April 18, 2023
Pam Harris
Episode 148

Ep 148: Not the Area Model

Math is Figure-Out-Able with Pam Harris

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Math is Figure-Out-Able with Pam Harris

Ep 148: Not the Area Model

Apr 18, 2023
Episode 148

Pam Harris

Establishing a strong understanding of the area model also prepares students for more advanced topics all the way into college! In this episode, Pam and Kim describe how useful the area model is as a foundation for further understanding throughout a student's learning career.

Talking Points:

- Using the area model builds intuition of the distributive property
- Using and understanding the area model for fraction multiplication
- Paradigm shift for teachers to think about what students know about whole numbers and apply it to fractions
- The area model as a tool, not something that is sketched for every problem
- Area model inspired graphic organizer for polynomial multiplication, polynomial division, and conics

See Episodes 146 and 147 for more discussion of the Area Model

Check out the new t-shirt at https://www.mathisfigureoutable.com/merchandise

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

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Establishing a strong understanding of the area model also prepares students for more advanced topics all the way into college! In this episode, Pam and Kim describe how useful the area model is as a foundation for further understanding throughout a student's learning career.

Talking Points:

- Using the area model builds intuition of the distributive property
- Using and understanding the area model for fraction multiplication
- Paradigm shift for teachers to think about what students know about whole numbers and apply it to fractions
- The area model as a tool, not something that is sketched for every problem
- Area model inspired graphic organizer for polynomial multiplication, polynomial division, and conics

See Episodes 146 and 147 for more discussion of the Area Model

Check out the new t-shirt at https://www.mathisfigureoutable.com/merchandise

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Pam:

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 students to think and reason like mathematicians. Not only are algorithms not really particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.

Kim:Today, we're going to be wrapping up our series on the area model and taking it a bit higher grade wise, as we examine how the area model impacts middle and high school.

Pam:Yeah, so area model Middle, High School. Let's keep going. So, first of all, teachers of younger students, please use the area model well. Check out the last two episodes. Please use it well because then we can take that and do marvelous things with it higher on, higher up, higher grades. One of the things is that because the chunking of the areas on an area model is based on the distributive property, we are for sure going to use that in higher math, and so we really want to have that good basis with area, so that we can take that intuition and we can build on it. One way we can do that is at the middle school level, where we have things like fractions and decimals, and we can help students kind of reason, build spatial reasoning with, say, fraction multiplication. So, for example, if I were to give some kind of lovely problem like 2 and a 1/2. So, 2 and 1/2 times 1 and 2/5. If that was the problem that we're going to deal with. When I originally, initially begin to start working with students with that problem, I would want to build some chunks of areas. I would want to use the area model to build intuition. So, I might say, "Hey, students? What would that look like? What are you thinking about when you see 2 and a 1/2 times 1 and 2/5?" And so, we might draw. I'm drawing on my paper right now something that's 2 long, and then 1/2 long by something that's 1 long, and then 2/5 long. Like, that sort of estimate there about how long that would be. And could we recognize that as an area that is 2 and a 1/2 by 1 and 2/5? And so, I might represent that, and then I might say, "What are some chunks you know?" And early on, as we're making sense of(unclear) fraction multiplication, I would expect students to cut that into that 2 and a 1/2 by 1 and 2/5 into probably 4 chunks where they might say, "Okay, I've got a 2. I cut the 2 and a 1/2 into 2, and 1/2. And I would cut the 1 and 2/5 into 1, and 2/5. And that's going to end up with kind of... It's going to be similar to what a Partial Product might look like, a Partial Place Value, Partial Product model might look like in whole numbers, where I've kind of have these 4 chunks I'm dealing with. And one of the chunks is that 2 by 1 chunk. So, the area of a 2 by 1 is 2. So, I've got a 2 by 1. In the middle that, I've got 2. I've also got a chunk that is that 1/2 by 1. And so, I'm thinking now students are going to have to think here, "What is the area of something that has a dimension of 1/2 by 1?" And I want students to actually have to think about that. Like, can they think about a 1 by 1, and then cut it in half to get a 1/2 by 1? Like, there's lots of different. I want students to grapple with that relationship, that the area of rectangle that's 1/2 by 1 is indeed 1/2. Then, I want them to grapple with this other chunk over there, that would be 2 by 2/5. What does it mean to have a rectangle that's 2 by 2/5? Is that 4/5? Like, what is twice 2/5's? Is that 4/5? And then, I have this little tiny rectangle in the bottom that would be 1/2 by 2/5. I'm looking at my drawing. It's not too... The scale's not too, too bad. But a 1/2 by 2/5? What in the world? Now, students really have to think about what the area of a 1/2 of 2/5 is. But I also want students thinking about a 1/2 of 2/5. Could they think about 1 by 2/5 to cut it in half to get 1/2 by 2/5? So, is the area of 1/2 by 2/5 just 1/2 of 2/5? Is that like 1/5? So, now in this rectangle, I've got 4 chunks of area. And they're not square by the way. Maybe I should have said that. This is not... 2 and a 1/2 by 1 and 2/5 is not square. It's rectangular. It's longer than it is wide. Then, I've got these rectangles in there that I'm going to add up. So, I've got a 2, and I've got an area of 2, an area of 1/2, an area 4/5, an area of 1/5. Well, now I want to strategically add those together. Because I don't know if you caught that, especially if you're listening as you're driving in the car. Ya'll, grab a pencil as we're doing this. I've got a rectangle that is four-fifths, an area four-fifths, and I've got a rectangle that has an area of one-fifth. Well, what's four-fifths and one-fifth? Is that just five-fifths? Bam! So, there's an area of 1. Then, I've also got that area of 2, so that's 3 so far plus that extra 1/2, and so we end up with 3 and a 1/2 is the area. Now, not all problems will end up where the area chunk so nicely, but I want to, I think, choose some strategic fraction multiplication problems and help really build that sense of spatial understanding.

