Have we convinced you to love ratio tables as much as we do yet? In this episode Pam and Kim tackle another model building Problem String to show what a powerful model a ratio table can be.
Talking Points:
Have we convinced you to love ratio tables as much as we do yet? In this episode Pam and Kim tackle another model building Problem String to show what a powerful model a ratio table can be.
Talking Points:
Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.
Kim:And I'm Kim.
Pam:And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But, ya'll, 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 keeps students from being the mathematicians they can be.
Kim:So, for the last two weeks, we've been talking about ratio tables and how powerful of a model they are. You may have noticed that both weeks we've done model building strings where we focus on a variety of strategies, kind of going in and out of the numbers. Sometimes we go a little bit over, and then back up. Sometimes we do 5 is half of 10. So, we're going to give another example today of a model building ratio table string. Also, don't forget to grab...
Pam:Yeah, because that's what the series is about, right? Is the ratio table?
Kim:Yeah.
Pam:Later, we'll talk more about strategy building, but for today, let's build that model some more. Sorry.
Kim:Yeah. That's okay. As we work through a Problem String, we'd love for you to grab a pencil and paper and practice your representation on a ratio table, and then post it to Pam's social media.
Pam:Yeah, then we can all share what it's looking like. Especially if you do it with students. We really want to see what your board looks like, what students are thinking. Love to see how that's going. We'll all just keep mentoring each other as we do more and more Real Math. Cool. So, Kim.
Kim:Yeah?
Pam:I'm going to give you some problems today.
Kim:Okay.
Pam:I just was at a friend's house. And we had spaghetti. And I think there's like the perfect ratio of pasta to sauce. Yes?
Kim:Mmhmm.
Pam:Like, I can't be drinking the sauce. That's too much sauce.
Kim:Yeah.
Pam:But it also... There's got to be enough sauce. Like, you have to. There's, like, this perfect. Well, I think that it's a perfect ratio. I have a big family. You know that. So, I make a lot of pasta, a lot of sauce, feed everybody. So, I think the perfect ratio is 8 cups of pasta to 5 cups of sauce. Mmm, it's making me hungry right now.
Kim:Okay, 8 cups of pasta...
Pam:Hey, by the way, do you prefer, like a red? Like, tomato sauce? Or do you prefer, like an Alfredo, white sauce?
Kim:Well, the white sauce is not spaghetti. That's fettuccine. So, if you're saying spaghetti, it's red sauce.
Pam:Well, just pasta. Can I just do pasta? Just pasta. Any kind of pasta.
Kim:Ah. They're both good.
Pam:Somebody said "past-ah" the other day. It's not "past-ah", right? It's pasta.
Kim:Pasta.
Pam:Pasta.
Kim:Alright.
Pam:Okay. Alright, so 8 to 5. I don't know.
Kim:Eight cups to 5 cups. Okay, I'm with you.
Pam:Eight cups to 5 cups. Okay. Now, let's say that your family is coming over. And I don't know, maybe we have to have our neighbors over too. Lots of people are coming over, and so I'm thinking I'm going to make 10 cups of sauce. If I make 10 cups of sauce, how many cups of pasta, so that I keep the ratio? If I keep it really tasty, how many cups of pasta would we need?
Kim:Yeah, I'm going to need 16. You're going to need 16 cups of pasta.
Pam:Because?
Kim:Well, you're doubling the amount of sauce, and so we need to double the amount of pasta.
Pam:Totally makes sense. But tonight, I'm just feeding my husband and myself. We're just barely empty nesters, and so I'm only going to make 1 cup of sauce. So, how many cups of pasta would be that yummy ratio?
Kim:Hang on.
Pam:You still there, Kim?
Kim:Yeah, I'm still here. I'm recording on my ratio table, so that it can be.
Pam:Oh yeah, well done. Oh, yeah because we're going to post it, so everybody can see what it looks like.
Kim:So, you only need... You said you had 1 cup of sauce?
Pam:Yeah, just 1 cup of sauce. I got to make pasta. How much pasta should I make?
Kim:Eight-fifths of a cup of pasta.
Pam:What? What does that even mean?
Kim:One and 3/5 cups of pasta.
Pam:One and 3/5 cups. Like Kim, if I go into my cups right now, I don't think I'm going to find three-fifths of a cup. Is there any other relationship? I mean, I like it. I like it. By the way, how did you get eight-fifths? Can you say that first?
Kim:So, I went from 5 cups of sauce to 1 cup of sauce by dividing by 5.
Pam:Okay.
Kim:And then, I went from 8 cups of pasta, also divided by 5, so the ratio would be the same.
Pam:So, that's how you got eight-fifths.
Kim:Yeah.
Pam:Eight divided by 5.
Kim:If you prefer, I could tell you it's 1.6 cups of pasta. I'm not sure if that's any better.
Pam:How did you get that?
Kim:So, if I went from 10 cups, the previous 10 cups of sauce to get to 1 cup. So, then. I'm sorry, sauce. 10 cups of sauce to 1 cup of sauce.
