# Ep 107: All About Decimals

July 05, 2022 Pam Harris Episode 107
Math is Figure-Out-Able with Pam Harris
Ep 107: All About Decimals

We've gotten a ton of questions about decimals. In this episode Pam and Kim discuss three different models and why they are great for questioning and building relationships, but may or may not be good for computation.
Talking Points:

• Base Ten Materials
• Exploding Dots by James Tanton
• Money
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 found a place where math is not about memorizing and 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 in this episode, we want to talk about something that many of you love or hate, or love to hate, and that is decimals.

Pam Harris:

Decimals. So we've done episode before on fractions-

Kim Montague:

Decimals, and percents. Yeah.

Pam Harris:

And let's dive into decimals. Like what about decimals? And specifically, we have gotten some questions and concerns, comments from listeners about three things. And we thought, ooh, these three things have to do with each other. Let's lump them in. Let's get at them in one episode, we can sort of pay attention. And so we'd ask you to consider today when we talked about these three things with decimals, kind of pause and consider the role that these three things play in your teaching of decimals. How do you handle them? What kind of role do they play? So let's kick this off. Go.

Kim Montague:

Yeah. Okay, so we got a message from Kathy, who's one of our Journey members, your membership site, implementation support.

Pam Harris:

Love our Journey members.

Kim Montague:

Yeah, we do. She congratulated us on our 100th episode, where we talked about place value. And one of the things that she mentioned, she sent a message and she said, "Hey, when using base 10 blocks, we begin by asking, 'Today, what's the value of small cube?' kind of like a legend on a map. And then once we knew that, we could identify the value of the rod flat and large cube for our work. And then when I taught sixth grade, the small cube could have represented a 1, 10, 100, 1/10, etc. And it depended on the day." She said, "I tried not attaching the name 10 to the rod, in hopes that the kids would know that it didn't always represent a 10. In general, I wanted the kids to learn that the models could represent different amounts. And it was important to find out what they were. I thought this would support flexible student thinking to build relationships between the base 10 blocks." And I think that's absolutely a fantastic suggestion. Because what she said-

Pam Harris:

Nicely done.

Kim Montague:

-at the very end, right was build relationships between the base 10 blocks. And she asked the question, because in our place value episode, we kind of railed a little bit on don't call them flat, rod, cube, unit, whatever we want to call them, or people are calling them.

Pam Harris:

They will give them these random names. Yeah.

Kim Montague:

And I think we at that time said, if you're going to call them something, at least call them 1, 10, 100. But Kathy is absolutely right to poke back a little bit and say, we need to consider the relationships between those models. And what is the unit of the day? Like, what does one represent? So I think that's something that is a consideration for all teachers. Hey, starting the day saying whether your first, second, your first introducing base 10 materials, if this represents one, then what might the value of this be? What might the value of this be? If holding something else up? What if this represents one, then -

Pam Harris:

And just because we've done probably some high school teachers on Well, when you're saying 'this', so for example, if you held up like the little unit cube, there's one little tiny cube not broken into anything. You could hold that up, and you could say, "If this represents one," then you could hold up the thing that has 10 of those in a line,

Kim Montague:

Yes.

Pam Harris:

You could say, "What would this represent?" But conversely, you could say, "What if this little tiny guy, this one unit cube, what if it represents 10, then what would this sort of rod looking thing?" We wouldn't call it that, right? You would hold it up.

Kim Montague:

Hold it up, yes.

Pam Harris:

"What do 10 of them, 10 of them stuck in a row here in a line, what would that represent? If if one of these was 10? Then what does this represent?" And then you could also do 10 of those 10s. So now that sometimes teachers will call that a flat, often, we sort of said at least call it a 100. But if the unit cube represented 10, then 10 of them would represent 100. So now I've got that sort of 10, that line of 10 cubes that sometimes people call a rod. That thing now represents 100. Well, if I've got 10 of those, that flat thing that's 10 of those rods, what would that represent? Now I'm using words to sort of help people kind of know what, we don't want to call them that. We just want to hold them up, then what would this represent? Because there's a lot of re-unitizing, that happens, and that really can be challenging, right? Oh, you just threw that word out. We might want to define that. Sorry. Keep going.

Kim Montague:

I was just gonna say that the point is the relationship that those models have with each other, it's a really nice manipulative, and you can do some nice questioning around it. But the questions can change. And really, if you're wanting kids to be flexible thinkers, they should change. We should, we shouldn't really be saying, this is a 'blank' for a number of years. And then surprise one day in fourth grade, we go no, now this is a 'blank'. We really should give kids the experience to wrap their heads around, if this, then this. Okay, well, now if this, then what?

Pam Harris:

Absolutely. And this idea of unitizing goes all the way back really young. The very first time we asked kids to think about representing three, like as soon as we say, (claps 3 times) three, and I hope you can hear my clapping, but that three claps represent, or can be represented by one word 'three', or one symbol, the numeral three. We're asking them to unitize taking a multiple of things and representing it with one thing. Then what you're describing is that we've got these materials that we might assume, "Oh, of course, that little one unit cube of course, that's one." And you put 10 of them together, "Of course, that's 10." By the way, so as you get to 10, you're re-unitizing in a different way, because now you've got 10 objects (claps 10 times), think that was 10, represented by two numerals, right?

