Being able to scale comfortably is important for both whole and decimal numbers. And just as there are more sophisticated ways to handle scaling with whole numbers, we can also be more sophisticated scaling decimals! In this episode Pam and Kim describe how we can help students become comfortable with decimal multiplication using their understanding of whole number multiplication and then scaling.
Pam Harris 00:00
Hey fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam.
Kim Montague 00:07
And I'm Kim.
Pam Harris 00:09
And you've found a place where math is not about memorizing or mimicking, waiting to be told or shown what to do. But 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 00:39
So if you haven't been tuning in lately, you've missed that we're talking about decimals and decimal multiplication. We answered some questions about how to help students make sense of decimals, and we're kind of think, things we think are not quite as helpful. In this episode, we are gonna dive into a subtle and important facet of decimal multiplication.
Pam Harris 01:00
So a few episodes ago, in fact, I believe it was Episode 106. So 106, in Episode 106, we talked about a subtle but important shift that we want to encourage students to make in whole number multiplication. And that subtle and important shift that we want students to make in whole number multiplication is that if students are really understanding multiplication, like say something like 40 times 17, kids will often start by finding 10 times 17 and 20 times 17, and then doubling that to get 40 times 17. So they sort of find the 10 times. Oh, that's a very handy, helpful thing. We want to do that. We want to encourage students to do that, at first, and then kind of scale up until they get to the number they were looking for, like 40 times 17. But then we want to make this important, but subtle shift, where we want to nudge kids to think about problems like 40 times 17, rather as four times 17, and then scale times 10. So if that's new to you go back and listen to episode 106. It's a really important, we do a lot of work with it, we get kind of in detail on it. But it's important to realize that we kind of have this strategy of scaling times 10, and then multiplying up, doubling up, multiplying up, adding things together until we get to the number we're looking for. That's kind of strategy one, or the more sophisticated strategy, where we kind of have this anticipatory thinking, we can look ahead, and we realized, since we need 40, we can find four times that thing, and then scale times 10. And we end up then with a bigger number times 10 at the end, versus your net first strategy, we're dealing with bigger numbers all the way along. Now we want both, y'all, we need both happening in students. You might be like, "Pam, I don't want to deal with those big numbers, I'd rather just deal with the small ones," which is kind of what the algorithm does. The algorithm only ever has to deal with digits. But we need kids to deal with those bigger numbers for a while, because that's what helps them develop place value. That's what helps them develop a sense of magnitude and what's actually happening. Don't get discouraged at that point if that takes them a little while to do all that thinking and reasoning because they're building their brains to really think multiplicatively. Then make the shift to get more efficient and sophisticated. Not the algorithm, but to where they are thinking more about sort of anticipatory thinking, if I'm trying to find 40 times 17, I'm gonna find four times 17. That's a smaller number to deal with. And then then, now I can just scale that times 10, bam, and I end up with the magnitude of the number that I needed at the end, but I wasn't dealing with that big number until the very end. So that can be kind of helpful. How does that relate to decimal multiplication? I know Kim's thinking right now. She's like, "Pam, is supposed to be about decimal multiplication." Okay, so what if we had a problem like point four times 17? Well, if you listen to last week's episode, where we really helped you think about and help students think about problems strings that could help them really deal with what it means to think about times point one or times 1/10. We want to do that first. So if we have that going, and we have students thinking about point four times 17, then they might think well, can I think about point one times 17, a 10th of 17. Now again, we have to do some work there so that this makes sense to students and have a feel for it. But we're sort of dividing by 10. So if I think about 1 seventeen, then to get to one divided by 10. So divided 17 by 10. And we're dealing with a 10th of 17 being 1.4, or help me 1.7 Sorry, so a 10th of 17 is 1.7 Now again, that beginning, the kid that's kind of getting a feel for this might then say to themselves, well, if I've got point one times 17, but I need point four times 17, then maybe I'll double that to get point two times 17. So that would be 3.4. And then maybe I'll add those together to get point three times 17. Or maybe I'll double the point two times 17, to get the point four times 17. So I'm gonna double 3.4, that's 6.8. And now I've been able to think about point four times 17, kind of in chunks that I understand. I understand a 10th of a number. And now I can just sort of scale that up until I kind of get to where I'm going, scale out that 10th to four tenths. And that will be a fine way for students to start solving some multiplication with decimal problems. But then just like we made the shift with whole numbers, we want to make the similar shift with decimal numbers. So when I see point four times 17, if I can sort of step outside the problem and think, using anticipatory thinking, kind of plan my attack, that I might recognize that I could do four times 17 and get that 68 and then scale times 1/10. So now I've got four times 17 is 68. And then times 1/10, is now that 6.8. So to say that one other way, I can either think about point four times 17. And I barely write it, I just wrote it down. I wrote 0.4 times 17. I can think about that as point one times 17 times four. And I might do that times four by doubling and then doubling again. I might just know, well, probably not in this case. But yeah, so I might like add up the point ones to get to the to get to the point for like, think of it that way. And that's kind of the beginning the first way. But then we want to help students make the transition to be thinking about it as four times 17. And then scale times point one or a 10th. So do the four times 17 first, and then scale it down. Now we actually deal with the bigger numbers first, and then the smaller numbers.