Pam:And how are you doing that? 10 to 1 is your?
Kim:Divided by 10.
Pam:Okay, so divided by 10.
Kim:So, then, 16 cups of pasta divided by 10 to give you 1.6 cups of pasta
Pam:And call me crazy, but I think that's easier to measure. Like, I would do 1 and a 1/2 cups of pasta, and then just a little bit more to get 1.6, 1.5 plus a little bit more. I don't know.
Kim:That's funny because I was actually intentionally leaving it in the fifths.
Pam:Well, so like, do you have a fifths cup measure?
Kim:I feel like I do. Do I not? I don't cook much, Pam. Come on.(unclear)
Pam:I'm pretty sure you do not have a fifths cup measure. Unless you're measuring in metric or something, but. Okay. Alright. totally cool.
Kim:Clearly, I don't cook a lot.
Pam:Okay, how about 9 cups of sauce? If I've got 9 cups of sauce, how many cups of pasta do I need?
Kim:Nine cups of sauce. Okay, so I'm going to go from 10 cups of sauce, minus 1 cup of sauce, to give me 9 cups of sauce.
Pam:Okay.
Kim:And here's where I'm happy to not be living in fraction land. So, I'm going to go 16 cups of pasta that went with the 10 cups of sauce, minus the 1.6 cups of pasta.
Pam:Mmhmm.
Kim:So, that's going to be 14.4 cups of pasta.
Pam:Nice, nice. So, 14.4 cups of pasta. 14 and almost a 1/2 a cup to 9 cups of sauce. And totally cool. Okay, but today I looked in my fridge, and I actually have 4 cups of pasta. I'd like to know how much sauce I need to make.
Kim:Hmm. 4 cups. So...
Pam:So, I think I just...
Kim:Four cups of sauce?
Pam:...flipped it a little bit, right?
Kim:What did you say? Did you say 4 cups of pasta?
Pam:Yeah, 4 cups of pasta.
Kim:Oh, okay. Alright. So, I only want 4 cups of pasta, which means I need 2 and a 1/2 cups of sauce. Because you told me when it's 8 cups of pasta, that's 5 cups of sauce. So, now, I need half as much pasta and half as much sauce.
Pam:Cool. So, 4 cups of pasta, 2 and a 1/2 cups of sauce. Nice. But now I just have 2 cups of pasta. Two cups of pasta. How much sauce should I make?
Kim:One point two five cups of sauce.
Pam:Because?
Kim:Because we just talked about 4 cups of pasta was 2 and a 1/2 cups of sauce, so now I need half again. Half the pasta and half the sauce.
Pam:Cool. 1 cup of pasta, Kim. Today, 1 cup of pasta. How much sauce? How many cups of sauce?
Kim:That's going to be 0.625. Is that right? Oh, yeah, 0.625. You said... Wait, wait, tell me again. Did you say 0.25? You said 1 cup of pasta?
Pam:1 cup of pasta.
Kim:Yeah. So, half the pasta, so I need to half the sauce.
Pam:Okay, what's half of?
Kim:Half of $1.25?
Pam:Half of $1.25.
Kim:Point six two five.
Pam:Point six two five. How are you thinking about that?
Kim:Half of $1.00 is $0.50.
Pam:Okay.
Kim:And half of $0.25 is 12 and a 1/2 cents. So, that's 62 and a half cents.
Pam:Thanks for doing that out for me. I was thinking about half of $1.20
Kim:Oh, that's nice.
Pam:That's $0.60, right? And then, I was getting stuck a little bit. Half of $0.05.
Kim:Yeah, it's weird.
Pam:Half of $0.05 is like, 2 and a half cents. So, there's another 62 and a half. Cool. So, let's see. What if... Back to sauce. What if I... No, sorry. Stay with pasta.
Kim:Okay.
Pam:What if I have 41.6 cups of pasta?
Kim:Oh, goodness, 41.6.
Pam:Feed the neighborhood.
Kim:Feed the neighborhood. Forty-one point six. Okay, bear with me. I'm going to go with... I knew 4 cups of pasta was going to go with 2 and a 1/2 cups of sauce. So, I actually wrote down 41.6, but I'm going to erase it because I need room for 40 cups. So, I'm writing in my ratio table 40 cups of pasta is going to go with 25 cups of sauce.
Pam:Oh, nice.
Kim:You said 41.6, right?
Pam:Correct.
Kim:So, then, I'm looking back at the 16 cups of pasta goes with 10 cups of sauce, and I'm going to scale that down, and I'm over here writing 1.6 cups of pasta goes with 1 cup. And now, I'm going to put those two together, the ones I just told you, the 40 cups and the 1.6 cups. And that is 26 cups of sauce.
Pam:Da-tah-dah.
Kim:That's alot of pasta.
Pam:And so, another way that I've heard students on the 41.6...Nicely done, I liked how you went from the 4. That was very effective. Not sure I've ever had kids do that. It was nice though...is that they'll go from 8 to 40. So, you liked to 40. They'll go from the 8 to 40 by scaling times 5.
