Episode #171: Division Through The Years – A Session With Yvette Lehman
Today you’ll hear from Yvette Lehman, an elementary educator from Windsor, Ontario. Yvette is a Primary and Junior Division Teacher Consultant for the Greater Essex County Board of Education and if the name seems familiar then you’re right, we have referenced Yvette’s work numerous times here on the podcast as she’s also a colleague of Kyle’s and a fantastic part of the Make Math Moments curriculum writing team!
During this session from the 2021 Make Math Moments Virtual Summit, you’ll hear about Yvette’s early experiences and struggles with mathematics which inspired her to dive deep into key elementary concepts to better conceptualize what was once only introduced to her procedurally. In particular, you’ll be taken on a journey through division from early fair sharing to proportional reasoning and solving two-step algebraic equations using context, strategies and models.
- Key elementary concepts of division from early fair sharing to proportional reasoning and solving two-step algebraic equations using context, strategies and models;
- How knowing the two types of division can help your students develop a deep conceptual understanding of the operation; and,
- Which lessons and tasks can I use to help my students explore the two types of division.
Yvette Lehman: So, this is my question, why does this matter? And some people sometimes push back and say like, "Aren't we just going to confuse people? Like, does it matter, the answer's the same?" And I'm like, "It does matter if we're saying that math can be seen and understood, then we have to be able to see its behaviors. We have to be able to see how the operator behaves and division behaves two ways for like very different purpose and both quotative and partitive reveal themselves in different ways throughout the curriculum.
So here is my next context for you... crosstalk
Kyle Pearce: Hey, hey there Math Moment Makers. Today, you are going to be hearing from Yvette Lehman, an elementary educator from Windsor, Ontario. Currently, Yvette is a primary junior division teacher consultant for the greater Essex County District School Board, and if that name seems kind of familiar, then you're right. We've not only referenced vets work here numerous times on the podcast, but she's also a colleague of mine. Yes, she works at greater Essex with myself and she is a fantastic part of the Make Math Moments curriculum writing team and I've got to tell you, I get some of my best awesome, awesome math learning through having some deep discussions about mathematics with Yvette.
Jon Orr: During this session from the 2021 Make Math Moments Virtual Summit, you'll hear about Yvette's early experiences and struggles with mathematics, which inspired her to dive deep into key elementary concepts to better conceptualize what was once only introduced to her procedurally. In particular, you'll be taken on a journey through division from early fair sharing to proportional reasoning and solving two-step algebraic equations using context strategies and models.
Kyle Pearce: Yes, my friends. Oh, and if you're wondering, yes, this is a recording from last virtual summit. So you might be thinking when's the next one? And yes, we're happy to announce that the 2022 Make Math Moments Virtual Summit will be coming to you for free, as always, on Saturday, November 19th and Sunday, November 20th, 2022 that's a lot of time for when this is going live.
Jon Orr: That's the future.
Kyle Pearce: Yes. It's a lot of time from when this goes live, but you're going to want to plug that date into your calendar and get over to the registration page to dive in.
Jon Orr: All right, let's go.
Kyle Pearce: Here. It is. Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pierce.
Jon Orr: And I'm John Orr. We are two math features from makemathmoments.com who together...
Kyle Pearce: With you, the community of math moment makers worldwide who want to build and deliver problem-based math lessons that spark curiosity,
Jon Orr: Fuel sense, making....
Kyle Pearce: And ignite your teacher moves. Welcome my friends to yet another weekly Make Math Moments episode on Making Math Moments That Matter podcast. Today, we've got kind of a special one here because we're bringing back-
Jon Orr: Yeah, it's a little different.
Kyle Pearce: Yeah, bringing back some of our past sessions from last November's virtual summit, getting you hopefully excited for this November summit that will be coming up. And today is a pretty awesome one about essentially the mathematical journey of division through one of our keynote presenters, Yvette Lehman. John, it's going to be great to listen to, but where would you best say the platform choice would be for those who are diving into this lesson here today or this episode, I should say?
Jon Orr: So this is a recording from the virtual conference that we run in November of every year for free, the Make Math Moments Virtual Summit and we chose this particular session for a couple of reasons. One, it's amazing, I always learn so much when I listen to Yvette, and then two is, even though it was a video session, a webinar, it has visuals attached to it, we chose it because it also has enough audio content that you actually can listen to it here on the podcast and still get so much out of it.
But if you're looking for that extra oomph or that extra push or that extra visual component, you can head on over to our YouTube channel, the links will be in the show notes paged for this episode if you're listening over in your podcast platform. YouTube's going to have the full video up there for this session, so you get all the visual references that Yvette might make or does make over there. But listening here, hey, you're still going to get a lot out of it. So let's not waste too much more time, Kyle, let's dive right in. Here comes Yvette in her session.
Kyle Pearce: All right.
Yvette Lehman: Thanks for the nice introduction and I first want to thank everybody for joining us this afternoon, especially a session around content. So this is a content session and it's my favorite content, which of course is division. So thank you for being here and just to jump things off, I just wanted to tell you a little bit about my journey, which will hopefully help explain why I love division so much. So I describe myself as a struggling math student. I was the type of student who excelled in a lot of areas. I did well generally in school, but I never understood anything that was happening in math class and I have a terrible memory.