Kim:Yeah.

Pam:What's actually happening with the problem? Yeah.

Kim:Yeah. You know, it's really nice because I think you can... As teachers, we're not bound by certain numbers, right? Like, we don't have to throw the ugliest ones up front. So, I love this problem because the chunks that you would see students maybe do place value chunks with, then doesn't leave them with horrible addition at the end. And you're relieving them from that struggle at the end to really think about the area.

Pam:Yeah, to focus on the area.

Kim:Mmhmm, yeah.

Pam:And that's a big paradigm shift, I think, for many math teachers is, I was sort of raised with this idea that we have to make sure that they have this general procedure because you better deal with these easier numbers now because you're going to need it for the harder numbers later. When in reality, it's not about easy and hard. It's about building students' brains to be able to reason, so that then they could judiciously choose what to do. It's such a different paradigm perspective. So, what could students do if they were looking at that 2 and a 1/2 by 1 and 2/5 if they don't just chunk it into those crazy pieces? We're not going to spend a lot of time here, but if you've listened to the episode from last week, could you Double and Halve? And I'm not actually going to do that. We're going to leave this to the readers. You know that line in textbooks used to say,"The proof is left to the reader." So, listeners, I would challenge you to take 2 and a 1/2 times 1 and 2/5, draw the array, and then think about doubling and halving and see what you do with that, see if you can find maybe some nicer chunks to deal with. But, Kim, I have a feeling that if I gave you 2 and a 1/2 times 1 and 2/5, you would abandon fractions. Am I right on that?

Kim:I would, I would.

Pam:Tell us about that.

Kim:Yeah, I would. So, 2 and a 1/2 and 1 and 2/5. So, 2 and a 1/2 is just 2.5 for me, and then 1 and 2/5 is one and four-tenths, 1.4.

Pam:Mmhmm.

Kim:So, then, I would. I would go Double/Halve, but I would start with 2.5 times 1.4, and then I would Double/Halve to get 5 times 7/10 or $0.70.

Pam:Nice, nice, nice, nice.

Kim:And I get the same 3.50.

Pam:Woah! So, pretty slick. Part of...

Kim:Oh, I said 3.50 because money. 3.5 Excuse me.

