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Multiplicative comparison is a crucial yet often overlooked concept in elementary mathematics. Many students in grades 4-6 struggle with fractions and multiplication, while those in grades 7-8 need a strong foundation to think algebraically. In this episode, we explore how understanding multiplicative comparison can unlock deeper mathematical reasoning and support students’ progression. When should we introduce it? How do we leverage it effectively? Join us as we break it down with real-world examples!

Key Takeaways:

• Understanding how it differs from additive comparison.

• A bridge between multiplication, fractions, and algebraic thinking.

• How a strong grasp of multiplicative comparison supports algebraic reasoning.

• Key moments to reinforce the concept in elementary math.

• Practical ways to help students develop this understanding through rich tasks and discussion.

Resources:

• How To Start The Year Off Right [Blog Post]
• Make Math Moments Online Workshop
• Assessment For Growth: A Blueprint Course For Standards Based Grading in Math Class
• Make Math Moments Problem Based Lessons & Units

Attention District Math Leaders:

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Looking to supplement your curriculum with problem based lessons and units? Make Math Moments Problem Based Lessons & Units

FULL TRANSCRIPT

Kyle Pearce: All right there math moment makers. Today you’ve got Yvette and I, and I’m Kyle by the way, and we are going to be digging into something that we are pretty passionate about. We actually did a lot of this learning together when we were doing our roles at Greater Essex here locally in Windsor Essex County. And we are going to make a claim here today that we’ve got a really, I’m gonna, I don’t wanna call it easy, but it seems simple. I’m gonna say it’s simple to think about. A simple concept, a simple idea that we could, if we spent more time honing in on this area, that it would help us in so many different places when it comes to math content knowledge. I know it’s bold, I know it’s brave, but Yvette, what is it that we’re gonna be talking about here today?

Yvette Lehman: Okay, I’m gonna tell you guys a quick story. So this topic, as Kyle mentioned, we have really invested in and we’ve recognized the power, but I had a reminder this week with my son, my only student right now, who I get to use for all the learning I’m doing, who is in fifth grade. yeah, he will, for sure, for sure.

Kyle Pearce: Well, your husband Scott’s also a student too, but he’s more like he’s like your teacher student, right? When you get to try things on him for PD, right?

Yvette Lehman: Luckily, luckily they’re both willing participants. But okay, so I’ve been thinking a lot about fluency strategies. And so Leto’s class right now they’re focusing on multiplication. And I have explicitly taught Leto and his teacher has explicitly taught partial products over addition where you’re decomposing one or both of the factors. We’re all familiar right with this strategy. But I’ve been noticing lately that Leto is

not always using partial products over addition. He’s actually using breaking it apart into factors. So yesterday, for example, he had to multiply 15 times seven and he told me he was like five times seven, 35 times three, 105.

And he does this all the time now. So if he’s multiplying by six, he multiplies by two and then he multiplies by three. If he’s multiplying by 12, he multiplies sometimes by two and then, but like he, like he is constantly decomposing when it makes sense, right? So if the number that he is multiplying by, if one of the factors can be decomposed into factors of two, three, four, five, that’s been his go-to strategy. And I thought to myself,

Okay, and I don’t want to get too in the weeds here, Kyle, but do want to hear something really fascinating about this? That strategy that he uses can be described, I would say, in one of two ways. You could say you’re breaking apart one of the factors into factors, but it’s also like tripling and thirding or five times and one fifth.

because when he takes the five and takes it away from the three, right? So he’s basically saying 15 is three times five, but I’m gonna take the five and multiply it by the seven. So now the 15 is a fifth the size, but the seven is five times greater.

Kyle Pearce: Right. Without even him. And now we don’t know what he’s thinking, but I think we can probably all guess that he’s probably not thinking about this explicitly. But just to say it another way, to make sure those who are listening, they’re on their walk right now, and they’re just like, boom, you just blew their mind. So really what it comes down to, and I think the most basic

introduction to this idea usually comes out in some form of number talks, math talks, that sort of thing, right, where we’re doing number strings or, or something along these lines where we look at doubling and having right this idea that it’s like, you know, if I was to like take half a number and then double it later, I will get back to the same place, right. And how helpful that can be in math talks. But what you’re saying is that he’s essentially using that same concept

as he’s decomposing these more challenging multiplication problems and really like factoring them like he’s factoring these numbers down. So taking six and when he takes six and then he basically haves six to get three uses the three to multiply by the original factor that he wanted to multiply by and he’s doing this in his head as well. I think you’re saying and then after taking the result of that multiplication product.

right, taking the product and then doubling it after. So it’s like he’s halved it and then he’s gonna double it back after. And it sounds like he’s got like quite a bit of fluency with this approach.