Kim Montague 07:29
And I think-
Pam Harris 07:29
What are you thinking about?
Kim Montague 07:29
Can I jump in for a second?
Pam Harris 07:30
Kim Montague 07:32
So if the kids are used to this idea of scaling up times 10. Thinking about the four times 17, and then scaling up to get 40 times 17. And they make sense of decimal place value and decimal relationships, then scaling down by 10, or scaling a 10th, isn't as far reaching as maybe it feels.
Pam Harris 08:00
As it might sound.
Kim Montague 08:01
When we're using, yeah, we're using some other methods.
Pam Harris 08:04
And one of the way I might put that is, if we've got students that are very, that we've done a lot of whole number work, so that they are doing that four times 17 and then multiplying by 10. And they're used to watching those numbers shift in place value where that zero comes in, then now to divide by 10 and having those numbers shift the other way in place value is not a far reach. We want to make sense of it. But it's not quite maybe as crazy as it might sound. You're like, "Pam, because just move the decimal." Or not, or they're actually thinking about the values.
Kim Montague 08:38
So that actually sounds like what I do when I'm thinking about a problem. Not always depending on the numbers of the problem. But if you were to give me a say a decimal times a decimal, I'm very often going to, like think outside the decimals, pull it back into what strategy do I want to use for whole numbers? And then I think about, okay, I have some version of the answer, some place value shift of the answer. And then I'll consider scaling down by 10, kind of reevaluating. Does that make sense? Scaling down by 10. Seeing if that makes sense. And so, you know, I know that it sounds, or it could sound to some people that, "Oh, it's just exactly what we tell kids to do in the algorithm. Do the steps and then move it, move, it move it. But it's not like that in your head.
Pam Harris 09:32
We could make a rhyme of that. Do the steps and move it, move it, move. No, no, don't do that. Don't do that. Sorry, interrupt.
Kim Montague 09:40
But I do stop along the way. Like with each shift, I stop and I say, "Is where I'm at in this moment, does it make sense for the numbers that then are in the problem." Then I would say again?
Pam Harris 09:55
Yeah, keep going. Keep going.
Kim Montague 09:56
I was gonna say for each shift, I stop, I reevaluate. I consider if it makes sense for the problem. And so it's not this mindless, like move over three places because the decimal has moved over three places in the problem or whatever.
Pam Harris 10:12
It's much more thinking about the values.
Kim Montague 10:15
Pam Harris 10:16
So I'm gonna suggest that my mom is from Switzerland. I say that just to say she grew up with the metric system. And when I can get her to talk to me about what's going on in her head as she does math, I began to realize that she does what you do in that when she scales, so just like in whole numbers if you were multiplying, like we were talking about before, if I was going to do something like 40 times 17, she would think about four times 17. And then she would think about, okay, so four times 17 is this number, then 40 times 17, I'm going to multiply by 10. But if she was doing 400, so hang on 680, I had to think about that for a second. So if four times 17 is 68, then 40 times 17, would be 680. But if she was doing 400 times 17. Kim, this is what I find interesting is she would not then say, "Oh, two, like, I know four times 17 is 68. But I'm doing 400 times 17. So move it move it." She wouldn't say, "Oh, then I'm gonna move the decimal point twice, because from four to 400 is two decimal places." She would literally say, "Four to 40. So that's 68 to 680. And then 440 to 400, that's 680 to 6800."