Pam:Oh, that's funny because I wrote down 3.50. Nice. So, ya'll, one of our points is we do believe that it is a great use of the area model to help students make sense of fraction multiplication, but then we don't necessarily want to get stuck there. Let's make sense of what's happening. We want students to make sense of the chunks. But then, we quickly want to go, "Hey, but let's chunk those chunks, let's group those groups, and let's use the strategies that we've developed for whole numbers. Can you Double and Halve? Well, let's see. Maybe not. Could you go a little Over? Well, let's check that out. Could we find 3 times 1 and 2/5, and then just take off a 1/2 of 1 and 2/5." Like a half of 1 and 2/5, ya'll. Half of 1. Half of 2/5. Little thing you might think about there. So, again, we might help students use the area model, use open arrays to chunk those chunks, group those groups in ways that are clever and efficient and they're making sense of area, they're building their spatial reasoning, they're building the sense of the distributive property. But then, don't get stuck in it. We don't want students to get stuck in the model. We want to, then, use everything else we know to solve things as efficient and clever as we can. Cool.

Kim:Right. Can I interrupt for just a second because I know we're talking about the area model? I think this is where sometimes this particular model gets a bad rap because the message that sometimes parents are hearing is, "Your child must draw an area model for every problem," and sometimes kids are like, "Not for that problem. That's not a four chunk, two chunk... Like whatever I'm thinking about, so now I have to draw this thing." And so, the area model should be a tool, and if we're just forcing kids to go back and sketch it out after they're already thinking about it, after they've already solved it, then it's... We're kind of missing the point a little bit.

Pam:Yeah, absolutely. It's sort of an effort that we that doesn't need to be expended. I'm sure there's a better way to say that. We don't want to force kids to do stuff they don't need. It's not going to be helpful. And so, yeah, we would encourage you to look at if you're ever saying, "Show your work," and you're like. Yes, I want to see what's happening in your brain. But to force kids to draw, especially if you're forcing them to draw the Place Value Partial Products all the time for every problem, that's not. Like, let's have a purpose. Let's make sure that it's a helpful purpose. Yeah, nicely said. I do agree with you that is one place where we are missing the boat a little bit, and wreaking... Not wreaking havoc but causing havoc. Like inviting controversy when we don't need it. Because it's not particularly helpful to force kids. And in fact, let me just be clear, when I'm using an area model, using it in class, often it's myself. I am actually representing. As the teacher, I'm representing student thinking. We'll do plenty of Problem Strings where students aren't drawing anything at all, but as they say the relationship they're thinking about, we are making that relationship visible, so that then we can point at it and talk about it, but we're not actually forcing students to draw at all. We will sometimes say to students, especially when we're doing area, we might say, "You know, like, area. We're really focused on area." Like, "What would that look like?" But we're not going to get stuck in it. Is that a way to say that? Cool.

Kim:Yeah.

