Ep 110: Thinking Through Decimal Multiplication

Pam Harris  00:00

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

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, reasoning, using mathematical relationships. We can mentor our students to be mathematicians as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rightly repeating steps actually keep students from being the mathematicians they can be.

Kim Montague  00:41

So in the last couple of episodes, we've been talking about decimal multiplication, and helping make sense of multiplication of decimals in kids' heads. So today, we're going to tackle some gnarly problems, and share what happens in our heads as we solve them.

Pam Harris  00:57

Because if we can build really reasoning with decimal multiplication, then we can get super flexible and have a lot of fun deciding kind of using that anticipatory thinking. What strategy do we want to use here? And then that's where math really becomes more and more Figure-Out-Able and fun and creative all at the same time. Alright, Kim, I got a question for you. 

Kim Montague  01:19

Yep. 

Pam Harris  01:21

Here is what we would call a gnarly problem. And I'm going to give it to you. And I'm going to ask you what comes to mind and just try to be as explicit as you can about all of the things. Even if you decide to do something, but you had considered something else, I want to hear what you had considered and then what you would like discarded and just really put your brain out there. Okay?

Kim Montague  01:42

Okay. Alright. 

Pam Harris  01:44

Alright. Here's the problem. Point seven, seven, or 77 hundredths times 2.4, or two and four tenths. Okay, 0.77 times 2.4.

Kim Montague  01:58

So you know, what's interesting is that in the last episode, I kind of mentioned that a lot of times with decimals I yank them out of, I yanked the digits out of the decimal place value. And I solve it however, want to solve it, and-

Pam Harris  02:13

With whole numbers.

Kim Montague  02:14

Then I consider the fact that I was really dealing with decimals. Umm, I'm kind of thinking about what I want to share here. And so maybe I'll start sharing and you could say, "No, no, I don't want to hear that strategy." But here's what I'm thinking. You know, I actually like percentages as well. So I kind of want to mix that in a little bit if that's okay. 

Pam Harris  02:39

Yeah. 

Kim Montague  02:40

So I want to think about 0.75 times 2.4. And so bear with me for a second because I know that 0.75 is the same thing as 75%. Yeah. And I know that 2.4 can be considered as $2.40. So I want 75% of $2.40. And that is $1.80 or 1.8.

Pam Harris  03:08

How do you know? How do you know it's $1.80? 

Kim Montague  03:10

Because 75%, so each 25% would represent 60 cents. So I need three of those 60 cents to be $1.80.

Pam Harris  03:21

To get three of those quarters is 75%. That's $1.80. Okay, cool. 

Kim Montague  03:25

So so far, I have 0.75 times 2.4 equals 1.8. So I'm actually pretty close. So now I want to think about-

Pam Harris  03:39

What are you thinking? Sorry, because the problem, and this is all audio, so I'm just gonna say it again, because the problem was 0.77 times 2.4. Right. Now you have 0.75 times 2.4. Okay, so sorry to interrupt.

Kim Montague  03:50

So I am 2% or 0.02. away. So I'm actually going to go two times 2.4 is 4.8. Because then I'm going to scale down, scale down. Does that make sense? So two times 2.4 is 4.8. So that means 0.2 times 2.4 is 0.48. And 0.02 times 2.4 is 0.048. And that's helpful because I can use the 0.75 and the 0.02 Each times 2.4. To get 0.77 times 2.4 is 1.848.

Pam Harris  04:39

Because 1.8 Plus that 0.048 is 1.848. Nice. 

Kim Montague  04:48

And the thing is, I don't question it. Like I um,

Pam Harris  04:53

You're not going off on your calculator right now, typing it in to make sure.

Kim Montague  04:55

No. 

Pam Harris  04:56

You're really confident in those values?

Kim Montague  04:57

No, because I used pieces that I knew. Uh huh. Yeah. 

Pam Harris  05:00

Yeah, yeah, pieces that you, and you walk through each, you're like get this. Yep, yep. And this, yep, yep. Yep. Nice. Nice.

Kim Montague  05:08

Nice. Is that what you would have done?

Pam Harris  05:10

No. 

Kim Montague  05:11

Oh? You wanna tell us?

Pam Harris  05:14

Yeah, yeah. So I'm gonna think a little bit about 2.4 times 0.77, I think. I think that's where I'm gonna go. Okay, I think that's where I'm gonna go. So I'm honestly in a ratio table, and I'm gonna think about 0.77. And I'm gonna then double that to get 2 point seven sevens. So I'm sort of thinking about 2 seventy-seven cents. So two times 77 cents would be $1.40, and 14 as $1.54. So was that kind of explicit how I just did that. 

Kim Montague  05:51

Yeah.

Pam Harris  05:51

I thought about 70 cents, and then two sevens. So 2 seventy-seven cents would be $1.54. And I still need, so the problem was 2.4 seventy-seven cents. Right now I've got 2 seventy-seven cents. So I need point four. So I'm gonna go ahead and double that to get 4 seventy-seven cents. So double 1.4? Well, I know double $1.50 is three. So that's going to be $3.08. A $1.50, $1.50, and then four cents and four cents. So four of the seventy-seven cents is $3.08. Now you might be like, "Pam, why do you need four of them?" Well, I don't, but I need point four of them. Because the problem was 2.4 seventy-seven cents. I've already got two of them. I had four of them. Four was $3.08. So point four is going to be a 10th of that. So a 10th of 4 is 0.4 and a 10th of 3.08 is 0.308. So now I've got 2 seventy-seven cents. And I've got 0.4 seventy-seven cents. Those corresponded to 1.54 and 0.308. And so when I add 1.54, or $1.54 to 30 cents with this eight tenths of a cent leftover. $1.54 and 30 cents is $1.84 with that 8 thousandths leftover. So I've got 1.848. Yeah, which is what you had Sorry, no one's surprised. I was actually, I had forgotten you did it. Because I was doing it. 