Yvette Lehman: Well, and what’s interesting is I don’t know that I would have picked up easily on what he was doing had I not read Jennifer Bay Williams and John San Giovanni’s book, Figuring Out Fluency, because in their strategies, they talk about the different ways to use partial products. And so the ones I would say that have been on our radar are like partial products over addition, partial products over subtraction. But they also talk about

breaking apart to multiply as in breaking apart one or both of the factors into factors. And so because I was aware, I knew this when he described his thinking, I could notice and name what he was doing. But again, he’s never been explicitly taught this as a strategy.

Kyle Pearce: Well, and that’s really interesting. I mean, that’s awesome, by the way, that you’re blessed to have a child that is able to sort of recognize these behaviors, is obviously curious enough to kind of test them. Like he must be testing them somehow, whether it’s in his head or otherwise. But here’s the sad reality, is that how many students are missing out on that experience because we’re not being intentional about this. And I want to.

you know, bring back I used to mention this quite a bit when we did PD on number fluency, and we’d talk about like the associative property, the communicative property and the distributive property. And, you know, in different curriculum or different standards around the world, these things pop up and they would like, you know, there’s just a simple definition of like what it is, you know, like, it’s like, hey, like, communicative, like you could just like,

You could swap around these two add-ins and everything’s good. Or I could do the same thing with factors. I can commute them. Well, what your son’s doing, as I envision this, so he factors this 6 down into two numbers. And he’s able, because of the commutative property, he’s able to then re-associate, use the associative property to re-associate part of 6, which is 3, to

multiply, and then he gets to multiply by the number that was originally essentially that he have now he’s got to double it back. Like you think about how these properties work and how poor a job I did in my classroom in trying to bring these properties to life. Like, it’s less about knowing the property or being able to like name the property, but more about like, how can these properties be helpful for these fluency strategies?

And really, think that’s kind of what we want to dig into here today is how can we emerge these ideas so that students have like, and I’m sorry, there’s not one person on this planet that’s going to say that what Leto just did is not a skill that he can use and lean on for the rest of his life. Everybody wants that for math, right? Can’t math be more about things that we will use in the real world, things that will be helpful to us later in life? And that strategy right there.

Kyle Pearce: I don’t know if I’ve explicitly used it before, but it’s definitely something that I’m going to try to be more intentional about and try to elicit, especially when working with children.

Yvette Lehman: That’s what I said to him. As I observed him using this, said to him, that’s not, wouldn’t have been my go-to, but now that I see you using it all the time, I think that it could help me be more efficient. Sure.

Kyle Pearce: That’s so awesome. love when a child is able to truly teach something. And here’s the funny part. I don’t know about Leto, but I do know this about my own kids and when I was in the classroom teaching. A lot of times, I would, you you try to play along so that the students are empowered, right? But sometimes they get to a point where they’re like, you’re just saying that. But what, like a time like this where you’re like, no, I’m serious.

I am serious that I never thought about that. It’s not that you didn’t know it could work like if he would have, if he told you about it, you’re like, yeah, that could work. But to be able to explicitly identify it and draw it out, it now brings me to this idea of, okay, based on this idea, we’re talking about multiplication in particular, let’s dig into the weeds a little bit on what are we?

What’s our call to action for folks here? And how can we as educators maybe not just draw this one idea out, but be able to strengthen all students’ understanding of multiplication and this very specific type of multiplication or this approach that we’re going to be using today in order to become more fluent and flexible with numbers?

Yvette Lehman: So as I reflected on this particular strategy or the flexibility of the efficiency, brought me, I tried to think about what experience has he had? How did he get to this place where he can see relationships with numbers? And it brought me back to a conversation like you said, we’ve had for many years, which is the need for an emphasis on multiplicative comparison, particularly in the junior grades.

