# Ep 109: Multiplying By One Tenth

July 19, 2022 Pam Harris Episode 109
Math is Figure-Out-Able with Pam Harris
Ep 109: Multiplying By One Tenth

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.
Talking Points:

• Reflecting on the more sophisticated strategy from Ep 106
• Applying knowledge of whole number multiplication to decimals
• It's more about reasoning than shifting the decimal
• Reasoning in small steps builds understanding vs. just counting the decimals
• Reasonableness comes from keeping kids in reasoning land
• Language to focus on and language to avoid
Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

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:

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:

Kim Montague:

And I think-

Pam Harris:

Kim Montague:

Can I jump in for a second?

Pam Harris:

Yeah.

Kim Montague:

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:

As it might sound.

Kim Montague:

When we're using, yeah, we're using some other methods.

Pam Harris:

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:

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:

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:

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:

Yeah, keep going. Keep going.

Kim Montague:

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:

It's much more thinking about the values. So I'm gonna suggest that my mom is from Switzerland.

Kim Montague:

Yeah. 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." It's one step at a time. One shift at a time.

Pam Harris:

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:

Sure.

Pam Harris:

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:

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:

Okay.

Kim Montague:

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:

Which would be 90 divided by 10.

Kim Montague:

Right.

Pam Harris:

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:

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:

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:

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:

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:

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:

To mean what, to mean what though. You won't say 10 times less to mean a tenth of.

Kim Montague:

Yes.

Pam Harris:

Yeah, divided by 10 or a 10th of. We cannot say 10 times less.

Kim Montague:

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:

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.