Kim Montague  07:39

You're thinking. 

Pam Harris  07:40

My own way. Yeah, I was actually thinking. 

Kim Montague  07:42

Yeah. Very cool.

Pam Harris  07:44

So, those are relationships. Like both of us decided to use different relationships that could sort of be helpful. I'm a little curious to wonder, what if I would have given you the problem 0.77 times 0.24? I wonder, would you have done anything different? Would that have struck you any different?

Kim Montague  08:06

Huh? That's a good question. I feel like, yeah, I feel like I might have thought about whole number relationships and then scaled down. Maybe I don't know, I still like the 75 cents. I still like, I feel like I still like 75% of 24.

Pam Harris  08:34

But that's a little different than 75% of 2.4.

Kim Montague  08:37

Yeah, so then I would say, well, I feel like I would say 75% of 24 is 18. And then scale the down to get 0.18.

Pam Harris  08:50

To get the 0.75. So the 0.75 times 24 would be 0.18. But the problem was 0.77 times 0.24. 

Kim Montague  09:00

Yeah. I'm just in my head thinking about would it be any different than what I had done before?

Pam Harris  09:08

I mean, so it's a little interesting to me that when I gave you 0.77 times 2.4, you're like, I'm thinking about 75% of 2.4. And then you found that extra 2% and added it back. It sounds to me like you have the same attack strategy. It's just that you would have been doing it with 24. 

Kim Montague  09:30

Yeah.

Pam Harris  09:31

And then scaled because you were actually supposed to do it with 0.24. That is a little different. I mean, both times you're gonna find 75% and 2%. But one time you stayed in the decimals. You found 75% of 2.4. And then 2% of 2.4. But if I had made it 0.77 times, 0.24, then you're like, "I'm thinking of 75% of 24." And I just knowing that you're gonna then have to scale divided by 100 at the end.

Kim Montague  10:05

You know what? I actually, I just, I was listening to you. I promise, but it was actually thinking.

Pam Harris  10:11

Your doing your own math. You were like so excited. Alright, alright, alright.

Kim Montague  10:14

I actually was like, No, I think I would think about it in whole numbers. And I was like, "Would I?" Because I love 24 for double half. And so I 

Pam Harris  10:22

Oh! 

Kim Montague  10:23

I think I would go, what I wrote on my paper just now with 77 times 24. And then below that I wrote 154 times 12. And then I was like, "Would I want to go there?" And then when I did the next, I did 308 times 6. And I was like, "Oh, yeah, cuz that's 1848. I know that."

Pam Harris  10:43

You can see. Hang on, you went so fast. You can see 308 times 6. How do you, just spell it out?

Kim Montague  10:50

It's just 300 times 6, and 8 times 6 is just 1848. Kind of like left to right-ish. 

Pam Harris  10:58

Okay. 

Kim Montague  10:59

And then because I had landed in whole numbers, I would go, "Okay, but I know that I have way too much of a place value shift. I gotta go one at a time. I definitely do." So then I would shift four place values to get 0.1848.

Pam Harris  11:23

But you're thinking that you'd probably do that one at a time? 

Kim Montague  11:26

Yes. 

Pam Harris  11:27

You're like, okay, so yeah. So then Kim, with that doubling and halving strategy, you could like lift 77 and 24 out of that problem, double and half, get this 1848. And then I could have given you 7.7 times 2.4. And with that, 1848, you could then reason that you would have to shift the decimal twice. Or I could have given you-

Kim Montague  11:47

Well I might have even bumped into that along the way is I was shifting originally, right? Because  when I'm considering shifting those place values, it's changing the problem.

Pam Harris  12:03

What do you mean, bump into what?  What do you mean by that?

Kim Montague  12:07

You said you might have given me 7.7 times 0.24. And because I was moving from 77 times 24, to 0.77 times 0.24. As I shifted the place value one at a time, I would have considered 7.7 times 0.24, 0.77 times... So I would have solved a lot of problems in just the shift itself.

Pam Harris  12:39

That's interesting. 

Kim Montague  12:40

Make sense? 

Pam Harris  12:40

I never thought about that. Yeah it does. Cool. Super cool. Alright, yall, also, that's a pretty gnarly problem, 0.77 times 2.4 that we did a few ways. I wonder, you might be thinking, "Pam, do we really want our students chewing on problems like this? Don't we just, don't we want them to sit and just do a bunch of mindless stuff, so they get the answer and they can move on?" I just took a deep breath. Because if that's what you're thinking, and I can understand why. I really can. I can empathize with what you're thinking about. Would you consider that what you just thought, or what you just said was, "Pam, can't we just have students do a bunch of mindless stuff and get an answer?" Well, if that's really what you want. That's the steps of the algorithm, right? We just want them to do the same thing every time and they just get an answer so they can move on. If that's really your goal, just hand them a calculator. There'll be more correct, there'll be more successful. If your goal is to do the same thing every time without developing this thinking and reasoning and magnitude and sense of size. And like Kim just said, answering all those questions at the same time. If your goal is just to do the same thing every time and get an answer, then why waste all that time that we do to get those steps of the algorithm in the first time? And then how many times are kids gonna say, "Miss, Miss? Is this where I line them up or where I move the decimal?" And I'm gonna go not, like it's not about whether I line them up or move a decimal. It's about thinking and reasoning and using relationships because we can. And when we do, then we build our brains to think mathematically and that is what mathematizing is all about, and we can do it. Y'all, go solve some problems, thinking or reasoning. 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 mathisFigureOutAble.com. Let's keep spreading the word that math is Figure-Out-Able