So I would learn the formula. And then a week later there would be a test and I would have no idea what formula was and people kind just told me that was okay and because I was doing well in other subjects, I was excelling in languages, teachers, my parents, would say to me like, "It's okay, you're just not a math person." and I believed completely them. I struggled through math the best I possibly could until I didn't have to take it anymore and I thought, "Thank goodness math is behind me forever," and I was free of this burden where I would sit in class day after day and not understand a single thing.
So fast forward I become a teacher and my parents are both educators, it felt like the right journey for me and I started teaching grade two and I'm feeling great. I'm in a grade two classroom. Math is finally back in my life, but in a way that is accessible for me and I can navigate. And then I find out that I'm being transitioned to grade 6 and not only is grade 6 a higher academic level, the math is more demanding, it's also a standardized testing grade here in Ontario.
I was working at the time in a school with a high socioeconomic community, traditionally high achieving students and I quickly realized within a week of instruction that the students were further in their math development than I was, and that it was a very scary feeling. Especially as a new teacher, I felt like I had to hide the fact that I was not... I just kept saying, "I'm just not a math person. Trust me, I have other great qualities. I speak two languages and I love the arts, but math is just not my thing," but it became pretty overwhelming pretty quickly. And I actually had my sister, who I considered a math person, she has a science background, tutoring me at night so that I could go in the next day and try to mimic procedures for my students to follow.
I did that for about two years and I survived. I had some great teaching partners who would jump in and teach my class for me when I was really struggling, but of course I was feeling very insecure about my ability to really support students in an area that I didn't understand myself.
So, as John mentioned, I work at Greater Essex, the same board as Kyle and he was a teacher consultant at the time, and about two, three years into teaching in grade six, I was asked to attend a PD session and I was actually selected as a math lead for my school, which I thought was kind of hilarious considering how insecure I was feeling about my math instruction. At that session, Kyle and the rest of the math team introduced us to the math proficiencies and when I started to understand the math proficiencies, of course at the time like many other districts, we were looking at the adding up document, looking at the research from the National Research Council and we were coming to understand what it meant to be procedural or to be fluent in mathematics.
As I started to investigate the five proficiencies, I very quickly realized that I didn't have any of them. I didn't have procedural fluency because I couldn't remember anything. I wasn't flexible with algorithms, I had to relearn them. Every single time I had to do it again. I had to relearn it essentially. I didn't have strategies. I felt very uncomfortable really when students were using different strategies in my classroom because I didn't understand them. So I wasn't able to be responsive or adaptive in the moment. And I definitely didn't have conceptual understanding and that I think was the really biggest barrier for me. I realized that because my memory is the way it is and I really can't remember things unless I understand them and I reflected on my own math experience and I started to wonder like, "Am I really not a math person or do I just not understand it yet?" And I had this moment where I realized math could be understood. Like I thought it was just a series of steps that we had to memorize.
So I left the session and I made a commitment to myself and I was fortunate to have a really great teaching partner and I said to her, when I got back, "All math can be understood," especially elementary math. Elementary math can be understood, and even on top of that, it can be seen, like it's visual. It actually has patterns and behaviors that can be reproduced pictorially or with concrete materials and I had no idea. So we made a commitment to ourselves and to our students from that day on that any concept that we were going to teach, we had to be able to represent it first, either concretely or through a pictorial representation and if we couldn't do it, that was an indicator that we didn't understand it.
So we basically set off to relearn everything we knew about math through teaching. Through the concepts that we were about to teach I realized everything can be seen. So then I really started to question are all of those concepts that I was afraid of really that difficult to understand? And I really love this quote from Cathy Fosnot. She is one of my go-to people. I love, love, love her work. In addition to support of my colleagues leveraging Cathy Fosnot's tools has been the greatest catalyst for my own personal learning, but in her book Conferring With Young Mathematicians, she says, "Perhaps it is not that the content is difficult, but the way we were taught about fractions," that's the concept that she's talking about, "was not helpful."
As soon as I read this, I was like, "Forget fractions. It could be fractions. It could be algebra. It could be area, whatever concept, perhaps it's not that the content is difficult, but that the way we were taught was not helpful." And I really wondered about that idea, and I wondered about it for myself as a struggling math student, but also for the students in front of me. Is the way that I'm teaching the content helpful and who is it helpful for? And if two weeks from now, a month from now, we haven't retained anything, did we learn anything?
So, as I mentioned, I started my journey and I realized this idea, and this was really such a pivotal moment in my own professional experience, when I realized that math can be seen and understood. In Ontario, we have these process expectations, problem solving, reasoning improving, reflecting, connecting, representing, communicating, selecting tools and strategies and I realized that none of those are possible if we don't first really understand and if we can't see the mathematics.