And I remember when I taught grade six, I administered Marion Small’s prime assessment. And, you know, one of the next steps that emerged for my students who were not demonstrating proficiency was to focus on multiplicative comparison, which is the idea that we need to be thinking about the relationships between numbers and moving away from an additive approach. And so when I think about our, you know, go to partial products, that’s an additive relationship.

for one of the factors. We’re saying, you know, 15 can be decomposed into 10 plus five. But what Leto is doing is he’s looking at 15 and he’s looking at a relationship that exists that’s multiplicative. Mckee knows that 15 is three times five and he can decompose it. And so sometimes we tend to, talked a lot about the progression from additive to multiplicative thinking and how so many students and probably a lot of adults get stuck in additive.

and how limiting that is when we’re stuck in additive. So what can we do to really, really strengthen our students’ ability to look at two quantities and compare them through a multiplicative lens where we’re looking at the relationship that exists? And I want you to kind of dig in, Kyle, and talk about, you know, the connection to fractions and how powerful spending time here is for that concept of fractions, which seems to be such a barrier for so many students.

Yeah, like if you think about it, I’ll use the number six again, know, one of the factors that you had in your example. So you have six. And really, there’s so much you have to know in order for this strategy to work, right? And when you think about it, when students go to actually factor a number like that, a lot of times they start using like guess and check. You know, like they start going like, well, you know, two times two.

is for, that doesn’t work, right? Sometimes they know it. It’s automatic. And I would say maybe with six here, that would be one of those cases where it’s automatic. imagine as that number gets larger, this gets a little bit more challenging. And if we’re just using guess and check, it can feel like an endless battle. So when we think about it as a fraction, though, if we actually start thinking about it and you start looking at larger numbers and saying, can I take this number and can I half it?

like can I have it like if I’m able to have it okay well I know that there’s this doubling having relationship available to me there right and ultimately when we’re utilizing this idea of fractions fractions are so multiplicative in in their nature that it allows you to kind of constantly be doing a multiplicative comparison and it also makes

operating with fractions a whole lot easier, especially when it comes to, multiplying or dividing them later, those in particular. But for me, as I look at numbers and if we think about it multiplicative instead of additively, what you’re getting is you’re getting this opportunity to see how the numbers relate to one another. you kind of triggered something in me when I think about partial products, because I love that strategy.

But in a way, it’s almost so easy. Because you can pick any numbers to work for distributive property. I could pick any numbers. They could be helpful or unhelpful. But I could take that number, and as long as I can decompose it into two numbers that the sum is equivalent to the original number, I’m good to go. Whereas here, you have much narrower opportunity of numbers to work with. And that fraction, that understanding of fractions and being able to

Kyle Pearce: these numbers multiplicatively, is like, it’s like if you can naturally feel like this is comfortable to you and fluent to you, it just unlocks the door to so many other opportunities to apply this strategy.

Yvette Lehman: Let’s talk about the number 15. Okay, so that was the one that Leto modeled this for, you know, for actually I said to him last night, I said to my husband, you know, you need to hear Leto’s strategy and let’s unpack mathematically what he’s doing here. And so let’s talk about 15. So if he knows that 15 can be decomposed into three times five, he also knows that a third of 15 is five and a fifth of 15 is three.

Kyle Pearce: And here’s the here’s the interesting part. He knows, but does he know, right? And that’s the interesting part.

Yvette Lehman: Well, and the thing is he does, like he actually does. that’s okay. So this goes back to, this is the whole conversation. Why does he know that? Because we talk about fractions all the time, constantly. Like we always describe things as fractions. So the other day he said to me, four students from my class are going to the chess tournament and there’s 18 students on the team. And I was like, that’s a, you know,

pretty big portion of the team coming from that one class. And he said, well, it’s about one fifth. And I said, well, can you be more precise? And he was like, well, it’s about two ninths. And I said, well, is two ninths more or less than one fifth? And he’s like, well, it’s more because it’s more than two tenths.

Kyle Pearce: Mmm, love it.

Yvette Lehman: So this brings me back to this is an idea to throw it to our community.

How often are we just talking about numbers in situations that come up outside of the math class? Constantly. Like we’re collecting money for a field trip. There is a certain number of students who are participating on a particular team. Are we constantly seeking opportunities to describe relationships that exist between quantities?

Kyle Pearce: Right. And I also want to push, too, that it’s like, and just in passing, you know? Because I think that is such a big difference. Like, we’ve all done it. We’ve all done it before, where we’re like, we take that idea, and then we bring it back to math class. And then we put the question up on the board, and then it becomes this formal thing. Right? And kids are trying to now mimic or do whatever it is they’re supposed to do. But just to have that conversation and to be able to, you know, like,

It’s pizza day at my kid’s school today. Think of the opportunity there to have a conversation. It might not be with everyone. It might just be with a handful of kids, right? The kids that you’re kind of hanging or walking around, and the kids are eating their pizza, and so forth. And there’s so many opportunities to have this conversation and just ask a thought-provoking question that many kids are probably going to be willing to at least answer.

