In today’s MathStratChat, Pam and Kim discuss the MathStratChat problem shared on social media on April 26, 2023.
Note: It’s more fun if you try to solve the problem, share it on social media, comment on others strategies, before you listen to Pam and Kim’s strategies.
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Hey, fellow mathematicians!Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris.Kim:
And I'm Kim Montague.Pam:
And you're listening to a MathStratChat episode. What's MathStratChat? Well, every Wednesday evening, I throw out a math problem on social media,and people from around the world chat about the strategies they use. We love to see everyone's thinking,Kim:
This Wednesday, our math problem was 4 and a 1/3 divided by 8. How would you solve this problem? Go ahead, and pause the podcast, and solve the problem any way you want. It was 4 and a1/3 divided by 8. Solve it, and then come on back to hear how Pam and I solved it.Pam:
Alright, Kim, I want to hear your thinking.Kim:
Okay.So, I saw the divided by 8, and I thought, "I'm going to scale down, Because I don't think it would be a lot of mental effort." So...Pam:
You mean like, you're going to think of it like a fraction,and then find an equivalent fraction where you're scaling down. Okay.Kim:
So, I actually drew a ratio table, and I wrote 4 and 1/3 in the first entry, kind of on the top of my ratio table, and I divided by 8 on the bottom. So,my first first entry on my ratio table says 4 and 1/3 bar line,whatever, 8. And then, in the next entry, I cut those in half.And so, I wrote 2 and a 1/6divided by 4.Pam:
Because half of 4 and 1/3is 2 and 1/6. Okay, cool.Kim:
Yep, yep, yep. So, then, I scaled down again by 2, and I wrote 1 and 1/12 divided by 2.Pam:
And I also kind of at that point, stopped and said, "Okay,wait 1 and 1/12 is the same as13/12." So, kind of right above that, I wrote 13/12. And then, I knew I wanted to scale down one more time. So, I was going to cut that the 13/12 in half. And at first I was like, "Oh, half of 13..."But then, I didn't love what that was going to do, and leave me with a fraction in the numerator. Which is okay, but I didn't really want to, so I thought about 13/12 Cut in half would be 13/24.So, that was my final answer.Pam:
Cool. And that's your final answer, 13/24, because then it's13/24 divided by 1.Kim:
13/24 divided by 1. Whoa!That's why you were finding an equivalent fraction the whole time because you wanted to find a fraction where you ended up with something divided by 1.Kim:
Cool. And I'll just note that when you got to 13/12divided by 2, I actually did think you were going to go 13divided by 2 and get six and a half 1/12's. Which is also equivalent to your 13/24, right?But you didn't want. You were like, "Eh, that's weird." And so, instead you divided the the denominator. But, yeah. Nice equivalence. Cool. Nice partitive approach. So, I've been playing with this quotitive approach for a reason. And I wanted to give a shout out to Julie Dixon. At the stroke of luck on Twitter. At the stroke of luck, Julie Dixon, for giving the problem actually from last week. Not exactly sure where I was. I think I might have been at the Florida State Conference last year when she gave the problem, I believe it was 4 and1/3 divided by 8, and she... Or is that the problem we're doing today?Kim:
That's the problem we're doing today.Pam:
3 and 1/3 divided by 4?Pam:
Yeah, I think it was 3 and a 1/3 divided by 4. And when she gave that problem, it got me really thinking about this idea of when beginning fraction division learners are... If we help them think about this fair sharing context, that it can really help students think about the meaning of fractions. So, in a problem, if we are really, if we're doing this fair sharing thing, then we get to some... We deal with a lot of unit fractions. And the definition of fractions being if we cut something into a certain number of equal pieces, then it is that kind of fraction. So, if we cut something into 4 equal pieces,then each share is 1/4. We get kind of piece. And so, that sort of plays out here, I think,again. So, for the way I was thinking about 4 and 1/3 divided by 8 is if I've got 4 and a 1/3,4 and 1/3 brownies, and I'm sharing it with 8 people. Well,I'm just going to share those 4brownies first. So, hey, there's8 of us. We're sharing 4brownies. Each of us get one-half. And we've dealt with all 4 of the whole brownies.Now, I have 1/3 of a brownie.And here's where this definition of fractions comes in. Like,what does it mean to take 1/3 of something and divide it into 8chunks? What does that mean? And kids have to really think about taking that unit fraction of1/3. And if I cut that, if I share that evenly 8 times, what kind of a piece do I get? And I would get 1/24. And how kids reason about that by maybe looking at the whole. And if I look at just the 1/3 out of that whole, and I cut that 1/3 into 8pieces, well, then I've got to cut each of the third in 8pieces. How many total pieces would we have? We'd have 24. And so, what's one of those pieces called? It's called 1/24. In other words, using division of fractions can help students actually understand fractions.So, it could be a beginning fraction task for students. Even though our standards might say that we're going to do division of fractions much later, we can use division of fractions younger to help students actually understand fractions.And so, I think that's kind of a brilliant thing. Very cool.Yeah. Nice.Kim:
So, you know, last week, I decided... This week, I decided to scale down. And last week, I think it was, when I scaled up.But I was just looking at the problem again, and realizing that had I scaled up this time on a ratio table, I could have gotten from 4 and a 1/3 divided by 8 pretty quickly to...Pam:
To an equivalent. And what do you say scale down or scale up, you really mean finding an equivalent fraction by scaling the numerator and the denominator up or down?Both Pam and Kim:
And so, you're saying if you would have scaled the numerator and denominator by something up instead of down(unclear)?Kim:
If I would have done times3, then 4 and a 1/3 would become13. And 8 would become 24. And very quickly we would have gotten to 13/24.Pam:
And Bam! You're at 13/24.Kim:
So, super strategy. And mathematicians often play with different relationships, and then look at numbers and let those numbers dictate what they do. But like you told me long ago, you might choose a strategy that's super good, but then play with relationships and find one that's even better. Nice mathematical reasoning. Good job.Kim:
Alright, friends. We can't wait to see your math strategy.And we'd love to know if it was like one of ours or something different. Represent your thinking, and take a picture of your work, and then tell the world on social media. While you're there, check out what other people did and comment on their thinking.Pam:
Yeah, and tag me on Twitter at @PWHarris.Or Instagram, PamHarris_math.And on Facebook, look for Math is Figure-Out-Able. And use the hashtag MathStratChat. And make sure you check out the next MathStratChat problem that we'll post every Wednesday around 7p.m. Central Time, and pop back here to hear how we're thinking about the problem. Ya'll, we love having you as part of the Math is Figure-Out-Able movement. Let's keep spreading the word that Math is Figure-Out-Able!