So as I said, I just really made this commitment to myself to start seeing it, to start representing it, to start actually observing the behaviors and as I did that, I learned all of these things I didn't know. There were so many relationships and connections that I made that I never knew existed. It's funny because I described that struggling math student, that very insecure young teacher, and now I'm a teacher consultant for our board and I support K-8 math and I actually do a little bit of work in secondary as well. And my mom said to me recently, "If you had told me 15 years ago that you would be supporting an entire system around math instruction, math content knowledge, I wouldn't have believed you," and I wouldn't have either.
I like to share my story because I know that there are other educators out there who feel the same way that I felt, that maybe your experience as well was that you didn't really understand and I feel like my journey of actually being able to see and understand the math has really opened my eyes to the beauty of mathematics and now it's by far my favorite subject, where I used to shy away from it. I would ask anybody, "Please, switch me math. I'll take your social studies, your science, your arts. I just don't want to teach math." Now I wish I could teach math all day. It's by far my favorite subject to teach because I find it fascinating. It's interesting, it's complex, it promotes thinking, so many things that I love to do. And as John mentioned to you, now I'm a writer for Math Makes Moments, I write some of the units and it's shocking, even to me that my journey has come so far.
So, as I mentioned, when I began this journey of seeing and understanding mathematics, the concept that probably held me back the most as a student has become my all time favorite concept. As a student, I didn't have my math facts and I couldn't remember the steps either. So the long division algorithm was really not accessible for me in any way, but as I have come to see the mathematics and understand the mathematics, I've realized all of these really, really interesting things about division and I would say of my learning journey, the concepts embedded in the behavior of division have opened up the most concepts in the elementary curriculum for me. And I hope that through the tasks that we're going to engage in together today, you'll be able to maybe see some of those connections as well.
So I am going to take you on a bit of a journey. We're going to start with some early concepts introduction to the behaviors of division, and then we'll move towards some more complex ideas towards the end of the presentation. So I'm going to take you through some tasks and these tasks are going to, as I mentioned, start with the really early behaviors of division, like probably about a grade four level, and then we're going to move towards intermediate towards the end, seven and eight.
So this is the first provocation and I always, when I write the units for Math Makes Moments, we always write them through context, like context to me is a really great way to see and understand the mathematics and what I really love about this approach is that we can root ourselves in the context, we can root ourselves in the visual to be able to make connections and draw conclusions.
So on this screen, you have 78 peas and those 78 peas are going to be sewn in 6 pots and the question is how many peas should be sewn in each pot? Of course, we want to ensure equal distribution. So I want you to think about this for a minute, and I'm not really interested in the answer. The answer is not that interesting to me. What's interesting to me is how it looks. So what do we know right now and what are we trying to find out?
So if I have 78 peas and I'm trying to sew them into 6 pots, how many peas per pot? And thinking about the students in your class or students that you've worked with at the elementary level, again, we're looking at, in Ontario here, like a grade three, four level, how might they go about modeling this? So I'm just going to give you a minute to reflect on it and then we're going to look at the solution and just kind of examine some of the behaviors of this division. So 78 peas, and we're going to fair share them or distribute them evenly amongst 6 pots. We're trying to find out how many peas per pot.
So hopefully you've had a chance to just kind think about this provocation and again, think about what we know and what we're trying reveal, and let's take a look at this video and I wonder if this kind of matched your thinking. So we have these 78 peas and we have 6 pots. So this is maybe a strategy that students will use. It's like every... like one at a time. Every pot's going to get a pea and I'm going to keep fair sharing them. Maybe students are even going 6 at a time, there we kind of doubled and we had those remaining peas. So we end up with this idea of 13 peas per pot.
So the division context here was 78 peas divided by 6 pots, reveals 13 peas per pot. It is essentially reveals a rate, which is pretty cool. I think about ratios and rates and how they're explored here in Ontario in the curriculum. It seems like very separate, it's like its own expectations, but really like we're leveraging division here. This is a very standard fair share context. We see this as early as primary. If have 8 marbles and I want to divide those 8 marbles by 4 friends. Very simple, fair share context, but what's really interesting is we've revealed a rate, 13 peas per pot, which is pretty cool.
Again, the strategies are going to vary. You're going to have 1 to 1 sharers, you're going to have skip counters, you're going to have students using some known facts. And of course, we always talk about this idea that we really want to make the math visual and concrete, but we don't want to stay there forever. We don't want students actually counting out the 78 piece forever and fair sharing them amongst the pots. Of course, we're going to want to move along the continuum until we get to a point that we're a little bit more abstract.
So we'd ask the students, it may not have even been clear to them. We have these 78 peas divided by 6 pots, which gave us 13 peas per pot. Down the road maybe this open array can be sufficient enough, students can see the peas living within this array and maybe they're using like a really cool strategy. My favorite strategy, which is partial products.
As a student who didn't know her math facts, I'm still not comfortable with my math facts. If you ask me right now to multiply 13 by 6, I would a panic. So I really, really love partial products and maybe this is a strategy that students will leverage over time through the support of building their fluency. But here are my 78 peas, here are my 6 pots. Well, let's put 10 in every pot that takes up 60. Let's put two more in each pot. That's 12. Oh, and then finally we have one pea left for every pot and now we have 13 peas in every pot.