Kim Montague:

Yeah.

Pam Harris:

We had numbers that were represented by one now they're represented by two. And so now, what we don't, what we're suggesting is that as Cathy Fosnot would say, the mathematics is not embodied in the manipulative. It's not just apparent and obvious, that we want to help students be flexible with those materials. So I say that well?

Kim Montague:

Yeah, you did. And really, we said that this is going to be an episode about decimals, right? And so I think a lot of times that early grades, we don't see kids struggle quite as much with, you know, base 10 materials, because we've only asked them to consider it 1, 10, 100, maybe 1000, right? We've only said this is this is the thing that they are, it's when we hit decimals, that all of a sudden, we've re-unitized. We've shifted what we asked kids to be considering. And it happens to be with decimals. And so we go, "Oh, they don't understand decimals." So it's not that the model is bad. It's probably some of the telling that we do with relation to base 10 materials. And we can just shift that a bit. And I think that's absolutely necessary.

Pam Harris:

So let's shift from telling, "Here's what they are," to asking, "If this cube?" And so if you think of the large cube that has 1000 of those little unit cubes in it. If this cube represents one, then what is this flat of 100 of those unit cubes? Or what is this rod of 10 of those unit cubes? How do those relate?

Kim Montague:

And we might have to supply the name tenth, hundredth, thousandth. But they certainly can tell us how many of them are needed to create one whole.

Pam Harris:

We could certainly figure it out and in the figuring in the deciding what those then how many of them are. And we can supply the name. As kids wrestle with those relationships, that's the power of the base 10 materials.

Kim Montague:

Right.

Pam Harris:

Kim Montague:

So what can we relate kids to?

Pam Harris:

Oh, yeah, what can we do with decimals? So a third thing that people will ask us about is, "Pam, what do you think about using money to help kids understand decimals?" And Kim, what's our answer to that?

Kim Montague:

Well, honestly, it's like the most relatable thing right? But people will say to us, "But kids don't know money." And to which we say, but they need to, right?

Pam Harris:

All the more reason. We must.

Kim Montague:

Yeah, we have to give kids experience with money for so many reasons other than just life. Life in general, they need some experience with money. But it is by far the most important relationship that they can have to decimals is thinking about money. We love having kids think about tenths as dimes, and hundredths as pennies and we can not let go-

Pam Harris:

Thousandths can be parts of pennies. Yeah, absolutely. Parts of pennies. Not a problem at all. Yeah, sorry to interrupt you.

Kim Montague:

No, it's okay. I was just gonna say that it gives us an anchor to recording decimals and recognizing decimals in written form.

Pam Harris:

Absolutely. Sometimes people will push back on me, especially higher math people will say, you know, like high school teachers, maybe even middle school teachers. You know, honestly, it's probably more middle school teachers than high school teachers will say, "No, no, no, it's too limited, money is is too limited, we have more places, more places in our system, than just tenths and hundredths. And so you know, it's too limited, money. Money will will limit kids in understanding decimals." To which I say, "But that doesn't mean that it's a bad starting place. And it doesn't mean that it's not a great comeback to place to help me think about the magnitudes." That as I get and dive and delve into decimals, money can be a very helpful way for me to go, "Wait, let me make sure this is reasonable." Like I'm thinking about, I might be thinking about liters or I might be thinking about parts of something that doesn't have anything to do with money, and I might actually transfer into money in my head to go, "Oh, yeah, okay, that's reasonable." Or whoa, that seems, it shouldn't be, it should be dimes on dollars, or it should be pennies, not dollars, or pennies on dimes. Like oh, like, ah, that sense of magnitude that we can build with money can be helpful in having that sense of reasonableness. So money. Yeah, bam. Yeah, sorry. Go ahead.

Kim Montague:

No, I was gonna say, I'm glad you just said that. Because there's plenty of times where I'm working with decimals. And it has nothing to do with money as the context. But I will step out of it and think about the problem that I've solved, just computation and go, "Would that be a reasonable amount of money? Yeah. Okay. Cool." And I know that I have made sense and thought through kind of the decimal placement well.

Pam Harris:

Yeah, which is really the magnitude, right? You have been looking at the numerals and saying to yourself, "Is it 154? Is it 1.54? Is it one 5.4?" Which is 15.4. I was just trying to move the decimal around. Like what makes sense, is it .154? What makes sense here and money might be a helpful anchor to making sense of those magnitudes? Absolutely.

Kim Montague:

So we tackled three separate things that people have asked us about. And I want to just talk a little bit about where this leads. So we got this email from Willow. And Willow 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." And she says, "Halves are easier to explain, but the concept does not always transfer because 0.3 is not 1/3." Absolutely not the same.

Pam Harris:

Sure enough. Good catch there. Good call. Point three is not 1/3, not equivalent to 1/3. Amen.

Kim Montague:

So stay tuned for next week when we fold in what we talked about this week into decimal multiplication.

Pam Harris:

So if you're excited for some decimal implications, stay tuned because that is next on the podcast docket. Y'all, thank you for tuning in and teaching more and more Real Math. To find out more 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