Math is Figure-Out-Able with Pam Harris

Ep 110: Thinking Through Decimal Multiplication

July 26, 2022 Pam Harris Episode 110
Math is Figure-Out-Able with Pam Harris
Ep 110: Thinking Through Decimal Multiplication
Show Notes Transcript

Multiplying decimals can be fun! In this episode Pam and Kim tackle several gnarly problems like 0.77 x 2.4, and have a blast doing it. 
Talking Points: 

  • Building reasoning allows students to be flexible and have fun with decimal multiplication 
  • Kim uses percent and money to solve 0.77 x 2.4 with confidence (no need for a calculator) 
  • Pam represents her thinking on a ratio table for 0.77 x 2.4 
  • Different strategies for 0.77 x 0.24? 
  • What about 7.7 x 0.24? 
  • Why is thinking and reasoning so important?
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 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:

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:

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:

Yep.

Pam Harris:

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:

Okay. Alright.

Pam Harris:

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:

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:

With whole numbers.

Kim Montague:

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:

Yeah.

Kim Montague:

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:

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

Kim Montague:

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

Pam Harris:

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

Kim Montague:

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:

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:

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:

Because 1.8 Plus that 0.048 is 1.848. Nice.

Kim Montague:

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

Pam Harris:

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

Kim Montague:

No.

Pam Harris:

You're really confident in those values?

Kim Montague:

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

Pam Harris:

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:

Nice. Is that what you would have done?

Pam Harris:

No.

Kim Montague:

Oh? You wanna tell us?

Pam Harris:

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. I thought about 70 cents, and then two sevens. So 2

Kim Montague:

Yeah. 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. You're thinking.

Pam Harris:

My own way. Yeah, I was actually thinking.

Kim Montague:

Yeah. Very cool.

Pam Harris:

So, those are relationships. Like both of us

Kim Montague:

Huh? That's a good question. I feel like, yeah, I decided to use different relationships that could sort of be helpful. I'm a little curious to wonder, what if I would have feel like I might have thought about whole number relationships given you the problem 0.77 times 0.24? I wonder, would you have done anything different? Would that have struck you any different? 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:

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

Kim Montague:

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:

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:

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

Pam Harris:

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:

Yeah.

Pam Harris:

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:

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

Pam Harris:

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

Kim Montague:

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:

Oh!

Kim Montague:

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:

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

Kim Montague:

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

Pam Harris:

Okay.

Kim Montague:

And then because I had landed in whole numbers, I

Pam Harris:

But you're thinking that you'd probably do that one would go, "Okay, but I know that I have way too much of a place at a time?

Kim Montague:

Yes. value shift. I gotta go one at a time. I definitely do." So then

Pam Harris:

You're like, okay, so yeah. So then Kim, with that I would shift four place values to get 0.1848. 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:

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:

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

Kim Montague:

You said you might have given me 7.7 times 0.24. mean by that? And because I was moving from 77 times 24, to 0.77 times 0.24. As Make sense?

Pam Harris:

That's interesting. I never thought about that. Yeah it does. Cool. I shifted the place value one at a time, I would have considered 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 7.7 times 0.24, 0.77 times... So I would have solved a lot of thinking, "Pam, do we really want our students chewing on problems like this? Don't we just, don't we want them to sit problems in just the shift itself. 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