So that was our first context and I want you to just maybe jot down on a piece of paper, what we knew and what we revealed. We knew the total number of peas. We knew the number of pots. What we revealed was how many per pot. I'm going to go be bold and I'm going to say that's a rate. And I'm saying it's a rate because it has a composed unit. It's 13 peas per pot. So that's why I'm going to be bold and say that is a rate.
Here on the screen is our second context. We have 96 peas, 88 here, 8 peas have already been deposited in this pot and in this context we have 96 peas, and I know that every single pot can sew 8 peas. Okay. So I have 96 altogether. If you want to write this information down, I have 96 peas, and I know that every single pot can hold 8 peas if I want them to grow and have enough space. If that's true, how many pots do I need? Okay. So again, let's focus on what is known. We know the total number of peas, we know how many peas per pot. We don't know how many pots and I want you to think about how you would go about solving this per particular context or how your students might go about solving it. Again, I don't really care about the answer. Of course, we want to get to being accurate, but what I really care about is the behavior, what is happening and how do I see it?
So in my mind right now, I'm picturing I can't draw the pots. I don't know how many there are, but I almost want to start like making groups of 8, making groups of 8, making groups of 8, which is interesting and it's interesting compared to the last ones. So let me give you a minute to think about how you would solve this or how your students would a gobo solving it and you have those 96 peas and you know that there are 8 peas in every pot. How many pots?
So maybe your solution or in your mind, if you didn't have time to fully draw it out, you were just kind of visualizing and again, I think that's such a strength for students. Like what are they seeing in their mind? When they hear this context, when they think about the behavior, what is happening? And I wonder if what you were seeing in your mind are these like groups of 8. I know there's 8, so I can make these like counts of 8, these iterations of 8. I'm going to go 8 at a time. So here we go. It's like, let's count these 8. So there's 8 peas. Okay, 8 more is 16. So maybe students are accessing this through repeated addition. Maybe they're accessing it through skip counting, but what we know in this is the number per pot.
So here we go again with our open array. What do you notice here? So here I have my 8 peas per pot. I know that, that's my known information. I know there are 96 altogether. Let's go ahead and be bold and say, I can fill 10 pots. I feel pretty good about that. Again, like I'm not super good with my math facts. In Ontario, I love this expectation, in grade three, they really, really promote mental math skills around the 2s, 5s and 10s. And I'm like, we can do so much with 2s, 5s and 10s if we're leveraging partial products. So I feel really good that I can fill at least 10 pots. So there we go, 10 pots are filled. That takes up 80 of my piece. I have 16 left over. Okay. That's 2 more pots of 9. There we go. There's 12 pots altogether. And you'll see if you're familiar with the flexible algorithm how that's really starting to emerge and no offense against those who are like really big supporters of long division algorithm. I am not. It's probably because I was a struggling student and it didn't work for me.
What I am a huge supporter of though, is getting to the flexible algorithm, like this idea that I can work with facts that I know, and I can just take away those groups and work with what I have left. I think that's a really, really effective strategy. Again, leveraging those partial products, leveraging those facts that we're super familiar with. So let's talk a little bit though about what is actually happening here. We had these two scenarios. We had our first context, which had... So here we go. This is again the same solution. We had these two contexts. The first context, if you remember the question I asked you, "What is known?" We knew the total number of pea and we knew the number of groups. In that context it was pots. What we didn't know was the size of the group. We didn't know the rate, how many per one? So our action was to take a total and share equally a given number of groups to determine how many per group. That was our first context.
In our second context, we had a total, we had 96 peas. What we did know was the size of the groups. I knew based on the planting instructions that I could put 8 peas per pot. So I knew the rate. I knew how many there's that composed compound unit again. What was unknown was the number of groups. So the action was to take a total and take like measured groups out of it to see how many. This is how our curriculum defines the two types of divisions. So in Ontario, if you go to the grade four curriculum, we have a revised curriculum for anybody who's coming to this session outside of Ontario, and they offer these really great teacher supports, where it's like, "Okay, we're going to introduce division, but hey, everybody, there are two types of division," and I had no idea. Again, being the student that I was, who saw basically mud when things were up on the board, I just thought division was, I don't even really know what I thought it was. I think I thought it was fair sharing. I knew that there were these sometimes scenarios that looked a little bit different, but I couldn't notice or name them. And I definitely wasn't representing them. So this was a huge, huge learning for me.
I remember the day this happened to me, that I realized there were two types. I was modeling division on the number line, super simple equation. It was like 12 divided by 4 and one group said, "Oh, that's easy. I take the number line, here's 12 and I divided into 4 parts, 4 equal parts, like cut it in half, cut it in half again, I have 4 parts, each one has 3." And then another student said, "No, no, I have 12. I take jumps of 4 and there are 3 parts," and I was like, "What is happening?"