Maybe not in a grade nine or 10 class, right? They’re too cool for school now, so you might have to get a little more creative there. But especially for students in these early formative years, there’s so many opportunities. Because you ask kids a question, it’s like they’re suckers for curiosity. They fall into the trap almost every time, right? Once they get thinking about it, and then they’re motivated to have that conversation. And what a great opportunity for them to build and to feel confident with these numbers.

Yvette Lehman: So here is our call to action. Kyle, today.

seeking out opportunities to explore relationships between quantities whenever they come up through ideally this multiplicative lens. And so I’m gonna talk, I just give one more example, you know, when we look at the number 20, are we confident, you know, that we can decode, we know that, you know, 20 is the result of let’s say four times five. So four times five, those are factors of 20.

But can we also describe that four is a fifth of 20 and five is a fourth of 20? That is the understanding that students really, really need to consolidate in the junior years.

Kyle Pearce: And let me guess it’s not through just a single lesson from the unit of study, right? And we’re talking like this, it has to happen and it has to be iterative and it has to be repetitive, repetitive from the idea over time where students have so many opportunities to get at it and try it and allow that connection to be made. It’s so critical. so I think we all have opportunities.

to improve and do more of it if we put ourselves up to the challenge. Set yourself a daily reminder at one point in the day to go walk up to a couple of kids or whatever it might be or in your household if it’s working with the kids. But ultimately, the more we remember and remind ourselves, I think, to have these conversations and to help them make these relationships, the beautiful part is the payoff comes in all aspects of mathematics. It’s not just in this one very specific scenario.

Yvette Lehman: Mm-hmm. Well, I was thinking about where does this live? You know, where does this idea exist? And I was thinking, you know, imagine you’re looking at a rectangle and you have the total area and you’re trying to find an unknown length. That’s where this lives. If you’re looking at, you know, solving an algebraic equation and there’s a multiplicative relationship with a variable, like that’s where this lives. And so when we think about what we’re setting students up for in the future,

like proportional reasoning is rooted in this understanding that there are these multiplicative relationships that exist between quantities and if they are obvious to us, it helps us know how to reason through a problem.

Kyle Pearce: Hmm. I love it. And you know, you think about even just unit conversion as, as maybe boring and mundane as that process can be. But when we start thinking about it as how many times bigger or what fraction of, when we think of it in that way, it starts to make more sense. You know, my, my son had said he used, he was talking about how tall someone was in his school. It was over a meter.

And he had said one point something meters, you I think he said 1.2 meters. He goes, you know, that’s probably like, I don’t know, like, a little bit more than three feet, you know, or whatever it was, you know, whatever. And obviously, this was a student in the class. So, you know, it’s like taking these opportunities to give them the opportunity to kind of use some of what they know, you know, and it’s like,

Okay, so a foot is about a third of a meter, right? Or it is a third of a yardstick if you’re in the US, right? So these are things that we can do and we can kind of promote for students. The more times they grapple with these ideas, I think the more clarity they gain in their own mind as they see these things and they see them in different contexts and at different times in the day and not just say out of a textbook or sitting in the seat.

Kyle Pearce: Well, math moment makers. Hopefully you found this conversation here with event or family really and let us the star of the show here today as you start thinking about multiplicative comparison and how we can get our students thinking more multiplicative Lee. Yes, it’s a fun word, but it’s definitely one worth getting familiar with.

As we have mentioned, there’s so many connections here, just not just in general fact fluency, but also connections to fractions, connections to measurement as we saw, unit conversions, all proportional reasoning. And of course, even in algebra, if we can strengthen this understanding of multiplicative thinking, students will become more confident in their math classes. And guess what? Everybody’s life gets a little bit easier. So if you haven’t yet,

head on over to makemathmoments.com forward slash report and check out our classroom math tree. You can go and take a short assessment and get yourself all dizzyed up by, I have no idea what I’m saying at the end. So we’re gonna record this part again. Head on over to makemathmoments.com forward slash report where you can take our classroom assessment. in order to understand what which parts of your classroom math tree are strong and which portions might need a little bit of work or attention. Once again, you can head on over to make math moments calm forward slash report and we look forward to seeing you sometime soon.

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