Pam:I'm glad you brought that up. Alright. Now, as we move from thinking about kind of middle school-y fractions, and decimals, and using an area model or an open array. We are definitely going to continue to build things on that basis. Area happens a lot in calculus. I mean, in calculus, we find the area under a curve. We can use that to find integration rates based on area. There's lots of things. I mean, just the definition of a derivative, Riemann sums and everything. Like, we often model that with area. So, as we think about a reason to use area models in the younger grades, we absolutely want to develop a sense of area. What we don't want is for kids to go, "Area? Perimeter? Mmm... Let's see one of them you add, one of them you multiply. Which ones this?" That's not. We want to develop a far more robust sense of what it means to measure linearly and what it means to measure that sort of two dimensional square unit, that span. And so that's super, super important. But then, we're going to kind of rely on that intuition a little bit to help motivate us using what I'm not going to now call an area model, what is going to become a bit more of a graphic organizer to do some other things. So, there is a distinction between an area model, and an array, and a graphic organizer, and we can use them at different times to do different things. So, let's just get some some clarity on where we can use the understanding that we build in the younger grades, the intuition and the real sense, the spatial reasoning that we build. Using an area model and an open array can then help us do some things with like, polynomial multiplication. What's... Yeah. Woah! Just got higher math. So, with polynomials if I'm looking at something like x minus 3 times x plus 2, sometimes people will say, "Oh, I'm going to use an area model to do that." Well, let's be clear. As soon as we've gotten negatives involved, we're really not measuring anymore. We don't really have negative lengths. We don't have negative area. That would be kind of like what a black hole or, I don't know, anti-matter. I mean, that's maybe off in space somewhere. As soon as we have negative numbers, we're not really having positive measurement anymore. But we can use the sense that we've sort of built about the distributive property and about kind of how we could take an area model and turn it into a graphic organizer to do things like, I could say,"Well, if I'm thinking about the product of x minus 3 times x plus 2, then I could kind of draw a graphic organizer. " And now, I am drawing a square, and I'm putting x on the left hand side, and then negative 3, sort of below that. And then across the top, I'm putting x, and then positive 2, and I am splitting that into 4 equal chunks. And I'm using this graphic organizer of this square cut into 4 equal chunks to help me think about the distributive property, to help me make sure I don't miss any parts. And I can think about x times x, and that gives me x^2, a product of x^2. And then, that negative 3 times x, and that gives me a product of negative 3x. And then, I've got x times 2 and I'm filling in the pieces of this graphic organizer. X times 2 is 2x, and then that final square in the graphic organizer, negative 3 times 2 is negative 6. And then, I could gather all those terms together. So, I had an x squared, a 2x, a negative 3x, and a negative 6. When I gather those together, I've got x^2. The negative 3x and the 2x is negative x, negative 1x. And then, I've got that negative 6 or minus 6. So, in other words, the product of the quantity x minus 3 times the quantity x plus 2 is x^2 minus x minus 6. And I can help find that, I can help keep track of the distributive property, using something that kind of resembles the area model that we built this intuition around with rectangles and area of whole numbers, and fractions, and decimals. And now I'm going to use that intuition to help me make sense of the distributive property with these variables in polynomial multiplication.

Kim:You're speaking my language, Pam, because that's...

Pam:Bam!

Kim:...exactly what Cooper is starting to work on.

Pam:Are you serious?

Kim:Yeah. And so, he is just kind of... There's no models, right, and so when we started(unclear)

Pam:That's unfortunate, that's unfortunate.

Kim:Well, yeah.

Pam:We're not liking there's no models. Okay, keep going.

Kim:So, I will sketch, "Oh, it's like this." And he looks at me, and he's like, "Okay..." and it's connecting to stuff that he's done before, which is really cool. You know?

Pam:Yeah. And that's the big point. So, you, when we say,"Know your content, know your kids," you obviously know your kid, and you know the content, and so you're looking at this higher math that he's doing, and you're like, "Oh, dude, we can base this on what we built before. Like, check it out. This is... Remember, this thing we've done before? Look how this relates to that." And that is exactly the point. So, middle school, high school teachers, I'm recommending that you don't call that graphic organizer an area model because it's not. It's not representing area, but it is representing the product of two factors. Just like earlier we used the area model to represent the product of the two dimensions of factors that were representing dimensions and area. We can sort of use that. Similarly, we can use the same kind of thing to think about polynomial division. There's actually a paper out there somewhere that says, "Students must learn long division with whole numbers because we're going to need it for our polynomial long division." To which I say, "Nope. No, we do not. We can absolutely use that same graphic organizer to reason about polynomial division and have far less opportunity for error because you don't have that crazy subtracting negatives that happens if..." And I know I'm talking straight to high school teachers right now. But, high school teachers, you know when you do polynomial long division, that once you've got that second row on there, and now you've got to subtract. You've got all these negatives floating around, and you're subtracting negatives. And I've heard you say to students, "Just put a plus sign, and then change all the signs." And, like,"Ahh!" I mean, that is a way to get correct answers, but not a lot of reasoning happening there. So, when students are doing polynomial long division, if you've done polynomial multiplication in that graphic organizer, you'll start to notice some patterns, like the like terms are happening in the diagonals. And if you notice that the diagonals are having when you're collecting like terms those are happening in the diagonals. When you undo that, then you can just reason about what you need, based on the part of the diagonal you have needs to add to that term. And I know I'm talking kind of all crazy. We'll do a whole podcast episode on polynomial long division at some point. But for today, I just want to note that if we use the area model well for whole numbers and decimals, then we can build into polynomial multiplication and polynomial division. (unclear)

Kim:I just want to...

Pam:Oh, keep going.

Kim:Go ahead.