I remember I called Kyle and this is when again, he was the teacher consultant and I was in the classroom. And I said, "Did you know, there were two types of division?" like as if people had been keeping this information from me and I'm so glad I figured this out. I'm so glad I used Cathy Fosnot's mini lessons to do my number talks because I am telling you understanding the behavior of these two types of division have allowed me to conceptualize and automatize so many other concepts, all the way up through like grade nine and 10 to the point that like, I can just look at things and know the answer because I understand the behaviors of these two types of division. That might seem a little farfetched, but I promise you that has been my journey. These two behaviors have revealed themselves often in the elementary curriculum and I just don't know... I definitely didn't know they were there before, I saw them as something completely separate.
So I want to try a little bit to prove it to you. If I haven't yet convinced you that the fact that these two types of division are for two different... like it's almost like it reveals something different, the known information differs. If I haven't proved to you yet that these are significant big ideas, I want to try to do that by upping the ante a little bit.
So in the Math Makes Moments units, I would not... So the unit that we just looked at is called Sewing Seeds. It's one of the earliest introductions to division. It's like pretty early junior where very friendly numbers, we're just really trying to notice and name the behaviors. I wouldn't jump to the one I'm going to show you now, there are units that come in between, but just to convince you that this might have some validity, I want to try to do something a little bit more cognitively demanding.
So this my question, why does this matter? And some people sometimes push back and say like, "Aren't we just going to confuse people? Does it matter, the answer is the same?" And I'm like, "It does matter. If we're saying that math can be seen and understood, then we have to be able to see its behaviors. We have to be able to see how the operator behaves," and division behaves in two ways for a very different purpose and both quotative and partitive reveal themselves in different ways throughout the curriculum.
So here is my next context for you. It's going to get a little bit more complicated, so I'm going to give you more time this time to draw this one out, and I strongly encourage you to draw it. Even if you are already at a point in your learning journey where you are in abstract land, I think that's awesome, I still want you to draw it. I think there's so much value in actually seeing the behaviors. So here's my scenario for you and again, we're going to start with context. This is Gabrielle's garden and she bought this sunflower from the gardener and at the time that she planted in the garden, it was 4 inches tall. So you might want to write down that information. So we had a sunflower plant in the garden day zero, here we are in the garden, it's 4 inches tall. The gardener told her that she should anticipate that this sunflower will grow 3 inches every week. So after planting it, he's like "Every week you're going to see it grow 3 inches." Okay?
So I'm going to jump to my next slide for you because this one we're not going to guess which sunflower she has. So here we are. Here's what we know. This is the information we have to start with. So maybe write this down, maybe draw yourself a diagram. You've planted this sunflower and it is 4 inches tall and it's going to grow 3 inches every single week from here on out until it reaches its maximum height. If that is true, then I can represent this scenario using this equation. I feel pretty good about this. The height of the sunflower is equal to the number of weeks it's been planted, multiplied by 3 inches because it's like every week it's going to grow through 3 inches. Plus I can't forget the initial height. It was already 4 inches when I planted it.
So if I feel good about this equation, that the height is equal to the number of weeks multiplied by 3, plus the original 4 Gabrielle's sunflower is now 40 inches tall. And I want you to figure out how many weeks it has been planted in her garden. So with the information that you've been given, can you model this solution? So can we determine the number of weeks by modeling this context? So I'm going to give you a few more minutes to be able to do this.
Okay. Hopefully you've had enough time just for the sake of time to just kind of try to represent what's happening here. So here we go. I wonder if your solution looks anything like this. So here we go. There's our 4 inches. That was when it was planted and we're going to add another week for every 3 inches. So we started at 4, then we added another 3, another 3, another 3, another 3, another 3, another 3, until we finally reached that height of 40 inches. And I can see students accessing this question that way. I know concretely where they actually take the 40 cubes and they actually model it. So if I start with 40, so imagine what we just saw there, I have the 40 cubes. Each cube represents an inch. I'm going to take the 4 right away. I'm like, "That was the initial height."
Then I'm going to make groups of 3 and I'm going to make those groups of 3 until I get to my total, which is those 36 inch extra inches or 40 inches altogether. Some students may want to represent it differently and maybe you did something like more of a linear model because the sunflower itself almost encourages this linear model. So this is every week. So every week we're getting those 3 inches, 3 inches, 3 inches, 3 inches. It goes all the way, we're counting weeks, we're counting inches and then we get to 36 and we're like, "Oh wait, we have to account for those 4 inches that we're there to start." So there's our 12 weeks plus the 4 inches that were there initially.
So my question to you is what we essentially did here, let me like talk this through backwards. How did we determine the number of weeks? How did we solve for weeks? We started with the 40 inches. We subtracted the 4 initial inches and then we used division because we took 36 and we counted out. We measured 3 inches, 3 inches, 3 inches, 3 inches, until we get landed on this idea of 12 weeks. So when we're solving this two step algebraic equation, we're leveraging one of the two types of division. This is a quotative context and I love this one because Kyle and I went into a grade seven or eight classroom a couple years ago and we were like, "We're going to make two step algebraic equations super visual. We are going to help them see it," and then we realized halfway through the lesson that every single context we gave them was partitive. It wasn't accounting for the entire other type of division that they are going to encounter when solving it in context.