Kim Montague 11:42
It's one step at a time. One shift at a time.
Pam Harris 11:45
One shift at a time, one times 10, one power of 10 at a time. And I said, "What?" And she's like, "You know, like, if I'm going from centimeters to kilometers, I don't, I go from centimeters to meters and meters to kilometers." And I was like, "Oh, that makes sense." Like she had made sense of the relationships. And so then she was able to bring that into multiplication. So Kim, I'm gonna give you an example. And I want you to if you don't mind, we share with everybody how you're thinking about a problem.
Kim Montague 12:15
Pam Harris 12:15
Like point six times 15. If I said 0.6 times 15. Can you just tell us what's going on in your head?
Kim Montague 12:23
Yeah, sure. So I would think about six times 15. And there are a variety of ways that I could do that. I actually happen to know six times 15 is 90.
Pam Harris 12:34
Kim Montague 12:35
So because six times 15 is 90, then point six times 15 is going to be a 10th of 90. So it's one shift, one divided by 10. So instead of 90, it's going to be nine.
Pam Harris 12:54
Which would be 90 divided by 10.
Kim Montague 12:56
Pam Harris 12:57
And so that's nine and that's how you would, yeah, nice, nice. Kim, what if it had been point zero six times 15?
Kim Montague 13:06
You know, it's funny, because I think I would probably take the in between step just like your mom talked about, I would have said I know six times 15 is 90. So point six would be nine. So point zero six would be point nine.
Pam Harris 13:22
Bam. And that thinking that sort of stopping and making sense like does that? Yep. And then doing the next place value shift. Yep. That's what we want to encourage in students.
Kim Montague 13:35
Yeah, I think all too often we go just count how many decimals there are? And then just go over to your answer and count how many decimals there are and kids have no concept of is it reasonable at all.
Pam Harris 13:48
Yeah, magnitude and size and what it means to multiply by a 10th. And then what it means to use that to think and reason. So you all, it starts with whole numbers, it starts with whole numbers and scaling up by multiplying by 10. And now that we're dealing with decimals, we're dividing by 10. We can divide by 10 and make sense of the magnitudes in the problem. What's actually happening with the sense of size and scale. It's not about just do the whole number algorithm and then just move the decimal and 'butt cheek'. No, no, no, no, it is really reasoning. When we complain, "Oh my gosh, that answer wasn't even reasonable." I would suggest it's because we took kids out of reasoning land to begin with. So we can help students make that same shift that we want to make it whole number multiplication. And we can do that with decimal multiplication, as well. Kim, I am remembering that I was listening to us very carefully in this podcast that we always said things like a 10th of or divided by 10.
Kim Montague 14:50
I was about to say something like that. Yeah. I don't know if I've told you this, but I'm, I have not always been as careful with my language. Um, with in regard to a 10th of, or you know, I have to really caution myself. But one thing I will never say is 10 times less. And I have to tell you, I was watching, I don't remember what show it was I was watching.
Pam Harris 15:15
To mean what, to mean what though. You won't say 10 times less to mean a tenth of.
Kim Montague 15:20
Pam Harris 15:20
Yeah, divided by 10 or a 10th of. We cannot say 10 times less.
Kim Montague 15:25
Pam, I was watching a show the other day, and there was a commercial. And it big, like bold letters across the screen, it was like 10 times less and I so remember, "No don't say that?" I don't know what it was. Which would be the worst, like, what's the better deal? Like you should go up to him and say, "I want your deal." Or is that, yeah. Anyway, so the language can be important that we're looking at a 10th of, or we're taking that number divided by 10. We're not subtracting off when we're thinking multiplicatively about relationships. Yes? Is that a way to say that? Yeah? Yeah, so listen, it's important for kids to make sense of what's happening first, and then we're going to get them more sophisticated, right? And then they're gonna be super flexible, and they can solve the problem in a way that makes the most sense to them. So next week, we're going to tackle some gnarly problems and share some of the flexibility that's happening in our heads.
Pam Harris 16:26
So stay tuned. It's going to be fantastic. And thank you for tuning in and teaching more and more Real Math. To find out about the Math is Figure-Out-Able movement. Visit mathisFigure-Out-Able.com. Let's keep spreading the word that math is Figure-Out-Able.