give them some experience thinking about doubling. But for today's podcast, we're going to run with it. So a 2.3 by four gives you an area of 9.2 sort of double what you had from the two. Nice. Last question. What is the area of a rectangle that is 2.3 by 4.2?

Kim Montague  18:15

Okay, you gave me the previous pieces of 2.3 times 4, and 2.3 times 0.2, so I'm going to add those together, which is 9.2 and 0.46 and so that is 9.66.

Pam Harris  18:34

And I would probably draw the rectangle, that would be 2.3. And I would maybe even add on to the one that we had that was 2.3 times four. And I would say, "How much bigger is this? It's just, oh, it's this little rectangle we had up here. It's this 2.3 by 0.2." (clicking noise) I'd draw like a dotted line. And I would like move it down there. And I would say, "Oh, there it is. It needs to be right here." And it would sort of tack on that little rectangle on the end and go, "Hmm, so I could think about a rectangle that is, think about the area of a rectangle that is 2.3 by 4.2 as a rectangle that is 2.3 by 4 with that little bit of a 2.3 by 0.2 tacked on over there." And notice how that's Smart Partial Products. We're not breaking it up into all four chunks, though we could and I would be okay with you doing some problems like that. But fairly quickly, y'all I want you to move to helping kids think about Smart Partial Products. What are some of the things that they know that they can use to help them that they don't have to just cut it all into all four sections? Also notice that I made a deal about how we are making sure the rectangles are in proportion. Here's what I absolutely am not suggesting is that for a problem like 2.3 times 4.2, that I'm drawing a square and I'm cutting that square to four equal pieces. And then I'm writing down a two on the dimension of whatever and then a point three and then across the top of four and a point two. And I'm pretending that like that that has anything to do with area. No, no, no. Like, let's let the magnitudes of those rectangles represent what they should. Let's have them be proportional. Y'all, if you're doing problems like this, kids are learning proportional reasoning. And one of the ways we're going to do that is to actually draw the area models proportional because then they actually can. There you go. Alright. So, Kim, we got to finish this podcast by answering Willow's question. So well, this question was, what is 0.3 times 0.7? 0.3 times 0.7? Well, Willow, I would expect that if students had done the work that we just did with Kim today on the podcast, we just did with everybody on the podcast, that students might look at that and say, "Hmm, point three times point seven, what do I know? Well, I bet I could find a 10th of seven, of point seven. I could find point one times point seven." So if I've got 70 cents, can I find a 10th of that? Point one of that 70 cents. And let's see, a 10th of 70 cents, is that seven cents? So that's 0.07. So if point one times point seven is seven cents, then I think a kid could think about three times that to get from point one to point three, they could multiply by three, and what's three times seven cents, 0.21. Now, you might say, "Pam, I don't know if kids will think about times three." They might be like, well, I got point one times point seven. But now I need, they might go point two, so they might double it and then add back. But still, either way, they've got maybe, what have they've got, seven cents, and they might get 14 cents, and then add the next seven cents to get 21 cents or 0.21. But I think that's how a beginning multiplier, who's actually understanding what's happening, to solve point three times point seven. Find point one first, and then scale it up to get the point three times point seven. What do you think, Willow? We are interested to know. So let us know. Let us know everybody what you kind of think about how we are multiplying decimals. Now again, this is how to start. So kids are actually understanding what's happening. You might be thinking, "Pam, we don't have time for that. Like, all I have time for is, you guys just use the algorithm and then butt cheek at the end." Because you might be used to only spending very little time on multiplying decimals. Because if kids are already multiplying with the algorithm, then you just kind of use the same algorithm, but then move the decimal at the end. We are suggesting, if you can get kids actually thinking and reasoning, using anything, if you get them actually thinking and reasoning, you will have to redo and undo less and that is where we're going to save you time. We're going to actually understand it here. Then we're going to get more sophisticated at it and we are going to take some time to do that. But then we can undo and redo less. How? How are we going to get more sophisticated? Well stay tuned. In the next episode we are going to get at that. We're gonna get at how we can get more sophisticated and efficient while maintaining kids actually understanding that. So y'all thanks for tuning in 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.