So again, why does this matter? Because if I'm going to actually see a behavior of this division, so if I'm going to be looking at H is equal to W multiplied by 3, plus 4, what is actually happening? Can I model the context? Can I understand the behavior of the operator? So we talk about this idea of division is the inverse operation to multiplication, but why? And in this scenario here, it's very obvious. I'm doing counts of 3. I have iterations of 3 and that's for true for quotative context. So that's one of the types of division that we're seeing reveal itself through this two step algebra equation.
And here's something a little bit more abstract, or maybe further along the continuum, where it's like, "You know what? I'm going to be bold again. I'm going to use those known facts," and it's like, "Well, I account for the 4, no problem, get rid of the four inches that were there to start. I know that 30 of those 36 inches came from 10 weeks because I know 10 measured groups of 3 is 30 and then if I have 6 left, well, that's just two more groups of three." So again, we're kind of moving down that developmental continuum towards that more flexible algorithm, a more, let's say, symbolic representation, a more abstract representation.
Here's my second context. Again, I'm going to give you a minute to write down what is known. What is known here? What do we know? All right. So here are 12 sunflowers. Gabrielle has really started her own business now. She has actually really taken to growing sunflowers in her garden to the point that she's growing enough that she's going to start selling them. So she is going to sell all 12 of these sunflowers. So she goes ahead and she sells one at a time and she starts bringing in some money. For every flower she sells, she's throwing money into the bank. So the bank is going up. Every time she makes a sale, you'll notice that she's selling the sunflowers for the same amount every time.
Here we go. This is the money she's earned. Oh, hold on, wait a minute. She actually... let's talk about profit. She has to deduct the amount she spent on the sunflowers to begin with. So she bought seeds, there was a cost associated with those seeds. So $63 actually represents the total profit. So she sold her 12 sunflowers. After she deducted her initial cost, she's left with $63 in the bank.
So this is what I can tell you. This is what I know for sure. I know that after deducted her expenses, Gabriela has $63 in the bank. I know that she had to deduct $9 because that's how much she spent to buy the seeds. I know that there were 12 sunflowers sold altogether.
So can we start to think about an equation that could be represented here? So I'm going to say that $63 is equal to 12 sunflowers multiplied by an unknown cost, an unknown value, minus 9. So I feel pretty good. I hope you feel okay with that too. I know I didn't give a lot of think time to think through that equation, but hopefully you believe me when I say that I have... I'm not going to show that screen yet, I have $63 left after deducting $9 and selling 12 of my sunflowers. What I don't know is the cost that she's charged per sunflower. I don't know how much she made per sunflower. So I want to think of that as an equation. And I'm going to ask you to try to model what is happening here. So I have 63 is equal to my 12 sunflower sold at an unknown cost, minus 9. And I wonder if you can try to model what's actually happening here. What are the behaviors of solving for this equation? So I'll give you a few minutes to think that through.
So I wonder in that very limited amount time that I gave you, if you thought about adding the $9 back. Were you like, "Well, I'm at 63, but I really want to know the total that she got for selling them." So you took your 63 and you added 9. You were like, "I'm going to add the nine back in. I now know this to be true, that 12 sunflowers sold at an unknown cost is equal to 72. I need to solve for one. I need to reveal a rate." Again, I love this behavior of this division. It's a super cool type of division that's like under, almost like we simplify it a little bit. I think that's great when we're introducing it early on, but it has so many really interesting behaviors.
So 72 is how much we got for 12. Well, how much do we pay for one? So we're going to divide 72 by 12, $72 divided by 12 pots. Our context here is partitive. The last time we took little jumps of 3, we're like 3, 3, 3, 3. Now we're taking our whole, and we're fair sharing it or partitioning it amongst those 12 sunflowers. So when we do $72 divided by 12 sunflowers, we end up with $6 per sunflower. Again, that compound unit, we're revealing a rate.
So in these two steps, algebraic equations, I tried to reveal how, like we have these two types of division happening all the time. I just don't know that we always know that they're happening, but if I'm saying that I believe, and I truly believe that math can be seen and understood, which one you're doing matters. So am I taking jumps of 12 or am I taking 12 parts to reveal a rate? And there's a big distinction between the behavior of those two types of division.
So I already mentioned this, that I feel wholeheartedly that these two behaviors understanding the difference between partitive and quotative division have opened up so much mathematics for me. I feel so much more confident engaging in conversations and developing context and modeling. I feel safe to be put on the spot because I trust I can make generalizations about this particular operator. And what's really cool is all the curricular connections.
So as I already mentioned, ratios and rates are embedded in division. Every single division sentence is a type of ratio, every single one. And I say that, I feel like I've investigated it enough, that it's one or the other. You either have a composed unit, which is where you have often two distinct units. So 6 apples, $3, and you're dividing through partitive division to reveal a rate. That's how John Vanderbil describes a composed unit. So one kind of structure of a ratio is a composed unit. When you have these two quantities that are married to each other and working in tandem. So it's like if it's going to be $6 for 3, then how much per one, how much for 12, like I can scale this in tandem. And when I take the two quantities within a composed unit and I divide them partitively, I reveal a rate.