Pam:Go ahead. Well, I'll finish.

Kim:I was... Okay.

Pam:Don't forget. I'm going to go. I'm going to go. High school teachers, the Problem Strings I've written in the advanced algebra Problem String book actually has you start with thinking about an area model with whole numbers into a graphic organizer with multiplying polynomials into a graphic organizer with then dividing polynomials. Okay, Kim, go.

Kim:Yeah. Well, I just want to give a shout out to my kid's former teacher because Luke recently was watching a video about long division, and I kind of walked up, and I was, "What are you doing?" And he said,"Oh, this was the video for today in math." And so I said,"Why? What?" And he said, "Well, we're about to start doing this other thing, and we needed to review long division first before we could do this new thing." And I was like, "Okay." And Luke's never done long division.

Pam:With whole numbers.

Kim:With whole numbers, right. So, he's always just kind of stayed under the radar and never done it. And so...

Pam:Meaning...sorry, let me just be clear...he's always reasoned through division. He hasn't done a series of steps that he memorized under a house trying to do long division.

Kim:Correct.

Pam:Keep going. Mmhmm.

Kim:Correct. And so, anyway. When he showed me a problem, the polynomial division problem, and I said, "Oh, this is a division problem. So, that's kinda like we know one of the factors, and..." I said area because I didn't know what else to say. So, now I'll do better, but. I said, "That's kind of..." And I said, "It's not really area, but it's kind of like, here's the product of that. So, all you're trying to figure out is the other dimension." And he was like, "Oh, that's it?" I was like, "Yeah."

Pam:Yeah, that's it. Yeah.

Kim:So, he goes on his merry way to just figure out and, you know, it's brilliant because he has a picture in his mind of the way you consider the two factors and the product, and the inverse of that is just what if I know one of the dimensions, then(unclear).

Pam:And you know the product.

Kim:Yeah.

Pam:Then, you could find the other factor. Oh, that's brilliant. Way to go, Luke! You know, you're reminding me of a story when my second was taking either second or third semester calculus. So, my number two was at university, and he's taking their second or third semester calculus. I can't remember which one. And he said, "Hey, Mom, I did your thing." And I was like,"What?" And he goes, "Yeah, did your thing today in calculus." I was like, "Oh, dude, send me a picture." And he sent me a picture of a full page. Because, you know, calculus problems kind of take a bit of work, right? So, there's one problem, took the full page, and he circled his little bit on the top. Because as part of this entire huge calculus problem, part of what he had to do was a little bit of polynomial division, and he had, literally, he had kind of that graphic organizer, looks like an area model kind of thing, and he had just a little bit. Like, he didn't even fill it all out because once he had put enough in it, then he just moved what he knew was going to happen to the party needed for the rest of the calculus problem.

Kim:Yep.

Pam:And he kind of put little smiley face next to that little part, and it was it was. Anyway, so, yeah. Because we can just use what we know to solve problems more and more advanced, if we actually own it and we actually are using relationships to solve problems. And I'll just mention as a last here, an application for higher math, that if we've got students really reasoning, using this graphic organizer, then we can kind of bring back area a little bit, at least a sense of area, and have a super nice way of completing the square. And so, completing the square is a thing that we do. And I think the best reason to complete the squares to be able to find the centers of conics, that if we have a conic section, and we're trying to like find the center of a circle, or a hyperbola, or an ellipse, then it can be really helpful to complete the square, so that we can tell where the the vertex is going to be, or the center is going to be. And that's a super nice. By the way, I'll just say it out loud, I don't really like completing the square for solving equations. I know. Someone's going to be like not happy about that, but I don't think it's a great time. I don't think it's a great time to bring in that. I just don't think we need it for solving equations. But I do think it's a fine thing that we can build. We can actually understand what's happening, completing the square. If we use this sense and understanding from an area model into the graphic organizer, we can literally help complete the square graphically physically, spatially, and it's a super nice way. So...

Kim:Yep.

Pam:...we like the area model.

Kim:It's so cool, right? So, teachers of younger grades, talking to you here. Let's use it really well in our grades, so that these teachers of older students can build on that intuition to do all of these really cool higher math things.

Pam:Absolutely. Alright, 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!

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