Kim:Mmhmm.
Pam:Five times 5 is the 25. And then, they'll add that 1.6. Nice. And in a huge way, what we just did was reason very similarly to how we were reasoning multiplicatively with multiplication two weeks ago. And multiplicatively with division in last week's episode. And in today's episode, we're reasoning proportionally about a non-unit ratio.
Kim:Yeah.
Pam:So, two weeks ago, we reasoned about... Golly, what was it? One to 72. One box had 72 smarties.
Kim:Yep.
Pam:And then, last week, we reasoned about 1 pack of water bottles had 12 water bottles.
Kim:Yep.
Pam:Those are the unit ratios because it's one to the thing. And today, we're reasoning about non-unit ratios. Eight cups to 5 cups, 8 to 5.That's a non-unit ratio, neither of those is 1. And we can sort of reason in and out, multiplying and adding.
Kim:Yeah.
Pam:As long as we keep the ratio consistent. So, we've said one of the hallmarks, or the definition of a ratio table, is that the ratios are equivalent. So, I'll just read a couple of ratios. We had 8 to 5; 16 to 10; 1.6 to 1; 14.4 to 9; 4 to 2.5. All of those ending with 41.6 to 26. All of those are equivalent ratios. So, that last problem that I asked you, I can actually write as I could write the ratio of 8 to 5 is equivalent to. And we didn't know the 26, right? So, 8 to 5 is equivalent to 41.6 to x, or whatever. And if I write that as a proportion, that could look like 8 over 41.6, or 8 to 41.6, equals 5 to x. 5 over x. And that problem could be solved with cross multiply and divide. Bleh. That would be what 41.6 times 5, and then divide that by 8? Or, we could reason proportionally like you did to solve that problem, and at the same time, build proportional reasoning.
Kim:Yeah.
Pam:Students actually get better, and their brains think more sophisticatedly, if we help them grapple with the relationships, not just give them ways to solve problems that use less sophisticated reasoning.
Kim:Yeah, so when Cooper, my younger, was in elementary school, you would sometimes put MathStratChat problems that were about solving proportions. And he loves to participate, right? And so, when we would post those problems, he would be like,"Oh...awesome..."
Pam:Oh, sadness.
Kim:And he would be a little sad. But when I said to him,"Oh, this is about scaling on a ratio table." He was like, "Oh, I could do that!" And it was like he knew what he could think about because he was used to scaling up and down in a ratio table. And all he had to know was, "I'm getting from 8 to 41.6? I can do that." And it's a nice introduction using a model that he has made sense of to introduce things that he hasn't even really tackled in school yet. I love that.
Pam:Yeah. That's super, super cool.
Kim:Yeah.
Pam:And I was just talking to our colleague Kristin Frang this morning. And we were talking about properties. And, Tad Watanabe has been sort of poking me on Twitter a little bit about properties. And so, eventually, we would definitely want to pull out that...you just mentioned scaling on a ratio table...that as we are scaling, that is an application of the associative property. That we're thinking about taking ,it's a multiplicative relationship. And we're thinking about as we scale, we're associating those factors together. And as we chunk things together. Like, if you take the 8 cups of pasta... Let me think on this one. If we took like 4 cups of pasta, and 2 cups of pasta, and added it together to get 6 cups of pasta.
Kim:Yep.
Pam:And so, we took the corresponding cups of sauce and added it together. That's more an application of the distributive property.
Kim:Yeah.
Pam:We sort of chunk the chunks together. And so, yes. We definitely eventually want to get to discussing properties and the basis of how all this works. So, teachers, we definitely need you to sort of go that direction, but the starting point is to ask good questions like the questions in a Problem String, represent, pull out the thinking of students, represent that thinking, make it visible, and help this tool become a tool, this ratio table become a tool for students to think and reason with. And then, we can dive a little bit deeper into the properties and what they're based on.
Kim:Yeah.
Pam:Super cool. We love the ratio table. Hey, let me mention one other thing. This week, the previous week, and the week previous to that. So, three episodes in a row, we've been talking about ratio tables. The strings that we have done, the Problem Strings that we've done in all of these three weeks have been model building strings.
Kim:Right.
Pam:We also at some point need to talk about what strategy building strings look like because it's not just about all the different ways. You'll notice that when Kim would solve it, I would say, "Ooh, and I can also think about it like this." Or last week, when I was solving it, Kim would say, "Ooh, and I could also think about it like that." When you're doing a model building string, we want to share and represent multiple strategies because we want to go in and out of the model, and we want to strengthen the sense of the model. But then, we want to also help students develop certain strategies using that model. And let's do that in a future episode. So, stay tuned for what a strategy building string could look like as students use the tool of a ratio table to solve problems.
Kim:So, you can check out what we did to record our thinking on a ratio table and share your thinking with us as well on social media.
Pam:Love to see what's going on as you're doing these Problem Strings with your students.
Kim:Yeah.
Pam:Hey, y'all, thanks 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!