The other type of ratio is a multiplicative comparison and Cathy Fosnot actually does a really, really cool unit it on this one. I was first introduced to multiplicative comparisons and I was like, "What a cool construct." In the grocery stamps and measuring strips there's a unit called Antonio. That was my very first introduction to this, and I really, really love this idea of a multiplicative comparison. When I have two of the same unit, I have 6 centimeters and 3 centimeters and I want to know like how many iterations of 3 are in 6? Well that's quotative because I'm saying to myself, "How many jumps of 3 inches are in 36 inches?" How many jumps of $12, let's say, a are in $72.
When I'm looking at that quotative context, where I have two quantities of the same unit and I'm dividing them through quotative division. So how many groups of this measured quantity are in that? I reveal like a scale factor or a multiplicative comparison between two quantities of the same unit. So again, I say sometimes proportional reasoning is looked at as these separate expectations, but to me I'm like, "Oh, it's just all division." It's all connected to these behaviors of the two types of division.
Let's talk about fractions, scary fractions. Fractions in our board anyways, have been... and I would say in the province of Ontario, based on our standardized testing, is an area of concern. And I was so, so thrilled to see, it was a day of rejoice when we saw the new revised curriculum and it said, "We're going to introduce all fractions in grade one, two and three in Ontario through fair sharing division." And in conferring with other magicians, Cathy also said that most researchers agree that starting instructional sequences with the development of fair sharing division is the best approach to introducing fractions and that if we actually introduce fractions in the way that I was introduced to fractions or the way I was teaching my students through this like part-whole relationship, we're actually going to reveal a ton of misconceptions.
So we can actually introduce fractions through division and a particular type of division. Not to say we're going to ignore quotative division, but a fair share context is a really, really great way to introduce fractions, through a context like I have 2 brownies divided, so partitive because you see here we have 2 distinct units. We're saying I have 2 brownies and 4 children, 4 parts, and I need to fair share the 2 brownies amongst the 4 parts, and I end up with, again, that rate, that compound unit of half of brownie per child. Why does this matter?
Let's talk about dividing fractions. I never would have thought in my life that I would be able to divide fractions without a single trick step or piece of paper, but now I know that I can use the behaviors of division to very strategically divide fractions through reasoning, without the use of any tips or tricks. I just leverage what I know about the behaviors, how dividing integers, one of my favorites, I don't have to memorize, "Oh, well this side, if I do this, and this is positive, and this is negative, then..." I don't have to memorize anything because as I mentioned, I'm not a very good memorizer, I can reason through it.
So when I think back to all those process expectations that we talked about, and I love the process expectations, and I think that this is what we want for our students. We want them to be problem solvers and be able to reason and prove and reflect and connect and represent and communicate and select tools and strategies, but in order to do that, we have to be able to understand and see. And for me, again, this being able to see the behaviors of this particular operator has been so eyeopening for me and that's why I really wanted to share that particular journey with you.
So for the time that I have left, I just want to show you a few more units. So of course, because division is my favorite topic, we have a joke in our department that I actually got banned from saying partitive and quotative for a while, because every time anybody would listen, I would be like, "Did you know that there are two types of division? Let me tell you about them." But I love division so much that, of course, when Kyle and John asked me to write some units, the very first units that I wanted to write were division units. So there are quite a few in the academy that I really, really love.
I want to highlight the fraction ones because I think they're pretty cool and again, would I have thought that I would be dividing fractions without the use of a trick or a formula or any procedure? No way, but now I know that I can do it through context. So here is a very simple context, I'm going to share with you for fractions. I think it's pretty cool. So the context here is that in 1/3 of the driveway, we used 1/4 of a bag of salt. So I'm going to have you write that down and think about what. So if 1/4 of a bag of salt, that's our total, our quantity, was used for a 1/3 of a group, how much do I need for the whole driveway?
So I think it's pretty straightforward. I'm like, "Well, that's pretty simple. If you use 1/4 of a bag for 1/3 of a driveway, then I just need like 1/4, 2/4, 3/4."It seems almost like silly how simple it is, but that's because I understand the behavior of partitive division. So if I take 1/4 of a bag divided by 1/3 part of the driveway, my rate that is revealed per one driveway is 3/4 bags per driveway. And again, I can take this context and really through the context, show how this particular operator behaves and I don't have to do any tricks that are going to maybe work for some and not all like, I love this idea.
So I don't only focus on supporting math. I also support reading in our district and I've been doing a lot of learning. Anybody else is in the reading realm right now knows there's a lot of learning to do around reading and I read recently something about this idea of casting a wider net. So like teaching a trick might work for some, but will it work for all? I love this idea too, that like this approach of teaching it through context, through behavior, through being able to see and understand will harm none. It's not going to hurt anybody. For those who already know the procedure, it's just going to build belief, it's going to let them trust the procedure. And that's oftentimes how I work. If I'm not sure about a behavior, I try it, I test it with what I know to really build that belief.
So here's an example of a dividing fractions context through partitive division. So I have 1/4 bags of salt divided by 1/3 driveway. Again like this is a ratio. That's pretty cool, right? So again, we're in the world of ratio and rates, but now we're working with fractions. I have a composed unit, salt and driveway. I imagine this is a proportional relationship where this is always going to be true, it's a 1/4 bag per 1/3 driveway. So that's true, that's my ratio and when I divide it through partitive division, I reveal this idea that it's 3/4 bags per driveway.
Here is my second fraction context and now this one is quotative. So in this scenario, I don't know the number of parts. I know the rate, I know the size that I want to measure out. So if the host intends to serve each guest at the party 1/5 of a pizza, how many servings can be made with the remaining 2 1/2 half pizzas? So think back to our early, early unit around peas where we were sharing out 8 peas in every single pot, we're just going to share out 1/5. So I'm going to see how many 1/5s I can make out of 2 1/2 pizzas. I'm going to visualize it. So I'm like, okay, I see five fifths, I see 10 fifths. I know that a whole other pizza would be another five fifths, but I only have half of pizza. So that's like 2 1/2 fifths. So I can make 12 and 1/2 parts. I can make 12 1/2 servings and we'll watch this quick little animation to prove it.
So again like that idea that I don't need any tricks. I can reason, I can prove, I can problem solve. So here are my servings because I know the size. That's why it's quotative and we're also looking at like pizza and pizza. So how many iterations of this part of pizza are in the whole? So you'll see all of my fifths and I can literally access this through counting. I don't need to memorize anything. I don't need to figure out why those trick work. I can just counts fifths. So 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12 and half a fifth, so 12 1/2. So what was the division that happened here? We divided 2 1/2 by 1/5.
So I hope that I built a little bit of belief in the fact that if I understand these behaviors, then I can access those process expectations of being able to connect and represent and strategically select models and strategies and I can prove, and I can justify because I understand how they work. I told you I was going to take you on a journey. What I really did is I gave you like an introduction and I jumped all the way to more of the intermediate concepts. There are units in between in the academy. So if you start with sewing seeds and you really start with just noticing and naming the behaviors of the division, then you can move into planting flowers, for example, which is a step along the developmental continuum. So there are steps in between that would get you to a point where you can almost automatize and demonstrate fluency with two step algebraic equations, with dividing fractions, dividing integers.
So my time is almost up now. So I'm going to hand it back over to our hosts today, but thank you so much for joining me and letting me talk to you for a whole hour about my favorite subject. So this was actually selfish for me. This whole hour was for my own enjoyment, but I really, really hope that you took something away or that I shared a little bit of my passion for division and that you may want to try one of these units with your students if you're in the classroom or with educators, if you're supporting educators in your system. Thanks everybody. And I'll pass over to Kyle.
Kyle Pearce: As always is both John and I learned so much from every single episode, but this one in particular, we really like from that content knowledge perspective. We talk a lot about how important content knowledge is in math and really, we always say, it's hard to know what you don't know, right? There was so much about math that I've been learning over these past few years and division is one of those concepts that I'm continually trying to deepen my own understanding of, and the more and more we dive into this concept, you start to see how it connects to so many different areas and how it really be a gatekeeper.
Jon Orr: Totally.
Kyle Pearce: A lot of people say that ratios and rates are the gatekeeper, but it's like division is almost a gatekeeper towards even getting to ratios and rates. So I hope you got something from that episode. Now, also, John, something that I don't want anyone walking away stressed out here because it's not something you walk away, someone just tells you about division and how complex it is and then all of a sudden it makes perfect sense.
What we hope this has done is sparked an interest to dive in deeper because there's still a lot of work that you'll need to do as well. Think of how it applies to your classroom, to your grade level, to the students you're working with and what can you do now to dive in and do more division problems yourself to try to reveal a better understanding between partitive division, quotative division, how does it apply to other parts of your curriculum? And my goodness, it would be awesome if we heard back from you, either commenting at the bottom of this podcast post on our website or on any of those social media platforms because we want to hear from you. Where are you at with your content knowledge and your understanding of division?
Would you like more on it or are you feeling pretty good? Like you're all buttoned up and ready to go after this episode with Yvette Lehman. I learned a ton. I've seen it before. I listen to it again, and I feel like I learned something new this time. So I'm hoping that you, friends at home, feel the same way.
Jon Orr: Yeah. Awesome stuff there, Kyle. Also, just a reminder to you, that was a video session that was in our 2021 Math Makes Moments Virtual Summit. If you're listening here on the podcast, don't forget that there was a complete video version of this session on the YouTube channel for free. Get on over to the YouTube channel, which is in the show notes page and you can watch or you can get the references. Also, all the activities and the references to this session are also on the show notes page, which is makemathmoments.com/episode171. Again, that's makemathmoments.com/episode171. Over there you'll also find our complete transcripts and the links to this session.
Kyle Pearce: All right, my friends, until next time I'm Kyle Pierce.
Jon Orr: And I'm John Orr
Kyle Pearce: Hi fives for us and a big high five to you.
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